diff --git a/.gitignore b/.gitignore --- a/.gitignore +++ b/.gitignore @@ -36,6 +36,8 @@ eg/factorial eg/fibonacci eg/fib01 +eg/higher +eg/id eg/ints eg/gcd eg/list @@ -61,6 +63,7 @@ bench/gps2 bench/lowtests bench/terpret +bench/weird proto/u-conjure test/expr test/conjurable diff --git a/Makefile b/Makefile --- a/Makefile +++ b/Makefile @@ -8,7 +8,7 @@ -i $(shell find /usr/share/doc/ghc/html/libraries -name template-haskell.haddock | tail -1) \ $(shell grep -q "Arch Linux" /etc/lsb-release && echo --optghc=-dynamic) \ | grep -v "^Warning: Couldn't find .haddock for export [A-Z]$$" -INSTALL_DEPS = leancheck express speculate +INSTALL_DEPS = leancheck express speculate template-haskell EG = \ eg/arith \ @@ -17,6 +17,8 @@ eg/factorial \ eg/fibonacci \ eg/fib01 \ + eg/higher \ + eg/id \ eg/ints \ eg/bools \ eg/gcd \ @@ -42,6 +44,7 @@ bench/gps \ bench/gps2 \ bench/terpret \ + bench/weird \ proto/u-conjure TESTS = \ diff --git a/TODO.md b/TODO.md --- a/TODO.md +++ b/TODO.md @@ -3,36 +3,31 @@ A non-exhaustive list of things TO DO for Conjure. +* Review `bench/redundants.out` and prune redundant candidates. + See sections below for ideas. -### for later -* pretty-print top-level if and case expressions? - -* consider not breaking arguments in some cases - (increased crossproduct of patterns). - but which cases? +## Prune modulo rewriting -* exclude magic numbers? e.g.: `1+1`? +The following is redundant, as only the second equation is necessary: -* allow conjuring from partially defined implementations? + foo 0 = 0 + foo x = x + x - partial :: ((Int -> Int) -> Int -> Int) -> (Int -> Int) - partial impl = f - where - f n = if n == 0 - then 1 - else impl f n +Another example is: -* consider discover orderings on arguments? + foo 0 = 1 + foo x = x + 1 -* consider leveraging lazyness somehow? - (related to allowing recursion under any lazy functions) +The tricky part is when dealing with multiple arguments: -* consider leveraging polymorphism somehow? + x ? 0 = x + x ? y = x + y -* consider allowing lambdas that introduce free variables? +One possible path would be to replace 0 in the second equation +then use the Theory to discover x + 0 is equal to 0. This file is part of Conjure, -(C) 2021-2022 Rudy Matela, +(C) 2021-2024 Rudy Matela, Distribued under the 3-Clause BSD license. diff --git a/bench/candidates.out b/bench/candidates.out --- a/bench/candidates.out +++ b/bench/candidates.out @@ -1,12054 +1,6739 @@ Candidates for: foo :: Int -> Int pruning with 15/35 rules - [3,0,7,0,20,0,75,0,297] direct candidates, 0 duplicates - [3,4,12,32,54,139,287,691,1713] pattern candidates, 0 duplicates - -rules: -x * y == x + y -x * y == y + x -x - x == 0 -x + 0 == x -0 + x == x -x - 0 == x -(x + y) + z == x + (y + z) -(x + y) + z == y + (x + z) -x - (y - z) == z + (x - y) -(x - y) - z == x - (y + z) -(x - y) - z == x - (z + y) -(x + y) - z == x + (y - z) -(x + y) - z == y + (x - z) -x + (y - x) == y -(x - y) + y == x -equations: -y + x == x + y -y + (x + z) == x + (y + z) -z + (x + y) == x + (y + z) -z + (y + x) == x + (y + z) -y + (x - z) == x + (y - z) -(x - z) + y == x + (y - z) -(z - y) + x == (x - y) + z -y - (x + y) == 0 - x -y - (y + x) == 0 - x -z - (y + z) == x - (x + y) -z - (y + z) == x - (y + x) -z - (z + y) == x - (x + y) -z - (z + y) == x - (y + x) -x + (0 - y) == x - y -(0 - y) + x == x - y -x - (x + 1) == 0 - 1 -x - (1 + x) == 0 - 1 -y - (y + 1) == x - (x + 1) -y - (y + 1) == x - (1 + x) -y - (1 + y) == x - (1 + x) - -direct candidates: - -foo x = x - -foo x = 0 - -foo x = 1 - -foo x = x + x - -foo x = x + 1 - -foo x = 1 + x - -foo x = 1 + 1 - -foo x = x - 1 - -foo x = 0 - x - -foo x = 0 - 1 - -foo x = (0 - 1) + x - -foo x = x + (x + x) - -foo x = x + (x + 1) - -foo x = x + (1 + x) - -foo x = x + (1 + 1) - -foo x = x + (x - 1) - -foo x = x + (0 - 1) - -foo x = 1 + (x + x) - -foo x = 1 + (x + 1) - -foo x = 1 + (1 + x) - -foo x = 1 + (1 + 1) - -foo x = 1 + (0 - x) - -foo x = x - (x + x) - -foo x = x - (x + 1) - -foo x = x - (1 + x) - -foo x = x - (1 + 1) - -foo x = 0 - (x + x) - -foo x = 0 - (x + 1) - -foo x = 0 - (1 + x) - -foo x = 0 - (1 + 1) - - -pattern candidates: - -foo x = x - -foo x = 0 - -foo x = 1 - -foo 0 = 0 -foo x = x - -foo 0 = 0 -foo x = 1 - -foo 0 = 1 -foo x = x - -foo 0 = 1 -foo x = 0 - -foo x = x + x - -foo x = x + 1 - -foo x = 1 + x - -foo x = 1 + 1 - -foo x = x - 1 - -foo x = 0 - x - -foo x = 0 - 1 - -foo x = 1 - x - -foo 1 = 0 -foo x = x - -foo 1 = 0 -foo x = 1 - -foo 1 = 1 -foo x = x - -foo 1 = 1 -foo x = 0 - -foo 0 = 0 -foo x = x + x - -foo 0 = 0 -foo x = x + 1 - -foo 0 = 0 -foo x = 1 + x - -foo 0 = 0 -foo x = 1 + 1 - -foo 0 = 0 -foo x = x - 1 - -foo 0 = 0 -foo x = 0 - x - -foo 0 = 0 -foo x = 0 - 1 - -foo 0 = 0 -foo x = 1 - x - -foo 0 = 1 -foo x = x + x - -foo 0 = 1 -foo x = x + 1 - -foo 0 = 1 -foo x = 1 + x - -foo 0 = 1 -foo x = 1 + 1 - -foo 0 = 1 -foo x = x - 1 - -foo 0 = 1 -foo x = 0 - x - -foo 0 = 1 -foo x = 0 - 1 - -foo 0 = 1 -foo x = 1 - x - -foo 0 = 1 + 1 -foo x = x - -foo 0 = 1 + 1 -foo x = 0 - -foo 0 = 1 + 1 -foo x = 1 - -foo 0 = 0 - 1 -foo x = x - -foo 0 = 0 - 1 -foo x = 0 - -foo 0 = 0 - 1 -foo x = 1 - -foo 0 = 0 -foo 1 = 0 -foo x = x - -foo 0 = 0 -foo 1 = 0 -foo x = 1 - -foo 0 = 0 -foo 1 = 1 -foo x = x - -foo 0 = 0 -foo 1 = 1 -foo x = 0 - -foo 0 = 0 -foo 1 = 1 -foo x = 1 - -foo 0 = 1 -foo 1 = 0 -foo x = x - -foo 0 = 1 -foo 1 = 0 -foo x = 0 - -foo 0 = 1 -foo 1 = 0 -foo x = 1 - -foo 0 = 1 -foo 1 = 1 -foo x = x - -foo 0 = 1 -foo 1 = 1 -foo x = 0 - -foo 0 = 0 -foo x = foo (x - 1) - -foo 0 = 0 -foo x = foo (1 - x) - -foo 0 = 1 -foo x = foo (x - 1) - -foo 0 = 1 -foo x = foo (1 - x) - -foo x = (x - 1) + x - -foo x = (0 - x) + 1 - -foo x = (0 - 1) + x - -foo x = (1 - x) + 1 - -foo x = x + (x + x) - -foo x = x + (x + 1) - -foo x = x + (1 + x) - -foo x = x + (1 + 1) - -foo x = x + (x - 1) - -foo x = x + (0 - 1) - -foo x = 1 + (x + x) - -foo x = 1 + (x + 1) - -foo x = 1 + (1 + x) - -foo x = 1 + (1 + 1) - -foo x = 1 + (0 - x) - -foo x = 1 + (1 - x) - -foo x = x - (x + x) - -foo x = x - (x + 1) - -foo x = x - (1 + x) - -foo x = x - (1 + 1) - -foo x = 0 - (x + x) - -foo x = 0 - (x + 1) - -foo x = 0 - (1 + x) - -foo x = 0 - (1 + 1) - -foo x = 1 - (x + x) - -foo x = 1 - (x + 1) - -foo x = 1 - (1 + x) - -foo x = 1 - (1 + 1) - -foo 1 = 0 -foo x = x + x - -foo 1 = 0 -foo x = x + 1 - -foo 1 = 0 -foo x = 1 + x - -foo 1 = 0 -foo x = 1 + 1 - -foo 1 = 0 -foo x = x - 1 - -foo 1 = 0 -foo x = 0 - x - -foo 1 = 0 -foo x = 0 - 1 - -foo 1 = 0 -foo x = 1 - x - -foo 1 = 1 -foo x = x + x - -foo 1 = 1 -foo x = x + 1 - -foo 1 = 1 -foo x = 1 + x - -foo 1 = 1 -foo x = 1 + 1 - -foo 1 = 1 -foo x = x - 1 - -foo 1 = 1 -foo x = 0 - x - -foo 1 = 1 -foo x = 0 - 1 - -foo 1 = 1 -foo x = 1 - x - -foo 1 = 1 + 1 -foo x = x - -foo 1 = 1 + 1 -foo x = 0 - -foo 1 = 1 + 1 -foo x = 1 - -foo 1 = 0 - 1 -foo x = x - -foo 1 = 0 - 1 -foo x = 0 - -foo 1 = 0 - 1 -foo x = 1 - -foo 1 = 0 -foo x = foo (x - 1) - -foo 1 = 0 -foo x = foo (1 - x) - -foo 1 = 1 -foo x = foo (x - 1) - -foo 1 = 1 -foo x = foo (1 - x) - -foo 0 = 0 -foo x = (x - 1) + x - -foo 0 = 0 -foo x = (0 - x) + 1 - -foo 0 = 0 -foo x = (0 - 1) + x - -foo 0 = 0 -foo x = (1 - x) + 1 - -foo 0 = 0 -foo x = x + (x + x) - -foo 0 = 0 -foo x = x + (x + 1) - -foo 0 = 0 -foo x = x + (1 + x) - -foo 0 = 0 -foo x = x + (1 + 1) - -foo 0 = 0 -foo x = x + (x - 1) - -foo 0 = 0 -foo x = x + (0 - 1) - -foo 0 = 0 -foo x = 1 + (x + x) - -foo 0 = 0 -foo x = 1 + (x + 1) - -foo 0 = 0 -foo x = 1 + (1 + x) - -foo 0 = 0 -foo x = 1 + (1 + 1) - -foo 0 = 0 -foo x = 1 + (0 - x) - -foo 0 = 0 -foo x = 1 + (1 - x) - -foo 0 = 0 -foo x = x - (x + x) - -foo 0 = 0 -foo x = x - (x + 1) - -foo 0 = 0 -foo x = x - (1 + x) - -foo 0 = 0 -foo x = x - (1 + 1) - -foo 0 = 0 -foo x = 0 - (x + x) - -foo 0 = 0 -foo x = 0 - (x + 1) - -foo 0 = 0 -foo x = 0 - (1 + x) - -foo 0 = 0 -foo x = 0 - (1 + 1) - -foo 0 = 0 -foo x = 1 - (x + x) - -foo 0 = 0 -foo x = 1 - (x + 1) - -foo 0 = 0 -foo x = 1 - (1 + x) - -foo 0 = 0 -foo x = 1 - (1 + 1) - -foo 0 = 1 -foo x = (x - 1) + x - -foo 0 = 1 -foo x = (0 - x) + 1 - -foo 0 = 1 -foo x = (0 - 1) + x - -foo 0 = 1 -foo x = (1 - x) + 1 - -foo 0 = 1 -foo x = x + (x + x) - -foo 0 = 1 -foo x = x + (x + 1) - -foo 0 = 1 -foo x = x + (1 + x) - -foo 0 = 1 -foo x = x + (1 + 1) - -foo 0 = 1 -foo x = x + (x - 1) - -foo 0 = 1 -foo x = x + (0 - 1) - -foo 0 = 1 -foo x = 1 + (x + x) - -foo 0 = 1 -foo x = 1 + (x + 1) - -foo 0 = 1 -foo x = 1 + (1 + x) - -foo 0 = 1 -foo x = 1 + (1 + 1) - -foo 0 = 1 -foo x = 1 + (0 - x) - -foo 0 = 1 -foo x = 1 + (1 - x) - -foo 0 = 1 -foo x = x - (x + x) - -foo 0 = 1 -foo x = x - (x + 1) - -foo 0 = 1 -foo x = x - (1 + x) - -foo 0 = 1 -foo x = x - (1 + 1) - -foo 0 = 1 -foo x = 0 - (x + x) - -foo 0 = 1 -foo x = 0 - (x + 1) - -foo 0 = 1 -foo x = 0 - (1 + x) - -foo 0 = 1 -foo x = 0 - (1 + 1) - -foo 0 = 1 -foo x = 1 - (x + x) - -foo 0 = 1 -foo x = 1 - (x + 1) - -foo 0 = 1 -foo x = 1 - (1 + x) - -foo 0 = 1 -foo x = 1 - (1 + 1) - -foo 0 = 1 + 1 -foo x = x + x - -foo 0 = 1 + 1 -foo x = x + 1 - -foo 0 = 1 + 1 -foo x = 1 + x - -foo 0 = 1 + 1 -foo x = x - 1 - -foo 0 = 1 + 1 -foo x = 0 - x - -foo 0 = 1 + 1 -foo x = 0 - 1 - -foo 0 = 1 + 1 -foo x = 1 - x - -foo 0 = 0 - 1 -foo x = x + x - -foo 0 = 0 - 1 -foo x = x + 1 - -foo 0 = 0 - 1 -foo x = 1 + x - -foo 0 = 0 - 1 -foo x = 1 + 1 - -foo 0 = 0 - 1 -foo x = x - 1 - -foo 0 = 0 - 1 -foo x = 0 - x - -foo 0 = 0 - 1 -foo x = 1 - x - -foo 0 = 1 + (1 + 1) -foo x = x - -foo 0 = 1 + (1 + 1) -foo x = 0 - -foo 0 = 1 + (1 + 1) -foo x = 1 - -foo 0 = 0 - (1 + 1) -foo x = x - -foo 0 = 0 - (1 + 1) -foo x = 0 - -foo 0 = 0 - (1 + 1) -foo x = 1 - -foo 0 = 1 - (1 + 1) -foo x = x - -foo 0 = 1 - (1 + 1) -foo x = 0 - -foo 0 = 1 - (1 + 1) -foo x = 1 - -foo 0 = 0 -foo 1 = 0 -foo x = x + x - -foo 0 = 0 -foo 1 = 0 -foo x = x + 1 - -foo 0 = 0 -foo 1 = 0 -foo x = 1 + x - -foo 0 = 0 -foo 1 = 0 -foo x = 1 + 1 - -foo 0 = 0 -foo 1 = 0 -foo x = x - 1 - -foo 0 = 0 -foo 1 = 0 -foo x = 0 - x - -foo 0 = 0 -foo 1 = 0 -foo x = 0 - 1 - -foo 0 = 0 -foo 1 = 0 -foo x = 1 - x - -foo 0 = 0 -foo 1 = 1 -foo x = x + x - -foo 0 = 0 -foo 1 = 1 -foo x = x + 1 - -foo 0 = 0 -foo 1 = 1 -foo x = 1 + x - -foo 0 = 0 -foo 1 = 1 -foo x = 1 + 1 - -foo 0 = 0 -foo 1 = 1 -foo x = x - 1 - -foo 0 = 0 -foo 1 = 1 -foo x = 0 - x - -foo 0 = 0 -foo 1 = 1 -foo x = 0 - 1 - -foo 0 = 0 -foo 1 = 1 -foo x = 1 - x - -foo 0 = 0 -foo 1 = 1 + 1 -foo x = x - -foo 0 = 0 -foo 1 = 1 + 1 -foo x = 0 - -foo 0 = 0 -foo 1 = 1 + 1 -foo x = 1 - -foo 0 = 0 -foo 1 = 0 - 1 -foo x = x - -foo 0 = 0 -foo 1 = 0 - 1 -foo x = 0 - -foo 0 = 0 -foo 1 = 0 - 1 -foo x = 1 - -foo 0 = 1 -foo 1 = 0 -foo x = x + x - -foo 0 = 1 -foo 1 = 0 -foo x = x + 1 - -foo 0 = 1 -foo 1 = 0 -foo x = 1 + x - -foo 0 = 1 -foo 1 = 0 -foo x = 1 + 1 - -foo 0 = 1 -foo 1 = 0 -foo x = x - 1 - -foo 0 = 1 -foo 1 = 0 -foo x = 0 - x - -foo 0 = 1 -foo 1 = 0 -foo x = 0 - 1 - -foo 0 = 1 -foo 1 = 0 -foo x = 1 - x - -foo 0 = 1 -foo 1 = 1 -foo x = x + x - -foo 0 = 1 -foo 1 = 1 -foo x = x + 1 - -foo 0 = 1 -foo 1 = 1 -foo x = 1 + x - -foo 0 = 1 -foo 1 = 1 -foo x = 1 + 1 - -foo 0 = 1 -foo 1 = 1 -foo x = x - 1 - -foo 0 = 1 -foo 1 = 1 -foo x = 0 - x - -foo 0 = 1 -foo 1 = 1 -foo x = 0 - 1 - -foo 0 = 1 -foo 1 = 1 -foo x = 1 - x - -foo 0 = 1 -foo 1 = 1 + 1 -foo x = x - -foo 0 = 1 -foo 1 = 1 + 1 -foo x = 0 - -foo 0 = 1 -foo 1 = 1 + 1 -foo x = 1 - -foo 0 = 1 -foo 1 = 0 - 1 -foo x = x - -foo 0 = 1 -foo 1 = 0 - 1 -foo x = 0 - -foo 0 = 1 -foo 1 = 0 - 1 -foo x = 1 - -foo 0 = 1 + 1 -foo 1 = 0 -foo x = x - -foo 0 = 1 + 1 -foo 1 = 0 -foo x = 0 - -foo 0 = 1 + 1 -foo 1 = 0 -foo x = 1 - -foo 0 = 1 + 1 -foo 1 = 1 -foo x = x - -foo 0 = 1 + 1 -foo 1 = 1 -foo x = 0 - -foo 0 = 1 + 1 -foo 1 = 1 -foo x = 1 - -foo 0 = 0 - 1 -foo 1 = 0 -foo x = x - -foo 0 = 0 - 1 -foo 1 = 0 -foo x = 0 - -foo 0 = 0 - 1 -foo 1 = 0 -foo x = 1 - -foo 0 = 0 - 1 -foo 1 = 1 -foo x = x - -foo 0 = 0 - 1 -foo 1 = 1 -foo x = 0 - -foo 0 = 0 - 1 -foo 1 = 1 -foo x = 1 - - -Candidates for: ? :: Int -> Int -> Int - pruning with 10/23 rules - [3,3,4,10,17,26,54,93,162] direct candidates, 0 duplicates - [3,10,40,116,338,953,2702,7505,20728] pattern candidates, 0 duplicates - -rules: -x * y == x + y -x * y == y + x -x + 0 == x -0 + x == x -dec (x + y) == x + dec y -dec (x + y) == y + dec x -dec (x + y) == dec x + y -dec (x + y) == dec y + x -(x + y) + z == x + (y + z) -(x + y) + z == y + (x + z) -equations: -y + x == x + y -y + dec x == x + dec y -dec x + y == x + dec y -dec y + x == dec x + y -x + dec 0 == dec x -dec 0 + x == dec x -y + (x + z) == x + (y + z) -z + (x + y) == x + (y + z) -z + (y + x) == x + (y + z) -y + dec (dec x) == x + dec (dec y) -dec (dec x) + y == x + dec (dec y) -x + dec (dec 0) == dec (dec x) -dec (dec 0) + x == dec (dec x) - -direct candidates: - -x ? y = x - -x ? y = y - -x ? y = 0 - -x ? y = dec x - -x ? y = dec y - -x ? y = dec 0 - -x ? y = x + x - -x ? y = x + y - -x ? y = y + x - -x ? y = y + y - -x ? y = dec x + y - -x ? y = dec y + x - -x ? y = dec 0 + x - -x ? y = dec 0 + y - -x ? y = x + dec x - -x ? y = x + dec y - -x ? y = x + dec 0 - -x ? y = y + dec x - -x ? y = y + dec y - -x ? y = y + dec 0 - -x ? y = x + (x + x) - -x ? y = x + (x + y) - -x ? y = x + (y + x) - -x ? y = x + (y + y) - -x ? y = y + (x + x) - -x ? y = y + (x + y) - -x ? y = y + (y + x) - -x ? y = y + (y + y) - -x ? y = dec x + dec x - -x ? y = dec x + dec y - -x ? y = dec x + dec 0 - -x ? y = dec y + dec x - -x ? y = dec y + dec y - -x ? y = dec y + dec 0 - -x ? y = dec 0 + dec x - -x ? y = dec 0 + dec y - -x ? y = dec 0 + dec 0 - -x ? y = x + (dec x + y) - -x ? y = x + (dec y + x) - -x ? y = x + (dec 0 + x) - -x ? y = x + (dec 0 + y) - -x ? y = x + (x + dec x) - -x ? y = x + (x + dec y) - -x ? y = x + (x + dec 0) - -x ? y = x + (y + dec x) - -x ? y = x + (y + dec y) - -x ? y = x + (y + dec 0) - -x ? y = y + (dec x + y) - -x ? y = y + (dec y + x) - -x ? y = y + (dec 0 + x) - -x ? y = y + (dec 0 + y) - -x ? y = y + (x + dec x) - -x ? y = y + (x + dec y) - -x ? y = y + (x + dec 0) - -x ? y = y + (y + dec x) - -x ? y = y + (y + dec y) - -x ? y = y + (y + dec 0) - -x ? y = dec x + (y + y) - -x ? y = dec y + (x + x) - -x ? y = dec 0 + (x + x) - -x ? y = dec 0 + (x + y) - -x ? y = dec 0 + (y + x) - -x ? y = dec 0 + (y + y) - - -pattern candidates: - -x ? y = x - -x ? y = y - -x ? y = 0 - -x ? y = dec x - -x ? y = dec y - -x ? y = dec 0 - -x ? 0 = x -x ? y = y - -x ? 0 = x -x ? y = 0 - -x ? 0 = 0 -x ? y = x - -x ? 0 = 0 -x ? y = y - -0 ? x = x -x ? y = 0 - -0 ? x = 0 -x ? y = x - -0 ? x = 0 -x ? y = y - -x ? y = x + x - -x ? y = x + y - -x ? y = y + x - -x ? y = y + y - -x ? y = dec (dec x) - -x ? y = dec (dec y) - -x ? y = dec (dec 0) - -x ? 0 = x -x ? y = dec x - -x ? 0 = x -x ? y = dec y - -x ? 0 = x -x ? y = dec 0 - -x ? 0 = 0 -x ? y = dec x - -x ? 0 = 0 -x ? y = dec y - -x ? 0 = 0 -x ? y = dec 0 - -x ? 0 = dec x -x ? y = x - -x ? 0 = dec x -x ? y = y - -x ? 0 = dec x -x ? y = 0 - -x ? 0 = dec 0 -x ? y = x - -x ? 0 = dec 0 -x ? y = y - -x ? 0 = dec 0 -x ? y = 0 - -0 ? x = x -x ? y = dec x - -0 ? x = x -x ? y = dec y - -0 ? x = x -x ? y = dec 0 - -0 ? x = 0 -x ? y = dec x - -0 ? x = 0 -x ? y = dec y - -0 ? x = 0 -x ? y = dec 0 - -0 ? x = dec x -x ? y = x - -0 ? x = dec x -x ? y = y - -0 ? x = dec x -x ? y = 0 - -0 ? x = dec 0 -x ? y = x - -0 ? x = dec 0 -x ? y = y - -0 ? x = dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = y - -x ? y = dec x + x - -x ? y = dec x + y - -x ? y = dec y + x - -x ? y = dec y + y - -x ? y = dec 0 + x - -x ? y = dec 0 + y - -x ? y = dec (dec (dec x)) - -x ? y = dec (dec (dec y)) - -x ? y = dec (dec (dec 0)) - -x ? y = x + dec x - -x ? y = x + dec y - -x ? y = x + dec 0 - -x ? y = y + dec x - -x ? y = y + dec y - -x ? y = y + dec 0 - -x ? 0 = x -x ? y = x + x - -x ? 0 = x -x ? y = x + y - -x ? 0 = x -x ? y = y + x - -x ? 0 = x -x ? y = y + y - -x ? 0 = x -x ? y = dec (dec x) - -x ? 0 = x -x ? y = dec (dec y) - -x ? 0 = x -x ? y = dec (dec 0) - -x ? 0 = 0 -x ? y = x + x - -x ? 0 = 0 -x ? y = x + y - -x ? 0 = 0 -x ? y = y + x - -x ? 0 = 0 -x ? y = y + y - -x ? 0 = 0 -x ? y = dec (dec x) - -x ? 0 = 0 -x ? y = dec (dec y) - -x ? 0 = 0 -x ? y = dec (dec 0) - -x ? 0 = dec x -x ? y = dec y - -x ? 0 = dec x -x ? y = dec 0 - -x ? 0 = dec 0 -x ? y = dec x - -x ? 0 = dec 0 -x ? y = dec y - -x ? 0 = x + x -x ? y = x - -x ? 0 = x + x -x ? y = y - -x ? 0 = x + x -x ? y = 0 - -x ? 0 = dec (dec x) -x ? y = x - -x ? 0 = dec (dec x) -x ? y = y - -x ? 0 = dec (dec x) -x ? y = 0 - -x ? 0 = dec (dec 0) -x ? y = x - -x ? 0 = dec (dec 0) -x ? y = y - -x ? 0 = dec (dec 0) -x ? y = 0 - -0 ? x = x -x ? y = x + x - -0 ? x = x -x ? y = x + y - -0 ? x = x -x ? y = y + x - -0 ? x = x -x ? y = y + y - -0 ? x = x -x ? y = dec (dec x) - -0 ? x = x -x ? y = dec (dec y) - -0 ? x = x -x ? y = dec (dec 0) - -0 ? x = 0 -x ? y = x + x - -0 ? x = 0 -x ? y = x + y - -0 ? x = 0 -x ? y = y + x - -0 ? x = 0 -x ? y = y + y - -0 ? x = 0 -x ? y = dec (dec x) - -0 ? x = 0 -x ? y = dec (dec y) - -0 ? x = 0 -x ? y = dec (dec 0) - -0 ? x = dec x -x ? y = dec 0 - -0 ? x = dec 0 -x ? y = dec x - -0 ? x = dec 0 -x ? y = dec y - -0 ? x = x + x -x ? y = x - -0 ? x = x + x -x ? y = y - -0 ? x = x + x -x ? y = 0 - -0 ? x = dec (dec x) -x ? y = x - -0 ? x = dec (dec x) -x ? y = y - -0 ? x = dec (dec x) -x ? y = 0 - -0 ? x = dec (dec 0) -x ? y = x - -0 ? x = dec (dec 0) -x ? y = y - -0 ? x = dec (dec 0) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = 0 - -x ? 0 = x -x ? y = x ? dec y - -x ? 0 = x -x ? y = y ? dec y - -x ? 0 = x -x ? y = 0 ? dec y - -x ? 0 = x -x ? y = dec x ? x - -x ? 0 = x -x ? y = dec x ? y - -x ? 0 = x -x ? y = dec x ? 0 - -x ? 0 = 0 -x ? y = x ? dec y - -x ? 0 = 0 -x ? y = y ? dec y - -x ? 0 = 0 -x ? y = 0 ? dec y - -x ? 0 = 0 -x ? y = dec x ? x - -x ? 0 = 0 -x ? y = dec x ? y - -x ? 0 = 0 -x ? y = dec x ? 0 - -0 ? x = x -x ? y = x ? dec y - -0 ? x = x -x ? y = y ? dec y - -0 ? x = x -x ? y = 0 ? dec y - -0 ? x = x -x ? y = dec x ? x - -0 ? x = x -x ? y = dec x ? y - -0 ? x = x -x ? y = dec x ? 0 - -0 ? x = 0 -x ? y = x ? dec y - -0 ? x = 0 -x ? y = y ? dec y - -0 ? x = 0 -x ? y = 0 ? dec y - -0 ? x = 0 -x ? y = dec x ? x - -0 ? x = 0 -x ? y = dec x ? y - -0 ? x = 0 -x ? y = dec x ? 0 - -x ? y = dec (dec x) + x - -x ? y = dec (dec x) + y - -x ? y = dec (dec y) + x - -x ? y = dec (dec y) + y - -x ? y = dec (dec 0) + x - -x ? y = dec (dec 0) + y - -x ? y = dec (dec (dec (dec x))) - -x ? y = dec (dec (dec (dec y))) - -x ? y = dec (dec (dec (dec 0))) - -x ? y = x + (x + x) - -x ? y = x + (x + y) - -x ? y = x + (y + x) - -x ? y = x + (y + y) - -x ? y = x + dec (dec x) - -x ? y = x + dec (dec y) - -x ? y = x + dec (dec 0) - -x ? y = y + (x + x) - -x ? y = y + (x + y) - -x ? y = y + (y + x) - -x ? y = y + (y + y) - -x ? y = y + dec (dec x) - -x ? y = y + dec (dec y) - -x ? y = y + dec (dec 0) - -x ? y = dec x + dec x - -x ? y = dec x + dec y - -x ? y = dec x + dec 0 - -x ? y = dec y + dec x - -x ? y = dec y + dec y - -x ? y = dec y + dec 0 - -x ? y = dec 0 + dec x - -x ? y = dec 0 + dec y - -x ? y = dec 0 + dec 0 - -x ? 0 = x -x ? y = dec x + x - -x ? 0 = x -x ? y = dec x + y - -x ? 0 = x -x ? y = dec y + x - -x ? 0 = x -x ? y = dec y + y - -x ? 0 = x -x ? y = dec 0 + x - -x ? 0 = x -x ? y = dec 0 + y - -x ? 0 = x -x ? y = dec (dec (dec x)) - -x ? 0 = x -x ? y = dec (dec (dec y)) - -x ? 0 = x -x ? y = dec (dec (dec 0)) - -x ? 0 = x -x ? y = x + dec x - -x ? 0 = x -x ? y = x + dec y - -x ? 0 = x -x ? y = x + dec 0 - -x ? 0 = x -x ? y = y + dec x - -x ? 0 = x -x ? y = y + dec y - -x ? 0 = x -x ? y = y + dec 0 - -x ? 0 = 0 -x ? y = dec x + x - -x ? 0 = 0 -x ? y = dec x + y - -x ? 0 = 0 -x ? y = dec y + x - -x ? 0 = 0 -x ? y = dec y + y - -x ? 0 = 0 -x ? y = dec 0 + x - -x ? 0 = 0 -x ? y = dec 0 + y - -x ? 0 = 0 -x ? y = dec (dec (dec x)) - -x ? 0 = 0 -x ? y = dec (dec (dec y)) - -x ? 0 = 0 -x ? y = dec (dec (dec 0)) - -x ? 0 = 0 -x ? y = x + dec x - -x ? 0 = 0 -x ? y = x + dec y - -x ? 0 = 0 -x ? y = x + dec 0 - -x ? 0 = 0 -x ? y = y + dec x - -x ? 0 = 0 -x ? y = y + dec y - -x ? 0 = 0 -x ? y = y + dec 0 - -x ? 0 = dec x -x ? y = x + x - -x ? 0 = dec x -x ? y = x + y - -x ? 0 = dec x -x ? y = y + x - -x ? 0 = dec x -x ? y = y + y - -x ? 0 = dec x -x ? y = dec (dec x) - -x ? 0 = dec x -x ? y = dec (dec y) - -x ? 0 = dec x -x ? y = dec (dec 0) - -x ? 0 = dec 0 -x ? y = x + x - -x ? 0 = dec 0 -x ? y = x + y - -x ? 0 = dec 0 -x ? y = y + x - -x ? 0 = dec 0 -x ? y = y + y - -x ? 0 = dec 0 -x ? y = dec (dec x) - -x ? 0 = dec 0 -x ? y = dec (dec y) - -x ? 0 = dec 0 -x ? y = dec (dec 0) - -x ? 0 = x + x -x ? y = dec x - -x ? 0 = x + x -x ? y = dec y - -x ? 0 = x + x -x ? y = dec 0 - -x ? 0 = dec (dec x) -x ? y = dec x - -x ? 0 = dec (dec x) -x ? y = dec y - -x ? 0 = dec (dec x) -x ? y = dec 0 - -x ? 0 = dec (dec 0) -x ? y = dec x - -x ? 0 = dec (dec 0) -x ? y = dec y - -x ? 0 = dec (dec 0) -x ? y = dec 0 - -x ? 0 = dec x + x -x ? y = x - -x ? 0 = dec x + x -x ? y = y - -x ? 0 = dec x + x -x ? y = 0 - -x ? 0 = dec 0 + x -x ? y = x - -x ? 0 = dec 0 + x -x ? y = y - -x ? 0 = dec 0 + x -x ? y = 0 - -x ? 0 = dec (dec (dec x)) -x ? y = x - -x ? 0 = dec (dec (dec x)) -x ? y = y - -x ? 0 = dec (dec (dec x)) -x ? y = 0 - -x ? 0 = dec (dec (dec 0)) -x ? y = x - -x ? 0 = dec (dec (dec 0)) -x ? y = y - -x ? 0 = dec (dec (dec 0)) -x ? y = 0 - -x ? 0 = x + dec x -x ? y = x - -x ? 0 = x + dec x -x ? y = y - -x ? 0 = x + dec x -x ? y = 0 - -x ? 0 = x + dec 0 -x ? y = x - -x ? 0 = x + dec 0 -x ? y = y - -x ? 0 = x + dec 0 -x ? y = 0 - -0 ? x = x -x ? y = dec x + x - -0 ? x = x -x ? y = dec x + y - -0 ? x = x -x ? y = dec y + x - -0 ? x = x -x ? y = dec y + y - -0 ? x = x -x ? y = dec 0 + x - -0 ? x = x -x ? y = dec 0 + y - -0 ? x = x -x ? y = dec (dec (dec x)) - -0 ? x = x -x ? y = dec (dec (dec y)) - -0 ? x = x -x ? y = dec (dec (dec 0)) - -0 ? x = x -x ? y = x + dec x - -0 ? x = x -x ? y = x + dec y - -0 ? x = x -x ? y = x + dec 0 - -0 ? x = x -x ? y = y + dec x - -0 ? x = x -x ? y = y + dec y - -0 ? x = x -x ? y = y + dec 0 - -0 ? x = 0 -x ? y = dec x + x - -0 ? x = 0 -x ? y = dec x + y - -0 ? x = 0 -x ? y = dec y + x - -0 ? x = 0 -x ? y = dec y + y - -0 ? x = 0 -x ? y = dec 0 + x - -0 ? x = 0 -x ? y = dec 0 + y - -0 ? x = 0 -x ? y = dec (dec (dec x)) - -0 ? x = 0 -x ? y = dec (dec (dec y)) - -0 ? x = 0 -x ? y = dec (dec (dec 0)) - -0 ? x = 0 -x ? y = x + dec x - -0 ? x = 0 -x ? y = x + dec y - -0 ? x = 0 -x ? y = x + dec 0 - -0 ? x = 0 -x ? y = y + dec x - -0 ? x = 0 -x ? y = y + dec y - -0 ? x = 0 -x ? y = y + dec 0 - -0 ? x = dec x -x ? y = x + x - -0 ? x = dec x -x ? y = x + y - -0 ? x = dec x -x ? y = y + x - -0 ? x = dec x -x ? y = y + y - -0 ? x = dec x -x ? y = dec (dec x) - -0 ? x = dec x -x ? y = dec (dec y) - -0 ? x = dec x -x ? y = dec (dec 0) - -0 ? x = dec 0 -x ? y = x + x - -0 ? x = dec 0 -x ? y = x + y - -0 ? x = dec 0 -x ? y = y + x - -0 ? x = dec 0 -x ? y = y + y - -0 ? x = dec 0 -x ? y = dec (dec x) - -0 ? x = dec 0 -x ? y = dec (dec y) - -0 ? x = dec 0 -x ? y = dec (dec 0) - -0 ? x = x + x -x ? y = dec x - -0 ? x = x + x -x ? y = dec y - -0 ? x = x + x -x ? y = dec 0 - -0 ? x = dec (dec x) -x ? y = dec x - -0 ? x = dec (dec x) -x ? y = dec y - -0 ? x = dec (dec x) -x ? y = dec 0 - -0 ? x = dec (dec 0) -x ? y = dec x - -0 ? x = dec (dec 0) -x ? y = dec y - -0 ? x = dec (dec 0) -x ? y = dec 0 - -0 ? x = dec x + x -x ? y = x - -0 ? x = dec x + x -x ? y = y - -0 ? x = dec x + x -x ? y = 0 - -0 ? x = dec 0 + x -x ? y = x - -0 ? x = dec 0 + x -x ? y = y - -0 ? x = dec 0 + x -x ? y = 0 - -0 ? x = dec (dec (dec x)) -x ? y = x - -0 ? x = dec (dec (dec x)) -x ? y = y - -0 ? x = dec (dec (dec x)) -x ? y = 0 - -0 ? x = dec (dec (dec 0)) -x ? y = x - -0 ? x = dec (dec (dec 0)) -x ? y = y - -0 ? x = dec (dec (dec 0)) -x ? y = 0 - -0 ? x = x + dec x -x ? y = x - -0 ? x = x + dec x -x ? y = y - -0 ? x = x + dec x -x ? y = 0 - -0 ? x = x + dec 0 -x ? y = x - -0 ? x = x + dec 0 -x ? y = y - -0 ? x = x + dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = x + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = x + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = y + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = y + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = x + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = x + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = y + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = y + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + x -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + x -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec x) -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec x) -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec x) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec 0) -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec 0) -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec 0) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = x + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = x + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = y + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = y + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = x + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = x + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = y + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = y + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + x -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + x -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec x) -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec x) -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec x) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec 0) -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec 0) -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec 0) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = x -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = x -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = x -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = x -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = 0 -x ? y = 0 - -x ? 0 = x -x ? y = x ? dec (dec y) - -x ? 0 = x -x ? y = y ? dec (dec y) - -x ? 0 = x -x ? y = 0 ? dec (dec y) - -x ? 0 = x -x ? y = dec x ? dec x - -x ? 0 = x -x ? y = dec x ? dec y - -x ? 0 = x -x ? y = dec x ? dec 0 - -x ? 0 = x -x ? y = dec y ? dec y - -x ? 0 = x -x ? y = dec 0 ? dec y - -x ? 0 = x -x ? y = dec (dec x) ? x - -x ? 0 = x -x ? y = dec (dec x) ? y - -x ? 0 = x -x ? y = dec (dec x) ? 0 - -x ? 0 = 0 -x ? y = x ? dec (dec y) - -x ? 0 = 0 -x ? y = y ? dec (dec y) - -x ? 0 = 0 -x ? y = 0 ? dec (dec y) - -x ? 0 = 0 -x ? y = dec x ? dec x - -x ? 0 = 0 -x ? y = dec x ? dec y - -x ? 0 = 0 -x ? y = dec x ? dec 0 - -x ? 0 = 0 -x ? y = dec y ? dec y - -x ? 0 = 0 -x ? y = dec 0 ? dec y - -x ? 0 = 0 -x ? y = dec (dec x) ? x - -x ? 0 = 0 -x ? y = dec (dec x) ? y - -x ? 0 = 0 -x ? y = dec (dec x) ? 0 - -0 ? x = x -x ? y = x ? dec (dec y) - -0 ? x = x -x ? y = y ? dec (dec y) - -0 ? x = x -x ? y = 0 ? dec (dec y) - -0 ? x = x -x ? y = dec x ? dec x - -0 ? x = x -x ? y = dec x ? dec y - -0 ? x = x -x ? y = dec x ? dec 0 - -0 ? x = x -x ? y = dec y ? dec y - -0 ? x = x -x ? y = dec 0 ? dec y - -0 ? x = x -x ? y = dec (dec x) ? x - -0 ? x = x -x ? y = dec (dec x) ? y - -0 ? x = x -x ? y = dec (dec x) ? 0 - -0 ? x = 0 -x ? y = x ? dec (dec y) - -0 ? x = 0 -x ? y = y ? dec (dec y) - -0 ? x = 0 -x ? y = 0 ? dec (dec y) - -0 ? x = 0 -x ? y = dec x ? dec x - -0 ? x = 0 -x ? y = dec x ? dec y - -0 ? x = 0 -x ? y = dec x ? dec 0 - -0 ? x = 0 -x ? y = dec y ? dec y - -0 ? x = 0 -x ? y = dec 0 ? dec y - -0 ? x = 0 -x ? y = dec (dec x) ? x - -0 ? x = 0 -x ? y = dec (dec x) ? y - -0 ? x = 0 -x ? y = dec (dec x) ? 0 - -x ? 0 = x -x ? y = dec (x ? dec y) - -x ? 0 = x -x ? y = dec (y ? dec y) - -x ? 0 = x -x ? y = dec (0 ? dec y) - -x ? 0 = x -x ? y = dec (dec x ? x) - -x ? 0 = x -x ? y = dec (dec x ? y) - -x ? 0 = x -x ? y = dec (dec x ? 0) - -x ? 0 = 0 -x ? y = dec (x ? dec y) - -x ? 0 = 0 -x ? y = dec (y ? dec y) - -x ? 0 = 0 -x ? y = dec (0 ? dec y) - -x ? 0 = 0 -x ? y = dec (dec x ? x) - -x ? 0 = 0 -x ? y = dec (dec x ? y) - -x ? 0 = 0 -x ? y = dec (dec x ? 0) - -x ? 0 = dec x -x ? y = x ? dec y - -x ? 0 = dec x -x ? y = y ? dec y - -x ? 0 = dec x -x ? y = 0 ? dec y - -x ? 0 = dec x -x ? y = dec x ? x - -x ? 0 = dec x -x ? y = dec x ? y - -x ? 0 = dec x -x ? y = dec x ? 0 - -x ? 0 = dec 0 -x ? y = x ? dec y - -x ? 0 = dec 0 -x ? y = y ? dec y - -x ? 0 = dec 0 -x ? y = 0 ? dec y - -x ? 0 = dec 0 -x ? y = dec x ? x - -x ? 0 = dec 0 -x ? y = dec x ? y - -x ? 0 = dec 0 -x ? y = dec x ? 0 - -0 ? x = x -x ? y = dec (x ? dec y) - -0 ? x = x -x ? y = dec (y ? dec y) - -0 ? x = x -x ? y = dec (0 ? dec y) - -0 ? x = x -x ? y = dec (dec x ? x) - -0 ? x = x -x ? y = dec (dec x ? y) - -0 ? x = x -x ? y = dec (dec x ? 0) - -0 ? x = 0 -x ? y = dec (x ? dec y) - -0 ? x = 0 -x ? y = dec (y ? dec y) - -0 ? x = 0 -x ? y = dec (0 ? dec y) - -0 ? x = 0 -x ? y = dec (dec x ? x) - -0 ? x = 0 -x ? y = dec (dec x ? y) - -0 ? x = 0 -x ? y = dec (dec x ? 0) - -0 ? x = dec x -x ? y = x ? dec y - -0 ? x = dec x -x ? y = y ? dec y - -0 ? x = dec x -x ? y = 0 ? dec y - -0 ? x = dec x -x ? y = dec x ? x - -0 ? x = dec x -x ? y = dec x ? y - -0 ? x = dec x -x ? y = dec x ? 0 - -0 ? x = dec 0 -x ? y = x ? dec y - -0 ? x = dec 0 -x ? y = y ? dec y - -0 ? x = dec 0 -x ? y = 0 ? dec y - -0 ? x = dec 0 -x ? y = dec x ? x - -0 ? x = dec 0 -x ? y = dec x ? y - -0 ? x = dec 0 -x ? y = dec x ? 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = x ? dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = y ? dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = 0 ? dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec x ? x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec x ? y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec x ? 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x ? x -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x ? 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x ? x -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x ? 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x ? x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x ? 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = x ? dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = y ? dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = 0 ? dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec x ? x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec x ? y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec x ? 0 - -0 ? 0 = 0 -0 ? x = x ? dec x -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 ? dec x -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = x ? dec x -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 ? dec x -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = x ? dec x -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 ? dec x -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x ? dec x -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 ? dec x -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = x ? dec x -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 ? dec x -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = x ? dec x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 ? dec x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = x ? dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = y ? dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = 0 ? dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec x ? x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec x ? y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec x ? 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x ? x -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x ? 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x ? x -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x ? 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x ? x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x ? 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = x ? dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = y ? dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = 0 ? dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec x ? x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec x ? y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec x ? 0 - -x ? y = dec (dec (dec x)) + x - -x ? y = dec (dec (dec x)) + y - -x ? y = dec (dec (dec y)) + x - -x ? y = dec (dec (dec y)) + y - -x ? y = dec (dec (dec 0)) + x - -x ? y = dec (dec (dec 0)) + y - -x ? y = dec (dec (dec (dec (dec x)))) - -x ? y = dec (dec (dec (dec (dec y)))) - -x ? y = dec (dec (dec (dec (dec 0)))) - -x ? y = x + (dec x + x) - -x ? y = x + (dec x + y) - -x ? y = x + (dec y + x) - -x ? y = x + (dec y + y) - -x ? y = x + (dec 0 + x) - -x ? y = x + (dec 0 + y) - -x ? y = x + dec (dec (dec x)) - -x ? y = x + dec (dec (dec y)) - -x ? y = x + dec (dec (dec 0)) - -x ? y = x + (x + dec x) - -x ? y = x + (x + dec y) - -x ? y = x + (x + dec 0) - -x ? y = x + (y + dec x) - -x ? y = x + (y + dec y) - -x ? y = x + (y + dec 0) - -x ? y = y + (dec x + x) - -x ? y = y + (dec x + y) - -x ? y = y + (dec y + x) - -x ? y = y + (dec y + y) - -x ? y = y + (dec 0 + x) - -x ? y = y + (dec 0 + y) - -x ? y = y + dec (dec (dec x)) - -x ? y = y + dec (dec (dec y)) - -x ? y = y + dec (dec (dec 0)) - -x ? y = y + (x + dec x) - -x ? y = y + (x + dec y) - -x ? y = y + (x + dec 0) - -x ? y = y + (y + dec x) - -x ? y = y + (y + dec y) - -x ? y = y + (y + dec 0) - -x ? y = dec x + (x + x) - -x ? y = dec x + (x + y) - -x ? y = dec x + (y + x) - -x ? y = dec x + (y + y) - -x ? y = dec x + dec (dec x) - -x ? y = dec x + dec (dec y) - -x ? y = dec x + dec (dec 0) - -x ? y = dec y + (x + x) - -x ? y = dec y + (x + y) - -x ? y = dec y + (y + x) - -x ? y = dec y + (y + y) - -x ? y = dec y + dec (dec x) - -x ? y = dec y + dec (dec y) - -x ? y = dec y + dec (dec 0) - -x ? y = dec 0 + (x + x) - -x ? y = dec 0 + (x + y) - -x ? y = dec 0 + (y + x) - -x ? y = dec 0 + (y + y) - -x ? y = dec 0 + dec (dec x) - -x ? y = dec 0 + dec (dec y) - -x ? y = dec 0 + dec (dec 0) - -x ? y = dec (dec x) + dec x - -x ? y = dec (dec x) + dec y - -x ? y = dec (dec x) + dec 0 - -x ? y = dec (dec y) + dec x - -x ? y = dec (dec y) + dec y - -x ? y = dec (dec y) + dec 0 - -x ? y = dec (dec 0) + dec x - -x ? y = dec (dec 0) + dec y - -x ? y = dec (dec 0) + dec 0 - -x ? 0 = x -x ? y = dec (dec x) + x - -x ? 0 = x -x ? y = dec (dec x) + y - -x ? 0 = x -x ? y = dec (dec y) + x - -x ? 0 = x -x ? y = dec (dec y) + y - -x ? 0 = x -x ? y = dec (dec 0) + x - -x ? 0 = x -x ? y = dec (dec 0) + y - -x ? 0 = x -x ? y = dec (dec (dec (dec x))) - -x ? 0 = x -x ? y = dec (dec (dec (dec y))) - -x ? 0 = x -x ? y = dec (dec (dec (dec 0))) - -x ? 0 = x -x ? y = x + (x + x) - -x ? 0 = x -x ? y = x + (x + y) - -x ? 0 = x -x ? y = x + (y + x) - -x ? 0 = x -x ? y = x + (y + y) - -x ? 0 = x -x ? y = x + dec (dec x) - -x ? 0 = x -x ? y = x + dec (dec y) - -x ? 0 = x -x ? y = x + dec (dec 0) - -x ? 0 = x -x ? y = y + (x + x) - -x ? 0 = x -x ? y = y + (x + y) - -x ? 0 = x -x ? y = y + (y + x) - -x ? 0 = x -x ? y = y + (y + y) - -x ? 0 = x -x ? y = y + dec (dec x) - -x ? 0 = x -x ? y = y + dec (dec y) - -x ? 0 = x -x ? y = y + dec (dec 0) - -x ? 0 = x -x ? y = dec x + dec x - -x ? 0 = x -x ? y = dec x + dec y - -x ? 0 = x -x ? y = dec x + dec 0 - -x ? 0 = x -x ? y = dec y + dec x - -x ? 0 = x -x ? y = dec y + dec y - -x ? 0 = x -x ? y = dec y + dec 0 - -x ? 0 = x -x ? y = dec 0 + dec x - -x ? 0 = x -x ? y = dec 0 + dec y - -x ? 0 = x -x ? y = dec 0 + dec 0 - -x ? 0 = 0 -x ? y = dec (dec x) + x - -x ? 0 = 0 -x ? y = dec (dec x) + y - -x ? 0 = 0 -x ? y = dec (dec y) + x - -x ? 0 = 0 -x ? y = dec (dec y) + y - -x ? 0 = 0 -x ? y = dec (dec 0) + x - -x ? 0 = 0 -x ? y = dec (dec 0) + y - -x ? 0 = 0 -x ? y = dec (dec (dec (dec x))) - -x ? 0 = 0 -x ? y = dec (dec (dec (dec y))) - -x ? 0 = 0 -x ? y = dec (dec (dec (dec 0))) - -x ? 0 = 0 -x ? y = x + (x + x) - -x ? 0 = 0 -x ? y = x + (x + y) - -x ? 0 = 0 -x ? y = x + (y + x) - -x ? 0 = 0 -x ? y = x + (y + y) - -x ? 0 = 0 -x ? y = x + dec (dec x) - -x ? 0 = 0 -x ? y = x + dec (dec y) - -x ? 0 = 0 -x ? y = x + dec (dec 0) - -x ? 0 = 0 -x ? y = y + (x + x) - -x ? 0 = 0 -x ? y = y + (x + y) - -x ? 0 = 0 -x ? y = y + (y + x) - -x ? 0 = 0 -x ? y = y + (y + y) - -x ? 0 = 0 -x ? y = y + dec (dec x) - -x ? 0 = 0 -x ? y = y + dec (dec y) - -x ? 0 = 0 -x ? y = y + dec (dec 0) - -x ? 0 = 0 -x ? y = dec x + dec x - -x ? 0 = 0 -x ? y = dec x + dec y - -x ? 0 = 0 -x ? y = dec x + dec 0 - -x ? 0 = 0 -x ? y = dec y + dec x - -x ? 0 = 0 -x ? y = dec y + dec y - -x ? 0 = 0 -x ? y = dec y + dec 0 - -x ? 0 = 0 -x ? y = dec 0 + dec x - -x ? 0 = 0 -x ? y = dec 0 + dec y - -x ? 0 = 0 -x ? y = dec 0 + dec 0 - -x ? 0 = dec x -x ? y = dec x + x - -x ? 0 = dec x -x ? y = dec x + y - -x ? 0 = dec x -x ? y = dec y + x - -x ? 0 = dec x -x ? y = dec y + y - -x ? 0 = dec x -x ? y = dec 0 + x - -x ? 0 = dec x -x ? y = dec 0 + y - -x ? 0 = dec x -x ? y = dec (dec (dec x)) - -x ? 0 = dec x -x ? y = dec (dec (dec y)) - -x ? 0 = dec x -x ? y = dec (dec (dec 0)) - -x ? 0 = dec x -x ? y = x + dec x - -x ? 0 = dec x -x ? y = x + dec y - -x ? 0 = dec x -x ? y = x + dec 0 - -x ? 0 = dec x -x ? y = y + dec x - -x ? 0 = dec x -x ? y = y + dec y - -x ? 0 = dec x -x ? y = y + dec 0 - -x ? 0 = dec 0 -x ? y = dec x + x - -x ? 0 = dec 0 -x ? y = dec x + y - -x ? 0 = dec 0 -x ? y = dec y + x - -x ? 0 = dec 0 -x ? y = dec y + y - -x ? 0 = dec 0 -x ? y = dec 0 + x - -x ? 0 = dec 0 -x ? y = dec 0 + y - -x ? 0 = dec 0 -x ? y = dec (dec (dec x)) - -x ? 0 = dec 0 -x ? y = dec (dec (dec y)) - -x ? 0 = dec 0 -x ? y = dec (dec (dec 0)) - -x ? 0 = dec 0 -x ? y = x + dec x - -x ? 0 = dec 0 -x ? y = x + dec y - -x ? 0 = dec 0 -x ? y = x + dec 0 - -x ? 0 = dec 0 -x ? y = y + dec x - -x ? 0 = dec 0 -x ? y = y + dec y - -x ? 0 = dec 0 -x ? y = y + dec 0 - -x ? 0 = x + x -x ? y = x + y - -x ? 0 = x + x -x ? y = y + x - -x ? 0 = x + x -x ? y = y + y - -x ? 0 = x + x -x ? y = dec (dec x) - -x ? 0 = x + x -x ? y = dec (dec y) - -x ? 0 = x + x -x ? y = dec (dec 0) - -x ? 0 = dec (dec x) -x ? y = x + x - -x ? 0 = dec (dec x) -x ? y = x + y - -x ? 0 = dec (dec x) -x ? y = y + x - -x ? 0 = dec (dec x) -x ? y = y + y - -x ? 0 = dec (dec x) -x ? y = dec (dec y) - -x ? 0 = dec (dec x) -x ? y = dec (dec 0) - -x ? 0 = dec (dec 0) -x ? y = x + x - -x ? 0 = dec (dec 0) -x ? y = x + y - -x ? 0 = dec (dec 0) -x ? y = y + x - -x ? 0 = dec (dec 0) -x ? y = y + y - -x ? 0 = dec (dec 0) -x ? y = dec (dec x) - -x ? 0 = dec (dec 0) -x ? y = dec (dec y) - -x ? 0 = dec x + x -x ? y = dec x - -x ? 0 = dec x + x -x ? y = dec y - -x ? 0 = dec x + x -x ? y = dec 0 - -x ? 0 = dec 0 + x -x ? y = dec x - -x ? 0 = dec 0 + x -x ? y = dec y - -x ? 0 = dec 0 + x -x ? y = dec 0 - -x ? 0 = dec (dec (dec x)) -x ? y = dec x - -x ? 0 = dec (dec (dec x)) -x ? y = dec y - -x ? 0 = dec (dec (dec x)) -x ? y = dec 0 - -x ? 0 = dec (dec (dec 0)) -x ? y = dec x - -x ? 0 = dec (dec (dec 0)) -x ? y = dec y - -x ? 0 = dec (dec (dec 0)) -x ? y = dec 0 - -x ? 0 = x + dec x -x ? y = dec x - -x ? 0 = x + dec x -x ? y = dec y - -x ? 0 = x + dec x -x ? y = dec 0 - -x ? 0 = x + dec 0 -x ? y = dec x - -x ? 0 = x + dec 0 -x ? y = dec y - -x ? 0 = x + dec 0 -x ? y = dec 0 - -x ? 0 = dec (dec x) + x -x ? y = x - -x ? 0 = dec (dec x) + x -x ? y = y - -x ? 0 = dec (dec x) + x -x ? y = 0 - -x ? 0 = dec (dec 0) + x -x ? y = x - -x ? 0 = dec (dec 0) + x -x ? y = y - -x ? 0 = dec (dec 0) + x -x ? y = 0 - -x ? 0 = dec (dec (dec (dec x))) -x ? y = x - -x ? 0 = dec (dec (dec (dec x))) -x ? y = y - -x ? 0 = dec (dec (dec (dec x))) -x ? y = 0 - -x ? 0 = dec (dec (dec (dec 0))) -x ? y = x - -x ? 0 = dec (dec (dec (dec 0))) -x ? y = y - -x ? 0 = dec (dec (dec (dec 0))) -x ? y = 0 - -x ? 0 = x + (x + x) -x ? y = x - -x ? 0 = x + (x + x) -x ? y = y - -x ? 0 = x + (x + x) -x ? y = 0 - -x ? 0 = x + dec (dec x) -x ? y = x - -x ? 0 = x + dec (dec x) -x ? y = y - -x ? 0 = x + dec (dec x) -x ? y = 0 - -x ? 0 = x + dec (dec 0) -x ? y = x - -x ? 0 = x + dec (dec 0) -x ? y = y - -x ? 0 = x + dec (dec 0) -x ? y = 0 - -x ? 0 = dec x + dec x -x ? y = x - -x ? 0 = dec x + dec x -x ? y = y - -x ? 0 = dec x + dec x -x ? y = 0 - -x ? 0 = dec x + dec 0 -x ? y = x - -x ? 0 = dec x + dec 0 -x ? y = y - -x ? 0 = dec x + dec 0 -x ? y = 0 - -x ? 0 = dec 0 + dec x -x ? y = x - -x ? 0 = dec 0 + dec x -x ? y = y - -x ? 0 = dec 0 + dec x -x ? y = 0 - -x ? 0 = dec 0 + dec 0 -x ? y = x - -x ? 0 = dec 0 + dec 0 -x ? y = y - -x ? 0 = dec 0 + dec 0 -x ? y = 0 - -0 ? x = x -x ? y = dec (dec x) + x - -0 ? x = x -x ? y = dec (dec x) + y - -0 ? x = x -x ? y = dec (dec y) + x - -0 ? x = x -x ? y = dec (dec y) + y - -0 ? x = x -x ? y = dec (dec 0) + x - -0 ? x = x -x ? y = dec (dec 0) + y - -0 ? x = x -x ? y = dec (dec (dec (dec x))) - -0 ? x = x -x ? y = dec (dec (dec (dec y))) - -0 ? x = x -x ? y = dec (dec (dec (dec 0))) - -0 ? x = x -x ? y = x + (x + x) - -0 ? x = x -x ? y = x + (x + y) - -0 ? x = x -x ? y = x + (y + x) - -0 ? x = x -x ? y = x + (y + y) - -0 ? x = x -x ? y = x + dec (dec x) - -0 ? x = x -x ? y = x + dec (dec y) - -0 ? x = x -x ? y = x + dec (dec 0) - -0 ? x = x -x ? y = y + (x + x) - -0 ? x = x -x ? y = y + (x + y) - -0 ? x = x -x ? y = y + (y + x) - -0 ? x = x -x ? y = y + (y + y) - -0 ? x = x -x ? y = y + dec (dec x) - -0 ? x = x -x ? y = y + dec (dec y) - -0 ? x = x -x ? y = y + dec (dec 0) - -0 ? x = x -x ? y = dec x + dec x - -0 ? x = x -x ? y = dec x + dec y - -0 ? x = x -x ? y = dec x + dec 0 - -0 ? x = x -x ? y = dec y + dec x - -0 ? x = x -x ? y = dec y + dec y - -0 ? x = x -x ? y = dec y + dec 0 - -0 ? x = x -x ? y = dec 0 + dec x - -0 ? x = x -x ? y = dec 0 + dec y - -0 ? x = x -x ? y = dec 0 + dec 0 - -0 ? x = 0 -x ? y = dec (dec x) + x - -0 ? x = 0 -x ? y = dec (dec x) + y - -0 ? x = 0 -x ? y = dec (dec y) + x - -0 ? x = 0 -x ? y = dec (dec y) + y - -0 ? x = 0 -x ? y = dec (dec 0) + x - -0 ? x = 0 -x ? y = dec (dec 0) + y - -0 ? x = 0 -x ? y = dec (dec (dec (dec x))) - -0 ? x = 0 -x ? y = dec (dec (dec (dec y))) - -0 ? x = 0 -x ? y = dec (dec (dec (dec 0))) - -0 ? x = 0 -x ? y = x + (x + x) - -0 ? x = 0 -x ? y = x + (x + y) - -0 ? x = 0 -x ? y = x + (y + x) - -0 ? x = 0 -x ? y = x + (y + y) - -0 ? x = 0 -x ? y = x + dec (dec x) - -0 ? x = 0 -x ? y = x + dec (dec y) - -0 ? x = 0 -x ? y = x + dec (dec 0) - -0 ? x = 0 -x ? y = y + (x + x) - -0 ? x = 0 -x ? y = y + (x + y) - -0 ? x = 0 -x ? y = y + (y + x) - -0 ? x = 0 -x ? y = y + (y + y) - -0 ? x = 0 -x ? y = y + dec (dec x) - -0 ? x = 0 -x ? y = y + dec (dec y) - -0 ? x = 0 -x ? y = y + dec (dec 0) - -0 ? x = 0 -x ? y = dec x + dec x - -0 ? x = 0 -x ? y = dec x + dec y - -0 ? x = 0 -x ? y = dec x + dec 0 - -0 ? x = 0 -x ? y = dec y + dec x - -0 ? x = 0 -x ? y = dec y + dec y - -0 ? x = 0 -x ? y = dec y + dec 0 - -0 ? x = 0 -x ? y = dec 0 + dec x - -0 ? x = 0 -x ? y = dec 0 + dec y - -0 ? x = 0 -x ? y = dec 0 + dec 0 - -0 ? x = dec x -x ? y = dec x + x - -0 ? x = dec x -x ? y = dec x + y - -0 ? x = dec x -x ? y = dec y + x - -0 ? x = dec x -x ? y = dec y + y - -0 ? x = dec x -x ? y = dec 0 + x - -0 ? x = dec x -x ? y = dec 0 + y - -0 ? x = dec x -x ? y = dec (dec (dec x)) - -0 ? x = dec x -x ? y = dec (dec (dec y)) - -0 ? x = dec x -x ? y = dec (dec (dec 0)) - -0 ? x = dec x -x ? y = x + dec x - -0 ? x = dec x -x ? y = x + dec y - -0 ? x = dec x -x ? y = x + dec 0 - -0 ? x = dec x -x ? y = y + dec x - -0 ? x = dec x -x ? y = y + dec y - -0 ? x = dec x -x ? y = y + dec 0 - -0 ? x = dec 0 -x ? y = dec x + x - -0 ? x = dec 0 -x ? y = dec x + y - -0 ? x = dec 0 -x ? y = dec y + x - -0 ? x = dec 0 -x ? y = dec y + y - -0 ? x = dec 0 -x ? y = dec 0 + x - -0 ? x = dec 0 -x ? y = dec 0 + y - -0 ? x = dec 0 -x ? y = dec (dec (dec x)) - -0 ? x = dec 0 -x ? y = dec (dec (dec y)) - -0 ? x = dec 0 -x ? y = dec (dec (dec 0)) - -0 ? x = dec 0 -x ? y = x + dec x - -0 ? x = dec 0 -x ? y = x + dec y - -0 ? x = dec 0 -x ? y = x + dec 0 - -0 ? x = dec 0 -x ? y = y + dec x - -0 ? x = dec 0 -x ? y = y + dec y - -0 ? x = dec 0 -x ? y = y + dec 0 - -0 ? x = x + x -x ? y = x + y - -0 ? x = x + x -x ? y = y + x - -0 ? x = x + x -x ? y = dec (dec x) - -0 ? x = x + x -x ? y = dec (dec y) - -0 ? x = x + x -x ? y = dec (dec 0) - -0 ? x = dec (dec x) -x ? y = x + x - -0 ? x = dec (dec x) -x ? y = x + y - -0 ? x = dec (dec x) -x ? y = y + x - -0 ? x = dec (dec x) -x ? y = y + y - -0 ? x = dec (dec x) -x ? y = dec (dec 0) - -0 ? x = dec (dec 0) -x ? y = x + x - -0 ? x = dec (dec 0) -x ? y = x + y - -0 ? x = dec (dec 0) -x ? y = y + x - -0 ? x = dec (dec 0) -x ? y = y + y - -0 ? x = dec (dec 0) -x ? y = dec (dec x) - -0 ? x = dec (dec 0) -x ? y = dec (dec y) - -0 ? x = dec x + x -x ? y = dec x - -0 ? x = dec x + x -x ? y = dec y - -0 ? x = dec x + x -x ? y = dec 0 - -0 ? x = dec 0 + x -x ? y = dec x - -0 ? x = dec 0 + x -x ? y = dec y - -0 ? x = dec 0 + x -x ? y = dec 0 - -0 ? x = dec (dec (dec x)) -x ? y = dec x - -0 ? x = dec (dec (dec x)) -x ? y = dec y - -0 ? x = dec (dec (dec x)) -x ? y = dec 0 - -0 ? x = dec (dec (dec 0)) -x ? y = dec x - -0 ? x = dec (dec (dec 0)) -x ? y = dec y - -0 ? x = dec (dec (dec 0)) -x ? y = dec 0 - -0 ? x = x + dec x -x ? y = dec x - -0 ? x = x + dec x -x ? y = dec y - -0 ? x = x + dec x -x ? y = dec 0 - -0 ? x = x + dec 0 -x ? y = dec x - -0 ? x = x + dec 0 -x ? y = dec y - -0 ? x = x + dec 0 -x ? y = dec 0 - -0 ? x = dec (dec x) + x -x ? y = x - -0 ? x = dec (dec x) + x -x ? y = y - -0 ? x = dec (dec x) + x -x ? y = 0 - -0 ? x = dec (dec 0) + x -x ? y = x - -0 ? x = dec (dec 0) + x -x ? y = y - -0 ? x = dec (dec 0) + x -x ? y = 0 - -0 ? x = dec (dec (dec (dec x))) -x ? y = x - -0 ? x = dec (dec (dec (dec x))) -x ? y = y - -0 ? x = dec (dec (dec (dec x))) -x ? y = 0 - -0 ? x = dec (dec (dec (dec 0))) -x ? y = x - -0 ? x = dec (dec (dec (dec 0))) -x ? y = y - -0 ? x = dec (dec (dec (dec 0))) -x ? y = 0 - -0 ? x = x + (x + x) -x ? y = x - -0 ? x = x + (x + x) -x ? y = y - -0 ? x = x + (x + x) -x ? y = 0 - -0 ? x = x + dec (dec x) -x ? y = x - -0 ? x = x + dec (dec x) -x ? y = y - -0 ? x = x + dec (dec x) -x ? y = 0 - -0 ? x = x + dec (dec 0) -x ? y = x - -0 ? x = x + dec (dec 0) -x ? y = y - -0 ? x = x + dec (dec 0) -x ? y = 0 - -0 ? x = dec x + dec x -x ? y = x - -0 ? x = dec x + dec x -x ? y = y - -0 ? x = dec x + dec x -x ? y = 0 - -0 ? x = dec x + dec 0 -x ? y = x - -0 ? x = dec x + dec 0 -x ? y = y - -0 ? x = dec x + dec 0 -x ? y = 0 - -0 ? x = dec 0 + dec x -x ? y = x - -0 ? x = dec 0 + dec x -x ? y = y - -0 ? x = dec 0 + dec x -x ? y = 0 - -0 ? x = dec 0 + dec 0 -x ? y = x - -0 ? x = dec 0 + dec 0 -x ? y = y - -0 ? x = dec 0 + dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec x + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec x + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec y + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec y + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec 0 + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec 0 + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec (dec (dec x)) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec (dec (dec y)) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = dec (dec (dec 0)) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = x + dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = x + dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = x + dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = y + dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = y + dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x -x ? y = y + dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec x + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec x + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec y + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec y + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec 0 + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec 0 + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec (dec (dec x)) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec (dec (dec y)) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec (dec (dec 0)) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = x + dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = x + dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = x + dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = y + dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = y + dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = y + dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = x + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = x + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = y + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = y + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = x + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = x + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = y + x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = y + y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec x) -x ? y = dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec x) -x ? y = dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec x) -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec 0) -x ? y = dec x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec 0) -x ? y = dec y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec 0) -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x + x -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x + x -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec x + x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 + x -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 + x -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec 0 + x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec (dec x)) -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec (dec x)) -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec (dec x)) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec (dec 0)) -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec (dec 0)) -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = dec (dec (dec 0)) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + dec x -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + dec x -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + dec x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + dec 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + dec 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = x + dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec x + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec x + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec y + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec y + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec 0 + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec 0 + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec (dec (dec x)) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec (dec (dec y)) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec (dec (dec 0)) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = x + dec x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = x + dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = x + dec 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = y + dec x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = y + dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = y + dec 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec x + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec x + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec y + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec y + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec 0 + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec 0 + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec (dec (dec x)) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec (dec (dec y)) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec (dec (dec 0)) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = x + dec x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = x + dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = x + dec 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = y + dec x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = y + dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = y + dec 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = x + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = x + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = y + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = y + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = x + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = x + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = y + x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = y + y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec x) -x ? y = dec x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec x) -x ? y = dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec x) -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec 0) -x ? y = dec x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec 0) -x ? y = dec y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec 0) -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x + x -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x + x -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec x + x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 + x -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 + x -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec 0 + x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec (dec x)) -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec (dec x)) -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec (dec x)) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec (dec 0)) -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec (dec 0)) -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = dec (dec (dec 0)) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + dec x -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + dec x -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + dec x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + dec 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + dec 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x + dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = x + x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = x + y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = y + x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = y + y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = x + x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = x + y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = y + x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = y + y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec 0 -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec 0 -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x + x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x + x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = x + x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec (dec x) -x ? y = x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec (dec x) -x ? y = y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec (dec x) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec (dec 0) -x ? y = x - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec (dec 0) -x ? y = y - -0 ? 0 = 0 -0 ? x = dec x -x ? 0 = dec (dec 0) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = x + x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = x + y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = y + x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = y + y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = x + x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = x + y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = y + x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = y + y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = dec (dec x) - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = dec (dec y) - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = dec (dec 0) - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec 0 -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec 0 -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x + x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x + x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = x + x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec (dec x) -x ? y = x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec (dec x) -x ? y = y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec (dec x) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec (dec 0) -x ? y = x - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec (dec 0) -x ? y = y - -0 ? 0 = 0 -0 ? x = dec 0 -x ? 0 = dec (dec 0) -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = dec x -x ? y = x - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = dec x -x ? y = y - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = x + x -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = dec x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = dec x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec x) -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = x -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = x -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = dec x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = dec x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec 0) -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec x + x -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec x + x -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec x + x -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec x + x -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec x + x -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec x + x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec 0 + x -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec 0 + x -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec 0 + x -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec 0 + x -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec 0 + x -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec 0 + x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec (dec x)) -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec (dec x)) -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec (dec x)) -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec (dec x)) -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec (dec x)) -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec (dec x)) -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec (dec 0)) -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec (dec 0)) -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec (dec 0)) -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = dec (dec (dec 0)) -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = dec (dec (dec 0)) -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = dec (dec (dec 0)) -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x + dec x -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = x + dec x -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = x + dec x -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x + dec x -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = x + dec x -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = x + dec x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x + dec 0 -x ? 0 = x -x ? y = x - -0 ? 0 = 0 -0 ? x = x + dec 0 -x ? 0 = x -x ? y = y - -0 ? 0 = 0 -0 ? x = x + dec 0 -x ? 0 = x -x ? y = 0 - -0 ? 0 = 0 -0 ? x = x + dec 0 -x ? 0 = 0 -x ? y = x - -0 ? 0 = 0 -0 ? x = x + dec 0 -x ? 0 = 0 -x ? y = y - -0 ? 0 = 0 -0 ? x = x + dec 0 -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = x + x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = x + y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = y + x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = y + y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = dec (dec x) - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = dec (dec y) - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x -x ? y = dec (dec 0) - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = x + x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = x + y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = y + x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = y + y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec (dec x) - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec (dec y) - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = 0 -x ? y = dec (dec 0) - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec x -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec x -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec x -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec 0 -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x + x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x + x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = x + x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec (dec x) -x ? y = x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec (dec x) -x ? y = y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec (dec x) -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec (dec 0) -x ? y = x - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec (dec 0) -x ? y = y - -0 ? 0 = dec 0 -0 ? x = x -x ? 0 = dec (dec 0) -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = x + x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = x + y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = y + x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = y + y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec (dec x) - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec (dec y) - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x -x ? y = dec (dec 0) - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = x + x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = x + y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = y + x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = y + y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec (dec x) - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec (dec y) - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = 0 -x ? y = dec (dec 0) - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec x -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec 0 -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x + x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x + x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = x + x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec (dec x) -x ? y = x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec (dec x) -x ? y = y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec (dec x) -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec (dec 0) -x ? y = x - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec (dec 0) -x ? y = y - -0 ? 0 = dec 0 -0 ? x = 0 -x ? 0 = dec (dec 0) -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = x -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = x -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = dec x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = dec x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec x -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = dec x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = dec x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec 0 -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = x + x -x ? 0 = x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = x + x -x ? 0 = x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = x + x -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = x + x -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = x + x -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec 0 -0 ? x = x + x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec (dec x) -x ? 0 = x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec (dec x) -x ? 0 = x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec (dec x) -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec (dec x) -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec (dec x) -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec (dec x) -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec (dec 0) -x ? 0 = x -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec (dec 0) -x ? 0 = x -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec (dec 0) -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec 0 -0 ? x = dec (dec 0) -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec 0 -0 ? x = dec (dec 0) -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec 0 -0 ? x = dec (dec 0) -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = x -x ? y = dec x - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = x -x ? y = dec y - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = dec x -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = dec x -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = x -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = x -x ? y = dec x - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = x -x ? y = dec y - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = x -x ? y = dec 0 - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = 0 -x ? y = dec x - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = 0 -x ? y = dec y - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = 0 -x ? y = dec 0 - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = dec x -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = dec x -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = dec x -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = dec 0 -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = dec 0 -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = 0 -x ? 0 = dec 0 -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = dec x -x ? 0 = x -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = dec x -x ? 0 = x -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = dec x -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = dec x -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = dec x -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = dec x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = dec 0 -x ? 0 = x -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = dec 0 -x ? 0 = x -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = dec 0 -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec (dec 0) -0 ? x = dec 0 -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec (dec 0) -0 ? x = dec 0 -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec (dec 0) -0 ? x = dec 0 -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec (dec (dec 0)) -0 ? x = x -x ? 0 = x -x ? y = x - -0 ? 0 = dec (dec (dec 0)) -0 ? x = x -x ? 0 = x -x ? y = y - -0 ? 0 = dec (dec (dec 0)) -0 ? x = x -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec (dec (dec 0)) -0 ? x = x -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec (dec (dec 0)) -0 ? x = x -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec (dec (dec 0)) -0 ? x = x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = dec (dec (dec 0)) -0 ? x = 0 -x ? 0 = x -x ? y = x - -0 ? 0 = dec (dec (dec 0)) -0 ? x = 0 -x ? 0 = x -x ? y = y - -0 ? 0 = dec (dec (dec 0)) -0 ? x = 0 -x ? 0 = x -x ? y = 0 - -0 ? 0 = dec (dec (dec 0)) -0 ? x = 0 -x ? 0 = 0 -x ? y = x - -0 ? 0 = dec (dec (dec 0)) -0 ? x = 0 -x ? 0 = 0 -x ? y = y - -0 ? 0 = dec (dec (dec 0)) -0 ? x = 0 -x ? 0 = 0 -x ? y = 0 - - -Candidates for: goo :: [Int] -> [Int] - pruning with 4/4 rules - [2,0,1,0,1,0,1,0,1] direct candidates, 0 duplicates - [2,1,2,3,4,7,10,17,26] pattern candidates, 0 duplicates - -rules: -xs ++ [] == xs -[] ++ xs == xs -(xs ++ ys) ++ zs == xs ++ (ys ++ zs) -(x:xs) ++ ys == x:(xs ++ ys) - -direct candidates: - -goo xs = xs - -goo xs = [] - -goo xs = xs ++ xs - -goo xs = xs ++ (xs ++ xs) - - -pattern candidates: - -goo xs = xs - -goo xs = [] - -goo [] = [] -goo (x:xs) = xs - -goo [] = [] -goo (x:xs) = goo xs - -goo xs = xs ++ xs - -goo [] = [] -goo (x:xs) = x:xs - -goo [] = [] -goo (x:xs) = [x] - -goo [] = [] -goo (x:xs) = xs ++ xs - -goo [] = [] -goo (x:xs) = x:goo xs - -goo [] = [] -goo (x:xs) = xs ++ goo xs - -goo [] = [] -goo (x:xs) = goo xs ++ xs - -goo xs = xs ++ (xs ++ xs) - -goo [] = [] -goo (x:xs) = goo xs ++ goo xs - -goo [] = [] -goo (x:xs) = x:x:xs - -goo [] = [] -goo (x:xs) = [x,x] - -goo [] = [] -goo (x:xs) = x:(xs ++ xs) - -goo [] = [] -goo (x:xs) = xs ++ (x:xs) - -goo [] = [] -goo (x:xs) = xs ++ [x] - -goo [] = [] -goo (x:xs) = xs ++ (xs ++ xs) - - -Candidates for: ?? :: [Int] -> [Int] -> [Int] - pruning with 4/4 rules - [3,0,4,0,8,0,16,0,32] direct candidates, 0 duplicates - [3,7,13,66,152,361,1394,2084,11820] pattern candidates, 0 duplicates - -rules: -xs ++ [] == xs -[] ++ xs == xs -(xs ++ ys) ++ zs == xs ++ (ys ++ zs) -(x:xs) ++ ys == x:(xs ++ ys) - -direct candidates: - -xs ?? ys = xs - -xs ?? ys = ys - -xs ?? ys = [] - -xs ?? ys = xs ++ xs - -xs ?? ys = xs ++ ys - -xs ?? ys = ys ++ xs - -xs ?? ys = ys ++ ys - -xs ?? ys = xs ++ (xs ++ xs) - -xs ?? ys = xs ++ (xs ++ ys) - -xs ?? ys = xs ++ (ys ++ xs) - -xs ?? ys = xs ++ (ys ++ ys) - -xs ?? ys = ys ++ (xs ++ xs) - -xs ?? ys = ys ++ (xs ++ ys) - -xs ?? ys = ys ++ (ys ++ xs) - -xs ?? ys = ys ++ (ys ++ ys) - - -pattern candidates: - -xs ?? ys = xs - -xs ?? ys = ys - -xs ?? ys = [] - -xs ?? [] = xs -xs ?? (x:ys) = ys - -xs ?? [] = xs -xs ?? (x:ys) = [] - -xs ?? [] = [] -xs ?? (x:ys) = xs - -xs ?? [] = [] -xs ?? (x:ys) = ys - -[] ?? xs = xs -(x:xs) ?? ys = [] - -[] ?? xs = [] -(x:xs) ?? ys = xs - -[] ?? xs = [] -(x:xs) ?? ys = ys - -xs ?? ys = xs ++ xs - -xs ?? ys = xs ++ ys - -xs ?? ys = ys ++ xs - -xs ?? ys = ys ++ ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys - -xs ?? [] = xs -xs ?? (x:ys) = xs ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = xs ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = xs ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = ys ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = ys ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = ys ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = [] ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = [] ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = xs ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = xs ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = xs ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = ys ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = ys ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = ys ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = [] ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = [] ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? [] - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? [] - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? [] - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? [] - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = x:xs - -xs ?? [] = xs -xs ?? (x:ys) = x:ys - -xs ?? [] = xs -xs ?? (x:ys) = [x] - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ xs - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ ys - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ xs - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ ys - -xs ?? [] = [] -xs ?? (x:ys) = x:xs - -xs ?? [] = [] -xs ?? (x:ys) = x:ys - -xs ?? [] = [] -xs ?? (x:ys) = [x] - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ xs - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ ys - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ xs - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ ys - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = xs - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = ys - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = [] - -[] ?? xs = xs -(x:xs) ?? ys = x:xs - -[] ?? xs = xs -(x:xs) ?? ys = x:ys - -[] ?? xs = xs -(x:xs) ?? ys = [x] - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ xs - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ ys - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ xs - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ ys - -[] ?? xs = [] -(x:xs) ?? ys = x:xs - -[] ?? xs = [] -(x:xs) ?? ys = x:ys - -[] ?? xs = [] -(x:xs) ?? ys = [x] - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ xs - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ ys - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ xs - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ ys - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = xs - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = ys - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs ?? [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys ?? [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs ?? xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs ?? [] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] ?? xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs ?? xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs ?? [] -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] ?? xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs ?? xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs ?? [] -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] ?? xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs ?? [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys ?? [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [] ?? xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [] ?? xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] ?? xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [] ?? xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [] ?? xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs ?? [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] ?? xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs ?? [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys ?? [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs ?? xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs ?? [] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] ?? xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs ?? xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs ?? [] -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] ?? xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs ?? xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs ?? [] -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] ?? xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs ?? [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys ?? ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys ?? [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] ?? xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] ?? ys - -xs ?? ys = xs ++ (xs ++ xs) - -xs ?? ys = xs ++ (xs ++ ys) - -xs ?? ys = xs ++ (ys ++ xs) - -xs ?? ys = xs ++ (ys ++ ys) - -xs ?? ys = ys ++ (xs ++ xs) - -xs ?? ys = ys ++ (xs ++ ys) - -xs ?? ys = ys ++ (ys ++ xs) - -xs ?? ys = ys ++ (ys ++ ys) - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = x:xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = x:ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [x] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = y:xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = y:ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [y] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs ++ xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs ++ ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys ++ xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys ++ ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = x:xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = x:ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [x] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = y:xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = y:ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [y] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs ++ xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs ++ ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys ++ xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys ++ ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = x:xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = x:xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = x:xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [x] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [x] -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [x] -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs ++ xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs ++ xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = xs ++ xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = x:xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = x:ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [x] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = y:xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = y:ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [y] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs ++ xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs ++ ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys ++ xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys ++ ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = x:xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = x:ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [x] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = y:xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = y:ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [y] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs ++ xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs ++ ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys ++ xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys ++ ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = x:xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = x:xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = x:xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [x] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [x] -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = [x] -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs ++ xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs ++ xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [] -(x:xs) ?? [] = xs ++ xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = x:xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = x:xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = x:xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = x:xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = x:xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = x:xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [x] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [x] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [x] -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = [x] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = [x] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = [x] -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs ++ xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs ++ xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs ++ xs -(x:xs) ?? [] = xs -(x:xs) ?? (y:ys) = [] - -[] ?? [] = [] -[] ?? (x:xs) = xs ++ xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = xs - -[] ?? [] = [] -[] ?? (x:xs) = xs ++ xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs ++ xs -(x:xs) ?? [] = [] -(x:xs) ?? (y:ys) = [] - -xs ?? [] = xs -xs ?? (x:ys) = (x:xs) ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = (x:xs) ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = (x:xs) ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = (x:ys) ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = (x:ys) ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = (x:ys) ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = [x] ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = [x] ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = [x] ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = (xs ++ xs) ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = (xs ++ xs) ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = (xs ++ xs) ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = (xs ++ ys) ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = (xs ++ ys) ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = (xs ++ ys) ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = (ys ++ xs) ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = (ys ++ xs) ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = (ys ++ xs) ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = (ys ++ ys) ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = (ys ++ ys) ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = (ys ++ ys) ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = (x:xs) ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = (x:xs) ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = (x:xs) ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = (x:ys) ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = (x:ys) ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = (x:ys) ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = [x] ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = [x] ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = [x] ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = (xs ++ xs) ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = (xs ++ xs) ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = (xs ++ xs) ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = (xs ++ ys) ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = (xs ++ ys) ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = (xs ++ ys) ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = (ys ++ xs) ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = (ys ++ xs) ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = (ys ++ xs) ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = (ys ++ ys) ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = (ys ++ ys) ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = (ys ++ ys) ?? [] - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? (x:xs) - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? (x:ys) - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? [x] - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? (xs ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? (xs ++ ys) - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? (ys ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? (ys ++ ys) - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? (x:xs) - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? (x:ys) - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? [x] - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? (xs ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? (xs ++ ys) - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? (ys ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? (ys ++ ys) - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? (x:xs) - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? (x:ys) - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? [x] - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? (xs ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? (xs ++ ys) - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? (ys ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? (ys ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? (x:xs) - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? (x:ys) - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? [x] - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? (xs ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? (xs ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? (ys ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? (ys ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? (x:xs) - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? (x:ys) - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? [x] - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? (xs ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? (xs ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? (ys ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? (ys ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? (x:xs) - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? (x:ys) - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? [x] - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? (xs ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? (xs ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? (ys ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? (ys ++ ys) - -xs ?? [] = xs -xs ?? (x:ys) = x:xs ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = x:xs ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = x:xs ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = x:ys ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = x:ys ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = x:ys ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = [x,] ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = [x,] ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ xs ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ xs ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ xs ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ ys ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ ys ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ ys ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ [] ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ [] ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ xs ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ xs ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ xs ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ ys ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ ys ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ ys ?? [] - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ [] ?? xs - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ [] ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = xs ?? xs ++ xs - -xs ?? [] = xs -xs ?? (x:ys) = xs ?? ys ++ xs - -xs ?? [] = xs -xs ?? (x:ys) = xs ?? [] ++ xs - -xs ?? [] = xs -xs ?? (x:ys) = ys ?? xs ++ xs - -xs ?? [] = xs -xs ?? (x:ys) = ys ?? ys ++ xs - -xs ?? [] = xs -xs ?? (x:ys) = ys ?? [] ++ xs - -xs ?? [] = xs -xs ?? (x:ys) = [] ?? xs ++ xs - -xs ?? [] = xs -xs ?? (x:ys) = [] ?? ys ++ xs - -xs ?? [] = xs -xs ?? (x:ys) = xs ?? xs ++ ys - -xs ?? [] = xs -xs ?? (x:ys) = xs ?? ys ++ ys - -xs ?? [] = xs -xs ?? (x:ys) = xs ?? [] ++ ys - -xs ?? [] = xs -xs ?? (x:ys) = ys ?? xs ++ ys - -xs ?? [] = xs -xs ?? (x:ys) = ys ?? ys ++ ys - -xs ?? [] = xs -xs ?? (x:ys) = ys ?? [] ++ ys - -xs ?? [] = xs -xs ?? (x:ys) = [] ?? xs ++ ys - -xs ?? [] = xs -xs ?? (x:ys) = [] ?? ys ++ ys - -xs ?? [] = [] -xs ?? (x:ys) = x:xs ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = x:xs ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = x:xs ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = x:ys ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = x:ys ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = x:ys ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = [x,] ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = [x,] ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ xs ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ xs ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ xs ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ ys ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ ys ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ ys ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ [] ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ [] ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ xs ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ xs ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ xs ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ ys ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ ys ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ ys ?? [] - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ [] ?? xs - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ [] ?? ys - -xs ?? [] = [] -xs ?? (x:ys) = xs ?? xs ++ xs - -xs ?? [] = [] -xs ?? (x:ys) = xs ?? ys ++ xs - -xs ?? [] = [] -xs ?? (x:ys) = xs ?? [] ++ xs - -xs ?? [] = [] -xs ?? (x:ys) = ys ?? xs ++ xs - -xs ?? [] = [] -xs ?? (x:ys) = ys ?? ys ++ xs - -xs ?? [] = [] -xs ?? (x:ys) = ys ?? [] ++ xs - -xs ?? [] = [] -xs ?? (x:ys) = [] ?? xs ++ xs - -xs ?? [] = [] -xs ?? (x:ys) = [] ?? ys ++ xs - -xs ?? [] = [] -xs ?? (x:ys) = xs ?? xs ++ ys - -xs ?? [] = [] -xs ?? (x:ys) = xs ?? ys ++ ys - -xs ?? [] = [] -xs ?? (x:ys) = xs ?? [] ++ ys - -xs ?? [] = [] -xs ?? (x:ys) = ys ?? xs ++ ys - -xs ?? [] = [] -xs ?? (x:ys) = ys ?? ys ++ ys - -xs ?? [] = [] -xs ?? (x:ys) = ys ?? [] ++ ys - -xs ?? [] = [] -xs ?? (x:ys) = [] ?? xs ++ ys - -xs ?? [] = [] -xs ?? (x:ys) = [] ?? ys ++ ys - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = xs ?? xs - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = xs ?? ys - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = xs ?? [] - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = ys ?? xs - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = ys ?? ys - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = ys ?? [] - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = [] ?? xs - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = [] ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = x:xs ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = x:xs ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = x:xs ?? [] - -[] ?? xs = xs -(x:xs) ?? ys = x:ys ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = x:ys ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = x:ys ?? [] - -[] ?? xs = xs -(x:xs) ?? ys = [x,] ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = [x,] ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ xs ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ xs ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ xs ?? [] - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ ys ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ ys ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ ys ?? [] - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ [] ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ [] ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ xs ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ xs ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ xs ?? [] - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ ys ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ ys ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ ys ?? [] - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ [] ?? xs - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ [] ?? ys - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? xs ++ xs - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? ys ++ xs - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? [] ++ xs - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? xs ++ xs - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? ys ++ xs - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? [] ++ xs - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? xs ++ xs - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? ys ++ xs - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? xs ++ ys - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? ys ++ ys - -[] ?? xs = xs -(x:xs) ?? ys = xs ?? [] ++ ys - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? xs ++ ys - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? ys ++ ys - -[] ?? xs = xs -(x:xs) ?? ys = ys ?? [] ++ ys - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? xs ++ ys - -[] ?? xs = xs -(x:xs) ?? ys = [] ?? ys ++ ys - -[] ?? xs = [] -(x:xs) ?? ys = x:xs ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = x:xs ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = x:xs ?? [] - -[] ?? xs = [] -(x:xs) ?? ys = x:ys ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = x:ys ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = x:ys ?? [] - -[] ?? xs = [] -(x:xs) ?? ys = [x,] ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = [x,] ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ xs ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ xs ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ xs ?? [] - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ ys ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ ys ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ ys ?? [] - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ [] ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ [] ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ xs ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ xs ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ xs ?? [] - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ ys ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ ys ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ ys ?? [] - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ [] ?? xs - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ [] ?? ys - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? xs ++ xs - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? ys ++ xs - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? [] ++ xs - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? xs ++ xs - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? ys ++ xs - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? [] ++ xs - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? xs ++ xs - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? ys ++ xs - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? xs ++ ys - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? ys ++ ys - -[] ?? xs = [] -(x:xs) ?? ys = xs ?? [] ++ ys - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? xs ++ ys - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? ys ++ ys - -[] ?? xs = [] -(x:xs) ?? ys = ys ?? [] ++ ys - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? xs ++ ys - -[] ?? xs = [] -(x:xs) ?? ys = [] ?? ys ++ ys - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = xs ?? xs - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = xs ?? ys - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = xs ?? [] - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = ys ?? xs - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = ys ?? ys - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = ys ?? [] - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = [] ?? xs - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = [] ?? ys - -xs ?? [] = xs -xs ?? (x:ys) = x:x:xs - -xs ?? [] = xs -xs ?? (x:ys) = x:x:ys - -xs ?? [] = xs -xs ?? (x:ys) = [x,x] - -xs ?? [] = xs -xs ?? (x:ys) = x:(xs ++ xs) - -xs ?? [] = xs -xs ?? (x:ys) = x:(xs ++ ys) - -xs ?? [] = xs -xs ?? (x:ys) = x:(ys ++ xs) - -xs ?? [] = xs -xs ?? (x:ys) = x:(ys ++ ys) - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ (x:xs) - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ (x:ys) - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ [x] - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ (xs ++ xs) - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ (xs ++ ys) - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ (ys ++ xs) - -xs ?? [] = xs -xs ?? (x:ys) = xs ++ (ys ++ ys) - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ (x:xs) - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ (x:ys) - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ [x] - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ (xs ++ xs) - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ (xs ++ ys) - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ (ys ++ xs) - -xs ?? [] = xs -xs ?? (x:ys) = ys ++ (ys ++ ys) - -xs ?? [] = [] -xs ?? (x:ys) = x:x:xs - -xs ?? [] = [] -xs ?? (x:ys) = x:x:ys - -xs ?? [] = [] -xs ?? (x:ys) = [x,x] - -xs ?? [] = [] -xs ?? (x:ys) = x:(xs ++ xs) - -xs ?? [] = [] -xs ?? (x:ys) = x:(xs ++ ys) - -xs ?? [] = [] -xs ?? (x:ys) = x:(ys ++ xs) - -xs ?? [] = [] -xs ?? (x:ys) = x:(ys ++ ys) - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ (x:xs) - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ (x:ys) - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ [x] - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ (xs ++ xs) - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ (xs ++ ys) - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ (ys ++ xs) - -xs ?? [] = [] -xs ?? (x:ys) = xs ++ (ys ++ ys) - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ (x:xs) - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ (x:ys) - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ [x] - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ (xs ++ xs) - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ (xs ++ ys) - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ (ys ++ xs) - -xs ?? [] = [] -xs ?? (x:ys) = ys ++ (ys ++ ys) - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = x:xs - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = x:ys - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = [x] - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = xs ++ ys - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = ys ++ xs - -xs ?? [] = xs ++ xs -xs ?? (x:ys) = ys ++ ys - -xs ?? [] = xs ++ (xs ++ xs) -xs ?? (x:ys) = xs - -xs ?? [] = xs ++ (xs ++ xs) -xs ?? (x:ys) = ys - -xs ?? [] = xs ++ (xs ++ xs) -xs ?? (x:ys) = [] - -[] ?? xs = xs -(x:xs) ?? ys = x:x:xs - -[] ?? xs = xs -(x:xs) ?? ys = x:x:ys - -[] ?? xs = xs -(x:xs) ?? ys = [x,x] - -[] ?? xs = xs -(x:xs) ?? ys = x:(xs ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = x:(xs ++ ys) - -[] ?? xs = xs -(x:xs) ?? ys = x:(ys ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = x:(ys ++ ys) - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ (x:xs) - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ (x:ys) - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ [x] - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ (xs ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ (xs ++ ys) - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ (ys ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = xs ++ (ys ++ ys) - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ (x:xs) - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ (x:ys) - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ [x] - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ (xs ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ (xs ++ ys) - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ (ys ++ xs) - -[] ?? xs = xs -(x:xs) ?? ys = ys ++ (ys ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = x:x:xs - -[] ?? xs = [] -(x:xs) ?? ys = x:x:ys - -[] ?? xs = [] -(x:xs) ?? ys = [x,x] - -[] ?? xs = [] -(x:xs) ?? ys = x:(xs ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = x:(xs ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = x:(ys ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = x:(ys ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ (x:xs) - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ (x:ys) - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ [x] - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ (xs ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ (xs ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ (ys ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = xs ++ (ys ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ (x:xs) - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ (x:ys) - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ [x] - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ (xs ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ (xs ++ ys) - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ (ys ++ xs) - -[] ?? xs = [] -(x:xs) ?? ys = ys ++ (ys ++ ys) - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = x:xs - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = x:ys - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = [x] - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = xs ++ ys - -[] ?? xs = xs ++ xs -(x:xs) ?? ys = ys ++ xs - -[] ?? xs = xs ++ (xs ++ xs) -(x:xs) ?? ys = xs - -[] ?? xs = xs ++ (xs ++ xs) -(x:xs) ?? ys = ys - -[] ?? xs = xs ++ (xs ++ xs) -(x:xs) ?? ys = [] - - -Candidates for: ton :: Bool -> Bool - pruning with 39/49 rules - [3,1,0,2,6,14,38,68,140] direct candidates, 0 duplicates - [3,3,0,0,0,0,0,0,0] pattern candidates, 0 duplicates - -rules: -not False == True -not True == False -p && p == p -p || p == p -not (not p) == p -p && False == False -p && True == p -False && p == False -True && p == p -p || False == p -p || True == True -False || p == p -True || p == True -not (p && q) == not p || not q -not (p && q) == not q || not p -not (p || q) == not p && not q -not (p || q) == not q && not p -p && not p == False -not p && p == False -p || not p == True -not p || p == True -(p && q) && r == p && (q && r) -(p && q) && r == q && (p && r) -(p || q) || r == p || (q || r) -(p || q) || r == q || (p || r) -p && (p && q) == p && q -p && (q && p) == p && q -p && (q && p) == q && p -p || (p || q) == p || q -p || (q || p) == p || q -p || (q || p) == q || p -p && (p || q) == p -p && (q || p) == p -(p || q) && p == p -(p || q) && q == q -p || p && q == p -p || q && p == p -p && q || p == p -p && q || q == q -equations: -q && p == p && q -q || p == p || q -q && (p && r) == p && (q && r) -r && (p && q) == p && (q && r) -r && (q && p) == p && (q && r) -q || (p || r) == p || (q || r) -r || (p || q) == p || (q || r) -r || (q || p) == p || (q || r) -(r || q) && p == p && (q || r) -r && q || p == p || q && r - -direct candidates: - -ton p = p - -ton p = False - -ton p = True - -ton p = not p - -ton p = p && ton p - -ton p = p || ton p - -ton p = p && ton (not p) - -ton p = p || ton (not p) - -ton p = p && not (ton p) - -ton p = not p && ton p - -ton p = p || not (ton p) - -ton p = not p || ton p - -ton p = p && not (ton (not p)) - -ton p = not p && ton (not p) - -ton p = p || not (ton (not p)) - -ton p = not p || ton (not p) - -ton p = p && (p && ton p) - -ton p = p && (ton p && p) - -ton p = p && (p || ton p) - -ton p = p && (ton p || p) - -ton p = not p && not (ton p) - -ton p = p || p && ton p - -ton p = p || ton p && p - -ton p = p || (p || ton p) - -ton p = p || (ton p || p) - -ton p = not p || not (ton p) - - -pattern candidates: - -ton p = p - -ton p = False - -ton p = True - -ton p = not p - -ton False = False -ton True = True - -ton False = True -ton True = False - - -Candidates for: &| :: Bool -> Bool -> Bool - pruning with 39/49 rules - [4,2,4,8,32,128,468,1576,4924] direct candidates, 0 duplicates - [4,14,26,8,4,48,60,280,592] pattern candidates, 0 duplicates - -rules: -not False == True -not True == False -p && p == p -p || p == p -not (not p) == p -p && False == False -p && True == p -False && p == False -True && p == p -p || False == p -p || True == True -False || p == p -True || p == True -not (p && q) == not p || not q -not (p && q) == not q || not p -not (p || q) == not p && not q -not (p || q) == not q && not p -p && not p == False -not p && p == False -p || not p == True -not p || p == True -(p && q) && r == p && (q && r) -(p && q) && r == q && (p && r) -(p || q) || r == p || (q || r) -(p || q) || r == q || (p || r) -p && (p && q) == p && q -p && (q && p) == p && q -p && (q && p) == q && p -p || (p || q) == p || q -p || (q || p) == p || q -p || (q || p) == q || p -p && (p || q) == p -p && (q || p) == p -(p || q) && p == p -(p || q) && q == q -p || p && q == p -p || q && p == p -p && q || p == p -p && q || q == q -equations: -q && p == p && q -q || p == p || q -q && (p && r) == p && (q && r) -r && (p && q) == p && (q && r) -r && (q && p) == p && (q && r) -q || (p || r) == p || (q || r) -r || (p || q) == p || (q || r) -r || (q || p) == p || (q || r) -(r || q) && p == p && (q || r) -r && q || p == p || q && r - -direct candidates: - -p &| q = p - -p &| q = q - -p &| q = False - -p &| q = True - -p &| q = not p - -p &| q = not q - -p &| q = p && q - -p &| q = q && p - -p &| q = p || q - -p &| q = q || p - -p &| q = not p && q - -p &| q = not q && p - -p &| q = not p || q - -p &| q = not q || p - -p &| q = p && not q - -p &| q = q && not p - -p &| q = p || not q - -p &| q = q || not p - -p &| q = not p && not q - -p &| q = not q && not p - -p &| q = not p || not q - -p &| q = not q || not p - -p &| q = p && p &| p - -p &| q = p && p &| q - -p &| q = p && p &| False - -p &| q = p && p &| True - -p &| q = p && q &| q - -p &| q = p && False &| q - -p &| q = p && True &| q - -p &| q = q && p &| p - -p &| q = q && p &| q - -p &| q = q && p &| False - -p &| q = q && p &| True - -p &| q = q && q &| q - -p &| q = q && False &| q - -p &| q = q && True &| q - -p &| q = p || p &| p - -p &| q = p || p &| q - -p &| q = p || p &| False - -p &| q = p || p &| True - -p &| q = p || q &| q - -p &| q = p || False &| q - -p &| q = p || True &| q - -p &| q = q || p &| p - -p &| q = q || p &| q - -p &| q = q || p &| False - -p &| q = q || p &| True - -p &| q = q || q &| q - -p &| q = q || False &| q - -p &| q = q || True &| q - -p &| q = p && (not p && q) - -p &| q = p && (not p || q) - -p &| q = p && (q && not p) - -p &| q = p && (q || not p) - -p &| q = q && (not q && p) - -p &| q = q && (not q || p) - -p &| q = q && (p && not q) - -p &| q = q && (p || not q) - -p &| q = p || not p && q - -p &| q = p || (not p || q) - -p &| q = p || q && not p - -p &| q = p || (q || not p) - -p &| q = q || not q && p - -p &| q = q || (not q || p) - -p &| q = q || p && not q - -p &| q = q || (p || not q) - -p &| q = not p && (p || q) - -p &| q = not p && (q || p) - -p &| q = not q && (p || q) - -p &| q = not q && (q || p) - -p &| q = not p || p && q - -p &| q = not p || q && p - -p &| q = not q || p && q - -p &| q = not q || q && p - -p &| q = (p || q) && not p - -p &| q = (p || q) && not q - -p &| q = (q || p) && not p - -p &| q = (q || p) && not q - -p &| q = p && q || not p - -p &| q = p && q || not q - -p &| q = q && p || not p - -p &| q = q && p || not q - -p &| q = p && p &| not p - -p &| q = p && p &| not q - -p &| q = p && q &| not q - -p &| q = p && False &| not q - -p &| q = p && True &| not q - -p &| q = p && not p &| p - -p &| q = p && not p &| q - -p &| q = p && not p &| False - -p &| q = p && not p &| True - -p &| q = p && not q &| q - -p &| q = q && p &| not p - -p &| q = q && p &| not q - -p &| q = q && q &| not q - -p &| q = q && False &| not q - -p &| q = q && True &| not q - -p &| q = q && not p &| p - -p &| q = q && not p &| q - -p &| q = q && not p &| False - -p &| q = q && not p &| True - -p &| q = q && not q &| q - -p &| q = p || p &| not p - -p &| q = p || p &| not q - -p &| q = p || q &| not q - -p &| q = p || False &| not q - -p &| q = p || True &| not q - -p &| q = p || not p &| p - -p &| q = p || not p &| q - -p &| q = p || not p &| False - -p &| q = p || not p &| True - -p &| q = p || not q &| q - -p &| q = q || p &| not p - -p &| q = q || p &| not q - -p &| q = q || q &| not q - -p &| q = q || False &| not q - -p &| q = q || True &| not q - -p &| q = q || not p &| p - -p &| q = q || not p &| q - -p &| q = q || not p &| False - -p &| q = q || not p &| True - -p &| q = q || not q &| q - -p &| q = p && not (p &| p) - -p &| q = p && not (p &| q) - -p &| q = p && not (p &| False) - -p &| q = p && not (p &| True) - -p &| q = p && not (q &| q) - -p &| q = p && not (False &| q) - -p &| q = p && not (True &| q) - -p &| q = q && not (p &| p) - -p &| q = q && not (p &| q) - -p &| q = q && not (p &| False) - -p &| q = q && not (p &| True) - -p &| q = q && not (q &| q) - -p &| q = q && not (False &| q) - -p &| q = q && not (True &| q) - -p &| q = not p && p &| p - -p &| q = not p && p &| q - -p &| q = not p && p &| False - -p &| q = not p && p &| True - -p &| q = not p && q &| q - -p &| q = not p && False &| q - -p &| q = not p && True &| q - -p &| q = not q && p &| p - -p &| q = not q && p &| q - -p &| q = not q && p &| False - -p &| q = not q && p &| True - -p &| q = not q && q &| q - -p &| q = not q && False &| q - -p &| q = not q && True &| q - -p &| q = p || not (p &| p) - -p &| q = p || not (p &| q) - -p &| q = p || not (p &| False) - -p &| q = p || not (p &| True) - -p &| q = p || not (q &| q) - -p &| q = p || not (False &| q) - -p &| q = p || not (True &| q) - -p &| q = q || not (p &| p) - -p &| q = q || not (p &| q) - -p &| q = q || not (p &| False) - -p &| q = q || not (p &| True) - -p &| q = q || not (q &| q) - -p &| q = q || not (False &| q) - -p &| q = q || not (True &| q) - -p &| q = not p || p &| p - -p &| q = not p || p &| q - -p &| q = not p || p &| False - -p &| q = not p || p &| True - -p &| q = not p || q &| q - -p &| q = not p || False &| q - -p &| q = not p || True &| q - -p &| q = not q || p &| p - -p &| q = not q || p &| q - -p &| q = not q || p &| False - -p &| q = not q || p &| True - -p &| q = not q || q &| q - -p &| q = not q || False &| q - -p &| q = not q || True &| q - - -pattern candidates: - -p &| q = p - -p &| q = q - -p &| q = False - -p &| q = True - -p &| q = not p - -p &| q = not q - -p &| False = p -p &| True = False - -p &| False = p -p &| True = True - -p &| False = False -p &| True = p - -p &| False = False -p &| True = True - -p &| False = True -p &| True = p - -p &| False = True -p &| True = False - -False &| p = p -True &| p = False - -False &| p = p -True &| p = True - -False &| p = False -True &| p = p - -False &| p = False -True &| p = True - -False &| p = True -True &| p = p - -False &| p = True -True &| p = False - -p &| q = p && q - -p &| q = q && p - -p &| q = p || q - -p &| q = q || p - -p &| False = p -p &| True = not p - -p &| False = False -p &| True = not p - -p &| False = True -p &| True = not p - -p &| False = not p -p &| True = p - -p &| False = not p -p &| True = False - -p &| False = not p -p &| True = True - -False &| p = p -True &| p = not p - -False &| p = False -True &| p = not p - -False &| p = True -True &| p = not p - -False &| p = not p -True &| p = p - -False &| p = not p -True &| p = False - -False &| p = not p -True &| p = True - -False &| False = False -False &| True = False -True &| False = False -True &| True = True - -False &| False = False -False &| True = False -True &| False = True -True &| True = False - -False &| False = False -False &| True = True -True &| False = False -True &| True = False - -False &| False = False -False &| True = True -True &| False = True -True &| True = False - -False &| False = False -False &| True = True -True &| False = True -True &| True = True - -False &| False = True -False &| True = False -True &| False = False -True &| True = False - -False &| False = True -False &| True = False -True &| False = False -True &| True = True - -False &| False = True -False &| True = False -True &| False = True -True &| True = True - -False &| False = True -False &| True = True -True &| False = False -True &| True = True - -False &| False = True -False &| True = True -True &| False = True -True &| True = False - -p &| q = not p && q - -p &| q = not q && p - -p &| q = not p || q - -p &| q = not q || p - -p &| q = p && not q - -p &| q = q && not p - -p &| q = p || not q - -p &| q = q || not p - -p &| q = not p && not q - -p &| q = not q && not p - -p &| q = not p || not q - -p &| q = not q || not p - -p &| q = (not p || q) && p - -p &| q = (not q || p) && q - -p &| q = (p || not q) && q - -p &| q = (q || not p) && p - -p &| q = not p && q || p - -p &| q = not q && p || q - -p &| q = p && not q || q - -p &| q = q && not p || p - -p &| q = p && (not p && q) - -p &| q = p && (not p || q) - -p &| q = p && (q && not p) - -p &| q = p && (q || not p) - -p &| q = q && (not q && p) - -p &| q = q && (not q || p) - -p &| q = q && (p && not q) - -p &| q = q && (p || not q) - -p &| q = p || not p && q - -p &| q = p || (not p || q) - -p &| q = p || q && not p - -p &| q = p || (q || not p) - -p &| q = q || not q && p - -p &| q = q || (not q || p) - -p &| q = q || p && not q - -p &| q = q || (p || not q) - -p &| q = not p && (p && q) - -p &| q = not p && (q && p) - -p &| q = not p && (p || q) - -p &| q = not p && (q || p) - -p &| q = not q && (p && q) - -p &| q = not q && (q && p) - -p &| q = not q && (p || q) - -p &| q = not q && (q || p) - -p &| q = not p || p && q - -p &| q = not p || q && p - -p &| q = not p || (p || q) - -p &| q = not p || (q || p) - -p &| q = not q || p && q - -p &| q = not q || q && p - -p &| q = not q || (p || q) - -p &| q = not q || (q || p) - -p &| q = (p || q) && not p - -p &| q = (p || q) && not q - -p &| q = (q || p) && not p - -p &| q = (q || p) && not q - -p &| q = p && q || not p - -p &| q = p && q || not q - -p &| q = q && p || not p - -p &| q = q && p || not q - - -Candidates for: gcd :: Int -> Int -> Int - pruning with 0/0 rules - [3,0,9,0,54,0,405,0,3402] direct candidates, 0 duplicates - [3,7,18,60,150,469,1245,3812,10634] pattern candidates, 0 duplicates - -no rules. - -direct candidates: - -gcd x y = x - -gcd x y = y - -gcd x y = 0 - -gcd x y = x `mod` x - -gcd x y = x `mod` y - -gcd x y = x `mod` 0 - -gcd x y = y `mod` x - -gcd x y = y `mod` y - -gcd x y = y `mod` 0 - -gcd x y = 0 `mod` x - -gcd x y = 0 `mod` y - -gcd x y = 0 `mod` 0 - -gcd x y = (x `mod` x) `mod` x - -gcd x y = (x `mod` x) `mod` y - -gcd x y = (x `mod` x) `mod` 0 - -gcd x y = (x `mod` y) `mod` x - -gcd x y = (x `mod` y) `mod` y - -gcd x y = (x `mod` y) `mod` 0 - -gcd x y = (x `mod` 0) `mod` x - -gcd x y = (x `mod` 0) `mod` y - -gcd x y = (x `mod` 0) `mod` 0 - -gcd x y = (y `mod` x) `mod` x - -gcd x y = (y `mod` x) `mod` y - -gcd x y = (y `mod` x) `mod` 0 - -gcd x y = (y `mod` y) `mod` x - -gcd x y = (y `mod` y) `mod` y - -gcd x y = (y `mod` y) `mod` 0 - -gcd x y = (y `mod` 0) `mod` x - -gcd x y = (y `mod` 0) `mod` y - -gcd x y = (y `mod` 0) `mod` 0 - -gcd x y = (0 `mod` x) `mod` x - -gcd x y = (0 `mod` x) `mod` y - -gcd x y = (0 `mod` x) `mod` 0 - -gcd x y = (0 `mod` y) `mod` x - -gcd x y = (0 `mod` y) `mod` y - -gcd x y = (0 `mod` y) `mod` 0 - -gcd x y = (0 `mod` 0) `mod` x - -gcd x y = (0 `mod` 0) `mod` y - -gcd x y = (0 `mod` 0) `mod` 0 - -gcd x y = x `mod` (x `mod` x) - -gcd x y = x `mod` (x `mod` y) - -gcd x y = x `mod` (x `mod` 0) - -gcd x y = x `mod` (y `mod` x) - -gcd x y = x `mod` (y `mod` y) - -gcd x y = x `mod` (y `mod` 0) - -gcd x y = x `mod` (0 `mod` x) - -gcd x y = x `mod` (0 `mod` y) - -gcd x y = x `mod` (0 `mod` 0) - -gcd x y = y `mod` (x `mod` x) - -gcd x y = y `mod` (x `mod` y) - -gcd x y = y `mod` (x `mod` 0) - -gcd x y = y `mod` (y `mod` x) - -gcd x y = y `mod` (y `mod` y) - -gcd x y = y `mod` (y `mod` 0) - -gcd x y = y `mod` (0 `mod` x) - -gcd x y = y `mod` (0 `mod` y) - -gcd x y = y `mod` (0 `mod` 0) - -gcd x y = 0 `mod` (x `mod` x) - -gcd x y = 0 `mod` (x `mod` y) - -gcd x y = 0 `mod` (x `mod` 0) - -gcd x y = 0 `mod` (y `mod` x) - -gcd x y = 0 `mod` (y `mod` y) - -gcd x y = 0 `mod` (y `mod` 0) - -gcd x y = 0 `mod` (0 `mod` x) - -gcd x y = 0 `mod` (0 `mod` y) - -gcd x y = 0 `mod` (0 `mod` 0) - - -pattern candidates: - -gcd x y = x - -gcd x y = y - -gcd x y = 0 - -gcd x 0 = x -gcd x y = y - -gcd x 0 = x -gcd x y = 0 - -gcd x 0 = 0 -gcd x y = x - -gcd x 0 = 0 -gcd x y = y - -gcd 0 x = x -gcd x y = 0 - -gcd 0 x = 0 -gcd x y = x - -gcd 0 x = 0 -gcd x y = y - -gcd x y = x `mod` x - -gcd x y = x `mod` y - -gcd x y = x `mod` 0 - -gcd x y = y `mod` x - -gcd x y = y `mod` y - -gcd x y = y `mod` 0 - -gcd x y = 0 `mod` x - -gcd x y = 0 `mod` y - -gcd x y = 0 `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = y - -gcd x 0 = x -gcd x y = x `mod` x - -gcd x 0 = x -gcd x y = x `mod` y - -gcd x 0 = x -gcd x y = x `mod` 0 - -gcd x 0 = x -gcd x y = y `mod` x - -gcd x 0 = x -gcd x y = y `mod` y - -gcd x 0 = x -gcd x y = y `mod` 0 - -gcd x 0 = x -gcd x y = 0 `mod` x - -gcd x 0 = x -gcd x y = 0 `mod` y - -gcd x 0 = x -gcd x y = 0 `mod` 0 - -gcd x 0 = 0 -gcd x y = x `mod` x - -gcd x 0 = 0 -gcd x y = x `mod` y - -gcd x 0 = 0 -gcd x y = x `mod` 0 - -gcd x 0 = 0 -gcd x y = y `mod` x - -gcd x 0 = 0 -gcd x y = y `mod` y - -gcd x 0 = 0 -gcd x y = y `mod` 0 - -gcd x 0 = 0 -gcd x y = 0 `mod` x - -gcd x 0 = 0 -gcd x y = 0 `mod` y - -gcd x 0 = 0 -gcd x y = 0 `mod` 0 - -gcd x 0 = x `mod` x -gcd x y = x - -gcd x 0 = x `mod` x -gcd x y = y - -gcd x 0 = x `mod` x -gcd x y = 0 - -gcd x 0 = x `mod` 0 -gcd x y = x - -gcd x 0 = x `mod` 0 -gcd x y = y - -gcd x 0 = x `mod` 0 -gcd x y = 0 - -gcd x 0 = 0 `mod` x -gcd x y = x - -gcd x 0 = 0 `mod` x -gcd x y = y - -gcd x 0 = 0 `mod` x -gcd x y = 0 - -gcd x 0 = 0 `mod` 0 -gcd x y = x - -gcd x 0 = 0 `mod` 0 -gcd x y = y - -gcd x 0 = 0 `mod` 0 -gcd x y = 0 - -gcd 0 x = x -gcd x y = x `mod` x - -gcd 0 x = x -gcd x y = x `mod` y - -gcd 0 x = x -gcd x y = x `mod` 0 - -gcd 0 x = x -gcd x y = y `mod` x - -gcd 0 x = x -gcd x y = y `mod` y - -gcd 0 x = x -gcd x y = y `mod` 0 - -gcd 0 x = x -gcd x y = 0 `mod` x - -gcd 0 x = x -gcd x y = 0 `mod` y - -gcd 0 x = x -gcd x y = 0 `mod` 0 - -gcd 0 x = 0 -gcd x y = x `mod` x - -gcd 0 x = 0 -gcd x y = x `mod` y - -gcd 0 x = 0 -gcd x y = x `mod` 0 - -gcd 0 x = 0 -gcd x y = y `mod` x - -gcd 0 x = 0 -gcd x y = y `mod` y - -gcd 0 x = 0 -gcd x y = y `mod` 0 - -gcd 0 x = 0 -gcd x y = 0 `mod` x - -gcd 0 x = 0 -gcd x y = 0 `mod` y - -gcd 0 x = 0 -gcd x y = 0 `mod` 0 - -gcd 0 x = x `mod` x -gcd x y = x - -gcd 0 x = x `mod` x -gcd x y = y - -gcd 0 x = x `mod` x -gcd x y = 0 - -gcd 0 x = x `mod` 0 -gcd x y = x - -gcd 0 x = x `mod` 0 -gcd x y = y - -gcd 0 x = x `mod` 0 -gcd x y = 0 - -gcd 0 x = 0 `mod` x -gcd x y = x - -gcd 0 x = 0 `mod` x -gcd x y = y - -gcd 0 x = 0 `mod` x -gcd x y = 0 - -gcd 0 x = 0 `mod` 0 -gcd x y = x - -gcd 0 x = 0 `mod` 0 -gcd x y = y - -gcd 0 x = 0 `mod` 0 -gcd x y = 0 - -gcd x y = (x `mod` x) `mod` x - -gcd x y = (x `mod` x) `mod` y - -gcd x y = (x `mod` x) `mod` 0 - -gcd x y = (x `mod` y) `mod` x - -gcd x y = (x `mod` y) `mod` y - -gcd x y = (x `mod` y) `mod` 0 - -gcd x y = (x `mod` 0) `mod` x - -gcd x y = (x `mod` 0) `mod` y - -gcd x y = (x `mod` 0) `mod` 0 - -gcd x y = (y `mod` x) `mod` x - -gcd x y = (y `mod` x) `mod` y - -gcd x y = (y `mod` x) `mod` 0 - -gcd x y = (y `mod` y) `mod` x - -gcd x y = (y `mod` y) `mod` y - -gcd x y = (y `mod` y) `mod` 0 - -gcd x y = (y `mod` 0) `mod` x - -gcd x y = (y `mod` 0) `mod` y - -gcd x y = (y `mod` 0) `mod` 0 - -gcd x y = (0 `mod` x) `mod` x - -gcd x y = (0 `mod` x) `mod` y - -gcd x y = (0 `mod` x) `mod` 0 - -gcd x y = (0 `mod` y) `mod` x - -gcd x y = (0 `mod` y) `mod` y - -gcd x y = (0 `mod` y) `mod` 0 - -gcd x y = (0 `mod` 0) `mod` x - -gcd x y = (0 `mod` 0) `mod` y - -gcd x y = (0 `mod` 0) `mod` 0 - -gcd x y = x `mod` (x `mod` x) - -gcd x y = x `mod` (x `mod` y) - -gcd x y = x `mod` (x `mod` 0) - -gcd x y = x `mod` (y `mod` x) - -gcd x y = x `mod` (y `mod` y) - -gcd x y = x `mod` (y `mod` 0) - -gcd x y = x `mod` (0 `mod` x) - -gcd x y = x `mod` (0 `mod` y) - -gcd x y = x `mod` (0 `mod` 0) - -gcd x y = y `mod` (x `mod` x) - -gcd x y = y `mod` (x `mod` y) - -gcd x y = y `mod` (x `mod` 0) - -gcd x y = y `mod` (y `mod` x) - -gcd x y = y `mod` (y `mod` y) - -gcd x y = y `mod` (y `mod` 0) - -gcd x y = y `mod` (0 `mod` x) - -gcd x y = y `mod` (0 `mod` y) - -gcd x y = y `mod` (0 `mod` 0) - -gcd x y = 0 `mod` (x `mod` x) - -gcd x y = 0 `mod` (x `mod` y) - -gcd x y = 0 `mod` (x `mod` 0) - -gcd x y = 0 `mod` (y `mod` x) - -gcd x y = 0 `mod` (y `mod` y) - -gcd x y = 0 `mod` (y `mod` 0) - -gcd x y = 0 `mod` (0 `mod` x) - -gcd x y = 0 `mod` (0 `mod` y) - -gcd x y = 0 `mod` (0 `mod` 0) - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = x `mod` x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = x `mod` y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = x `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = y `mod` x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = y `mod` y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = y `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = 0 `mod` x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = 0 `mod` y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = 0 `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = x `mod` x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = x `mod` y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = x `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = y `mod` x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = y `mod` y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = y `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = 0 `mod` x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = 0 `mod` y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = 0 `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x `mod` x -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x `mod` x -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x `mod` x -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x `mod` 0 -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x `mod` 0 -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = x `mod` 0 -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 `mod` x -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 `mod` x -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 `mod` x -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 `mod` 0 -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 `mod` 0 -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = x -gcd x 0 = 0 `mod` 0 -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = x `mod` x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = x `mod` y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = x `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = y `mod` x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = y `mod` y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = y `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = 0 `mod` x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = 0 `mod` y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = 0 `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = x `mod` x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = x `mod` y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = x `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = y `mod` x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = y `mod` y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = y `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = 0 `mod` x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = 0 `mod` y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = 0 `mod` 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x `mod` x -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x `mod` x -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x `mod` x -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x `mod` 0 -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x `mod` 0 -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = x `mod` 0 -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 `mod` x -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 `mod` x -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 `mod` x -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 `mod` 0 -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 `mod` 0 -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = 0 -gcd x 0 = 0 `mod` 0 -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = x `mod` x -gcd x 0 = x -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = x `mod` x -gcd x 0 = x -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = x `mod` x -gcd x 0 = x -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = x `mod` x -gcd x 0 = 0 -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = x `mod` x -gcd x 0 = 0 -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = x `mod` x -gcd x 0 = 0 -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = x `mod` 0 -gcd x 0 = x -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = x `mod` 0 -gcd x 0 = x -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = x `mod` 0 -gcd x 0 = x -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = x `mod` 0 -gcd x 0 = 0 -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = x `mod` 0 -gcd x 0 = 0 -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = x `mod` 0 -gcd x 0 = 0 -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` x -gcd x 0 = x -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` x -gcd x 0 = x -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` x -gcd x 0 = x -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` x -gcd x 0 = 0 -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` x -gcd x 0 = 0 -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` x -gcd x 0 = 0 -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` 0 -gcd x 0 = x -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` 0 -gcd x 0 = x -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` 0 -gcd x 0 = x -gcd x y = 0 - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` 0 -gcd x 0 = 0 -gcd x y = x - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` 0 -gcd x 0 = 0 -gcd x y = y - -gcd 0 0 = 0 -gcd 0 x = 0 `mod` 0 -gcd x 0 = 0 -gcd x y = 0 - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = x - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = y - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = x -gcd x 0 = x -gcd x y = 0 - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = x - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = y - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = x -gcd x 0 = 0 -gcd x y = 0 - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = x - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = y - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = 0 -gcd x 0 = x -gcd x y = 0 - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = x - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = y - -gcd 0 0 = 0 `mod` 0 -gcd 0 x = 0 -gcd x 0 = 0 -gcd x y = 0 - -gcd x 0 = x -gcd x y = gcd x (x `mod` y) - -gcd x 0 = x -gcd x y = gcd x (y `mod` x) - -gcd x 0 = x -gcd x y = gcd x (y `mod` y) - -gcd x 0 = x -gcd x y = gcd x (0 `mod` y) - -gcd x 0 = x -gcd x y = gcd y (x `mod` y) - -gcd x 0 = x -gcd x y = gcd y (y `mod` x) - -gcd x 0 = x -gcd x y = gcd y (y `mod` y) - -gcd x 0 = x -gcd x y = gcd y (0 `mod` y) - -gcd x 0 = x -gcd x y = gcd 0 (x `mod` y) - -gcd x 0 = x -gcd x y = gcd 0 (y `mod` x) - -gcd x 0 = x -gcd x y = gcd 0 (y `mod` y) - -gcd x 0 = x -gcd x y = gcd 0 (0 `mod` y) - -gcd x 0 = x -gcd x y = gcd (x `mod` x) x - -gcd x 0 = x -gcd x y = gcd (x `mod` x) y - -gcd x 0 = x -gcd x y = gcd (x `mod` x) 0 - -gcd x 0 = x -gcd x y = gcd (x `mod` y) x - -gcd x 0 = x -gcd x y = gcd (x `mod` y) y - -gcd x 0 = x -gcd x y = gcd (x `mod` y) 0 - -gcd x 0 = x -gcd x y = gcd (y `mod` x) x - -gcd x 0 = x -gcd x y = gcd (y `mod` x) y - -gcd x 0 = x -gcd x y = gcd (y `mod` x) 0 - -gcd x 0 = x -gcd x y = gcd (0 `mod` x) x - -gcd x 0 = x -gcd x y = gcd (0 `mod` x) y - -gcd x 0 = x -gcd x y = gcd (0 `mod` x) 0 - -gcd x 0 = 0 -gcd x y = gcd x (x `mod` y) - -gcd x 0 = 0 -gcd x y = gcd x (y `mod` x) - -gcd x 0 = 0 -gcd x y = gcd x (y `mod` y) - -gcd x 0 = 0 -gcd x y = gcd x (0 `mod` y) - -gcd x 0 = 0 -gcd x y = gcd y (x `mod` y) - -gcd x 0 = 0 -gcd x y = gcd y (y `mod` x) - -gcd x 0 = 0 -gcd x y = gcd y (y `mod` y) - -gcd x 0 = 0 -gcd x y = gcd y (0 `mod` y) - -gcd x 0 = 0 -gcd x y = gcd 0 (x `mod` y) - -gcd x 0 = 0 -gcd x y = gcd 0 (y `mod` x) - -gcd x 0 = 0 -gcd x y = gcd 0 (y `mod` y) - -gcd x 0 = 0 -gcd x y = gcd 0 (0 `mod` y) - -gcd x 0 = 0 -gcd x y = gcd (x `mod` x) x - -gcd x 0 = 0 -gcd x y = gcd (x `mod` x) y - -gcd x 0 = 0 -gcd x y = gcd (x `mod` x) 0 - -gcd x 0 = 0 -gcd x y = gcd (x `mod` y) x - -gcd x 0 = 0 -gcd x y = gcd (x `mod` y) y - -gcd x 0 = 0 -gcd x y = gcd (x `mod` y) 0 - -gcd x 0 = 0 -gcd x y = gcd (y `mod` x) x - -gcd x 0 = 0 -gcd x y = gcd (y `mod` x) y - -gcd x 0 = 0 -gcd x y = gcd (y `mod` x) 0 - -gcd x 0 = 0 -gcd x y = gcd (0 `mod` x) x - -gcd x 0 = 0 -gcd x y = gcd (0 `mod` x) y - -gcd x 0 = 0 -gcd x y = gcd (0 `mod` x) 0 - -gcd 0 x = x -gcd x y = gcd x (x `mod` y) - -gcd 0 x = x -gcd x y = gcd x (y `mod` x) - -gcd 0 x = x -gcd x y = gcd x (y `mod` y) - -gcd 0 x = x -gcd x y = gcd x (0 `mod` y) - -gcd 0 x = x -gcd x y = gcd y (x `mod` y) - -gcd 0 x = x -gcd x y = gcd y (y `mod` x) - -gcd 0 x = x -gcd x y = gcd y (y `mod` y) - -gcd 0 x = x -gcd x y = gcd y (0 `mod` y) - -gcd 0 x = x -gcd x y = gcd 0 (x `mod` y) - -gcd 0 x = x -gcd x y = gcd 0 (y `mod` x) - -gcd 0 x = x -gcd x y = gcd 0 (y `mod` y) - -gcd 0 x = x -gcd x y = gcd 0 (0 `mod` y) - -gcd 0 x = x -gcd x y = gcd (x `mod` x) x - -gcd 0 x = x -gcd x y = gcd (x `mod` x) y - -gcd 0 x = x -gcd x y = gcd (x `mod` x) 0 - -gcd 0 x = x -gcd x y = gcd (x `mod` y) x - -gcd 0 x = x -gcd x y = gcd (x `mod` y) y - -gcd 0 x = x -gcd x y = gcd (x `mod` y) 0 - -gcd 0 x = x -gcd x y = gcd (y `mod` x) x - -gcd 0 x = x -gcd x y = gcd (y `mod` x) y - -gcd 0 x = x -gcd x y = gcd (y `mod` x) 0 - -gcd 0 x = x -gcd x y = gcd (0 `mod` x) x - -gcd 0 x = x -gcd x y = gcd (0 `mod` x) y - -gcd 0 x = x -gcd x y = gcd (0 `mod` x) 0 - -gcd 0 x = 0 -gcd x y = gcd x (x `mod` y) - -gcd 0 x = 0 -gcd x y = gcd x (y `mod` x) - -gcd 0 x = 0 -gcd x y = gcd x (y `mod` y) - -gcd 0 x = 0 -gcd x y = gcd x (0 `mod` y) - -gcd 0 x = 0 -gcd x y = gcd y (x `mod` y) - -gcd 0 x = 0 -gcd x y = gcd y (y `mod` x) - -gcd 0 x = 0 -gcd x y = gcd y (y `mod` y) - -gcd 0 x = 0 -gcd x y = gcd y (0 `mod` y) - -gcd 0 x = 0 -gcd x y = gcd 0 (x `mod` y) - -gcd 0 x = 0 -gcd x y = gcd 0 (y `mod` x) - -gcd 0 x = 0 -gcd x y = gcd 0 (y `mod` y) - -gcd 0 x = 0 -gcd x y = gcd 0 (0 `mod` y) - -gcd 0 x = 0 -gcd x y = gcd (x `mod` x) x - -gcd 0 x = 0 -gcd x y = gcd (x `mod` x) y - -gcd 0 x = 0 -gcd x y = gcd (x `mod` x) 0 - -gcd 0 x = 0 -gcd x y = gcd (x `mod` y) x - -gcd 0 x = 0 -gcd x y = gcd (x `mod` y) y - -gcd 0 x = 0 -gcd x y = gcd (x `mod` y) 0 - -gcd 0 x = 0 -gcd x y = gcd (y `mod` x) x - -gcd 0 x = 0 -gcd x y = gcd (y `mod` x) y - -gcd 0 x = 0 -gcd x y = gcd (y `mod` x) 0 - -gcd 0 x = 0 -gcd x y = gcd (0 `mod` x) x - -gcd 0 x = 0 -gcd x y = gcd (0 `mod` x) y - -gcd 0 x = 0 -gcd x y = gcd (0 `mod` x) 0 - -gcd x 0 = x -gcd x y = (x `mod` x) `mod` x - -gcd x 0 = x -gcd x y = (x `mod` x) `mod` y - -gcd x 0 = x -gcd x y = (x `mod` x) `mod` 0 - -gcd x 0 = x -gcd x y = (x `mod` y) `mod` x - -gcd x 0 = x -gcd x y = (x `mod` y) `mod` y - -gcd x 0 = x -gcd x y = (x `mod` y) `mod` 0 - -gcd x 0 = x -gcd x y = (x `mod` 0) `mod` x - -gcd x 0 = x -gcd x y = (x `mod` 0) `mod` y - -gcd x 0 = x -gcd x y = (x `mod` 0) `mod` 0 - -gcd x 0 = x -gcd x y = (y `mod` x) `mod` x - -gcd x 0 = x -gcd x y = (y `mod` x) `mod` y - -gcd x 0 = x -gcd x y = (y `mod` x) `mod` 0 - -gcd x 0 = x -gcd x y = (y `mod` y) `mod` x - -gcd x 0 = x -gcd x y = (y `mod` y) `mod` y - -gcd x 0 = x -gcd x y = (y `mod` y) `mod` 0 - -gcd x 0 = x -gcd x y = (y `mod` 0) `mod` x - -gcd x 0 = x -gcd x y = (y `mod` 0) `mod` y - -gcd x 0 = x -gcd x y = (y `mod` 0) `mod` 0 - -gcd x 0 = x -gcd x y = (0 `mod` x) `mod` x - -gcd x 0 = x -gcd x y = (0 `mod` x) `mod` y - -gcd x 0 = x -gcd x y = (0 `mod` x) `mod` 0 - -gcd x 0 = x -gcd x y = (0 `mod` y) `mod` x - -gcd x 0 = x -gcd x y = (0 `mod` y) `mod` y - -gcd x 0 = x -gcd x y = (0 `mod` y) `mod` 0 - -gcd x 0 = x -gcd x y = (0 `mod` 0) `mod` x - -gcd x 0 = x -gcd x y = (0 `mod` 0) `mod` y - -gcd x 0 = x -gcd x y = (0 `mod` 0) `mod` 0 - -gcd x 0 = x -gcd x y = x `mod` (x `mod` x) - -gcd x 0 = x -gcd x y = x `mod` (x `mod` y) - -gcd x 0 = x -gcd x y = x `mod` (x `mod` 0) - -gcd x 0 = x -gcd x y = x `mod` (y `mod` x) - -gcd x 0 = x -gcd x y = x `mod` (y `mod` y) - -gcd x 0 = x -gcd x y = x `mod` (y `mod` 0) - -gcd x 0 = x -gcd x y = x `mod` (0 `mod` x) - -gcd x 0 = x -gcd x y = x `mod` (0 `mod` y) - -gcd x 0 = x -gcd x y = x `mod` (0 `mod` 0) - -gcd x 0 = x -gcd x y = y `mod` (x `mod` x) - -gcd x 0 = x -gcd x y = y `mod` (x `mod` y) - -gcd x 0 = x -gcd x y = y `mod` (x `mod` 0) - -gcd x 0 = x -gcd x y = y `mod` (y `mod` x) - -gcd x 0 = x -gcd x y = y `mod` (y `mod` y) - -gcd x 0 = x -gcd x y = y `mod` (y `mod` 0) - -gcd x 0 = x -gcd x y = y `mod` (0 `mod` x) - -gcd x 0 = x -gcd x y = y `mod` (0 `mod` y) - -gcd x 0 = x -gcd x y = y `mod` (0 `mod` 0) - -gcd x 0 = x -gcd x y = 0 `mod` (x `mod` x) - -gcd x 0 = x -gcd x y = 0 `mod` (x `mod` y) - -gcd x 0 = x -gcd x y = 0 `mod` (x `mod` 0) - -gcd x 0 = x -gcd x y = 0 `mod` (y `mod` x) - -gcd x 0 = x -gcd x y = 0 `mod` (y `mod` y) - -gcd x 0 = x -gcd x y = 0 `mod` (y `mod` 0) - -gcd x 0 = x -gcd x y = 0 `mod` (0 `mod` x) - -gcd x 0 = x -gcd x y = 0 `mod` (0 `mod` y) - -gcd x 0 = x -gcd x y = 0 `mod` (0 `mod` 0) - -gcd x 0 = 0 -gcd x y = (x `mod` x) `mod` x - -gcd x 0 = 0 -gcd x y = (x `mod` x) `mod` y - -gcd x 0 = 0 -gcd x y = (x `mod` x) `mod` 0 - -gcd x 0 = 0 -gcd x y = (x `mod` y) `mod` x - -gcd x 0 = 0 -gcd x y = (x `mod` y) `mod` y - -gcd x 0 = 0 -gcd x y = (x `mod` y) `mod` 0 - -gcd x 0 = 0 -gcd x y = (x `mod` 0) `mod` x - -gcd x 0 = 0 -gcd x y = (x `mod` 0) `mod` y - -gcd x 0 = 0 -gcd x y = (x `mod` 0) `mod` 0 - -gcd x 0 = 0 -gcd x y = (y `mod` x) `mod` x - -gcd x 0 = 0 -gcd x y = (y `mod` x) `mod` y - -gcd x 0 = 0 -gcd x y = (y `mod` x) `mod` 0 - -gcd x 0 = 0 -gcd x y = (y `mod` y) `mod` x - -gcd x 0 = 0 -gcd x y = (y `mod` y) `mod` y - -gcd x 0 = 0 -gcd x y = (y `mod` y) `mod` 0 - -gcd x 0 = 0 -gcd x y = (y `mod` 0) `mod` x - -gcd x 0 = 0 -gcd x y = (y `mod` 0) `mod` y - -gcd x 0 = 0 -gcd x y = (y `mod` 0) `mod` 0 - -gcd x 0 = 0 -gcd x y = (0 `mod` x) `mod` x - -gcd x 0 = 0 -gcd x y = (0 `mod` x) `mod` y - -gcd x 0 = 0 -gcd x y = (0 `mod` x) `mod` 0 - -gcd x 0 = 0 -gcd x y = (0 `mod` y) `mod` x - -gcd x 0 = 0 -gcd x y = (0 `mod` y) `mod` y - -gcd x 0 = 0 -gcd x y = (0 `mod` y) `mod` 0 - -gcd x 0 = 0 -gcd x y = (0 `mod` 0) `mod` x - -gcd x 0 = 0 -gcd x y = (0 `mod` 0) `mod` y - -gcd x 0 = 0 -gcd x y = (0 `mod` 0) `mod` 0 - -gcd x 0 = 0 -gcd x y = x `mod` (x `mod` x) - -gcd x 0 = 0 -gcd x y = x `mod` (x `mod` y) - -gcd x 0 = 0 -gcd x y = x `mod` (x `mod` 0) - -gcd x 0 = 0 -gcd x y = x `mod` (y `mod` x) - -gcd x 0 = 0 -gcd x y = x `mod` (y `mod` y) - -gcd x 0 = 0 -gcd x y = x `mod` (y `mod` 0) - -gcd x 0 = 0 -gcd x y = x `mod` (0 `mod` x) - -gcd x 0 = 0 -gcd x y = x `mod` (0 `mod` y) - -gcd x 0 = 0 -gcd x y = x `mod` (0 `mod` 0) - -gcd x 0 = 0 -gcd x y = y `mod` (x `mod` x) - -gcd x 0 = 0 -gcd x y = y `mod` (x `mod` y) - -gcd x 0 = 0 -gcd x y = y `mod` (x `mod` 0) - -gcd x 0 = 0 -gcd x y = y `mod` (y `mod` x) - -gcd x 0 = 0 -gcd x y = y `mod` (y `mod` y) - -gcd x 0 = 0 -gcd x y = y `mod` (y `mod` 0) - -gcd x 0 = 0 -gcd x y = y `mod` (0 `mod` x) - -gcd x 0 = 0 -gcd x y = y `mod` (0 `mod` y) - -gcd x 0 = 0 -gcd x y = y `mod` (0 `mod` 0) - -gcd x 0 = 0 -gcd x y = 0 `mod` (x `mod` x) - -gcd x 0 = 0 -gcd x y = 0 `mod` (x `mod` y) - -gcd x 0 = 0 -gcd x y = 0 `mod` (x `mod` 0) - -gcd x 0 = 0 -gcd x y = 0 `mod` (y `mod` x) - -gcd x 0 = 0 -gcd x y = 0 `mod` (y `mod` y) - -gcd x 0 = 0 -gcd x y = 0 `mod` (y `mod` 0) - -gcd x 0 = 0 -gcd x y = 0 `mod` (0 `mod` x) - -gcd x 0 = 0 -gcd x y = 0 `mod` (0 `mod` y) - -gcd x 0 = 0 -gcd x y = 0 `mod` (0 `mod` 0) - -gcd x 0 = x `mod` x -gcd x y = x `mod` y - -gcd x 0 = x `mod` x -gcd x y = x `mod` 0 - -gcd x 0 = x `mod` x -gcd x y = y `mod` x - -gcd x 0 = x `mod` x -gcd x y = y `mod` y - -gcd x 0 = x `mod` x -gcd x y = y `mod` 0 - -gcd x 0 = x `mod` x -gcd x y = 0 `mod` x - -gcd x 0 = x `mod` x -gcd x y = 0 `mod` y - -gcd x 0 = x `mod` x -gcd x y = 0 `mod` 0 - -gcd x 0 = x `mod` 0 -gcd x y = x `mod` x - -gcd x 0 = x `mod` 0 -gcd x y = x `mod` y - -gcd x 0 = x `mod` 0 -gcd x y = y `mod` x - -gcd x 0 = x `mod` 0 -gcd x y = y `mod` y - -gcd x 0 = x `mod` 0 -gcd x y = y `mod` 0 - -gcd x 0 = x `mod` 0 -gcd x y = 0 `mod` x - -gcd x 0 = x `mod` 0 -gcd x y = 0 `mod` y - -gcd x 0 = x `mod` 0 -gcd x y = 0 `mod` 0 - -gcd x 0 = 0 `mod` x -gcd x y = x `mod` x - -gcd x 0 = 0 `mod` x -gcd x y = x `mod` y - -gcd x 0 = 0 `mod` x -gcd x y = x `mod` 0 - -gcd x 0 = 0 `mod` x -gcd x y = y `mod` x - -gcd x 0 = 0 `mod` x -gcd x y = y `mod` y - -gcd x 0 = 0 `mod` x -gcd x y = y `mod` 0 - -gcd x 0 = 0 `mod` x -gcd x y = 0 `mod` y - -gcd x 0 = 0 `mod` x -gcd x y = 0 `mod` 0 - -gcd x 0 = 0 `mod` 0 -gcd x y = x `mod` x - -gcd x 0 = 0 `mod` 0 -gcd x y = x `mod` y - -gcd x 0 = 0 `mod` 0 -gcd x y = x `mod` 0 - -gcd x 0 = 0 `mod` 0 -gcd x y = y `mod` x - -gcd x 0 = 0 `mod` 0 -gcd x y = y `mod` y - -gcd x 0 = 0 `mod` 0 -gcd x y = y `mod` 0 - -gcd x 0 = 0 `mod` 0 -gcd x y = 0 `mod` x - -gcd x 0 = 0 `mod` 0 -gcd x y = 0 `mod` y - -gcd x 0 = (x `mod` x) `mod` x -gcd x y = x - -gcd x 0 = (x `mod` x) `mod` x -gcd x y = y - -gcd x 0 = (x `mod` x) `mod` x -gcd x y = 0 - -gcd x 0 = (x `mod` x) `mod` 0 -gcd x y = x - -gcd x 0 = (x `mod` x) `mod` 0 -gcd x y = y - -gcd x 0 = (x `mod` x) `mod` 0 -gcd x y = 0 - -gcd x 0 = (x `mod` 0) `mod` x -gcd x y = x - -gcd x 0 = (x `mod` 0) `mod` x -gcd x y = y - -gcd x 0 = (x `mod` 0) `mod` x -gcd x y = 0 - -gcd x 0 = (x `mod` 0) `mod` 0 -gcd x y = x - -gcd x 0 = (x `mod` 0) `mod` 0 -gcd x y = y - -gcd x 0 = (x `mod` 0) `mod` 0 -gcd x y = 0 - -gcd x 0 = (0 `mod` x) `mod` x -gcd x y = x - -gcd x 0 = (0 `mod` x) `mod` x -gcd x y = y - -gcd x 0 = (0 `mod` x) `mod` x -gcd x y = 0 - -gcd x 0 = (0 `mod` x) `mod` 0 -gcd x y = x - -gcd x 0 = (0 `mod` x) `mod` 0 -gcd x y = y - -gcd x 0 = (0 `mod` x) `mod` 0 -gcd x y = 0 - -gcd x 0 = (0 `mod` 0) `mod` x -gcd x y = x - -gcd x 0 = (0 `mod` 0) `mod` x -gcd x y = y - -gcd x 0 = (0 `mod` 0) `mod` x -gcd x y = 0 - -gcd x 0 = (0 `mod` 0) `mod` 0 -gcd x y = x - -gcd x 0 = (0 `mod` 0) `mod` 0 -gcd x y = y - -gcd x 0 = (0 `mod` 0) `mod` 0 -gcd x y = 0 - -gcd x 0 = x `mod` (x `mod` x) -gcd x y = x - -gcd x 0 = x `mod` (x `mod` x) -gcd x y = y - -gcd x 0 = x `mod` (x `mod` x) -gcd x y = 0 - -gcd x 0 = x `mod` (x `mod` 0) -gcd x y = x - -gcd x 0 = x `mod` (x `mod` 0) -gcd x y = y - -gcd x 0 = x `mod` (x `mod` 0) -gcd x y = 0 - -gcd x 0 = x `mod` (0 `mod` x) -gcd x y = x - -gcd x 0 = x `mod` (0 `mod` x) -gcd x y = y - -gcd x 0 = x `mod` (0 `mod` x) -gcd x y = 0 - -gcd x 0 = x `mod` (0 `mod` 0) -gcd x y = x - -gcd x 0 = x `mod` (0 `mod` 0) -gcd x y = y - -gcd x 0 = x `mod` (0 `mod` 0) -gcd x y = 0 - -gcd x 0 = 0 `mod` (x `mod` x) -gcd x y = x - -gcd x 0 = 0 `mod` (x `mod` x) -gcd x y = y - -gcd x 0 = 0 `mod` (x `mod` x) -gcd x y = 0 - -gcd x 0 = 0 `mod` (x `mod` 0) -gcd x y = x - -gcd x 0 = 0 `mod` (x `mod` 0) -gcd x y = y - -gcd x 0 = 0 `mod` (x `mod` 0) -gcd x y = 0 - -gcd x 0 = 0 `mod` (0 `mod` x) -gcd x y = x - -gcd x 0 = 0 `mod` (0 `mod` x) -gcd x y = y - -gcd x 0 = 0 `mod` (0 `mod` x) -gcd x y = 0 - -gcd x 0 = 0 `mod` (0 `mod` 0) -gcd x y = x - -gcd x 0 = 0 `mod` (0 `mod` 0) -gcd x y = y - -gcd x 0 = 0 `mod` (0 `mod` 0) -gcd x y = 0 - -gcd 0 x = x -gcd x y = (x `mod` x) `mod` x - -gcd 0 x = x -gcd x y = (x `mod` x) `mod` y - -gcd 0 x = x -gcd x y = (x `mod` x) `mod` 0 - -gcd 0 x = x -gcd x y = (x `mod` y) `mod` x - -gcd 0 x = x -gcd x y = (x `mod` y) `mod` y - -gcd 0 x = x -gcd x y = (x `mod` y) `mod` 0 - -gcd 0 x = x -gcd x y = (x `mod` 0) `mod` x - -gcd 0 x = x -gcd x y = (x `mod` 0) `mod` y - -gcd 0 x = x -gcd x y = (x `mod` 0) `mod` 0 - -gcd 0 x = x -gcd x y = (y `mod` x) `mod` x - -gcd 0 x = x -gcd x y = (y `mod` x) `mod` y - -gcd 0 x = x -gcd x y = (y `mod` x) `mod` 0 - -gcd 0 x = x -gcd x y = (y `mod` y) `mod` x - -gcd 0 x = x -gcd x y = (y `mod` y) `mod` y - -gcd 0 x = x -gcd x y = (y `mod` y) `mod` 0 - -gcd 0 x = x -gcd x y = (y `mod` 0) `mod` x - -gcd 0 x = x -gcd x y = (y `mod` 0) `mod` y - -gcd 0 x = x -gcd x y = (y `mod` 0) `mod` 0 - -gcd 0 x = x -gcd x y = (0 `mod` x) `mod` x - -gcd 0 x = x -gcd x y = (0 `mod` x) `mod` y - -gcd 0 x = x -gcd x y = (0 `mod` x) `mod` 0 - -gcd 0 x = x -gcd x y = (0 `mod` y) `mod` x - -gcd 0 x = x -gcd x y = (0 `mod` y) `mod` y - -gcd 0 x = x -gcd x y = (0 `mod` y) `mod` 0 - -gcd 0 x = x -gcd x y = (0 `mod` 0) `mod` x - -gcd 0 x = x -gcd x y = (0 `mod` 0) `mod` y - -gcd 0 x = x -gcd x y = (0 `mod` 0) `mod` 0 - -gcd 0 x = x -gcd x y = x `mod` (x `mod` x) - -gcd 0 x = x -gcd x y = x `mod` (x `mod` y) - -gcd 0 x = x -gcd x y = x `mod` (x `mod` 0) - -gcd 0 x = x -gcd x y = x `mod` (y `mod` x) - -gcd 0 x = x -gcd x y = x `mod` (y `mod` y) - -gcd 0 x = x -gcd x y = x `mod` (y `mod` 0) - -gcd 0 x = x -gcd x y = x `mod` (0 `mod` x) - -gcd 0 x = x -gcd x y = x `mod` (0 `mod` y) - -gcd 0 x = x -gcd x y = x `mod` (0 `mod` 0) - -gcd 0 x = x -gcd x y = y `mod` (x `mod` x) - -gcd 0 x = x -gcd x y = y `mod` (x `mod` y) - -gcd 0 x = x -gcd x y = y `mod` (x `mod` 0) - -gcd 0 x = x -gcd x y = y `mod` (y `mod` x) - -gcd 0 x = x -gcd x y = y `mod` (y `mod` y) - -gcd 0 x = x -gcd x y = y `mod` (y `mod` 0) - -gcd 0 x = x -gcd x y = y `mod` (0 `mod` x) - -gcd 0 x = x -gcd x y = y `mod` (0 `mod` y) - -gcd 0 x = x -gcd x y = y `mod` (0 `mod` 0) - -gcd 0 x = x -gcd x y = 0 `mod` (x `mod` x) - -gcd 0 x = x -gcd x y = 0 `mod` (x `mod` y) - -gcd 0 x = x -gcd x y = 0 `mod` (x `mod` 0) - -gcd 0 x = x -gcd x y = 0 `mod` (y `mod` x) - -gcd 0 x = x -gcd x y = 0 `mod` (y `mod` y) - -gcd 0 x = x -gcd x y = 0 `mod` (y `mod` 0) - -gcd 0 x = x -gcd x y = 0 `mod` (0 `mod` x) - -gcd 0 x = x -gcd x y = 0 `mod` (0 `mod` y) - -gcd 0 x = x -gcd x y = 0 `mod` (0 `mod` 0) - -gcd 0 x = 0 -gcd x y = (x `mod` x) `mod` x - -gcd 0 x = 0 -gcd x y = (x `mod` x) `mod` y - -gcd 0 x = 0 -gcd x y = (x `mod` x) `mod` 0 - -gcd 0 x = 0 -gcd x y = (x `mod` y) `mod` x - -gcd 0 x = 0 -gcd x y = (x `mod` y) `mod` y - -gcd 0 x = 0 -gcd x y = (x `mod` y) `mod` 0 - -gcd 0 x = 0 -gcd x y = (x `mod` 0) `mod` x - -gcd 0 x = 0 -gcd x y = (x `mod` 0) `mod` y - -gcd 0 x = 0 -gcd x y = (x `mod` 0) `mod` 0 - -gcd 0 x = 0 -gcd x y = (y `mod` x) `mod` x - -gcd 0 x = 0 -gcd x y = (y `mod` x) `mod` y - -gcd 0 x = 0 -gcd x y = (y `mod` x) `mod` 0 - -gcd 0 x = 0 -gcd x y = (y `mod` y) `mod` x - -gcd 0 x = 0 -gcd x y = (y `mod` y) `mod` y - -gcd 0 x = 0 -gcd x y = (y `mod` y) `mod` 0 - -gcd 0 x = 0 -gcd x y = (y `mod` 0) `mod` x - -gcd 0 x = 0 -gcd x y = (y `mod` 0) `mod` y - -gcd 0 x = 0 -gcd x y = (y `mod` 0) `mod` 0 - -gcd 0 x = 0 -gcd x y = (0 `mod` x) `mod` x - -gcd 0 x = 0 -gcd x y = (0 `mod` x) `mod` y - -gcd 0 x = 0 -gcd x y = (0 `mod` x) `mod` 0 - -gcd 0 x = 0 -gcd x y = (0 `mod` y) `mod` x - -gcd 0 x = 0 -gcd x y = (0 `mod` y) `mod` y - -gcd 0 x = 0 -gcd x y = (0 `mod` y) `mod` 0 - -gcd 0 x = 0 -gcd x y = (0 `mod` 0) `mod` x - -gcd 0 x = 0 -gcd x y = (0 `mod` 0) `mod` y - -gcd 0 x = 0 -gcd x y = (0 `mod` 0) `mod` 0 - -gcd 0 x = 0 -gcd x y = x `mod` (x `mod` x) - -gcd 0 x = 0 -gcd x y = x `mod` (x `mod` y) - -gcd 0 x = 0 -gcd x y = x `mod` (x `mod` 0) - -gcd 0 x = 0 -gcd x y = x `mod` (y `mod` x) - -gcd 0 x = 0 -gcd x y = x `mod` (y `mod` y) - -gcd 0 x = 0 -gcd x y = x `mod` (y `mod` 0) - -gcd 0 x = 0 -gcd x y = x `mod` (0 `mod` x) - -gcd 0 x = 0 -gcd x y = x `mod` (0 `mod` y) - -gcd 0 x = 0 -gcd x y = x `mod` (0 `mod` 0) - -gcd 0 x = 0 -gcd x y = y `mod` (x `mod` x) - -gcd 0 x = 0 -gcd x y = y `mod` (x `mod` y) - -gcd 0 x = 0 -gcd x y = y `mod` (x `mod` 0) - -gcd 0 x = 0 -gcd x y = y `mod` (y `mod` x) - -gcd 0 x = 0 -gcd x y = y `mod` (y `mod` y) - -gcd 0 x = 0 -gcd x y = y `mod` (y `mod` 0) - -gcd 0 x = 0 -gcd x y = y `mod` (0 `mod` x) - -gcd 0 x = 0 -gcd x y = y `mod` (0 `mod` y) - -gcd 0 x = 0 -gcd x y = y `mod` (0 `mod` 0) - -gcd 0 x = 0 -gcd x y = 0 `mod` (x `mod` x) - -gcd 0 x = 0 -gcd x y = 0 `mod` (x `mod` y) - -gcd 0 x = 0 -gcd x y = 0 `mod` (x `mod` 0) - -gcd 0 x = 0 -gcd x y = 0 `mod` (y `mod` x) - -gcd 0 x = 0 -gcd x y = 0 `mod` (y `mod` y) - -gcd 0 x = 0 -gcd x y = 0 `mod` (y `mod` 0) - -gcd 0 x = 0 -gcd x y = 0 `mod` (0 `mod` x) - -gcd 0 x = 0 -gcd x y = 0 `mod` (0 `mod` y) - -gcd 0 x = 0 -gcd x y = 0 `mod` (0 `mod` 0) - -gcd 0 x = x `mod` x -gcd x y = x `mod` y - -gcd 0 x = x `mod` x -gcd x y = x `mod` 0 - -gcd 0 x = x `mod` x -gcd x y = y `mod` x - -gcd 0 x = x `mod` x -gcd x y = y `mod` 0 - -gcd 0 x = x `mod` x -gcd x y = 0 `mod` x - -gcd 0 x = x `mod` x -gcd x y = 0 `mod` y - -gcd 0 x = x `mod` x -gcd x y = 0 `mod` 0 - -gcd 0 x = x `mod` 0 -gcd x y = x `mod` x - -gcd 0 x = x `mod` 0 -gcd x y = x `mod` y - -gcd 0 x = x `mod` 0 -gcd x y = y `mod` x - -gcd 0 x = x `mod` 0 -gcd x y = y `mod` y - -gcd 0 x = x `mod` 0 -gcd x y = 0 `mod` x - -gcd 0 x = x `mod` 0 -gcd x y = 0 `mod` y - -gcd 0 x = x `mod` 0 -gcd x y = 0 `mod` 0 - -gcd 0 x = 0 `mod` x -gcd x y = x `mod` x - -gcd 0 x = 0 `mod` x -gcd x y = x `mod` y - -gcd 0 x = 0 `mod` x -gcd x y = x `mod` 0 - -gcd 0 x = 0 `mod` x -gcd x y = y `mod` x - -gcd 0 x = 0 `mod` x -gcd x y = y `mod` y - -gcd 0 x = 0 `mod` x -gcd x y = y `mod` 0 - -gcd 0 x = 0 `mod` x -gcd x y = 0 `mod` 0 - -gcd 0 x = 0 `mod` 0 -gcd x y = x `mod` x - -gcd 0 x = 0 `mod` 0 -gcd x y = x `mod` y - -gcd 0 x = 0 `mod` 0 -gcd x y = x `mod` 0 - -gcd 0 x = 0 `mod` 0 -gcd x y = y `mod` x - -gcd 0 x = 0 `mod` 0 -gcd x y = y `mod` y - -gcd 0 x = 0 `mod` 0 -gcd x y = y `mod` 0 - -gcd 0 x = 0 `mod` 0 -gcd x y = 0 `mod` x - -gcd 0 x = 0 `mod` 0 -gcd x y = 0 `mod` y - -gcd 0 x = (x `mod` x) `mod` x -gcd x y = x - -gcd 0 x = (x `mod` x) `mod` x -gcd x y = y - -gcd 0 x = (x `mod` x) `mod` x -gcd x y = 0 - -gcd 0 x = (x `mod` x) `mod` 0 -gcd x y = x - -gcd 0 x = (x `mod` x) `mod` 0 -gcd x y = y - -gcd 0 x = (x `mod` x) `mod` 0 -gcd x y = 0 - -gcd 0 x = (x `mod` 0) `mod` x -gcd x y = x - -gcd 0 x = (x `mod` 0) `mod` x -gcd x y = y - -gcd 0 x = (x `mod` 0) `mod` x -gcd x y = 0 - -gcd 0 x = (x `mod` 0) `mod` 0 -gcd x y = x - -gcd 0 x = (x `mod` 0) `mod` 0 -gcd x y = y - -gcd 0 x = (x `mod` 0) `mod` 0 -gcd x y = 0 - -gcd 0 x = (0 `mod` x) `mod` x -gcd x y = x - -gcd 0 x = (0 `mod` x) `mod` x -gcd x y = y - -gcd 0 x = (0 `mod` x) `mod` x -gcd x y = 0 - -gcd 0 x = (0 `mod` x) `mod` 0 -gcd x y = x - -gcd 0 x = (0 `mod` x) `mod` 0 -gcd x y = y - -gcd 0 x = (0 `mod` x) `mod` 0 -gcd x y = 0 - -gcd 0 x = (0 `mod` 0) `mod` x -gcd x y = x - -gcd 0 x = (0 `mod` 0) `mod` x -gcd x y = y - -gcd 0 x = (0 `mod` 0) `mod` x -gcd x y = 0 - -gcd 0 x = (0 `mod` 0) `mod` 0 -gcd x y = x - -gcd 0 x = (0 `mod` 0) `mod` 0 -gcd x y = y - -gcd 0 x = (0 `mod` 0) `mod` 0 -gcd x y = 0 - -gcd 0 x = x `mod` (x `mod` x) -gcd x y = x - -gcd 0 x = x `mod` (x `mod` x) -gcd x y = y - -gcd 0 x = x `mod` (x `mod` x) -gcd x y = 0 - -gcd 0 x = x `mod` (x `mod` 0) -gcd x y = x - -gcd 0 x = x `mod` (x `mod` 0) -gcd x y = y - -gcd 0 x = x `mod` (x `mod` 0) -gcd x y = 0 - -gcd 0 x = x `mod` (0 `mod` x) -gcd x y = x - -gcd 0 x = x `mod` (0 `mod` x) -gcd x y = y - -gcd 0 x = x `mod` (0 `mod` x) -gcd x y = 0 - -gcd 0 x = x `mod` (0 `mod` 0) -gcd x y = x - -gcd 0 x = x `mod` (0 `mod` 0) -gcd x y = y - -gcd 0 x = x `mod` (0 `mod` 0) -gcd x y = 0 - -gcd 0 x = 0 `mod` (x `mod` x) -gcd x y = x - -gcd 0 x = 0 `mod` (x `mod` x) -gcd x y = y - -gcd 0 x = 0 `mod` (x `mod` x) -gcd x y = 0 - -gcd 0 x = 0 `mod` (x `mod` 0) -gcd x y = x - -gcd 0 x = 0 `mod` (x `mod` 0) -gcd x y = y - -gcd 0 x = 0 `mod` (x `mod` 0) -gcd x y = 0 - -gcd 0 x = 0 `mod` (0 `mod` x) -gcd x y = x - -gcd 0 x = 0 `mod` (0 `mod` x) -gcd x y = y - -gcd 0 x = 0 `mod` (0 `mod` x) -gcd x y = 0 - -gcd 0 x = 0 `mod` (0 `mod` 0) -gcd x y = x - -gcd 0 x = 0 `mod` (0 `mod` 0) -gcd x y = y - -gcd 0 x = 0 `mod` (0 `mod` 0) + [3,0,7,0,19,0,90,0,454] direct candidates, 0 duplicates + [3,3,8,13,29,52,139,212,629] pattern candidates, 0 duplicates + +rules: +x * y == x + y +x * y == y + x +x - x == 0 +x + 0 == x +0 + x == x +x - 0 == x +(x + y) + z == x + (y + z) +(x + y) + z == y + (x + z) +x - (y - z) == z + (x - y) +(x - y) - z == x - (y + z) +(x - y) - z == x - (z + y) +(x + y) - z == x + (y - z) +(x + y) - z == y + (x - z) +x + (y - x) == y +(x - y) + y == x +equations: +y + x == x + y +y + (x + z) == x + (y + z) +z + (x + y) == x + (y + z) +z + (y + x) == x + (y + z) +y + (x - z) == x + (y - z) +(x - z) + y == x + (y - z) +(z - y) + x == (x - y) + z +y - (x + y) == 0 - x +y - (y + x) == 0 - x +z - (y + z) == x - (x + y) +z - (y + z) == x - (y + x) +z - (z + y) == x - (x + y) +z - (z + y) == x - (y + x) +x + (0 - y) == x - y +(0 - y) + x == x - y +x - (x + 1) == 0 - 1 +x - (1 + x) == 0 - 1 +y - (y + 1) == x - (x + 1) +y - (y + 1) == x - (1 + x) +y - (1 + y) == x - (1 + x) + +direct candidates: + +foo x = x + +foo x = 0 + +foo x = 1 + +foo x = x + x + +foo x = x + 1 + +foo x = 1 + 1 + +foo x = x - 1 + +foo x = 0 - x + +foo x = 0 - 1 + +foo x = 1 - x + +foo x = x + (x + x) + +foo x = x + (x + 1) + +foo x = x + (1 + 1) + +foo x = x + (x - 1) + +foo x = x + (0 - 1) + +foo x = 1 + (x + x) + +foo x = 1 + (x + 1) + +foo x = 1 + (1 + 1) + +foo x = 1 + (0 - x) + +foo x = 1 + (1 - x) + +foo x = x - (x + x) + +foo x = x - (x + 1) + +foo x = x - (1 + 1) + +foo x = 0 - (x + x) + +foo x = 0 - (x + 1) + +foo x = 0 - (1 + 1) + +foo x = 1 - (x + x) + +foo x = 1 - (x + 1) + +foo x = 1 - (1 + 1) + + +pattern candidates: + +foo x = x + +foo x = 0 + +foo x = 1 + +foo 0 = 0 +foo x = 1 + +foo 0 = 1 +foo x = x + +foo 0 = 1 +foo x = 0 + +foo x = x + x + +foo x = x + 1 + +foo x = x - 1 + +foo x = 0 - x + +foo x = 1 - x + +foo 1 = 0 +foo x = x + +foo 1 = 0 +foo x = 1 + +foo 1 = 1 +foo x = 0 + +foo 0 = 0 +foo x = x + x + +foo 0 = 0 +foo x = x + 1 + +foo 0 = 0 +foo x = x - 1 + +foo 0 = 0 +foo x = 0 - x + +foo 0 = 0 +foo x = 1 - x + +foo 0 = 1 +foo x = x + x + +foo 0 = 1 +foo x = x + 1 + +foo 0 = 1 +foo x = x - 1 + +foo 0 = 1 +foo x = 0 - x + +foo 0 = 1 +foo x = 1 - x + +foo 0 = 0 +foo 1 = 0 +foo x = 1 + +foo 0 = 1 +foo 1 = 0 +foo x = x + +foo 0 = 1 +foo 1 = 1 +foo x = 0 + +foo 0 = 0 +foo x = foo (x - 1) + +foo 0 = 0 +foo x = foo (0 - x) + +foo 0 = 0 +foo x = foo (1 - x) + +foo 0 = 1 +foo x = foo (x - 1) + +foo 0 = 1 +foo x = foo (0 - x) + +foo 0 = 1 +foo x = foo (1 - x) + +foo x = x + (x + x) + +foo x = x + (x + 1) + +foo x = x + (x - 1) + +foo x = 1 + (x + x) + +foo x = 1 + (x + 1) + +foo x = 1 + (0 - x) + +foo x = 1 + (1 - x) + +foo x = x - (x + x) + +foo x = x - (x + 1) + +foo x = 0 - (x + x) + +foo x = 0 - (x + 1) + +foo x = 1 - (x + x) + +foo x = 1 - (x + 1) + +foo 1 = 0 +foo x = x + x + +foo 1 = 0 +foo x = x + 1 + +foo 1 = 0 +foo x = x - 1 + +foo 1 = 0 +foo x = 0 - x + +foo 1 = 0 +foo x = 1 - x + +foo 1 = 1 +foo x = x + x + +foo 1 = 1 +foo x = x + 1 + +foo 1 = 1 +foo x = x - 1 + +foo 1 = 1 +foo x = 0 - x + +foo 1 = 1 +foo x = 1 - x + +foo 1 = 0 +foo x = foo (x - 1) + +foo 1 = 0 +foo x = foo (0 - x) + +foo 1 = 0 +foo x = foo (1 - x) + +foo 1 = 1 +foo x = foo (x - 1) + +foo 1 = 1 +foo x = foo (0 - x) + +foo 1 = 1 +foo x = foo (1 - x) + +foo 0 = 0 +foo x = x + (x + x) + +foo 0 = 0 +foo x = x + (x + 1) + +foo 0 = 0 +foo x = x + (x - 1) + +foo 0 = 0 +foo x = 1 + (x + x) + +foo 0 = 0 +foo x = 1 + (x + 1) + +foo 0 = 0 +foo x = 1 + (0 - x) + +foo 0 = 0 +foo x = 1 + (1 - x) + +foo 0 = 0 +foo x = x - (x + x) + +foo 0 = 0 +foo x = x - (x + 1) + +foo 0 = 0 +foo x = 0 - (x + x) + +foo 0 = 0 +foo x = 0 - (x + 1) + +foo 0 = 0 +foo x = 1 - (x + x) + +foo 0 = 0 +foo x = 1 - (x + 1) + +foo 0 = 1 +foo x = x + (x + x) + +foo 0 = 1 +foo x = x + (x + 1) + +foo 0 = 1 +foo x = x + (x - 1) + +foo 0 = 1 +foo x = 1 + (x + x) + +foo 0 = 1 +foo x = 1 + (x + 1) + +foo 0 = 1 +foo x = 1 + (0 - x) + +foo 0 = 1 +foo x = 1 + (1 - x) + +foo 0 = 1 +foo x = x - (x + x) + +foo 0 = 1 +foo x = x - (x + 1) + +foo 0 = 1 +foo x = 0 - (x + x) + +foo 0 = 1 +foo x = 0 - (x + 1) + +foo 0 = 1 +foo x = 1 - (x + x) + +foo 0 = 1 +foo x = 1 - (x + 1) + +foo 0 = 0 +foo 1 = 0 +foo x = x + x + +foo 0 = 0 +foo 1 = 0 +foo x = x + 1 + +foo 0 = 0 +foo 1 = 0 +foo x = x - 1 + +foo 0 = 0 +foo 1 = 0 +foo x = 0 - x + +foo 0 = 0 +foo 1 = 0 +foo x = 1 - x + +foo 0 = 0 +foo 1 = 1 +foo x = x + x + +foo 0 = 0 +foo 1 = 1 +foo x = x + 1 + +foo 0 = 0 +foo 1 = 1 +foo x = x - 1 + +foo 0 = 0 +foo 1 = 1 +foo x = 0 - x + +foo 0 = 0 +foo 1 = 1 +foo x = 1 - x + +foo 0 = 1 +foo 1 = 0 +foo x = x + x + +foo 0 = 1 +foo 1 = 0 +foo x = x + 1 + +foo 0 = 1 +foo 1 = 0 +foo x = x - 1 + +foo 0 = 1 +foo 1 = 0 +foo x = 0 - x + +foo 0 = 1 +foo 1 = 0 +foo x = 1 - x + +foo 0 = 1 +foo 1 = 1 +foo x = x + x + +foo 0 = 1 +foo 1 = 1 +foo x = x + 1 + +foo 0 = 1 +foo 1 = 1 +foo x = x - 1 + +foo 0 = 1 +foo 1 = 1 +foo x = 0 - x + +foo 0 = 1 +foo 1 = 1 +foo x = 1 - x + + +Candidates for: ? :: Int -> Int -> Int + pruning with 10/23 rules + [3,3,6,9,21,39,87,180,390] direct candidates, 0 duplicates + [3,7,22,53,129,313,747,1743,4077] pattern candidates, 0 duplicates + +rules: +x * y == x + y +x * y == y + x +x + 0 == x +0 + x == x +dec (x + y) == x + dec y +dec (x + y) == y + dec x +dec (x + y) == dec x + y +dec (x + y) == dec y + x +(x + y) + z == x + (y + z) +(x + y) + z == y + (x + z) +equations: +y + x == x + y +y + dec x == x + dec y +dec x + y == x + dec y +dec y + x == dec x + y +x + dec 0 == dec x +dec 0 + x == dec x +y + (x + z) == x + (y + z) +z + (x + y) == x + (y + z) +z + (y + x) == x + (y + z) +y + dec (dec x) == x + dec (dec y) +dec (dec x) + y == x + dec (dec y) +x + dec (dec 0) == dec (dec x) +dec (dec 0) + x == dec (dec x) + +direct candidates: + +x ? y = x + +x ? y = y + +x ? y = 0 + +x ? y = dec x + +x ? y = dec y + +x ? y = dec 0 + +x ? y = x + x + +x ? y = x + y + +x ? y = y + y + +x ? y = dec (dec x) + +x ? y = dec (dec y) + +x ? y = dec (dec 0) + +x ? y = dec (dec (dec x)) + +x ? y = dec (dec (dec y)) + +x ? y = dec (dec (dec 0)) + +x ? y = x + dec x + +x ? y = x + dec y + +x ? y = x + dec 0 + +x ? y = y + dec x + +x ? y = y + dec y + +x ? y = y + dec 0 + +x ? y = dec (dec (dec (dec x))) + +x ? y = dec (dec (dec (dec y))) + +x ? y = dec (dec (dec (dec 0))) + +x ? y = x + (x + x) + +x ? y = x + (x + y) + +x ? y = x + (y + y) + +x ? y = x + dec (dec x) + +x ? y = x + dec (dec y) + +x ? y = x + dec (dec 0) + +x ? y = y + (x + x) + +x ? y = y + (x + y) + +x ? y = y + (y + y) + +x ? y = y + dec (dec x) + +x ? y = y + dec (dec y) + +x ? y = y + dec (dec 0) + +x ? y = dec x + dec x + +x ? y = dec x + dec y + +x ? y = dec x + dec 0 + +x ? y = dec y + dec y + +x ? y = dec y + dec 0 + +x ? y = dec 0 + dec 0 + +x ? y = dec (dec (dec (dec (dec x)))) + +x ? y = dec (dec (dec (dec (dec y)))) + +x ? y = dec (dec (dec (dec (dec 0)))) + +x ? y = x + dec (dec (dec x)) + +x ? y = x + dec (dec (dec y)) + +x ? y = x + dec (dec (dec 0)) + +x ? y = x + (x + dec x) + +x ? y = x + (x + dec y) + +x ? y = x + (x + dec 0) + +x ? y = x + (y + dec x) + +x ? y = x + (y + dec y) + +x ? y = x + (y + dec 0) + +x ? y = y + dec (dec (dec x)) + +x ? y = y + dec (dec (dec y)) + +x ? y = y + dec (dec (dec 0)) + +x ? y = y + (x + dec x) + +x ? y = y + (x + dec y) + +x ? y = y + (x + dec 0) + +x ? y = y + (y + dec x) + +x ? y = y + (y + dec y) + +x ? y = y + (y + dec 0) + +x ? y = dec x + (x + x) + +x ? y = dec x + (x + y) + +x ? y = dec x + (y + y) + +x ? y = dec x + dec (dec x) + +x ? y = dec x + dec (dec y) + +x ? y = dec x + dec (dec 0) + +x ? y = dec y + (x + x) + +x ? y = dec y + (x + y) + +x ? y = dec y + (y + y) + +x ? y = dec y + dec (dec x) + +x ? y = dec y + dec (dec y) + +x ? y = dec y + dec (dec 0) + +x ? y = dec 0 + (x + x) + +x ? y = dec 0 + (x + y) + +x ? y = dec 0 + (y + y) + +x ? y = dec 0 + dec (dec x) + +x ? y = dec 0 + dec (dec y) + +x ? y = dec 0 + dec (dec 0) + + +pattern candidates: + +x ? y = x + +x ? y = y + +x ? y = 0 + +x ? y = dec x + +x ? y = dec y + +x ? 0 = x +x ? y = y + +x ? 0 = x +x ? y = 0 + +x ? 0 = 0 +x ? y = x + +0 ? x = x +x ? y = 0 + +0 ? x = 0 +x ? y = y + +x ? y = x + x + +x ? y = x + y + +x ? y = y + y + +x ? y = dec (dec x) + +x ? y = dec (dec y) + +x ? 0 = x +x ? y = dec x + +x ? 0 = x +x ? y = dec y + +x ? 0 = 0 +x ? y = dec x + +x ? 0 = 0 +x ? y = dec y + +x ? 0 = dec x +x ? y = x + +x ? 0 = dec x +x ? y = y + +x ? 0 = dec x +x ? y = 0 + +0 ? x = x +x ? y = dec x + +0 ? x = x +x ? y = dec y + +0 ? x = 0 +x ? y = dec x + +0 ? x = 0 +x ? y = dec y + +0 ? x = dec x +x ? y = x + +0 ? x = dec x +x ? y = y + +0 ? x = dec x +x ? y = 0 + +0 ? x = x +x ? 0 = x +x ? y = 0 + +0 ? x = x +x ? 0 = 0 +x ? y = x + +0 ? x = 0 +x ? 0 = x +x ? y = y + +x ? y = dec (dec (dec x)) + +x ? y = dec (dec (dec y)) + +x ? y = x + dec x + +x ? y = x + dec y + +x ? y = y + dec x + +x ? y = y + dec y + +x ? 0 = x +x ? y = x + x + +x ? 0 = x +x ? y = x + y + +x ? 0 = x +x ? y = y + y + +x ? 0 = x +x ? y = dec (dec x) + +x ? 0 = x +x ? y = dec (dec y) + +x ? 0 = 0 +x ? y = x + x + +x ? 0 = 0 +x ? y = x + y + +x ? 0 = 0 +x ? y = y + y + +x ? 0 = 0 +x ? y = dec (dec x) + +x ? 0 = 0 +x ? y = dec (dec y) + +x ? 0 = dec x +x ? y = dec y + +x ? 0 = x + x +x ? y = x + +x ? 0 = x + x +x ? y = y + +x ? 0 = x + x +x ? y = 0 + +x ? 0 = dec (dec x) +x ? y = x + +x ? 0 = dec (dec x) +x ? y = y + +x ? 0 = dec (dec x) +x ? y = 0 + +0 ? x = x +x ? y = x + x + +0 ? x = x +x ? y = x + y + +0 ? x = x +x ? y = y + y + +0 ? x = x +x ? y = dec (dec x) + +0 ? x = x +x ? y = dec (dec y) + +0 ? x = 0 +x ? y = x + x + +0 ? x = 0 +x ? y = x + y + +0 ? x = 0 +x ? y = y + y + +0 ? x = 0 +x ? y = dec (dec x) + +0 ? x = 0 +x ? y = dec (dec y) + +0 ? x = x + x +x ? y = x + +0 ? x = x + x +x ? y = y + +0 ? x = x + x +x ? y = 0 + +0 ? x = dec (dec x) +x ? y = x + +0 ? x = dec (dec x) +x ? y = y + +0 ? x = dec (dec x) +x ? y = 0 + +0 ? x = x +x ? 0 = x +x ? y = dec x + +0 ? x = x +x ? 0 = x +x ? y = dec y + +0 ? x = x +x ? 0 = 0 +x ? y = dec x + +0 ? x = x +x ? 0 = 0 +x ? y = dec y + +0 ? x = x +x ? 0 = dec x +x ? y = x + +0 ? x = x +x ? 0 = dec x +x ? y = 0 + +0 ? x = 0 +x ? 0 = x +x ? y = dec x + +0 ? x = 0 +x ? 0 = x +x ? y = dec y + +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x + +0 ? x = 0 +x ? 0 = 0 +x ? y = dec y + +0 ? x = 0 +x ? 0 = dec x +x ? y = y + +0 ? x = dec x +x ? 0 = x +x ? y = y + +0 ? x = dec x +x ? 0 = x +x ? y = 0 + +0 ? x = dec x +x ? 0 = 0 +x ? y = x + +x ? 0 = x +x ? y = x ? dec y + +x ? 0 = x +x ? y = y ? dec y + +x ? 0 = x +x ? y = 0 ? dec y + +x ? 0 = x +x ? y = dec x ? x + +x ? 0 = x +x ? y = dec x ? y + +x ? 0 = 0 +x ? y = x ? dec y + +x ? 0 = 0 +x ? y = y ? dec y + +x ? 0 = 0 +x ? y = 0 ? dec y + +x ? 0 = 0 +x ? y = dec x ? x + +x ? 0 = 0 +x ? y = dec x ? y + +0 ? x = x +x ? y = x ? dec y + +0 ? x = x +x ? y = y ? dec y + +0 ? x = x +x ? y = dec x ? x + +0 ? x = x +x ? y = dec x ? y + +0 ? x = x +x ? y = dec x ? 0 + +0 ? x = 0 +x ? y = x ? dec y + +0 ? x = 0 +x ? y = y ? dec y + +0 ? x = 0 +x ? y = dec x ? x + +0 ? x = 0 +x ? y = dec x ? y + +0 ? x = 0 +x ? y = dec x ? 0 + +x ? y = dec (dec (dec (dec x))) + +x ? y = dec (dec (dec (dec y))) + +x ? y = x + (x + x) + +x ? y = x + (x + y) + +x ? y = x + (y + y) + +x ? y = x + dec (dec x) + +x ? y = x + dec (dec y) + +x ? y = y + (x + x) + +x ? y = y + (x + y) + +x ? y = y + (y + y) + +x ? y = y + dec (dec x) + +x ? y = y + dec (dec y) + +x ? y = dec x + dec x + +x ? y = dec x + dec y + +x ? y = dec y + dec y + +x ? 0 = x +x ? y = dec (dec (dec x)) + +x ? 0 = x +x ? y = dec (dec (dec y)) + +x ? 0 = x +x ? y = x + dec x + +x ? 0 = x +x ? y = x + dec y + +x ? 0 = x +x ? y = y + dec x + +x ? 0 = x +x ? y = y + dec y + +x ? 0 = 0 +x ? y = dec (dec (dec x)) + +x ? 0 = 0 +x ? y = dec (dec (dec y)) + +x ? 0 = 0 +x ? y = x + dec x + +x ? 0 = 0 +x ? y = x + dec y + +x ? 0 = 0 +x ? y = y + dec x + +x ? 0 = 0 +x ? y = y + dec y + +x ? 0 = dec x +x ? y = x + x + +x ? 0 = dec x +x ? y = x + y + +x ? 0 = dec x +x ? y = y + y + +x ? 0 = dec x +x ? y = dec (dec x) + +x ? 0 = dec x +x ? y = dec (dec y) + +x ? 0 = x + x +x ? y = dec x + +x ? 0 = x + x +x ? y = dec y + +x ? 0 = dec (dec x) +x ? y = dec x + +x ? 0 = dec (dec x) +x ? y = dec y + +x ? 0 = dec (dec (dec x)) +x ? y = x + +x ? 0 = dec (dec (dec x)) +x ? y = y + +x ? 0 = dec (dec (dec x)) +x ? y = 0 + +x ? 0 = x + dec x +x ? y = x + +x ? 0 = x + dec x +x ? y = y + +x ? 0 = x + dec x +x ? y = 0 + +0 ? x = x +x ? y = dec (dec (dec x)) + +0 ? x = x +x ? y = dec (dec (dec y)) + +0 ? x = x +x ? y = x + dec x + +0 ? x = x +x ? y = x + dec y + +0 ? x = x +x ? y = y + dec x + +0 ? x = x +x ? y = y + dec y + +0 ? x = 0 +x ? y = dec (dec (dec x)) + +0 ? x = 0 +x ? y = dec (dec (dec y)) + +0 ? x = 0 +x ? y = x + dec x + +0 ? x = 0 +x ? y = x + dec y + +0 ? x = 0 +x ? y = y + dec x + +0 ? x = 0 +x ? y = y + dec y + +0 ? x = dec x +x ? y = x + x + +0 ? x = dec x +x ? y = x + y + +0 ? x = dec x +x ? y = y + y + +0 ? x = dec x +x ? y = dec (dec x) + +0 ? x = dec x +x ? y = dec (dec y) + +0 ? x = x + x +x ? y = dec x + +0 ? x = x + x +x ? y = dec y + +0 ? x = dec (dec x) +x ? y = dec x + +0 ? x = dec (dec x) +x ? y = dec y + +0 ? x = dec (dec (dec x)) +x ? y = x + +0 ? x = dec (dec (dec x)) +x ? y = y + +0 ? x = dec (dec (dec x)) +x ? y = 0 + +0 ? x = x + dec x +x ? y = x + +0 ? x = x + dec x +x ? y = y + +0 ? x = x + dec x +x ? y = 0 + +0 ? x = x +x ? 0 = x +x ? y = x + x + +0 ? x = x +x ? 0 = x +x ? y = x + y + +0 ? x = x +x ? 0 = x +x ? y = y + y + +0 ? x = x +x ? 0 = x +x ? y = dec (dec x) + +0 ? x = x +x ? 0 = x +x ? y = dec (dec y) + +0 ? x = x +x ? 0 = 0 +x ? y = x + x + +0 ? x = x +x ? 0 = 0 +x ? y = x + y + +0 ? x = x +x ? 0 = 0 +x ? y = y + y + +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec x) + +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec y) + +0 ? x = x +x ? 0 = dec x +x ? y = dec y + +0 ? x = x +x ? 0 = x + x +x ? y = x + +0 ? x = x +x ? 0 = x + x +x ? y = 0 + +0 ? x = x +x ? 0 = dec (dec x) +x ? y = x + +0 ? x = x +x ? 0 = dec (dec x) +x ? y = 0 + +0 ? x = 0 +x ? 0 = x +x ? y = x + x + +0 ? x = 0 +x ? 0 = x +x ? y = x + y + +0 ? x = 0 +x ? 0 = x +x ? y = y + y + +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec x) + +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec y) + +0 ? x = 0 +x ? 0 = 0 +x ? y = x + x + +0 ? x = 0 +x ? 0 = 0 +x ? y = x + y + +0 ? x = 0 +x ? 0 = 0 +x ? y = y + y + +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec x) + +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec y) + +0 ? x = 0 +x ? 0 = dec x +x ? y = dec y + +0 ? x = 0 +x ? 0 = x + x +x ? y = y + +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = y + +0 ? x = dec x +x ? 0 = x +x ? y = dec x + +0 ? x = dec x +x ? 0 = 0 +x ? y = dec x + +0 ? x = dec x +x ? 0 = dec x +x ? y = x + +0 ? x = dec x +x ? 0 = dec x +x ? y = y + +0 ? x = dec x +x ? 0 = dec x +x ? y = 0 + +0 ? x = x + x +x ? 0 = x +x ? y = y + +0 ? x = x + x +x ? 0 = x +x ? y = 0 + +0 ? x = x + x +x ? 0 = 0 +x ? y = x + +0 ? x = dec (dec x) +x ? 0 = x +x ? y = y + +0 ? x = dec (dec x) +x ? 0 = x +x ? y = 0 + +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x +x ? y = dec x + +x ? 0 = x +x ? y = x ? dec (dec y) + +x ? 0 = x +x ? y = y ? dec (dec y) + +x ? 0 = x +x ? y = 0 ? dec (dec y) + +x ? 0 = x +x ? y = dec x ? dec x + +x ? 0 = x +x ? y = dec x ? dec y + +x ? 0 = x +x ? y = dec y ? dec y + +x ? 0 = x +x ? y = dec (dec x) ? x + +x ? 0 = x +x ? y = dec (dec x) ? y + +x ? 0 = 0 +x ? y = x ? dec (dec y) + +x ? 0 = 0 +x ? y = y ? dec (dec y) + +x ? 0 = 0 +x ? y = 0 ? dec (dec y) + +x ? 0 = 0 +x ? y = dec x ? dec x + +x ? 0 = 0 +x ? y = dec x ? dec y + +x ? 0 = 0 +x ? y = dec y ? dec y + +x ? 0 = 0 +x ? y = dec (dec x) ? x + +x ? 0 = 0 +x ? y = dec (dec x) ? y + +0 ? x = x +x ? y = x ? dec (dec y) + +0 ? x = x +x ? y = y ? dec (dec y) + +0 ? x = x +x ? y = dec x ? dec x + +0 ? x = x +x ? y = dec x ? dec y + +0 ? x = x +x ? y = dec y ? dec y + +0 ? x = x +x ? y = dec (dec x) ? x + +0 ? x = x +x ? y = dec (dec x) ? y + +0 ? x = x +x ? y = dec (dec x) ? 0 + +0 ? x = 0 +x ? y = x ? dec (dec y) + +0 ? x = 0 +x ? y = y ? dec (dec y) + +0 ? x = 0 +x ? y = dec x ? dec x + +0 ? x = 0 +x ? y = dec x ? dec y + +0 ? x = 0 +x ? y = dec y ? dec y + +0 ? x = 0 +x ? y = dec (dec x) ? x + +0 ? x = 0 +x ? y = dec (dec x) ? y + +0 ? x = 0 +x ? y = dec (dec x) ? 0 + +x ? 0 = x +x ? y = dec (x ? dec y) + +x ? 0 = x +x ? y = dec (y ? dec y) + +x ? 0 = x +x ? y = dec (0 ? dec y) + +x ? 0 = x +x ? y = dec (dec x ? x) + +x ? 0 = x +x ? y = dec (dec x ? y) + +x ? 0 = 0 +x ? y = dec (x ? dec y) + +x ? 0 = 0 +x ? y = dec (y ? dec y) + +x ? 0 = 0 +x ? y = dec (0 ? dec y) + +x ? 0 = 0 +x ? y = dec (dec x ? x) + +x ? 0 = 0 +x ? y = dec (dec x ? y) + +x ? 0 = dec x +x ? y = x ? dec y + +x ? 0 = dec x +x ? y = y ? dec y + +x ? 0 = dec x +x ? y = 0 ? dec y + +x ? 0 = dec x +x ? y = dec x ? x + +x ? 0 = dec x +x ? y = dec x ? y + +0 ? x = x +x ? y = dec (x ? dec y) + +0 ? x = x +x ? y = dec (y ? dec y) + +0 ? x = x +x ? y = dec (dec x ? x) + +0 ? x = x +x ? y = dec (dec x ? y) + +0 ? x = x +x ? y = dec (dec x ? 0) + +0 ? x = 0 +x ? y = dec (x ? dec y) + +0 ? x = 0 +x ? y = dec (y ? dec y) + +0 ? x = 0 +x ? y = dec (dec x ? x) + +0 ? x = 0 +x ? y = dec (dec x ? y) + +0 ? x = 0 +x ? y = dec (dec x ? 0) + +0 ? x = dec x +x ? y = x ? dec y + +0 ? x = dec x +x ? y = y ? dec y + +0 ? x = dec x +x ? y = dec x ? x + +0 ? x = dec x +x ? y = dec x ? y + +0 ? x = dec x +x ? y = dec x ? 0 + +0 ? x = x +x ? 0 = x +x ? y = x ? dec y + +0 ? x = x +x ? 0 = x +x ? y = y ? dec y + +0 ? x = x +x ? 0 = x +x ? y = dec x ? x + +0 ? x = x +x ? 0 = x +x ? y = dec x ? y + +0 ? x = x +x ? 0 = 0 +x ? y = x ? dec y + +0 ? x = x +x ? 0 = 0 +x ? y = y ? dec y + +0 ? x = x +x ? 0 = 0 +x ? y = dec x ? x + +0 ? x = x +x ? 0 = 0 +x ? y = dec x ? y + +0 ? x = 0 +x ? 0 = x +x ? y = x ? dec y + +0 ? x = 0 +x ? 0 = x +x ? y = y ? dec y + +0 ? x = 0 +x ? 0 = x +x ? y = dec x ? x + +0 ? x = 0 +x ? 0 = x +x ? y = dec x ? y + +0 ? x = 0 +x ? 0 = 0 +x ? y = x ? dec y + +0 ? x = 0 +x ? 0 = 0 +x ? y = y ? dec y + +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x ? x + +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x ? y + +x ? y = dec (dec (dec (dec (dec x)))) + +x ? y = dec (dec (dec (dec (dec y)))) + +x ? y = x + dec (dec (dec x)) + +x ? y = x + dec (dec (dec y)) + +x ? y = x + (x + dec x) + +x ? y = x + (x + dec y) + +x ? y = x + (y + dec x) + +x ? y = x + (y + dec y) + +x ? y = y + dec (dec (dec x)) + +x ? y = y + dec (dec (dec y)) + +x ? y = y + (x + dec x) + +x ? y = y + (x + dec y) + +x ? y = y + (y + dec x) + +x ? y = y + (y + dec y) + +x ? y = dec x + (x + x) + +x ? y = dec x + (x + y) + +x ? y = dec x + (y + y) + +x ? y = dec x + dec (dec x) + +x ? y = dec x + dec (dec y) + +x ? y = dec y + (x + x) + +x ? y = dec y + (x + y) + +x ? y = dec y + (y + y) + +x ? y = dec y + dec (dec x) + +x ? y = dec y + dec (dec y) + +x ? 0 = x +x ? y = dec (dec (dec (dec x))) + +x ? 0 = x +x ? y = dec (dec (dec (dec y))) + +x ? 0 = x +x ? y = x + (x + x) + +x ? 0 = x +x ? y = x + (x + y) + +x ? 0 = x +x ? y = x + (y + y) + +x ? 0 = x +x ? y = x + dec (dec x) + +x ? 0 = x +x ? y = x + dec (dec y) + +x ? 0 = x +x ? y = y + (x + x) + +x ? 0 = x +x ? y = y + (x + y) + +x ? 0 = x +x ? y = y + (y + y) + +x ? 0 = x +x ? y = y + dec (dec x) + +x ? 0 = x +x ? y = y + dec (dec y) + +x ? 0 = x +x ? y = dec x + dec x + +x ? 0 = x +x ? y = dec x + dec y + +x ? 0 = x +x ? y = dec y + dec y + +x ? 0 = 0 +x ? y = dec (dec (dec (dec x))) + +x ? 0 = 0 +x ? y = dec (dec (dec (dec y))) + +x ? 0 = 0 +x ? y = x + (x + x) + +x ? 0 = 0 +x ? y = x + (x + y) + +x ? 0 = 0 +x ? y = x + (y + y) + +x ? 0 = 0 +x ? y = x + dec (dec x) + +x ? 0 = 0 +x ? y = x + dec (dec y) + +x ? 0 = 0 +x ? y = y + (x + x) + +x ? 0 = 0 +x ? y = y + (x + y) + +x ? 0 = 0 +x ? y = y + (y + y) + +x ? 0 = 0 +x ? y = y + dec (dec x) + +x ? 0 = 0 +x ? y = y + dec (dec y) + +x ? 0 = 0 +x ? y = dec x + dec x + +x ? 0 = 0 +x ? y = dec x + dec y + +x ? 0 = 0 +x ? y = dec y + dec y + +x ? 0 = dec x +x ? y = dec (dec (dec x)) + +x ? 0 = dec x +x ? y = dec (dec (dec y)) + +x ? 0 = dec x +x ? y = x + dec x + +x ? 0 = dec x +x ? y = x + dec y + +x ? 0 = dec x +x ? y = y + dec x + +x ? 0 = dec x +x ? y = y + dec y + +x ? 0 = x + x +x ? y = x + y + +x ? 0 = x + x +x ? y = y + y + +x ? 0 = x + x +x ? y = dec (dec x) + +x ? 0 = x + x +x ? y = dec (dec y) + +x ? 0 = dec (dec x) +x ? y = x + x + +x ? 0 = dec (dec x) +x ? y = x + y + +x ? 0 = dec (dec x) +x ? y = y + y + +x ? 0 = dec (dec x) +x ? y = dec (dec y) + +x ? 0 = dec (dec (dec x)) +x ? y = dec x + +x ? 0 = dec (dec (dec x)) +x ? y = dec y + +x ? 0 = x + dec x +x ? y = dec x + +x ? 0 = x + dec x +x ? y = dec y + +x ? 0 = dec (dec (dec (dec x))) +x ? y = x + +x ? 0 = dec (dec (dec (dec x))) +x ? y = y + +x ? 0 = dec (dec (dec (dec x))) +x ? y = 0 + +x ? 0 = x + (x + x) +x ? y = x + +x ? 0 = x + (x + x) +x ? y = y + +x ? 0 = x + (x + x) +x ? y = 0 + +x ? 0 = x + dec (dec x) +x ? y = x + +x ? 0 = x + dec (dec x) +x ? y = y + +x ? 0 = x + dec (dec x) +x ? y = 0 + +x ? 0 = dec x + dec x +x ? y = x + +x ? 0 = dec x + dec x +x ? y = y + +x ? 0 = dec x + dec x +x ? y = 0 + +0 ? x = x +x ? y = dec (dec (dec (dec x))) + +0 ? x = x +x ? y = dec (dec (dec (dec y))) + +0 ? x = x +x ? y = x + (x + x) + +0 ? x = x +x ? y = x + (x + y) + +0 ? x = x +x ? y = x + (y + y) + +0 ? x = x +x ? y = x + dec (dec x) + +0 ? x = x +x ? y = x + dec (dec y) + +0 ? x = x +x ? y = y + (x + x) + +0 ? x = x +x ? y = y + (x + y) + +0 ? x = x +x ? y = y + (y + y) + +0 ? x = x +x ? y = y + dec (dec x) + +0 ? x = x +x ? y = y + dec (dec y) + +0 ? x = x +x ? y = dec x + dec x + +0 ? x = x +x ? y = dec x + dec y + +0 ? x = x +x ? y = dec y + dec y + +0 ? x = 0 +x ? y = dec (dec (dec (dec x))) + +0 ? x = 0 +x ? y = dec (dec (dec (dec y))) + +0 ? x = 0 +x ? y = x + (x + x) + +0 ? x = 0 +x ? y = x + (x + y) + +0 ? x = 0 +x ? y = x + (y + y) + +0 ? x = 0 +x ? y = x + dec (dec x) + +0 ? x = 0 +x ? y = x + dec (dec y) + +0 ? x = 0 +x ? y = y + (x + x) + +0 ? x = 0 +x ? y = y + (x + y) + +0 ? x = 0 +x ? y = y + (y + y) + +0 ? x = 0 +x ? y = y + dec (dec x) + +0 ? x = 0 +x ? y = y + dec (dec y) + +0 ? x = 0 +x ? y = dec x + dec x + +0 ? x = 0 +x ? y = dec x + dec y + +0 ? x = 0 +x ? y = dec y + dec y + +0 ? x = dec x +x ? y = dec (dec (dec x)) + +0 ? x = dec x +x ? y = dec (dec (dec y)) + +0 ? x = dec x +x ? y = x + dec x + +0 ? x = dec x +x ? y = x + dec y + +0 ? x = dec x +x ? y = y + dec x + +0 ? x = dec x +x ? y = y + dec y + +0 ? x = x + x +x ? y = x + y + +0 ? x = x + x +x ? y = dec (dec x) + +0 ? x = x + x +x ? y = dec (dec y) + +0 ? x = dec (dec x) +x ? y = x + x + +0 ? x = dec (dec x) +x ? y = x + y + +0 ? x = dec (dec x) +x ? y = y + y + +0 ? x = dec (dec (dec x)) +x ? y = dec x + +0 ? x = dec (dec (dec x)) +x ? y = dec y + +0 ? x = x + dec x +x ? y = dec x + +0 ? x = x + dec x +x ? y = dec y + +0 ? x = dec (dec (dec (dec x))) +x ? y = x + +0 ? x = dec (dec (dec (dec x))) +x ? y = y + +0 ? x = dec (dec (dec (dec x))) +x ? y = 0 + +0 ? x = x + (x + x) +x ? y = x + +0 ? x = x + (x + x) +x ? y = y + +0 ? x = x + (x + x) +x ? y = 0 + +0 ? x = x + dec (dec x) +x ? y = x + +0 ? x = x + dec (dec x) +x ? y = y + +0 ? x = x + dec (dec x) +x ? y = 0 + +0 ? x = dec x + dec x +x ? y = x + +0 ? x = dec x + dec x +x ? y = y + +0 ? x = dec x + dec x +x ? y = 0 + +0 ? x = x +x ? 0 = x +x ? y = dec (dec (dec x)) + +0 ? x = x +x ? 0 = x +x ? y = dec (dec (dec y)) + +0 ? x = x +x ? 0 = x +x ? y = x + dec x + +0 ? x = x +x ? 0 = x +x ? y = x + dec y + +0 ? x = x +x ? 0 = x +x ? y = y + dec x + +0 ? x = x +x ? 0 = x +x ? y = y + dec y + +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec (dec x)) + +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec (dec y)) + +0 ? x = x +x ? 0 = 0 +x ? y = x + dec x + +0 ? x = x +x ? 0 = 0 +x ? y = x + dec y + +0 ? x = x +x ? 0 = 0 +x ? y = y + dec x + +0 ? x = x +x ? 0 = 0 +x ? y = y + dec y + +0 ? x = x +x ? 0 = dec x +x ? y = x + x + +0 ? x = x +x ? 0 = dec x +x ? y = x + y + +0 ? x = x +x ? 0 = dec x +x ? y = y + y + +0 ? x = x +x ? 0 = dec x +x ? y = dec (dec x) + +0 ? x = x +x ? 0 = dec x +x ? y = dec (dec y) + +0 ? x = x +x ? 0 = x + x +x ? y = dec x + +0 ? x = x +x ? 0 = x + x +x ? y = dec y + +0 ? x = x +x ? 0 = dec (dec x) +x ? y = dec x + +0 ? x = x +x ? 0 = dec (dec x) +x ? y = dec y + +0 ? x = x +x ? 0 = dec (dec (dec x)) +x ? y = x + +0 ? x = x +x ? 0 = dec (dec (dec x)) +x ? y = 0 + +0 ? x = x +x ? 0 = x + dec x +x ? y = x + +0 ? x = x +x ? 0 = x + dec x +x ? y = 0 + +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec (dec x)) + +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec (dec y)) + +0 ? x = 0 +x ? 0 = x +x ? y = x + dec x + +0 ? x = 0 +x ? 0 = x +x ? y = x + dec y + +0 ? x = 0 +x ? 0 = x +x ? y = y + dec x + +0 ? x = 0 +x ? 0 = x +x ? y = y + dec y + +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec (dec x)) + +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec (dec y)) + +0 ? x = 0 +x ? 0 = 0 +x ? y = x + dec x + +0 ? x = 0 +x ? 0 = 0 +x ? y = x + dec y + +0 ? x = 0 +x ? 0 = 0 +x ? y = y + dec x + +0 ? x = 0 +x ? 0 = 0 +x ? y = y + dec y + +0 ? x = 0 +x ? 0 = dec x +x ? y = x + x + +0 ? x = 0 +x ? 0 = dec x +x ? y = x + y + +0 ? x = 0 +x ? 0 = dec x +x ? y = y + y + +0 ? x = 0 +x ? 0 = dec x +x ? y = dec (dec x) + +0 ? x = 0 +x ? 0 = dec x +x ? y = dec (dec y) + +0 ? x = 0 +x ? 0 = x + x +x ? y = dec x + +0 ? x = 0 +x ? 0 = x + x +x ? y = dec y + +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = dec x + +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = dec y + +0 ? x = 0 +x ? 0 = dec (dec (dec x)) +x ? y = y + +0 ? x = 0 +x ? 0 = x + dec x +x ? y = y + +0 ? x = dec x +x ? 0 = x +x ? y = x + x + +0 ? x = dec x +x ? 0 = x +x ? y = x + y + +0 ? x = dec x +x ? 0 = x +x ? y = y + y + +0 ? x = dec x +x ? 0 = x +x ? y = dec (dec x) + +0 ? x = dec x +x ? 0 = x +x ? y = dec (dec y) + +0 ? x = dec x +x ? 0 = 0 +x ? y = x + x + +0 ? x = dec x +x ? 0 = 0 +x ? y = x + y + +0 ? x = dec x +x ? 0 = 0 +x ? y = y + y + +0 ? x = dec x +x ? 0 = 0 +x ? y = dec (dec x) + +0 ? x = dec x +x ? 0 = 0 +x ? y = dec (dec y) + +0 ? x = dec x +x ? 0 = x + x +x ? y = x + +0 ? x = dec x +x ? 0 = x + x +x ? y = y + +0 ? x = dec x +x ? 0 = x + x +x ? y = 0 + +0 ? x = dec x +x ? 0 = dec (dec x) +x ? y = x + +0 ? x = dec x +x ? 0 = dec (dec x) +x ? y = y + +0 ? x = dec x +x ? 0 = dec (dec x) +x ? y = 0 + +0 ? x = x + x +x ? 0 = x +x ? y = dec x + +0 ? x = x + x +x ? 0 = x +x ? y = dec y + +0 ? x = x + x +x ? 0 = 0 +x ? y = dec x + +0 ? x = x + x +x ? 0 = 0 +x ? y = dec y + +0 ? x = x + x +x ? 0 = dec x +x ? y = x + +0 ? x = x + x +x ? 0 = dec x +x ? y = y + +0 ? x = x + x +x ? 0 = dec x +x ? y = 0 + +0 ? x = dec (dec x) +x ? 0 = x +x ? y = dec x + +0 ? x = dec (dec x) +x ? 0 = x +x ? y = dec y + +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = dec x + +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = dec y + +0 ? x = dec (dec x) +x ? 0 = dec x +x ? y = x + +0 ? x = dec (dec x) +x ? 0 = dec x +x ? y = y + +0 ? x = dec (dec x) +x ? 0 = dec x +x ? y = 0 + +0 ? x = dec (dec (dec x)) +x ? 0 = x +x ? y = y + +0 ? x = dec (dec (dec x)) +x ? 0 = x +x ? y = 0 + +0 ? x = dec (dec (dec x)) +x ? 0 = 0 +x ? y = x + +0 ? x = x + dec x +x ? 0 = x +x ? y = y + +0 ? x = x + dec x +x ? 0 = x +x ? y = 0 + +0 ? x = x + dec x +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x +x ? y = x + x + +0 ? 0 = 0 +0 ? x = dec x +x ? y = x + y + +0 ? 0 = 0 +0 ? x = dec x +x ? y = y + y + +0 ? 0 = 0 +0 ? x = dec x +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = dec x +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = x + x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = x + x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? y = dec y + + +Candidates for: goo :: [Int] -> [Int] + pruning with 4/4 rules + [2,0,1,0,1,0,1,0,1] direct candidates, 0 duplicates + [2,1,2,3,4,7,10,17,26] pattern candidates, 0 duplicates + +rules: +xs ++ [] == xs +[] ++ xs == xs +(xs ++ ys) ++ zs == xs ++ (ys ++ zs) +(x:xs) ++ ys == x:(xs ++ ys) + +direct candidates: + +goo xs = xs + +goo xs = [] + +goo xs = xs ++ xs + +goo xs = xs ++ (xs ++ xs) + + +pattern candidates: + +goo xs = xs + +goo xs = [] + +goo [] = [] +goo (x:xs) = xs + +goo [] = [] +goo (x:xs) = goo xs + +goo xs = xs ++ xs + +goo [] = [] +goo (x:xs) = x:xs + +goo [] = [] +goo (x:xs) = [x] + +goo [] = [] +goo (x:xs) = xs ++ xs + +goo [] = [] +goo (x:xs) = x:goo xs + +goo [] = [] +goo (x:xs) = xs ++ goo xs + +goo [] = [] +goo (x:xs) = goo xs ++ xs + +goo xs = xs ++ (xs ++ xs) + +goo [] = [] +goo (x:xs) = goo xs ++ goo xs + +goo [] = [] +goo (x:xs) = x:x:xs + +goo [] = [] +goo (x:xs) = [x,x] + +goo [] = [] +goo (x:xs) = x:(xs ++ xs) + +goo [] = [] +goo (x:xs) = xs ++ (x:xs) + +goo [] = [] +goo (x:xs) = xs ++ [x] + +goo [] = [] +goo (x:xs) = xs ++ (xs ++ xs) + + +Candidates for: ?? :: [Int] -> [Int] -> [Int] + pruning with 4/4 rules + [3,0,4,0,8,0,16,0,32] direct candidates, 0 duplicates + [3,7,15,59,139,331,987,2016,6970] pattern candidates, 0 duplicates + +rules: +xs ++ [] == xs +[] ++ xs == xs +(xs ++ ys) ++ zs == xs ++ (ys ++ zs) +(x:xs) ++ ys == x:(xs ++ ys) + +direct candidates: + +xs ?? ys = xs + +xs ?? ys = ys + +xs ?? ys = [] + +xs ?? ys = xs ++ xs + +xs ?? ys = xs ++ ys + +xs ?? ys = ys ++ xs + +xs ?? ys = ys ++ ys + +xs ?? ys = xs ++ (xs ++ xs) + +xs ?? ys = xs ++ (xs ++ ys) + +xs ?? ys = xs ++ (ys ++ xs) + +xs ?? ys = xs ++ (ys ++ ys) + +xs ?? ys = ys ++ (xs ++ xs) + +xs ?? ys = ys ++ (xs ++ ys) + +xs ?? ys = ys ++ (ys ++ xs) + +xs ?? ys = ys ++ (ys ++ ys) + + +pattern candidates: + +xs ?? ys = xs + +xs ?? ys = ys + +xs ?? ys = [] + +xs ?? [] = xs +xs ?? (x:ys) = ys + +xs ?? [] = xs +xs ?? (x:ys) = [] + +xs ?? [] = [] +xs ?? (x:ys) = xs + +xs ?? [] = [] +xs ?? (x:ys) = ys + +[] ?? xs = xs +(x:xs) ?? ys = [] + +[] ?? xs = [] +(x:xs) ?? ys = xs + +[] ?? xs = [] +(x:xs) ?? ys = ys + +xs ?? ys = xs ++ xs + +xs ?? ys = xs ++ ys + +xs ?? ys = ys ++ xs + +xs ?? ys = ys ++ ys + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = [] + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = x:xs + +xs ?? [] = xs +xs ?? (x:ys) = x:ys + +xs ?? [] = xs +xs ?? (x:ys) = [x] + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = x:xs + +xs ?? [] = [] +xs ?? (x:ys) = x:ys + +xs ?? [] = [] +xs ?? (x:ys) = [x] + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = [] + +[] ?? xs = xs +(x:xs) ?? ys = x:xs + +[] ?? xs = xs +(x:xs) ?? ys = x:ys + +[] ?? xs = xs +(x:xs) ?? ys = [x] + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = x:xs + +[] ?? xs = [] +(x:xs) ?? ys = x:ys + +[] ?? xs = [] +(x:xs) ?? ys = [x] + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = xs + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? xs + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? ys + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? [] + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? xs + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? ys + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? [] + +[] ?? xs = xs +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = xs + +[] ?? xs = xs +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = xs + +[] ?? xs = xs +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = ys + +[] ?? xs = xs +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = ys + +[] ?? xs = xs +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = [] + +[] ?? xs = xs +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = [] + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? xs + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? ys + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? [] + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? xs + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? ys + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? [] + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? xs + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? ys + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? [] + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? xs + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? ys + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? [] + +[] ?? xs = [] +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = xs + +[] ?? xs = [] +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = xs + +[] ?? xs = [] +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = ys + +[] ?? xs = [] +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = ys + +[] ?? xs = [] +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = [] + +[] ?? xs = [] +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = [] + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? xs + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? ys + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? [] + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? xs + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? ys + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = xs ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = xs ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = xs ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = ys ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = ys ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = ys ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = [] ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = [] ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? ys = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? ys = xs + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? ys = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? ys = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? ys = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? ys = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? ys = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? ys = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? ys = [] + +xs ?? ys = xs ++ (xs ++ xs) + +xs ?? ys = xs ++ (xs ++ ys) + +xs ?? ys = xs ++ (ys ++ xs) + +xs ?? ys = xs ++ (ys ++ ys) + +xs ?? ys = ys ++ (xs ++ xs) + +xs ?? ys = ys ++ (xs ++ ys) + +xs ?? ys = ys ++ (ys ++ xs) + +xs ?? ys = ys ++ (ys ++ ys) + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = x:xs + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = x:ys + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [x] + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = y:xs + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = y:ys + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [y] + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ++ xs + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ++ ys + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ++ xs + +[] ?? xs = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ++ ys + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = x:xs + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = x:ys + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [x] + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = y:xs + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = y:ys + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [y] + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ++ xs + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ++ ys + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ++ xs + +[] ?? xs = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ++ ys + +[] ?? xs = xs +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = xs + +[] ?? xs = xs +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = ys + +[] ?? xs = xs +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = [] + +[] ?? xs = xs +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = xs + +[] ?? xs = xs +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = ys + +[] ?? xs = xs +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = [] + +[] ?? xs = xs +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = xs + +[] ?? xs = xs +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = ys + +[] ?? xs = xs +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = [] + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = x:xs + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = x:ys + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [x] + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = y:xs + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = y:ys + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [y] + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ++ xs + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ++ ys + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ++ xs + +[] ?? xs = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ++ ys + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = x:xs + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = x:ys + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [x] + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = y:xs + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = y:ys + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [y] + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ++ xs + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ++ ys + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ++ xs + +[] ?? xs = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ++ ys + +[] ?? xs = [] +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = xs + +[] ?? xs = [] +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = ys + +[] ?? xs = [] +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = [] + +[] ?? xs = [] +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = xs + +[] ?? xs = [] +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = ys + +[] ?? xs = [] +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = [] + +[] ?? xs = [] +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = xs + +[] ?? xs = [] +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = ys + +[] ?? xs = [] +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = [] + +[] ?? xs = xs ++ xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? xs = xs ++ xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? xs = xs ++ xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? xs = xs ++ xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = x:xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = x:ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = [x] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = xs ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = xs ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = ys ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? ys = ys ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? ys = xs + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? ys = ys + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? ys = [] + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? ys = xs + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? ys = ys + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? ys = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? ys = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? ys = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? ys = [] + +xs ?? [] = xs +xs ?? (x:ys) = (x:xs) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (x:xs) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (x:ys) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (x:ys) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = [x] ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = [x] ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (xs ++ xs) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (xs ++ xs) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (xs ++ ys) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (xs ++ ys) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (ys ++ xs) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (ys ++ xs) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (ys ++ ys) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (ys ++ ys) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (x:xs) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (x:xs) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (x:ys) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (x:ys) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = [x] ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = [x] ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (xs ++ xs) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (xs ++ xs) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (xs ++ ys) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (xs ++ ys) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (ys ++ xs) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (ys ++ xs) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (ys ++ ys) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (ys ++ ys) ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (x:xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (x:ys) + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? [x] + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (ys ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (x:xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (x:ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? [x] + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (x:xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (x:ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? [x] + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (x:xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (x:ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? [x] + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (ys ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = x:xs ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = x:xs ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = x:ys ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = x:ys ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = x:[] ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = x:[] ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ xs ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ xs ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ ys ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ ys ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ [] ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ [] ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ xs ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ xs ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ ys ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ ys ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ [] ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ [] ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? xs ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? ys ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? xs ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? ys ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? xs ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? ys ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? xs ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? ys ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? xs ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? ys ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? xs ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? ys ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = x:xs ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = x:xs ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = x:ys ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = x:ys ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = x:[] ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = x:[] ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ xs ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ xs ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ ys ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ ys ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ [] ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ [] ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ xs ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ xs ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ ys ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ ys ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ [] ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ [] ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? xs ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? ys ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? xs ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? ys ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? xs ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? ys ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? xs ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? ys ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? xs ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? ys ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? xs ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? ys ++ ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = xs ?? xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = xs ?? ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys ?? xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys ?? ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = [] ?? xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = [] ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = x:xs ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = x:xs ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = x:xs ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = x:ys ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = x:ys ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = x:ys ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ xs ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ xs ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ xs ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ ys ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ ys ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ ys ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ xs ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ xs ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ xs ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ ys ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ ys ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ ys ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? xs ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? ys ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? [] ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? xs ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? ys ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? [] ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? xs ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? ys ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? [] ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? xs ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? ys ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? [] ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = x:xs ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = x:xs ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = x:xs ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = x:ys ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = x:ys ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = x:ys ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ xs ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ xs ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ xs ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ ys ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ ys ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ ys ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ xs ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ xs ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ xs ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ ys ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ ys ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ ys ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? xs ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? ys ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? [] ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? xs ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? ys ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? [] ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? xs ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? ys ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? [] ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? xs ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? ys ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? [] ++ ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = xs ?? xs + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = xs ?? ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = xs ?? [] + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = ys ?? xs + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = ys ?? ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = ys ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] ?? xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] ?? xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +xs ?? [] = xs +xs ?? (x:ys) = x:x:xs + +xs ?? [] = xs +xs ?? (x:ys) = x:x:ys + +xs ?? [] = xs +xs ?? (x:ys) = [x,x] + +xs ?? [] = xs +xs ?? (x:ys) = x:(xs ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = x:(xs ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = x:(ys ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = x:(ys ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (x:xs) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (x:ys) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ [x] + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (xs ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (xs ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (ys ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (ys ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (x:xs) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (x:ys) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ [x] + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (xs ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (xs ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (ys ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (ys ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = x:x:xs + +xs ?? [] = [] +xs ?? (x:ys) = x:x:ys + +xs ?? [] = [] +xs ?? (x:ys) = [x,x] + +xs ?? [] = [] +xs ?? (x:ys) = x:(xs ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = x:(xs ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = x:(ys ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = x:(ys ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (x:xs) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (x:ys) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ [x] + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (xs ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (xs ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (ys ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (ys ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (x:xs) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (x:ys) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ [x] + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (xs ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (xs ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (ys ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (ys ++ ys) + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = x:xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = x:ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = [x] + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = xs ++ ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys ++ xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys ++ ys + +xs ?? [] = xs ++ (xs ++ xs) +xs ?? (x:ys) = xs + +xs ?? [] = xs ++ (xs ++ xs) +xs ?? (x:ys) = ys + +xs ?? [] = xs ++ (xs ++ xs) +xs ?? (x:ys) = [] + +[] ?? xs = xs +(x:xs) ?? ys = x:x:xs + +[] ?? xs = xs +(x:xs) ?? ys = x:x:ys + +[] ?? xs = xs +(x:xs) ?? ys = [x,x] + +[] ?? xs = xs +(x:xs) ?? ys = x:(xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = x:(xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = x:(ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = x:(ys ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (x:xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (x:ys) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ [x] + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (ys ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (x:xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (x:ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ [x] + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = x:x:xs + +[] ?? xs = [] +(x:xs) ?? ys = x:x:ys + +[] ?? xs = [] +(x:xs) ?? ys = [x,x] + +[] ?? xs = [] +(x:xs) ?? ys = x:(xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = x:(xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = x:(ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = x:(ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (x:xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (x:ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ [x] + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (x:xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (x:ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ [x] + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (ys ++ ys) + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = x:xs + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = x:ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = [x] + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = xs ++ ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = ys ++ xs + +[] ?? xs = xs ++ (xs ++ xs) +(x:xs) ?? ys = xs + +[] ?? xs = xs ++ (xs ++ xs) +(x:xs) ?? ys = ys + +[] ?? xs = xs ++ (xs ++ xs) +(x:xs) ?? ys = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = x:xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = x:ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [x] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = y:xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = y:ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [y] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + + +Candidates for: ton :: Bool -> Bool + pruning with 39/49 rules + [3,1,0,2,6,10,22,40,76] direct candidates, 0 duplicates + [3,3,0,0,0,0,0,0,0] pattern candidates, 0 duplicates + +rules: +not False == True +not True == False +p && p == p +p || p == p +not (not p) == p +p && False == False +p && True == p +False && p == False +True && p == p +p || False == p +p || True == True +False || p == p +True || p == True +not (p && q) == not p || not q +not (p && q) == not q || not p +not (p || q) == not p && not q +not (p || q) == not q && not p +p && not p == False +not p && p == False +p || not p == True +not p || p == True +(p && q) && r == p && (q && r) +(p && q) && r == q && (p && r) +(p || q) || r == p || (q || r) +(p || q) || r == q || (p || r) +p && (p && q) == p && q +p && (q && p) == p && q +p && (q && p) == q && p +p || (p || q) == p || q +p || (q || p) == p || q +p || (q || p) == q || p +p && (p || q) == p +p && (q || p) == p +(p || q) && p == p +(p || q) && q == q +p || p && q == p +p || q && p == p +p && q || p == p +p && q || q == q +equations: +q && p == p && q +q || p == p || q +q && (p && r) == p && (q && r) +r && (p && q) == p && (q && r) +r && (q && p) == p && (q && r) +q || (p || r) == p || (q || r) +r || (p || q) == p || (q || r) +r || (q || p) == p || (q || r) +(r || q) && p == p && (q || r) +r && q || p == p || q && r + +direct candidates: + +ton p = p + +ton p = False + +ton p = True + +ton p = not p + +ton p = p && ton p + +ton p = p || ton p + +ton p = p && ton (not p) + +ton p = p || ton (not p) + +ton p = p && not (ton p) + +ton p = not p && ton p + +ton p = p || not (ton p) + +ton p = not p || ton p + +ton p = p && not (ton (not p)) + +ton p = not p && ton (not p) + +ton p = p || not (ton (not p)) + +ton p = not p || ton (not p) + +ton p = p && (p && ton p) + +ton p = p && (p || ton p) + +ton p = not p && not (ton p) + +ton p = p || p && ton p + +ton p = p || (p || ton p) + +ton p = not p || not (ton p) + + +pattern candidates: + +ton p = p + +ton p = False + +ton p = True + +ton p = not p + +ton False = False +ton True = True + +ton False = True +ton True = False + + +Candidates for: &| :: Bool -> Bool -> Bool + pruning with 39/49 rules + [4,2,2,4,30,112,342,1032,2852] direct candidates, 0 duplicates + [4,14,26,10,2,16,18,60,120] pattern candidates, 0 duplicates + +rules: +not False == True +not True == False +p && p == p +p || p == p +not (not p) == p +p && False == False +p && True == p +False && p == False +True && p == p +p || False == p +p || True == True +False || p == p +True || p == True +not (p && q) == not p || not q +not (p && q) == not q || not p +not (p || q) == not p && not q +not (p || q) == not q && not p +p && not p == False +not p && p == False +p || not p == True +not p || p == True +(p && q) && r == p && (q && r) +(p && q) && r == q && (p && r) +(p || q) || r == p || (q || r) +(p || q) || r == q || (p || r) +p && (p && q) == p && q +p && (q && p) == p && q +p && (q && p) == q && p +p || (p || q) == p || q +p || (q || p) == p || q +p || (q || p) == q || p +p && (p || q) == p +p && (q || p) == p +(p || q) && p == p +(p || q) && q == q +p || p && q == p +p || q && p == p +p && q || p == p +p && q || q == q +equations: +q && p == p && q +q || p == p || q +q && (p && r) == p && (q && r) +r && (p && q) == p && (q && r) +r && (q && p) == p && (q && r) +q || (p || r) == p || (q || r) +r || (p || q) == p || (q || r) +r || (q || p) == p || (q || r) +(r || q) && p == p && (q || r) +r && q || p == p || q && r + +direct candidates: + +p &| q = p + +p &| q = q + +p &| q = False + +p &| q = True + +p &| q = not p + +p &| q = not q + +p &| q = p && q + +p &| q = p || q + +p &| q = p && not q + +p &| q = q && not p + +p &| q = p || not q + +p &| q = q || not p + +p &| q = not p && not q + +p &| q = not p || not q + +p &| q = p && p &| p + +p &| q = p && p &| q + +p &| q = p && p &| False + +p &| q = p && p &| True + +p &| q = p && q &| q + +p &| q = p && False &| q + +p &| q = p && True &| q + +p &| q = q && p &| p + +p &| q = q && p &| q + +p &| q = q && p &| False + +p &| q = q && p &| True + +p &| q = q && q &| q + +p &| q = q && False &| q + +p &| q = q && True &| q + +p &| q = p || p &| p + +p &| q = p || p &| q + +p &| q = p || p &| False + +p &| q = p || p &| True + +p &| q = p || q &| q + +p &| q = p || False &| q + +p &| q = p || True &| q + +p &| q = q || p &| p + +p &| q = q || p &| q + +p &| q = q || p &| False + +p &| q = q || p &| True + +p &| q = q || q &| q + +p &| q = q || False &| q + +p &| q = q || True &| q + +p &| q = p && (q && not p) + +p &| q = p && (q || not p) + +p &| q = q && (p && not q) + +p &| q = q && (p || not q) + +p &| q = p || q && not p + +p &| q = p || (q || not p) + +p &| q = q || p && not q + +p &| q = q || (p || not q) + +p &| q = not p && (p && q) + +p &| q = not p && (p || q) + +p &| q = not q && (p && q) + +p &| q = not q && (p || q) + +p &| q = not p || p && q + +p &| q = not p || (p || q) + +p &| q = not q || p && q + +p &| q = not q || (p || q) + +p &| q = p && p &| not p + +p &| q = p && p &| not q + +p &| q = p && q &| not q + +p &| q = p && False &| not q + +p &| q = p && True &| not q + +p &| q = p && not p &| p + +p &| q = p && not p &| q + +p &| q = p && not p &| False + +p &| q = p && not p &| True + +p &| q = p && not q &| q + +p &| q = q && p &| not p + +p &| q = q && p &| not q + +p &| q = q && q &| not q + +p &| q = q && False &| not q + +p &| q = q && True &| not q + +p &| q = q && not p &| p + +p &| q = q && not p &| q + +p &| q = q && not p &| False + +p &| q = q && not p &| True + +p &| q = q && not q &| q + +p &| q = p || p &| not p + +p &| q = p || p &| not q + +p &| q = p || q &| not q + +p &| q = p || False &| not q + +p &| q = p || True &| not q + +p &| q = p || not p &| p + +p &| q = p || not p &| q + +p &| q = p || not p &| False + +p &| q = p || not p &| True + +p &| q = p || not q &| q + +p &| q = q || p &| not p + +p &| q = q || p &| not q + +p &| q = q || q &| not q + +p &| q = q || False &| not q + +p &| q = q || True &| not q + +p &| q = q || not p &| p + +p &| q = q || not p &| q + +p &| q = q || not p &| False + +p &| q = q || not p &| True + +p &| q = q || not q &| q + +p &| q = p && not (p &| p) + +p &| q = p && not (p &| q) + +p &| q = p && not (p &| False) + +p &| q = p && not (p &| True) + +p &| q = p && not (q &| q) + +p &| q = p && not (False &| q) + +p &| q = p && not (True &| q) + +p &| q = q && not (p &| p) + +p &| q = q && not (p &| q) + +p &| q = q && not (p &| False) + +p &| q = q && not (p &| True) + +p &| q = q && not (q &| q) + +p &| q = q && not (False &| q) + +p &| q = q && not (True &| q) + +p &| q = not p && p &| p + +p &| q = not p && p &| q + +p &| q = not p && p &| False + +p &| q = not p && p &| True + +p &| q = not p && q &| q + +p &| q = not p && False &| q + +p &| q = not p && True &| q + +p &| q = not q && p &| p + +p &| q = not q && p &| q + +p &| q = not q && p &| False + +p &| q = not q && p &| True + +p &| q = not q && q &| q + +p &| q = not q && False &| q + +p &| q = not q && True &| q + +p &| q = p || not (p &| p) + +p &| q = p || not (p &| q) + +p &| q = p || not (p &| False) + +p &| q = p || not (p &| True) + +p &| q = p || not (q &| q) + +p &| q = p || not (False &| q) + +p &| q = p || not (True &| q) + +p &| q = q || not (p &| p) + +p &| q = q || not (p &| q) + +p &| q = q || not (p &| False) + +p &| q = q || not (p &| True) + +p &| q = q || not (q &| q) + +p &| q = q || not (False &| q) + +p &| q = q || not (True &| q) + +p &| q = not p || p &| p + +p &| q = not p || p &| q + +p &| q = not p || p &| False + +p &| q = not p || p &| True + +p &| q = not p || q &| q + +p &| q = not p || False &| q + +p &| q = not p || True &| q + +p &| q = not q || p &| p + +p &| q = not q || p &| q + +p &| q = not q || p &| False + +p &| q = not q || p &| True + +p &| q = not q || q &| q + +p &| q = not q || False &| q + +p &| q = not q || True &| q + + +pattern candidates: + +p &| q = p + +p &| q = q + +p &| q = False + +p &| q = True + +p &| q = not p + +p &| q = not q + +p &| False = p +p &| True = False + +p &| False = p +p &| True = True + +p &| False = False +p &| True = p + +p &| False = False +p &| True = True + +p &| False = True +p &| True = p + +p &| False = True +p &| True = False + +False &| p = p +True &| p = False + +False &| p = p +True &| p = True + +False &| p = False +True &| p = p + +False &| p = False +True &| p = True + +False &| p = True +True &| p = p + +False &| p = True +True &| p = False + +p &| q = p && q + +p &| q = p || q + +p &| False = p +p &| True = not p + +p &| False = False +p &| True = not p + +p &| False = True +p &| True = not p + +p &| False = not p +p &| True = p + +p &| False = not p +p &| True = False + +p &| False = not p +p &| True = True + +False &| p = p +True &| p = not p + +False &| p = False +True &| p = not p + +False &| p = True +True &| p = not p + +False &| p = not p +True &| p = p + +False &| p = not p +True &| p = False + +False &| p = not p +True &| p = True + +False &| p = p +True &| False = False +True &| True = True + +False &| p = p +True &| False = True +True &| True = False + +False &| p = False +True &| False = False +True &| True = True + +False &| p = False +True &| False = True +True &| True = False + +False &| p = True +True &| False = False +True &| True = True + +False &| p = True +True &| False = True +True &| True = False + +False &| False = False +False &| True = True +True &| p = p + +False &| False = False +False &| True = True +True &| p = False + +False &| False = False +False &| True = True +True &| p = True + +False &| False = True +False &| True = False +True &| p = p + +False &| False = True +False &| True = False +True &| p = False + +False &| False = True +False &| True = False +True &| p = True + +p &| q = p && not q + +p &| q = q && not p + +p &| q = p || not q + +p &| q = q || not p + +False &| p = not p +True &| False = False +True &| True = True + +False &| p = not p +True &| False = True +True &| True = False + +False &| False = False +False &| True = True +True &| p = not p + +False &| False = True +False &| True = False +True &| p = not p + +False &| False = False +False &| True = True +True &| False = True +True &| True = False + +False &| False = True +False &| True = False +True &| False = False +True &| True = True + +p &| q = not p && not q + +p &| q = not p || not q + +p &| q = p && (q && not p) + +p &| q = p && (q || not p) + +p &| q = q && (p && not q) + +p &| q = q && (p || not q) + +p &| q = p || q && not p + +p &| q = p || (q || not p) + +p &| q = q || p && not q + +p &| q = q || (p || not q) + +p &| q = not p && (p && q) + +p &| q = not p && (p || q) + +p &| q = not q && (p && q) + +p &| q = not q && (p || q) + +p &| q = not p || p && q + +p &| q = not p || (p || q) + +p &| q = not q || p && q + +p &| q = not q || (p || q) + + +Candidates for: gcd :: Int -> Int -> Int + pruning with 0/0 rules + [3,0,9,0,54,0,405,0,3402] direct candidates, 0 duplicates + [3,5,11,50,98,359,786,2836,6723] pattern candidates, 0 duplicates + +no rules. + +direct candidates: + +gcd x y = x + +gcd x y = y + +gcd x y = 0 + +gcd x y = x `mod` x + +gcd x y = x `mod` y + +gcd x y = x `mod` 0 + +gcd x y = y `mod` x + +gcd x y = y `mod` y + +gcd x y = y `mod` 0 + +gcd x y = 0 `mod` x + +gcd x y = 0 `mod` y + +gcd x y = 0 `mod` 0 + +gcd x y = (x `mod` x) `mod` x + +gcd x y = (x `mod` x) `mod` y + +gcd x y = (x `mod` x) `mod` 0 + +gcd x y = (x `mod` y) `mod` x + +gcd x y = (x `mod` y) `mod` y + +gcd x y = (x `mod` y) `mod` 0 + +gcd x y = (x `mod` 0) `mod` x + +gcd x y = (x `mod` 0) `mod` y + +gcd x y = (x `mod` 0) `mod` 0 + +gcd x y = (y `mod` x) `mod` x + +gcd x y = (y `mod` x) `mod` y + +gcd x y = (y `mod` x) `mod` 0 + +gcd x y = (y `mod` y) `mod` x + +gcd x y = (y `mod` y) `mod` y + +gcd x y = (y `mod` y) `mod` 0 + +gcd x y = (y `mod` 0) `mod` x + +gcd x y = (y `mod` 0) `mod` y + +gcd x y = (y `mod` 0) `mod` 0 + +gcd x y = (0 `mod` x) `mod` x + +gcd x y = (0 `mod` x) `mod` y + +gcd x y = (0 `mod` x) `mod` 0 + +gcd x y = (0 `mod` y) `mod` x + +gcd x y = (0 `mod` y) `mod` y + +gcd x y = (0 `mod` y) `mod` 0 + +gcd x y = (0 `mod` 0) `mod` x + +gcd x y = (0 `mod` 0) `mod` y + +gcd x y = (0 `mod` 0) `mod` 0 + +gcd x y = x `mod` (x `mod` x) + +gcd x y = x `mod` (x `mod` y) + +gcd x y = x `mod` (x `mod` 0) + +gcd x y = x `mod` (y `mod` x) + +gcd x y = x `mod` (y `mod` y) + +gcd x y = x `mod` (y `mod` 0) + +gcd x y = x `mod` (0 `mod` x) + +gcd x y = x `mod` (0 `mod` y) + +gcd x y = x `mod` (0 `mod` 0) + +gcd x y = y `mod` (x `mod` x) + +gcd x y = y `mod` (x `mod` y) + +gcd x y = y `mod` (x `mod` 0) + +gcd x y = y `mod` (y `mod` x) + +gcd x y = y `mod` (y `mod` y) + +gcd x y = y `mod` (y `mod` 0) + +gcd x y = y `mod` (0 `mod` x) + +gcd x y = y `mod` (0 `mod` y) + +gcd x y = y `mod` (0 `mod` 0) + +gcd x y = 0 `mod` (x `mod` x) + +gcd x y = 0 `mod` (x `mod` y) + +gcd x y = 0 `mod` (x `mod` 0) + +gcd x y = 0 `mod` (y `mod` x) + +gcd x y = 0 `mod` (y `mod` y) + +gcd x y = 0 `mod` (y `mod` 0) + +gcd x y = 0 `mod` (0 `mod` x) + +gcd x y = 0 `mod` (0 `mod` y) + +gcd x y = 0 `mod` (0 `mod` 0) + + +pattern candidates: + +gcd x y = x + +gcd x y = y + +gcd x y = 0 + +gcd x 0 = x +gcd x y = y + +gcd x 0 = x +gcd x y = 0 + +gcd x 0 = 0 +gcd x y = x + +gcd 0 x = x +gcd x y = 0 + +gcd 0 x = 0 +gcd x y = y + +gcd x y = x `mod` x + +gcd x y = x `mod` y + +gcd x y = x `mod` 0 + +gcd x y = y `mod` x + +gcd x y = y `mod` y + +gcd x y = y `mod` 0 + +gcd x y = 0 `mod` x + +gcd x y = 0 `mod` y + +gcd 0 x = x +gcd x 0 = x +gcd x y = 0 + +gcd 0 x = x +gcd x 0 = 0 +gcd x y = x + +gcd 0 x = 0 +gcd x 0 = x +gcd x y = y + +gcd x 0 = x +gcd x y = x `mod` x + +gcd x 0 = x +gcd x y = x `mod` y + +gcd x 0 = x +gcd x y = x `mod` 0 + +gcd x 0 = x +gcd x y = y `mod` x + +gcd x 0 = x +gcd x y = y `mod` y + +gcd x 0 = x +gcd x y = y `mod` 0 + +gcd x 0 = x +gcd x y = 0 `mod` x + +gcd x 0 = x +gcd x y = 0 `mod` y + +gcd x 0 = 0 +gcd x y = x `mod` x + +gcd x 0 = 0 +gcd x y = x `mod` y + +gcd x 0 = 0 +gcd x y = x `mod` 0 + +gcd x 0 = 0 +gcd x y = y `mod` x + +gcd x 0 = 0 +gcd x y = y `mod` y + +gcd x 0 = 0 +gcd x y = y `mod` 0 + +gcd x 0 = 0 +gcd x y = 0 `mod` x + +gcd x 0 = 0 +gcd x y = 0 `mod` y + +gcd x 0 = x `mod` x +gcd x y = x + +gcd x 0 = x `mod` x +gcd x y = y + +gcd x 0 = x `mod` x +gcd x y = 0 + +gcd x 0 = x `mod` 0 +gcd x y = x + +gcd x 0 = x `mod` 0 +gcd x y = y + +gcd x 0 = x `mod` 0 +gcd x y = 0 + +gcd x 0 = 0 `mod` x +gcd x y = x + +gcd x 0 = 0 `mod` x +gcd x y = y + +gcd x 0 = 0 `mod` x +gcd x y = 0 + +gcd 0 x = x +gcd x y = x `mod` x + +gcd 0 x = x +gcd x y = x `mod` y + +gcd 0 x = x +gcd x y = x `mod` 0 + +gcd 0 x = x +gcd x y = y `mod` x + +gcd 0 x = x +gcd x y = y `mod` y + +gcd 0 x = x +gcd x y = y `mod` 0 + +gcd 0 x = x +gcd x y = 0 `mod` x + +gcd 0 x = x +gcd x y = 0 `mod` y + +gcd 0 x = 0 +gcd x y = x `mod` x + +gcd 0 x = 0 +gcd x y = x `mod` y + +gcd 0 x = 0 +gcd x y = x `mod` 0 + +gcd 0 x = 0 +gcd x y = y `mod` x + +gcd 0 x = 0 +gcd x y = y `mod` y + +gcd 0 x = 0 +gcd x y = y `mod` 0 + +gcd 0 x = 0 +gcd x y = 0 `mod` x + +gcd 0 x = 0 +gcd x y = 0 `mod` y + +gcd 0 x = x `mod` x +gcd x y = x + +gcd 0 x = x `mod` x +gcd x y = y + +gcd 0 x = x `mod` x +gcd x y = 0 + +gcd 0 x = x `mod` 0 +gcd x y = x + +gcd 0 x = x `mod` 0 +gcd x y = y + +gcd 0 x = x `mod` 0 +gcd x y = 0 + +gcd 0 x = 0 `mod` x +gcd x y = x + +gcd 0 x = 0 `mod` x +gcd x y = y + +gcd 0 x = 0 `mod` x +gcd x y = 0 + +gcd x y = (x `mod` x) `mod` x + +gcd x y = (x `mod` x) `mod` y + +gcd x y = (x `mod` x) `mod` 0 + +gcd x y = (x `mod` y) `mod` x + +gcd x y = (x `mod` y) `mod` y + +gcd x y = (x `mod` y) `mod` 0 + +gcd x y = (x `mod` 0) `mod` x + +gcd x y = (x `mod` 0) `mod` y + +gcd x y = (x `mod` 0) `mod` 0 + +gcd x y = (y `mod` x) `mod` x + +gcd x y = (y `mod` x) `mod` y + +gcd x y = (y `mod` x) `mod` 0 + +gcd x y = (y `mod` y) `mod` x + +gcd x y = (y `mod` y) `mod` y + +gcd x y = (y `mod` y) `mod` 0 + +gcd x y = (y `mod` 0) `mod` x + +gcd x y = (y `mod` 0) `mod` y + +gcd x y = (y `mod` 0) `mod` 0 + +gcd x y = (0 `mod` x) `mod` x + +gcd x y = (0 `mod` x) `mod` y + +gcd x y = (0 `mod` x) `mod` 0 + +gcd x y = (0 `mod` y) `mod` x + +gcd x y = (0 `mod` y) `mod` y + +gcd x y = (0 `mod` y) `mod` 0 + +gcd x y = x `mod` (x `mod` x) + +gcd x y = x `mod` (x `mod` y) + +gcd x y = x `mod` (x `mod` 0) + +gcd x y = x `mod` (y `mod` x) + +gcd x y = x `mod` (y `mod` y) + +gcd x y = x `mod` (y `mod` 0) + +gcd x y = x `mod` (0 `mod` x) + +gcd x y = x `mod` (0 `mod` y) + +gcd x y = y `mod` (x `mod` x) + +gcd x y = y `mod` (x `mod` y) + +gcd x y = y `mod` (x `mod` 0) + +gcd x y = y `mod` (y `mod` x) + +gcd x y = y `mod` (y `mod` y) + +gcd x y = y `mod` (y `mod` 0) + +gcd x y = y `mod` (0 `mod` x) + +gcd x y = y `mod` (0 `mod` y) + +gcd x y = 0 `mod` (x `mod` x) + +gcd x y = 0 `mod` (x `mod` y) + +gcd x y = 0 `mod` (x `mod` 0) + +gcd x y = 0 `mod` (y `mod` x) + +gcd x y = 0 `mod` (y `mod` y) + +gcd x y = 0 `mod` (y `mod` 0) + +gcd x y = 0 `mod` (0 `mod` x) + +gcd x y = 0 `mod` (0 `mod` y) + +gcd 0 x = x +gcd x 0 = x +gcd x y = x `mod` x + +gcd 0 x = x +gcd x 0 = x +gcd x y = x `mod` y + +gcd 0 x = x +gcd x 0 = x +gcd x y = x `mod` 0 + +gcd 0 x = x +gcd x 0 = x +gcd x y = y `mod` x + +gcd 0 x = x +gcd x 0 = x +gcd x y = y `mod` y + +gcd 0 x = x +gcd x 0 = x +gcd x y = y `mod` 0 + +gcd 0 x = x +gcd x 0 = x +gcd x y = 0 `mod` x + +gcd 0 x = x +gcd x 0 = x +gcd x y = 0 `mod` y + +gcd 0 x = x +gcd x 0 = 0 +gcd x y = x `mod` x + +gcd 0 x = x +gcd x 0 = 0 +gcd x y = x `mod` y + +gcd 0 x = x +gcd x 0 = 0 +gcd x y = x `mod` 0 + +gcd 0 x = x +gcd x 0 = 0 +gcd x y = y `mod` x + +gcd 0 x = x +gcd x 0 = 0 +gcd x y = y `mod` y + +gcd 0 x = x +gcd x 0 = 0 +gcd x y = y `mod` 0 + +gcd 0 x = x +gcd x 0 = 0 +gcd x y = 0 `mod` x + +gcd 0 x = x +gcd x 0 = 0 +gcd x y = 0 `mod` y + +gcd 0 x = x +gcd x 0 = x `mod` x +gcd x y = x + +gcd 0 x = x +gcd x 0 = x `mod` x +gcd x y = 0 + +gcd 0 x = x +gcd x 0 = x `mod` 0 +gcd x y = x + +gcd 0 x = x +gcd x 0 = x `mod` 0 +gcd x y = 0 + +gcd 0 x = x +gcd x 0 = 0 `mod` x +gcd x y = x + +gcd 0 x = x +gcd x 0 = 0 `mod` x +gcd x y = 0 + +gcd 0 x = 0 +gcd x 0 = x +gcd x y = x `mod` x + +gcd 0 x = 0 +gcd x 0 = x +gcd x y = x `mod` y + +gcd 0 x = 0 +gcd x 0 = x +gcd x y = x `mod` 0 + +gcd 0 x = 0 +gcd x 0 = x +gcd x y = y `mod` x + +gcd 0 x = 0 +gcd x 0 = x +gcd x y = y `mod` y + +gcd 0 x = 0 +gcd x 0 = x +gcd x y = y `mod` 0 + +gcd 0 x = 0 +gcd x 0 = x +gcd x y = 0 `mod` x + +gcd 0 x = 0 +gcd x 0 = x +gcd x y = 0 `mod` y + +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = x `mod` x + +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = x `mod` y + +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = x `mod` 0 + +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = y `mod` x + +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = y `mod` y + +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = y `mod` 0 + +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = 0 `mod` x + +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = 0 `mod` y + +gcd 0 x = 0 +gcd x 0 = x `mod` x +gcd x y = y + +gcd 0 x = 0 +gcd x 0 = x `mod` 0 +gcd x y = y + +gcd 0 x = 0 +gcd x 0 = 0 `mod` x +gcd x y = y + +gcd 0 x = x `mod` x +gcd x 0 = x +gcd x y = y + +gcd 0 x = x `mod` x +gcd x 0 = x +gcd x y = 0 + +gcd 0 x = x `mod` x +gcd x 0 = 0 +gcd x y = x + +gcd 0 x = x `mod` 0 +gcd x 0 = x +gcd x y = y + +gcd 0 x = x `mod` 0 +gcd x 0 = x +gcd x y = 0 + +gcd 0 x = x `mod` 0 +gcd x 0 = 0 +gcd x y = x + +gcd 0 x = 0 `mod` x +gcd x 0 = x +gcd x y = y + +gcd 0 x = 0 `mod` x +gcd x 0 = x +gcd x y = 0 + +gcd 0 x = 0 `mod` x +gcd x 0 = 0 +gcd x y = x + +gcd x 0 = x +gcd x y = gcd x (x `mod` y) + +gcd x 0 = x +gcd x y = gcd x (y `mod` y) + +gcd x 0 = x +gcd x y = gcd x (0 `mod` y) + +gcd x 0 = x +gcd x y = gcd y (x `mod` y) + +gcd x 0 = x +gcd x y = gcd y (y `mod` y) + +gcd x 0 = x +gcd x y = gcd y (0 `mod` y) + +gcd x 0 = x +gcd x y = gcd 0 (x `mod` y) + +gcd x 0 = x +gcd x y = gcd 0 (y `mod` y) + +gcd x 0 = x +gcd x y = gcd 0 (0 `mod` y) + +gcd x 0 = x +gcd x y = gcd (x `mod` x) x + +gcd x 0 = x +gcd x y = gcd (x `mod` x) y + +gcd x 0 = x +gcd x y = gcd (y `mod` x) x + +gcd x 0 = x +gcd x y = gcd (y `mod` x) y + +gcd x 0 = x +gcd x y = gcd (0 `mod` x) x + +gcd x 0 = x +gcd x y = gcd (0 `mod` x) y + +gcd x 0 = 0 +gcd x y = gcd x (x `mod` y) + +gcd x 0 = 0 +gcd x y = gcd x (y `mod` y) + +gcd x 0 = 0 +gcd x y = gcd x (0 `mod` y) + +gcd x 0 = 0 +gcd x y = gcd y (x `mod` y) + +gcd x 0 = 0 +gcd x y = gcd y (y `mod` y) + +gcd x 0 = 0 +gcd x y = gcd y (0 `mod` y) + +gcd x 0 = 0 +gcd x y = gcd 0 (x `mod` y) + +gcd x 0 = 0 +gcd x y = gcd 0 (y `mod` y) + +gcd x 0 = 0 +gcd x y = gcd 0 (0 `mod` y) + +gcd x 0 = 0 +gcd x y = gcd (x `mod` x) x + +gcd x 0 = 0 +gcd x y = gcd (x `mod` x) y + +gcd x 0 = 0 +gcd x y = gcd (y `mod` x) x + +gcd x 0 = 0 +gcd x y = gcd (y `mod` x) y + +gcd x 0 = 0 +gcd x y = gcd (0 `mod` x) x + +gcd x 0 = 0 +gcd x y = gcd (0 `mod` x) y + +gcd 0 x = x +gcd x y = gcd x (x `mod` y) + +gcd 0 x = x +gcd x y = gcd x (y `mod` y) + +gcd 0 x = x +gcd x y = gcd x (0 `mod` y) + +gcd 0 x = x +gcd x y = gcd y (x `mod` y) + +gcd 0 x = x +gcd x y = gcd y (y `mod` y) + +gcd 0 x = x +gcd x y = gcd y (0 `mod` y) + +gcd 0 x = x +gcd x y = gcd (x `mod` x) x + +gcd 0 x = x +gcd x y = gcd (x `mod` x) y + +gcd 0 x = x +gcd x y = gcd (x `mod` x) 0 + +gcd 0 x = x +gcd x y = gcd (y `mod` x) x + +gcd 0 x = x +gcd x y = gcd (y `mod` x) y + +gcd 0 x = x +gcd x y = gcd (y `mod` x) 0 + +gcd 0 x = x +gcd x y = gcd (0 `mod` x) x + +gcd 0 x = x +gcd x y = gcd (0 `mod` x) y + +gcd 0 x = x +gcd x y = gcd (0 `mod` x) 0 + +gcd 0 x = 0 +gcd x y = gcd x (x `mod` y) + +gcd 0 x = 0 +gcd x y = gcd x (y `mod` y) + +gcd 0 x = 0 +gcd x y = gcd x (0 `mod` y) + +gcd 0 x = 0 +gcd x y = gcd y (x `mod` y) + +gcd 0 x = 0 +gcd x y = gcd y (y `mod` y) + +gcd 0 x = 0 +gcd x y = gcd y (0 `mod` y) + +gcd 0 x = 0 +gcd x y = gcd (x `mod` x) x + +gcd 0 x = 0 +gcd x y = gcd (x `mod` x) y + +gcd 0 x = 0 +gcd x y = gcd (x `mod` x) 0 + +gcd 0 x = 0 +gcd x y = gcd (y `mod` x) x + +gcd 0 x = 0 +gcd x y = gcd (y `mod` x) y + +gcd 0 x = 0 +gcd x y = gcd (y `mod` x) 0 + +gcd 0 x = 0 +gcd x y = gcd (0 `mod` x) x + +gcd 0 x = 0 +gcd x y = gcd (0 `mod` x) y + +gcd 0 x = 0 +gcd x y = gcd (0 `mod` x) 0 + +gcd x 0 = x +gcd x y = (x `mod` x) `mod` x + +gcd x 0 = x +gcd x y = (x `mod` x) `mod` y + +gcd x 0 = x +gcd x y = (x `mod` x) `mod` 0 + +gcd x 0 = x +gcd x y = (x `mod` y) `mod` x + +gcd x 0 = x +gcd x y = (x `mod` y) `mod` y + +gcd x 0 = x +gcd x y = (x `mod` y) `mod` 0 + +gcd x 0 = x +gcd x y = (x `mod` 0) `mod` x + +gcd x 0 = x +gcd x y = (x `mod` 0) `mod` y + +gcd x 0 = x +gcd x y = (x `mod` 0) `mod` 0 + +gcd x 0 = x +gcd x y = (y `mod` x) `mod` x + +gcd x 0 = x +gcd x y = (y `mod` x) `mod` y + +gcd x 0 = x +gcd x y = (y `mod` x) `mod` 0 + +gcd x 0 = x +gcd x y = (y `mod` y) `mod` x + +gcd x 0 = x +gcd x y = (y `mod` y) `mod` y + +gcd x 0 = x +gcd x y = (y `mod` y) `mod` 0 + +gcd x 0 = x +gcd x y = (y `mod` 0) `mod` x + +gcd x 0 = x +gcd x y = (y `mod` 0) `mod` y + +gcd x 0 = x +gcd x y = (y `mod` 0) `mod` 0 + +gcd x 0 = x +gcd x y = (0 `mod` x) `mod` x + +gcd x 0 = x +gcd x y = (0 `mod` x) `mod` y + +gcd x 0 = x +gcd x y = (0 `mod` x) `mod` 0 + +gcd x 0 = x +gcd x y = (0 `mod` y) `mod` x + +gcd x 0 = x +gcd x y = (0 `mod` y) `mod` y + +gcd x 0 = x +gcd x y = (0 `mod` y) `mod` 0 + +gcd x 0 = x +gcd x y = x `mod` (x `mod` x) + +gcd x 0 = x +gcd x y = x `mod` (x `mod` y) + +gcd x 0 = x +gcd x y = x `mod` (x `mod` 0) + +gcd x 0 = x +gcd x y = x `mod` (y `mod` x) + +gcd x 0 = x +gcd x y = x `mod` (y `mod` y) + +gcd x 0 = x +gcd x y = x `mod` (y `mod` 0) + +gcd x 0 = x +gcd x y = x `mod` (0 `mod` x) + +gcd x 0 = x +gcd x y = x `mod` (0 `mod` y) + +gcd x 0 = x +gcd x y = y `mod` (x `mod` x) + +gcd x 0 = x +gcd x y = y `mod` (x `mod` y) + +gcd x 0 = x +gcd x y = y `mod` (x `mod` 0) + +gcd x 0 = x +gcd x y = y `mod` (y `mod` x) + +gcd x 0 = x +gcd x y = y `mod` (y `mod` y) + +gcd x 0 = x +gcd x y = y `mod` (y `mod` 0) + +gcd x 0 = x +gcd x y = y `mod` (0 `mod` x) + +gcd x 0 = x +gcd x y = y `mod` (0 `mod` y) + +gcd x 0 = x +gcd x y = 0 `mod` (x `mod` x) + +gcd x 0 = x +gcd x y = 0 `mod` (x `mod` y) + +gcd x 0 = x +gcd x y = 0 `mod` (x `mod` 0) + +gcd x 0 = x +gcd x y = 0 `mod` (y `mod` x) + +gcd x 0 = x +gcd x y = 0 `mod` (y `mod` y) + +gcd x 0 = x +gcd x y = 0 `mod` (y `mod` 0) + +gcd x 0 = x +gcd x y = 0 `mod` (0 `mod` x) + +gcd x 0 = x +gcd x y = 0 `mod` (0 `mod` y) + +gcd x 0 = 0 +gcd x y = (x `mod` x) `mod` x + +gcd x 0 = 0 +gcd x y = (x `mod` x) `mod` y + +gcd x 0 = 0 +gcd x y = (x `mod` x) `mod` 0 + +gcd x 0 = 0 +gcd x y = (x `mod` y) `mod` x + +gcd x 0 = 0 +gcd x y = (x `mod` y) `mod` y + +gcd x 0 = 0 +gcd x y = (x `mod` y) `mod` 0 + +gcd x 0 = 0 +gcd x y = (x `mod` 0) `mod` x + +gcd x 0 = 0 +gcd x y = (x `mod` 0) `mod` y + +gcd x 0 = 0 +gcd x y = (x `mod` 0) `mod` 0 + +gcd x 0 = 0 +gcd x y = (y `mod` x) `mod` x + +gcd x 0 = 0 +gcd x y = (y `mod` x) `mod` y + +gcd x 0 = 0 +gcd x y = (y `mod` x) `mod` 0 + +gcd x 0 = 0 +gcd x y = (y `mod` y) `mod` x + +gcd x 0 = 0 +gcd x y = (y `mod` y) `mod` y + +gcd x 0 = 0 +gcd x y = (y `mod` y) `mod` 0 + +gcd x 0 = 0 +gcd x y = (y `mod` 0) `mod` x + +gcd x 0 = 0 +gcd x y = (y `mod` 0) `mod` y + +gcd x 0 = 0 +gcd x y = (y `mod` 0) `mod` 0 + +gcd x 0 = 0 +gcd x y = (0 `mod` x) `mod` x + +gcd x 0 = 0 +gcd x y = (0 `mod` x) `mod` y + +gcd x 0 = 0 +gcd x y = (0 `mod` x) `mod` 0 + +gcd x 0 = 0 +gcd x y = (0 `mod` y) `mod` x + +gcd x 0 = 0 +gcd x y = (0 `mod` y) `mod` y + +gcd x 0 = 0 +gcd x y = (0 `mod` y) `mod` 0 + +gcd x 0 = 0 +gcd x y = x `mod` (x `mod` x) + +gcd x 0 = 0 +gcd x y = x `mod` (x `mod` y) + +gcd x 0 = 0 +gcd x y = x `mod` (x `mod` 0) + +gcd x 0 = 0 +gcd x y = x `mod` (y `mod` x) + +gcd x 0 = 0 +gcd x y = x `mod` (y `mod` y) + +gcd x 0 = 0 +gcd x y = x `mod` (y `mod` 0) + +gcd x 0 = 0 +gcd x y = x `mod` (0 `mod` x) + +gcd x 0 = 0 +gcd x y = x `mod` (0 `mod` y) + +gcd x 0 = 0 +gcd x y = y `mod` (x `mod` x) + +gcd x 0 = 0 +gcd x y = y `mod` (x `mod` y) + +gcd x 0 = 0 +gcd x y = y `mod` (x `mod` 0) + +gcd x 0 = 0 +gcd x y = y `mod` (y `mod` x) + +gcd x 0 = 0 +gcd x y = y `mod` (y `mod` y) + +gcd x 0 = 0 +gcd x y = y `mod` (y `mod` 0) + +gcd x 0 = 0 +gcd x y = y `mod` (0 `mod` x) + +gcd x 0 = 0 +gcd x y = y `mod` (0 `mod` y) + +gcd x 0 = 0 +gcd x y = 0 `mod` (x `mod` x) + +gcd x 0 = 0 +gcd x y = 0 `mod` (x `mod` y) + +gcd x 0 = 0 +gcd x y = 0 `mod` (x `mod` 0) + +gcd x 0 = 0 +gcd x y = 0 `mod` (y `mod` x) + +gcd x 0 = 0 +gcd x y = 0 `mod` (y `mod` y) + +gcd x 0 = 0 +gcd x y = 0 `mod` (y `mod` 0) + +gcd x 0 = 0 +gcd x y = 0 `mod` (0 `mod` x) + +gcd x 0 = 0 +gcd x y = 0 `mod` (0 `mod` y) + +gcd x 0 = x `mod` x +gcd x y = x `mod` y + +gcd x 0 = x `mod` x +gcd x y = x `mod` 0 + +gcd x 0 = x `mod` x +gcd x y = y `mod` x + +gcd x 0 = x `mod` x +gcd x y = y `mod` y + +gcd x 0 = x `mod` x +gcd x y = y `mod` 0 + +gcd x 0 = x `mod` x +gcd x y = 0 `mod` x + +gcd x 0 = x `mod` x +gcd x y = 0 `mod` y + +gcd x 0 = x `mod` 0 +gcd x y = x `mod` x + +gcd x 0 = x `mod` 0 +gcd x y = y `mod` x + +gcd x 0 = x `mod` 0 +gcd x y = y `mod` y + +gcd x 0 = x `mod` 0 +gcd x y = y `mod` 0 + +gcd x 0 = x `mod` 0 +gcd x y = 0 `mod` x + +gcd x 0 = x `mod` 0 +gcd x y = 0 `mod` y + +gcd x 0 = 0 `mod` x +gcd x y = x `mod` x + +gcd x 0 = 0 `mod` x +gcd x y = x `mod` y + +gcd x 0 = 0 `mod` x +gcd x y = x `mod` 0 + +gcd x 0 = 0 `mod` x +gcd x y = y `mod` y + +gcd x 0 = 0 `mod` x +gcd x y = y `mod` 0 + +gcd x 0 = 0 `mod` x +gcd x y = 0 `mod` y + +gcd x 0 = (x `mod` x) `mod` x +gcd x y = x + +gcd x 0 = (x `mod` x) `mod` x +gcd x y = y + +gcd x 0 = (x `mod` x) `mod` x +gcd x y = 0 + +gcd x 0 = (x `mod` x) `mod` 0 +gcd x y = x + +gcd x 0 = (x `mod` x) `mod` 0 +gcd x y = y + +gcd x 0 = (x `mod` x) `mod` 0 +gcd x y = 0 + +gcd x 0 = (x `mod` 0) `mod` x +gcd x y = x + +gcd x 0 = (x `mod` 0) `mod` x +gcd x y = y + +gcd x 0 = (x `mod` 0) `mod` x +gcd x y = 0 + +gcd x 0 = (x `mod` 0) `mod` 0 +gcd x y = x + +gcd x 0 = (x `mod` 0) `mod` 0 +gcd x y = y + +gcd x 0 = (x `mod` 0) `mod` 0 +gcd x y = 0 + +gcd x 0 = (0 `mod` x) `mod` x +gcd x y = x + +gcd x 0 = (0 `mod` x) `mod` x +gcd x y = y + +gcd x 0 = (0 `mod` x) `mod` x +gcd x y = 0 + +gcd x 0 = (0 `mod` x) `mod` 0 +gcd x y = x + +gcd x 0 = (0 `mod` x) `mod` 0 +gcd x y = y + +gcd x 0 = (0 `mod` x) `mod` 0 +gcd x y = 0 + +gcd x 0 = x `mod` (x `mod` x) +gcd x y = x + +gcd x 0 = x `mod` (x `mod` x) +gcd x y = y + +gcd x 0 = x `mod` (x `mod` x) +gcd x y = 0 + +gcd x 0 = x `mod` (x `mod` 0) +gcd x y = x + +gcd x 0 = x `mod` (x `mod` 0) +gcd x y = y + +gcd x 0 = x `mod` (x `mod` 0) +gcd x y = 0 + +gcd x 0 = x `mod` (0 `mod` x) +gcd x y = x + +gcd x 0 = x `mod` (0 `mod` x) +gcd x y = y + +gcd x 0 = x `mod` (0 `mod` x) +gcd x y = 0 + +gcd x 0 = 0 `mod` (x `mod` x) +gcd x y = x + +gcd x 0 = 0 `mod` (x `mod` x) +gcd x y = y + +gcd x 0 = 0 `mod` (x `mod` x) +gcd x y = 0 + +gcd x 0 = 0 `mod` (x `mod` 0) +gcd x y = x + +gcd x 0 = 0 `mod` (x `mod` 0) +gcd x y = y + +gcd x 0 = 0 `mod` (x `mod` 0) +gcd x y = 0 + +gcd x 0 = 0 `mod` (0 `mod` x) +gcd x y = x + +gcd x 0 = 0 `mod` (0 `mod` x) +gcd x y = y + +gcd x 0 = 0 `mod` (0 `mod` x) +gcd x y = 0 + +gcd 0 x = x +gcd x y = (x `mod` x) `mod` x + +gcd 0 x = x +gcd x y = (x `mod` x) `mod` y + +gcd 0 x = x +gcd x y = (x `mod` x) `mod` 0 + +gcd 0 x = x +gcd x y = (x `mod` y) `mod` x + +gcd 0 x = x +gcd x y = (x `mod` y) `mod` y + +gcd 0 x = x +gcd x y = (x `mod` y) `mod` 0 + +gcd 0 x = x +gcd x y = (x `mod` 0) `mod` x + +gcd 0 x = x +gcd x y = (x `mod` 0) `mod` y + +gcd 0 x = x +gcd x y = (x `mod` 0) `mod` 0 + +gcd 0 x = x +gcd x y = (y `mod` x) `mod` x + +gcd 0 x = x +gcd x y = (y `mod` x) `mod` y + +gcd 0 x = x +gcd x y = (y `mod` x) `mod` 0 + +gcd 0 x = x +gcd x y = (y `mod` y) `mod` x + +gcd 0 x = x +gcd x y = (y `mod` y) `mod` y + +gcd 0 x = x +gcd x y = (y `mod` y) `mod` 0 + +gcd 0 x = x +gcd x y = (y `mod` 0) `mod` x + +gcd 0 x = x +gcd x y = (y `mod` 0) `mod` y + +gcd 0 x = x +gcd x y = (y `mod` 0) `mod` 0 + +gcd 0 x = x +gcd x y = (0 `mod` x) `mod` x + +gcd 0 x = x +gcd x y = (0 `mod` x) `mod` y + +gcd 0 x = x +gcd x y = (0 `mod` x) `mod` 0 + +gcd 0 x = x +gcd x y = (0 `mod` y) `mod` x + +gcd 0 x = x +gcd x y = (0 `mod` y) `mod` y + +gcd 0 x = x +gcd x y = (0 `mod` y) `mod` 0 + +gcd 0 x = x +gcd x y = x `mod` (x `mod` x) + +gcd 0 x = x +gcd x y = x `mod` (x `mod` y) + +gcd 0 x = x +gcd x y = x `mod` (x `mod` 0) + +gcd 0 x = x +gcd x y = x `mod` (y `mod` x) + +gcd 0 x = x +gcd x y = x `mod` (y `mod` y) + +gcd 0 x = x +gcd x y = x `mod` (y `mod` 0) + +gcd 0 x = x +gcd x y = x `mod` (0 `mod` x) + +gcd 0 x = x +gcd x y = x `mod` (0 `mod` y) + +gcd 0 x = x +gcd x y = y `mod` (x `mod` x) + +gcd 0 x = x +gcd x y = y `mod` (x `mod` y) + +gcd 0 x = x +gcd x y = y `mod` (x `mod` 0) + +gcd 0 x = x +gcd x y = y `mod` (y `mod` x) + +gcd 0 x = x +gcd x y = y `mod` (y `mod` y) + +gcd 0 x = x +gcd x y = y `mod` (y `mod` 0) + +gcd 0 x = x +gcd x y = y `mod` (0 `mod` x) + +gcd 0 x = x +gcd x y = y `mod` (0 `mod` y) + +gcd 0 x = x +gcd x y = 0 `mod` (x `mod` x) + +gcd 0 x = x +gcd x y = 0 `mod` (x `mod` y) + +gcd 0 x = x +gcd x y = 0 `mod` (x `mod` 0) + +gcd 0 x = x +gcd x y = 0 `mod` (y `mod` x) + +gcd 0 x = x +gcd x y = 0 `mod` (y `mod` y) + +gcd 0 x = x +gcd x y = 0 `mod` (y `mod` 0) + +gcd 0 x = x +gcd x y = 0 `mod` (0 `mod` x) + +gcd 0 x = x +gcd x y = 0 `mod` (0 `mod` y) + +gcd 0 x = 0 +gcd x y = (x `mod` x) `mod` x + +gcd 0 x = 0 +gcd x y = (x `mod` x) `mod` y + +gcd 0 x = 0 +gcd x y = (x `mod` x) `mod` 0 + +gcd 0 x = 0 +gcd x y = (x `mod` y) `mod` x + +gcd 0 x = 0 +gcd x y = (x `mod` y) `mod` y + +gcd 0 x = 0 +gcd x y = (x `mod` y) `mod` 0 + +gcd 0 x = 0 +gcd x y = (x `mod` 0) `mod` x + +gcd 0 x = 0 +gcd x y = (x `mod` 0) `mod` y + +gcd 0 x = 0 +gcd x y = (x `mod` 0) `mod` 0 + +gcd 0 x = 0 +gcd x y = (y `mod` x) `mod` x + +gcd 0 x = 0 +gcd x y = (y `mod` x) `mod` y + +gcd 0 x = 0 +gcd x y = (y `mod` x) `mod` 0 + +gcd 0 x = 0 +gcd x y = (y `mod` y) `mod` x + +gcd 0 x = 0 +gcd x y = (y `mod` y) `mod` y + +gcd 0 x = 0 +gcd x y = (y `mod` y) `mod` 0 + +gcd 0 x = 0 +gcd x y = (y `mod` 0) `mod` x + +gcd 0 x = 0 +gcd x y = (y `mod` 0) `mod` y + +gcd 0 x = 0 +gcd x y = (y `mod` 0) `mod` 0 + +gcd 0 x = 0 +gcd x y = (0 `mod` x) `mod` x + +gcd 0 x = 0 +gcd x y = (0 `mod` x) `mod` y + +gcd 0 x = 0 +gcd x y = (0 `mod` x) `mod` 0 + +gcd 0 x = 0 +gcd x y = (0 `mod` y) `mod` x + +gcd 0 x = 0 +gcd x y = (0 `mod` y) `mod` y + +gcd 0 x = 0 +gcd x y = (0 `mod` y) `mod` 0 + +gcd 0 x = 0 +gcd x y = x `mod` (x `mod` x) + +gcd 0 x = 0 +gcd x y = x `mod` (x `mod` y) + +gcd 0 x = 0 +gcd x y = x `mod` (x `mod` 0) + +gcd 0 x = 0 +gcd x y = x `mod` (y `mod` x) + +gcd 0 x = 0 +gcd x y = x `mod` (y `mod` y) + +gcd 0 x = 0 +gcd x y = x `mod` (y `mod` 0) + +gcd 0 x = 0 +gcd x y = x `mod` (0 `mod` x) + +gcd 0 x = 0 +gcd x y = x `mod` (0 `mod` y) + +gcd 0 x = 0 +gcd x y = y `mod` (x `mod` x) + +gcd 0 x = 0 +gcd x y = y `mod` (x `mod` y) + +gcd 0 x = 0 +gcd x y = y `mod` (x `mod` 0) + +gcd 0 x = 0 +gcd x y = y `mod` (y `mod` x) + +gcd 0 x = 0 +gcd x y = y `mod` (y `mod` y) + +gcd 0 x = 0 +gcd x y = y `mod` (y `mod` 0) + +gcd 0 x = 0 +gcd x y = y `mod` (0 `mod` x) + +gcd 0 x = 0 +gcd x y = y `mod` (0 `mod` y) + +gcd 0 x = 0 +gcd x y = 0 `mod` (x `mod` x) + +gcd 0 x = 0 +gcd x y = 0 `mod` (x `mod` y) + +gcd 0 x = 0 +gcd x y = 0 `mod` (x `mod` 0) + +gcd 0 x = 0 +gcd x y = 0 `mod` (y `mod` x) + +gcd 0 x = 0 +gcd x y = 0 `mod` (y `mod` y) + +gcd 0 x = 0 +gcd x y = 0 `mod` (y `mod` 0) + +gcd 0 x = 0 +gcd x y = 0 `mod` (0 `mod` x) + +gcd 0 x = 0 +gcd x y = 0 `mod` (0 `mod` y) + +gcd 0 x = x `mod` x +gcd x y = x `mod` y + +gcd 0 x = x `mod` x +gcd x y = x `mod` 0 + +gcd 0 x = x `mod` x +gcd x y = y `mod` x + +gcd 0 x = x `mod` x +gcd x y = y `mod` 0 + +gcd 0 x = x `mod` x +gcd x y = 0 `mod` x + +gcd 0 x = x `mod` x +gcd x y = 0 `mod` y + +gcd 0 x = x `mod` 0 +gcd x y = x `mod` x + +gcd 0 x = x `mod` 0 +gcd x y = x `mod` y + +gcd 0 x = x `mod` 0 +gcd x y = y `mod` y + +gcd 0 x = x `mod` 0 +gcd x y = 0 `mod` x + +gcd 0 x = x `mod` 0 +gcd x y = 0 `mod` y + +gcd 0 x = 0 `mod` x +gcd x y = x `mod` x + +gcd 0 x = 0 `mod` x +gcd x y = x `mod` 0 + +gcd 0 x = 0 `mod` x +gcd x y = y `mod` x + +gcd 0 x = 0 `mod` x +gcd x y = y `mod` y + +gcd 0 x = 0 `mod` x +gcd x y = y `mod` 0 + +gcd 0 x = (x `mod` x) `mod` x +gcd x y = x + +gcd 0 x = (x `mod` x) `mod` x +gcd x y = y + +gcd 0 x = (x `mod` x) `mod` x +gcd x y = 0 + +gcd 0 x = (x `mod` x) `mod` 0 +gcd x y = x + +gcd 0 x = (x `mod` x) `mod` 0 +gcd x y = y + +gcd 0 x = (x `mod` x) `mod` 0 +gcd x y = 0 + +gcd 0 x = (x `mod` 0) `mod` x +gcd x y = x + +gcd 0 x = (x `mod` 0) `mod` x +gcd x y = y + +gcd 0 x = (x `mod` 0) `mod` x +gcd x y = 0 + +gcd 0 x = (x `mod` 0) `mod` 0 +gcd x y = x + +gcd 0 x = (x `mod` 0) `mod` 0 +gcd x y = y + +gcd 0 x = (x `mod` 0) `mod` 0 +gcd x y = 0 + +gcd 0 x = (0 `mod` x) `mod` x +gcd x y = x + +gcd 0 x = (0 `mod` x) `mod` x +gcd x y = y + +gcd 0 x = (0 `mod` x) `mod` x +gcd x y = 0 + +gcd 0 x = (0 `mod` x) `mod` 0 +gcd x y = x + +gcd 0 x = (0 `mod` x) `mod` 0 +gcd x y = y + +gcd 0 x = (0 `mod` x) `mod` 0 +gcd x y = 0 + +gcd 0 x = x `mod` (x `mod` x) +gcd x y = x + +gcd 0 x = x `mod` (x `mod` x) +gcd x y = y + +gcd 0 x = x `mod` (x `mod` x) +gcd x y = 0 + +gcd 0 x = x `mod` (x `mod` 0) +gcd x y = x + +gcd 0 x = x `mod` (x `mod` 0) +gcd x y = y + +gcd 0 x = x `mod` (x `mod` 0) +gcd x y = 0 + +gcd 0 x = x `mod` (0 `mod` x) +gcd x y = x + +gcd 0 x = x `mod` (0 `mod` x) +gcd x y = y + +gcd 0 x = x `mod` (0 `mod` x) +gcd x y = 0 + +gcd 0 x = 0 `mod` (x `mod` x) +gcd x y = x + +gcd 0 x = 0 `mod` (x `mod` x) +gcd x y = y + +gcd 0 x = 0 `mod` (x `mod` x) +gcd x y = 0 + +gcd 0 x = 0 `mod` (x `mod` 0) +gcd x y = x + +gcd 0 x = 0 `mod` (x `mod` 0) +gcd x y = y + +gcd 0 x = 0 `mod` (x `mod` 0) +gcd x y = 0 + +gcd 0 x = 0 `mod` (0 `mod` x) +gcd x y = x + +gcd 0 x = 0 `mod` (0 `mod` x) +gcd x y = y + +gcd 0 x = 0 `mod` (0 `mod` x) gcd x y = 0 diff --git a/bench/gps.out b/bench/gps.out --- a/bench/gps.out +++ b/bench/gps.out @@ -4,9 +4,9 @@ -- looking through 1 candidates of size 1 -- looking through 1 candidates of size 2 -- looking through 1 candidates of size 3 --- looking through 2 candidates of size 4 +-- looking through 1 candidates of size 4 -- tested 4 candidates -gps1 x y = fromIntegral x + y +gps1 x y = y + fromIntegral x gps2 :: Int -> Maybe [Char] -- testing 6 combinations of argument values @@ -14,7 +14,7 @@ -- looking through 1 candidates of size 1 -- looking through 2 candidates of size 2 -- looking through 3 candidates of size 3 --- looking through 3 candidates of size 4 +-- looking through 2 candidates of size 4 -- looking through 2 candidates of size 5 -- looking through 0 candidates of size 6 -- looking through 32 candidates of size 7 @@ -24,8 +24,10 @@ -- looking through 16 candidates of size 11 -- looking through 512 candidates of size 12 -- looking through 1792 candidates of size 13 --- tested 1204 candidates -gps2 x = if 2000 <= x then Just "large" else (if x < 1000 then Just "small" else Nothing) +-- tested 1203 candidates +gps2 x = if 2000 <= x + then Just "large" + else if x < 1000 then Just "small" else Nothing gps3 :: Int -> Int -> Int -> [Int] -- testing 2 combinations of argument values @@ -35,10 +37,10 @@ -- looking through 0 candidates of size 3 -- looking through 64 candidates of size 4 -- looking through 0 candidates of size 5 --- looking through 1536 candidates of size 6 +-- looking through 1248 candidates of size 6 -- looking through 0 candidates of size 7 --- looking through 26858 candidates of size 8 --- tested 19480 candidates +-- looking through 18627 candidates of size 8 +-- tested 12358 candidates gps3 x y z = enumFromThenTo x (x + z) (y - 1) gps3 :: Int -> Int -> Int -> [Int] @@ -48,11 +50,11 @@ -- looking through 0 candidates of size 2 -- looking through 3 candidates of size 3 -- looking through 0 candidates of size 4 --- looking through 18 candidates of size 5 +-- looking through 15 candidates of size 5 -- looking through 0 candidates of size 6 --- looking through 108 candidates of size 7 +-- looking through 81 candidates of size 7 -- looking through 36 candidates of size 8 --- tested 166 candidates +-- tested 136 candidates cannot conjure gps4 :: [Char] -> [Char] -> [Char] -> Bool @@ -67,9 +69,9 @@ -- looking through 0 candidates of size 7 -- looking through 0 candidates of size 8 -- looking through 162 candidates of size 9 --- looking through 16 candidates of size 10 --- looking through 30 candidates of size 11 --- tested 187 candidates +-- looking through 26 candidates of size 10 +-- looking through 15 candidates of size 11 +-- tested 197 candidates gps4 cs ds es = length cs < length ds && length ds < length es gps5 :: [Char] -> [Char] @@ -88,12 +90,12 @@ -- testing 9 combinations of argument values -- pruning with 16/18 rules -- looking through 4 candidates of size 1 --- looking through 3 candidates of size 2 --- looking through 27 candidates of size 3 --- looking through 32 candidates of size 4 --- looking through 506 candidates of size 5 --- looking through 530 candidates of size 6 --- tested 1102 candidates +-- looking through 2 candidates of size 2 +-- looking through 14 candidates of size 3 +-- looking through 17 candidates of size 4 +-- looking through 188 candidates of size 5 +-- looking through 200 candidates of size 6 +-- tested 425 candidates cannot conjure gps7 :: [Char] -> ([Char],Int) @@ -119,6 +121,7 @@ -- looking through 0 candidates of size 1 -- looking through 0 candidates of size 2 -- looking through 0 candidates of size 3 +-- looking through 0 candidates of size 4 -- tested 0 candidates cannot conjure @@ -130,12 +133,12 @@ -- looking through 4 candidates of size 3 -- looking through 4 candidates of size 4 -- looking through 10 candidates of size 5 --- looking through 24 candidates of size 6 +-- looking through 21 candidates of size 6 -- looking through 34 candidates of size 7 -- looking through 83 candidates of size 8 -- looking through 143 candidates of size 9 --- looking through 259 candidates of size 10 --- tested 328 candidates +-- looking through 256 candidates of size 10 +-- tested 325 candidates gps9 x = filter even (filter (x >) (map sq [1..x])) wallisNext :: Ratio Integer -> Ratio Integer @@ -146,10 +149,10 @@ -- looking through 3 candidates of size 3 -- looking through 15 candidates of size 4 -- looking through 4 candidates of size 5 --- looking through 118 candidates of size 6 +-- looking through 86 candidates of size 6 -- looking through 5 candidates of size 7 --- looking through 1001 candidates of size 8 --- tested 1015 candidates +-- looking through 537 candidates of size 8 +-- tested 588 candidates wallisNext (x % y) = (y + 1) % (x + 1) wallisNext :: Ratio Integer -> Ratio Integer @@ -160,10 +163,10 @@ -- looking through 4 candidates of size 3 -- looking through 16 candidates of size 4 -- looking through 3 candidates of size 5 --- looking through 103 candidates of size 6 +-- looking through 71 candidates of size 6 -- looking through 5 candidates of size 7 --- looking through 801 candidates of size 8 --- tested 832 candidates +-- looking through 393 candidates of size 8 +-- tested 451 candidates wallisNext (x % y) = (y + 1) % (x + 1) gps10 :: Int -> Ratio Integer @@ -210,16 +213,16 @@ -- pruning with 13/27 rules -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 --- looking through 5 candidates of size 3 --- looking through 16 candidates of size 4 --- looking through 26 candidates of size 5 --- looking through 71 candidates of size 6 --- looking through 160 candidates of size 7 --- looking through 432 candidates of size 8 --- looking through 1166 candidates of size 9 --- looking through 3187 candidates of size 10 --- looking through 9282 candidates of size 11 --- tested 10413 candidates +-- looking through 4 candidates of size 3 +-- looking through 11 candidates of size 4 +-- looking through 22 candidates of size 5 +-- looking through 51 candidates of size 6 +-- looking through 133 candidates of size 7 +-- looking through 318 candidates of size 8 +-- looking through 938 candidates of size 9 +-- looking through 2418 candidates of size 10 +-- looking through 7229 candidates of size 11 +-- tested 8246 candidates gps12 xs = (length xs - fromJust (findIndex (0 ==) (reverse xs))) - 1 gps13 :: [Ratio Integer] -> Ratio Integer @@ -229,11 +232,11 @@ -- looking through 1 candidates of size 2 -- looking through 3 candidates of size 3 -- looking through 10 candidates of size 4 --- looking through 15 candidates of size 5 --- looking through 56 candidates of size 6 --- looking through 125 candidates of size 7 --- looking through 449 candidates of size 8 --- tested 317 candidates +-- looking through 14 candidates of size 5 +-- looking through 51 candidates of size 6 +-- looking through 105 candidates of size 7 +-- looking through 365 candidates of size 8 +-- tested 271 candidates gps13 qs = foldr (+) 0 qs / fromIntegral (length qs) odd :: Int -> Bool @@ -241,11 +244,11 @@ -- pruning with 12/13 rules -- looking through 0 candidates of size 1 -- looking through 0 candidates of size 2 --- looking through 7 candidates of size 3 --- looking through 7 candidates of size 4 --- looking through 83 candidates of size 5 --- tested 29 candidates -odd x = x `mod` 2 /= 0 +-- looking through 4 candidates of size 3 +-- looking through 4 candidates of size 4 +-- looking through 44 candidates of size 5 +-- tested 25 candidates +odd x = 0 /= x `mod` 2 gps14 :: [Int] -> Int -- testing 3 combinations of argument values @@ -262,24 +265,24 @@ -- pruning with 39/58 rules -- looking through 3 candidates of size 1 -- looking through 9 candidates of size 2 --- looking through 7 candidates of size 3 --- looking through 58 candidates of size 4 --- looking through 66 candidates of size 5 --- looking through 600 candidates of size 6 --- looking through 1034 candidates of size 7 --- tested 749 candidates +-- looking through 3 candidates of size 3 +-- looking through 27 candidates of size 4 +-- looking through 30 candidates of size 5 +-- looking through 258 candidates of size 6 +-- looking through 474 candidates of size 7 +-- tested 381 candidates gps14 [] = 0 -gps14 (x:xs) = x `mod` 2 + gps14 xs +gps14 (x:xs) = x + gps14 xs `mod` 2 gps15 :: [Int] -> [Int] -> Bool -- testing 5 combinations of argument values -- pruning with 3/7 rules -- looking through 0 candidates of size 1 -- looking through 0 candidates of size 2 --- looking through 2 candidates of size 3 --- looking through 8 candidates of size 4 --- tested 4 candidates -gps15 xs ys = reverse xs == ys +-- looking through 1 candidates of size 3 +-- looking through 4 candidates of size 4 +-- tested 3 candidates +gps15 xs ys = xs == reverse ys gps16 :: [Char] -> [Char] -> Bool -- testing 6 combinations of argument values @@ -296,33 +299,32 @@ -- testing 4 combinations of argument values -- pruning with 27/65 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 11 candidates of size 3 --- looking through 29 candidates of size 4 --- looking through 60 candidates of size 5 --- looking through 145 candidates of size 6 --- looking through 501 candidates of size 7 --- looking through 1098 candidates of size 8 --- looking through 4673 candidates of size 9 --- tested 1852 candidates +-- looking through 3 candidates of size 2 +-- looking through 7 candidates of size 3 +-- looking through 13 candidates of size 4 +-- looking through 35 candidates of size 5 +-- looking through 61 candidates of size 6 +-- looking through 219 candidates of size 7 +-- looking through 354 candidates of size 8 +-- looking through 1431 candidates of size 9 +-- tested 729 candidates gps17 0 = 0 -gps17 x = x * x + gps17 (x - 1) +gps17 x = gps17 (x - 1) + x * x gps18 :: [Int] -> [Int] -> [Int] -- testing 3 combinations of argument values -- pruning with 2/6 rules -- looking through 3 candidates of size 1 -- looking through 7 candidates of size 2 --- looking through 9 candidates of size 3 --- looking through 44 candidates of size 4 --- looking through 116 candidates of size 5 --- looking through 92 candidates of size 6 --- looking through 787 candidates of size 7 --- looking through 220 candidates of size 8 --- looking through 4632 candidates of size 9 --- tested 3924 candidates -gps18 [] [] = [] -gps18 [] (x:xs) = xs +-- looking through 11 candidates of size 3 +-- looking through 37 candidates of size 4 +-- looking through 98 candidates of size 5 +-- looking through 106 candidates of size 6 +-- looking through 440 candidates of size 7 +-- looking through 428 candidates of size 8 +-- looking through 1950 candidates of size 9 +-- tested 2031 candidates +gps18 [] xs = xs gps18 (x:xs) [] = xs gps18 (x:xs) (y:ys) = x + y:gps18 xs ys @@ -331,9 +333,9 @@ -- pruning with 2/7 rules -- looking through 2 candidates of size 1 -- looking through 1 candidates of size 2 --- looking through 0 candidates of size 3 --- looking through 12 candidates of size 4 --- tested 13 candidates +-- looking through 1 candidates of size 3 +-- looking through 11 candidates of size 4 +-- tested 14 candidates gps18 xs ys = zipWith (+) xs ys gps19 :: Int -> [Char] -> [Char] @@ -396,7 +398,9 @@ -- looking through 20409 candidates of size 14 -- tested 27901 candidates pig1 "" = "ay" -pig1 (c:cs) = if isVowel c then (c:cs) ++ "ay" else cs ++ (c:"ay") +pig1 (c:cs) = if isVowel c + then (c:cs) ++ "ay" + else cs ++ (c:"ay") gps20c :: [Char] -> [Char] -- pruning with 1/1 rules @@ -432,10 +436,10 @@ -- looking through 2 candidates of size 3 -- looking through 0 candidates of size 4 -- looking through 0 candidates of size 5 --- looking through 4 candidates of size 6 +-- looking through 3 candidates of size 6 -- tested 5 candidates gps22 "" = 0 -gps22 (c:cs) = scrabble1 c + gps22 cs +gps22 (c:cs) = gps22 cs + scrabble1 c gps23 :: [Char] -> ([(Int,Int)],Int,Double) -- pruning with 0/0 rules @@ -451,13 +455,13 @@ -- looking through 2 candidates of size 2 -- looking through 4 candidates of size 3 -- looking through 3 candidates of size 4 --- looking through 9 candidates of size 5 --- looking through 19 candidates of size 6 --- looking through 40 candidates of size 7 --- looking through 97 candidates of size 8 --- looking through 286 candidates of size 9 --- looking through 739 candidates of size 10 --- tested 673 candidates +-- looking through 8 candidates of size 5 +-- looking through 15 candidates of size 6 +-- looking through 31 candidates of size 7 +-- looking through 77 candidates of size 8 +-- looking through 203 candidates of size 9 +-- looking through 487 candidates of size 10 +-- tested 485 candidates gps24 cs = chr (ord ' ' + sum (map ord cs) `mod` 64) gps25 :: Int -> [Int] @@ -487,47 +491,49 @@ -- looking through 0 candidates of size 7 -- looking through 0 candidates of size 8 -- looking through 0 candidates of size 9 --- looking through 180 candidates of size 10 +-- looking through 90 candidates of size 10 -- looking through 2592 candidates of size 11 --- tested 1579 candidates -gps27 x y z = if x < y then (if y < z then y else z) else x +-- tested 1489 candidates +gps27 x y z = if x < y + then if y < z then y else z + else x gps27b :: Int -> Int -> Int -> Int -- testing 3 combinations of argument values -- pruning with 20/30 rules -- looking through 3 candidates of size 1 -- looking through 0 candidates of size 2 --- looking through 12 candidates of size 3 +-- looking through 6 candidates of size 3 -- looking through 0 candidates of size 4 --- looking through 36 candidates of size 5 --- tested 16 candidates -gps27b x y z = min (max x y) z +-- looking through 12 candidates of size 5 +-- tested 15 candidates +gps27b x y z = min z (max x y) gps28 :: Int -> Int -> Int -> Int -> Int -- testing 5 combinations of argument values -- pruning with 6/10 rules -- looking through 4 candidates of size 1 -- looking through 0 candidates of size 2 --- looking through 12 candidates of size 3 +-- looking through 6 candidates of size 3 -- looking through 0 candidates of size 4 --- looking through 24 candidates of size 5 +-- looking through 12 candidates of size 5 -- looking through 0 candidates of size 6 --- looking through 72 candidates of size 7 --- tested 44 candidates +-- looking through 36 candidates of size 7 +-- tested 25 candidates gps28 x y z x' = x `min` (y `min` (z `min` x')) gps29 :: [Char] -> Int -- pruning with 7/11 rules -- looking through 2 candidates of size 1 -- looking through 2 candidates of size 2 --- looking through 3 candidates of size 3 --- looking through 4 candidates of size 4 --- looking through 6 candidates of size 5 --- looking through 10 candidates of size 6 --- looking through 18 candidates of size 7 --- looking through 33 candidates of size 8 --- looking through 65 candidates of size 9 --- tested 86 candidates +-- looking through 2 candidates of size 3 +-- looking through 0 candidates of size 4 +-- looking through 2 candidates of size 5 +-- looking through 6 candidates of size 6 +-- looking through 10 candidates of size 7 +-- looking through 10 candidates of size 8 +-- looking through 24 candidates of size 9 +-- tested 40 candidates gps29 "" = 0 -gps29 (c:cs) = (if isVowel c then 1 else 0) + gps29 cs +gps29 (c:cs) = gps29 cs + (if isVowel c then 1 else 0) diff --git a/bench/gps2.hs b/bench/gps2.hs --- a/bench/gps2.hs +++ b/bench/gps2.hs @@ -280,7 +280,7 @@ -- gps8g x xs = head $ filter ((x ==) . uncurry (+)) $ liftA2 (,) xs xs gps8c :: IO () -gps8c = conjure "gps" gps8p [] +gps8c = conjure "gps8" gps8p [] -- GPSB#9: Fizz Buzz (CW) @@ -302,24 +302,36 @@ -- probably unreachable performance-wise gps9c :: IO () -gps9c = conjure "gps" gps9p [] +gps9c = conjure "gps9" gps9p [] gps10p :: [Int] -> Int +gps10p [0] = -2 +gps10p [3] = -1 +gps10p [6] = 0 +gps10p [9] = 1 +gps10p [0,3] = -3 +gps10p [3,0] = -3 +gps10p [3,2,1] = -5 gps10p [1,2,3] = -5 gps10p [10,20,30] = 13 -- does not do rounding gps10g :: [Int] -> Int -gps10g xs = sum $ map (\x -> x `div` 3 - 2) xs --- TODO: reachable with explicit fold? --- gps10g [] = 0 --- gps10g (x:xs) = x`div`3 - 2 : gps10g xs --- ?? +gps10g [] = 0 +gps10g (x:xs) = (x `div` 3 - 2) + gps10g xs -- unreachable due to lambda gps10c :: IO () -gps10c = conjure "gps" gps9p [] +gps10c = conjure "gps10" gps10p + [ pr (0 :: Int) + , pr (1 :: Int) + , pr (2 :: Int) + , pr (3 :: Int) + , prim "`div`" (div :: Int -> Int -> Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) + ] gps11p :: Int -> Int -> Int diff --git a/bench/gps2.out b/bench/gps2.out --- a/bench/gps2.out +++ b/bench/gps2.out @@ -32,10 +32,10 @@ -- testing 6 combinations of argument values -- pruning with 8/9 rules -- looking through 4 candidates of size 1 --- looking through 25 candidates of size 2 --- looking through 237 candidates of size 3 --- looking through 433 candidates of size 4 --- tested 699 candidates +-- looking through 24 candidates of size 2 +-- looking through 75 candidates of size 3 +-- looking through 234 candidates of size 4 +-- tested 337 candidates cannot conjure gps2 :: Double -> Double -> Int -> Double @@ -43,11 +43,11 @@ -- pruning with 2/6 rules -- looking through 2 candidates of size 1 -- looking through 2 candidates of size 2 --- looking through 18 candidates of size 3 --- looking through 70 candidates of size 4 --- looking through 316 candidates of size 5 --- looking through 1440 candidates of size 6 --- tested 1848 candidates +-- looking through 17 candidates of size 3 +-- looking through 62 candidates of size 4 +-- looking through 266 candidates of size 5 +-- looking through 1154 candidates of size 6 +-- tested 1503 candidates cannot conjure gps3 :: [Char] -> Int @@ -80,14 +80,14 @@ -- looking through 2 candidates of size 1 -- looking through 1 candidates of size 2 -- looking through 2 candidates of size 3 --- looking through 10 candidates of size 4 +-- looking through 8 candidates of size 4 -- looking through 6 candidates of size 5 --- looking through 60 candidates of size 6 +-- looking through 46 candidates of size 6 -- looking through 26 candidates of size 7 --- looking through 538 candidates of size 8 +-- looking through 400 candidates of size 8 -- looking through 134 candidates of size 9 --- looking through 5296 candidates of size 10 --- tested 2698 candidates +-- looking through 3914 candidates of size 10 +-- tested 1552 candidates tell [] x = [] tell (x:xs) y = y `div` x:tell xs (y `mod` x) @@ -108,13 +108,13 @@ -- looking through 5 candidates of size 3 -- looking through 9 candidates of size 4 -- looking through 45 candidates of size 5 --- looking through 92 candidates of size 6 --- looking through 438 candidates of size 7 --- looking through 944 candidates of size 8 --- looking through 4413 candidates of size 9 --- looking through 9649 candidates of size 10 --- looking through 44336 candidates of size 11 --- tested 32852 candidates +-- looking through 82 candidates of size 6 +-- looking through 428 candidates of size 7 +-- looking through 887 candidates of size 8 +-- looking through 4354 candidates of size 9 +-- looking through 9260 candidates of size 10 +-- looking through 43875 candidates of size 11 +-- tested 32327 candidates gps5 x = tell [25,10,5,1] x gps6 :: [Int] -> Int @@ -133,7 +133,7 @@ -- tested 0 candidates cannot conjure -gps :: Int -> [Int] -> (Int,Int) +gps8 :: Int -> [Int] -> (Int,Int) -- testing 3 combinations of argument values -- pruning with 0/0 rules -- looking through 0 candidates of size 1 @@ -141,30 +141,39 @@ -- tested 0 candidates cannot conjure -gps :: Int -> [Char] +gps9 :: Int -> [Char] -- testing 7 combinations of argument values -- pruning with 0/0 rules -- looking through 0 candidates of size 1 -- tested 0 candidates cannot conjure -gps :: Int -> [Char] +gps10 :: [Int] -> Int -- testing 7 combinations of argument values --- pruning with 0/0 rules --- looking through 0 candidates of size 1 --- tested 0 candidates -cannot conjure +-- pruning with 67/100 rules +-- looking through 4 candidates of size 1 +-- looking through 16 candidates of size 2 +-- looking through 4 candidates of size 3 +-- looking through 68 candidates of size 4 +-- looking through 80 candidates of size 5 +-- looking through 1084 candidates of size 6 +-- looking through 1556 candidates of size 7 +-- looking through 20600 candidates of size 8 +-- looking through 35632 candidates of size 9 +-- tested 29542 candidates +gps10 [] = 0 +gps10 (x:xs) = gps10 xs + (x `div` 3 - 2) gcd :: Int -> Int -> Int -- testing 11 combinations of argument values -- pruning with 0/0 rules -- looking through 3 candidates of size 1 --- looking through 7 candidates of size 2 --- looking through 18 candidates of size 3 --- looking through 60 candidates of size 4 --- looking through 150 candidates of size 5 --- looking through 469 candidates of size 6 --- tested 243 candidates +-- looking through 5 candidates of size 2 +-- looking through 11 candidates of size 3 +-- looking through 50 candidates of size 4 +-- looking through 98 candidates of size 5 +-- looking through 359 candidates of size 6 +-- tested 171 candidates gcd x 0 = x gcd x y = gcd y (x `mod` y) @@ -196,7 +205,9 @@ -- looking through 67 candidates of size 12 -- tested 83 candidates gps13_leaders [] = [] -gps13_leaders (x:xs) = if all (x >) xs then x:gps13_leaders xs else gps13_leaders xs +gps13_leaders (x:xs) = if all (x >) xs + then x:gps13_leaders xs + else gps13_leaders xs gps14_luhn :: [Int] -> Int -- pruning with 0/0 rules @@ -228,7 +239,9 @@ -- looking through 6293 candidates of size 12 -- tested 3977 candidates gps16_middle "" = "" -gps16_middle (c:cs) = if length cs <= 1 then c:cs else gps16_middle (init cs) +gps16_middle (c:cs) = if length cs <= 1 + then c:cs + else gps16_middle (init cs) gps17_pds :: [Int] -> Int -- testing 5 combinations of argument values @@ -237,8 +250,8 @@ -- looking through 2 candidates of size 2 -- looking through 2 candidates of size 3 -- looking through 1 candidates of size 4 --- looking through 5 candidates of size 5 --- tested 11 candidates +-- looking through 3 candidates of size 5 +-- tested 9 candidates cannot conjure gps18_price :: [Double] -> [Double] -> Double @@ -246,23 +259,23 @@ -- pruning with 26/35 rules -- looking through 2 candidates of size 1 -- looking through 8 candidates of size 2 --- looking through 64 candidates of size 3 --- looking through 72 candidates of size 4 --- looking through 1193 candidates of size 5 --- looking through 714 candidates of size 6 --- tested 2053 candidates +-- looking through 32 candidates of size 3 +-- looking through 86 candidates of size 4 +-- looking through 412 candidates of size 5 +-- looking through 1072 candidates of size 6 +-- tested 1612 candidates cannot conjure gps19_snowday :: Int -> Double -> Double -> Double -> Double -- testing 7 combinations of argument values -- pruning with 6/19 rules -- looking through 3 candidates of size 1 --- looking through 0 candidates of size 2 --- looking through 24 candidates of size 3 --- looking through 150 candidates of size 4 --- looking through 546 candidates of size 5 --- looking through 3540 candidates of size 6 --- tested 4263 candidates +-- looking through 4 candidates of size 2 +-- looking through 22 candidates of size 3 +-- looking through 118 candidates of size 4 +-- looking through 318 candidates of size 5 +-- looking through 1936 candidates of size 6 +-- tested 2401 candidates cannot conjure gps20 :: [Char] -> Bool @@ -284,7 +297,9 @@ -- looking through 0 candidates of size 7 -- looking through 4 candidates of size 8 -- tested 3 candidates -spin cs = if length cs >= 5 then reverse cs else cs +spin cs = if length cs >= 5 + then reverse cs + else cs gps21_spinwords :: [Char] -> [Char] -- pruning with 16/16 rules @@ -304,15 +319,15 @@ -- looking through 0 candidates of size 2 -- looking through 2 candidates of size 3 -- looking through 3 candidates of size 4 --- looking through 17 candidates of size 5 --- looking through 14 candidates of size 6 --- looking through 103 candidates of size 7 +-- looking through 14 candidates of size 5 +-- looking through 13 candidates of size 6 +-- looking through 94 candidates of size 7 -- looking through 104 candidates of size 8 --- looking through 901 candidates of size 9 --- looking through 935 candidates of size 10 --- looking through 9226 candidates of size 11 +-- looking through 856 candidates of size 9 +-- looking through 929 candidates of size 10 +-- looking through 8914 candidates of size 11 -- looking through 9686 candidates of size 12 --- tested 20992 candidates +-- tested 20616 candidates cannot conjure gps22 :: Int -> [Char] @@ -324,18 +339,18 @@ gps23 :: [Char] -> [Char] -> [Char] -> [Char] -- pruning with 0/0 rules -- looking through 3 candidates of size 1 --- looking through 5 candidates of size 2 --- looking through 9 candidates of size 3 --- looking through 0 candidates of size 4 --- looking through 162 candidates of size 5 --- looking through 612 candidates of size 6 --- looking through 0 candidates of size 7 --- looking through 0 candidates of size 8 --- looking through 3168 candidates of size 9 --- looking through 0 candidates of size 10 --- looking through 0 candidates of size 11 --- looking through 5184 candidates of size 12 --- tested 9143 candidates +-- looking through 9 candidates of size 2 +-- looking through 31 candidates of size 3 +-- looking through 36 candidates of size 4 +-- looking through 187 candidates of size 5 +-- looking through 627 candidates of size 6 +-- looking through 944 candidates of size 7 +-- looking through 863 candidates of size 8 +-- looking through 951 candidates of size 9 +-- looking through 3902 candidates of size 10 +-- looking through 6758 candidates of size 11 +-- looking through 3276 candidates of size 12 +-- tested 17587 candidates cannot conjure gps24 :: [Char] -> Twitter @@ -355,17 +370,19 @@ -- looking through 483 candidates of size 12 -- tested 936 candidates gps24 "" = Empty -gps24 (c:cs) = if 140 > length cs then Tweet (length (c:cs)) else TooMany +gps24 (c:cs) = if 140 > length cs + then Tweet (length (c:cs)) + else TooMany gps25 :: [Double] -> [Double] -> Double -- testing 6 combinations of argument values -- pruning with 31/59 rules -- looking through 2 candidates of size 1 -- looking through 9 candidates of size 2 --- looking through 81 candidates of size 3 --- looking through 312 candidates of size 4 --- looking through 2249 candidates of size 5 --- looking through 9419 candidates of size 6 --- tested 12072 candidates +-- looking through 49 candidates of size 3 +-- looking through 211 candidates of size 4 +-- looking through 1063 candidates of size 5 +-- looking through 4717 candidates of size 6 +-- tested 6051 candidates cannot conjure diff --git a/bench/ill-hit.out b/bench/ill-hit.out --- a/bench/ill-hit.out +++ b/bench/ill-hit.out @@ -3,10 +3,10 @@ -- pruning with 14/25 rules -- looking through 2 candidates of size 1 -- looking through 3 candidates of size 2 --- looking through 4 candidates of size 3 --- looking through 10 candidates of size 4 --- looking through 19 candidates of size 5 --- tested 20 candidates +-- looking through 3 candidates of size 3 +-- looking through 7 candidates of size 4 +-- looking through 11 candidates of size 5 +-- tested 16 candidates sum [] = 0 sum (x:xs) = x + sum xs @@ -14,10 +14,10 @@ -- pruning with 14/25 rules -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 --- looking through 6 candidates of size 3 --- looking through 20 candidates of size 4 --- looking through 34 candidates of size 5 --- tested 34 candidates +-- looking through 5 candidates of size 3 +-- looking through 12 candidates of size 4 +-- looking through 18 candidates of size 5 +-- tested 25 candidates sum [] = 0 sum (x:xs) = x + sum xs diff --git a/bench/lowtests.out b/bench/lowtests.out --- a/bench/lowtests.out +++ b/bench/lowtests.out @@ -22,10 +22,10 @@ -- looking through 12 candidates of size 7 -- looking through 21 candidates of size 8 -- looking through 8 candidates of size 9 --- looking through 0 candidates of size 10 --- looking through 0 candidates of size 11 --- looking through 0 candidates of size 12 --- tested 47 candidates +-- looking through 6 candidates of size 10 +-- looking through 30 candidates of size 11 +-- looking through 45 candidates of size 12 +-- tested 128 candidates cannot conjure subset :: [Int] -> [Int] -> Bool diff --git a/bench/p12.out b/bench/p12.out --- a/bench/p12.out +++ b/bench/p12.out @@ -3,12 +3,12 @@ -- testing 6 combinations of argument values -- pruning with 67/100 rules -- looking through 3 candidates of size 1 --- looking through 5 candidates of size 2 --- looking through 14 candidates of size 3 --- looking through 39 candidates of size 4 --- looking through 128 candidates of size 5 --- looking through 497 candidates of size 6 --- tested 200 candidates +-- looking through 3 candidates of size 2 +-- looking through 6 candidates of size 3 +-- looking through 15 candidates of size 4 +-- looking through 43 candidates of size 5 +-- looking through 152 candidates of size 6 +-- tested 77 candidates factorial 0 = 1 factorial x = x * factorial (dec x) diff --git a/bench/redundants.hs b/bench/redundants.hs --- a/bench/redundants.hs +++ b/bench/redundants.hs @@ -8,6 +8,7 @@ import Conjure.Engine import Conjure.Defn import Conjure.Utils +import Test.LeanCheck.Tiers (discardT) -- | This function prints redundant candidates. @@ -34,7 +35,9 @@ numRedundant = numCandidates - numUnique classes = classifyBy (equalModuloTesting maxTests maxEvalRecursions nm f) candidates candidates = concat css - css = take n css' + css = take n + . discardT isRedundantByIntroduction -- additional pruning rule + $ css' (css', thy) = candidateDefnsC args nm f ps -- Conjure uses this for listing candidates nRules = length (rules thy) nREs = length (equations thy) + nRules diff --git a/bench/redundants.out b/bench/redundants.out --- a/bench/redundants.out +++ b/bench/redundants.out @@ -1,2117 +1,953 @@ Redundant candidates for: foo :: Int -> Int pruning with 15/35 rules - [3,4,12,32,54] candidates - 68/105 unique candidates - 37/105 redundant candidates - -rules: -x * y == x + y -x * y == y + x -x - x == 0 -x + 0 == x -0 + x == x -x - 0 == x -(x + y) + z == x + (y + z) -(x + y) + z == y + (x + z) -x - (y - z) == z + (x - y) -(x - y) - z == x - (y + z) -(x - y) - z == x - (z + y) -(x + y) - z == x + (y - z) -(x + y) - z == y + (x - z) -x + (y - x) == y -(x - y) + y == x -equations: -y + x == x + y -y + (x + z) == x + (y + z) -z + (x + y) == x + (y + z) -z + (y + x) == x + (y + z) -y + (x - z) == x + (y - z) -(x - z) + y == x + (y - z) -(z - y) + x == (x - y) + z -y - (x + y) == 0 - x -y - (y + x) == 0 - x -z - (y + z) == x - (x + y) -z - (y + z) == x - (y + x) -z - (z + y) == x - (x + y) -z - (z + y) == x - (y + x) -x + (0 - y) == x - y -(0 - y) + x == x - y -x - (x + 1) == 0 - 1 -x - (1 + x) == 0 - 1 -y - (y + 1) == x - (x + 1) -y - (y + 1) == x - (1 + x) -y - (1 + y) == x - (1 + x) - -class of 4 equivalent candidates: - - foo x = x - - foo 0 = 0 - foo x = x - - foo 1 = 1 - foo x = x - - foo 0 = 0 - foo 1 = 1 - foo x = x - - -class of 2 equivalent candidates: - - foo 0 = 0 - foo x = 1 - - foo 0 = 0 - foo 1 = 1 - foo x = 1 - - -class of 2 equivalent candidates: - - foo 0 = 1 - foo x = x - - foo 0 = 1 - foo 1 = 1 - foo x = x - - -class of 2 equivalent candidates: - - foo 0 = 1 - foo x = 0 - - foo 0 = 1 - foo 1 = 0 - foo x = 0 - - -class of 2 equivalent candidates: - - foo x = x + x - - foo 0 = 0 - foo x = x + x - - -class of 4 equivalent candidates: - - foo x = x + 1 - - foo x = 1 + x - - foo 0 = 1 - foo x = x + 1 - - foo 0 = 1 - foo x = 1 + x - - -class of 4 equivalent candidates: - - foo x = x - 1 - - foo x = (0 - 1) + x - - foo x = x + (0 - 1) - - foo 1 = 0 - foo x = x - 1 - - -class of 5 equivalent candidates: - - foo x = 0 - x - - foo 0 = 0 - foo x = 0 - x - - foo x = x - (x + x) - - foo x = 1 - (x + 1) - - foo x = 1 - (1 + x) - - -class of 4 equivalent candidates: - - foo x = 0 - 1 - - foo x = x - (x + 1) - - foo x = x - (1 + x) - - foo x = 1 - (1 + 1) - - -class of 5 equivalent candidates: - - foo x = 1 - x - - foo 0 = 1 - foo x = 1 - x - - foo x = (0 - x) + 1 - - foo x = 1 + (0 - x) - - foo 1 = 0 - foo x = 1 - x - - -class of 2 equivalent candidates: - - foo 1 = 0 - foo x = x - - foo 0 = 0 - foo 1 = 0 - foo x = x - - -class of 2 equivalent candidates: - - foo 1 = 0 - foo x = 1 - - foo 0 = 1 - foo 1 = 0 - foo x = 1 - - -class of 2 equivalent candidates: - - foo 1 = 1 - foo x = 0 - - foo 0 = 0 - foo 1 = 1 - foo x = 0 - - -class of 2 equivalent candidates: - - foo 0 = 0 - foo x = x + 1 - - foo 0 = 0 - foo x = 1 + x - - -class of 2 equivalent candidates: - - foo x = (x - 1) + x - - foo x = x + (x - 1) - - -class of 2 equivalent candidates: - - foo x = (1 - x) + 1 - - foo x = 1 + (1 - x) - - -class of 3 equivalent candidates: - - foo x = x + (x + 1) - - foo x = x + (1 + x) - - foo x = 1 + (x + x) - - -class of 3 equivalent candidates: - - foo x = x + (1 + 1) - - foo x = 1 + (x + 1) - - foo x = 1 + (1 + x) - - -class of 2 equivalent candidates: - - foo x = 0 - (x + 1) - - foo x = 0 - (1 + x) - - -class of 2 equivalent candidates: - - foo 1 = 0 - foo x = x + 1 - - foo 1 = 0 - foo x = 1 + x - - -class of 2 equivalent candidates: - - foo 1 = 1 - foo x = x + 1 - - foo 1 = 1 - foo x = 1 + x - - - -Redundant candidates for: ? :: Int -> Int -> Int - pruning with 10/23 rules - [3,10,40,116,338] candidates - 363/507 unique candidates - 144/507 redundant candidates - -rules: -x * y == x + y -x * y == y + x -x + 0 == x -0 + x == x -dec (x + y) == x + dec y -dec (x + y) == y + dec x -dec (x + y) == dec x + y -dec (x + y) == dec y + x -(x + y) + z == x + (y + z) -(x + y) + z == y + (x + z) -equations: -y + x == x + y -y + dec x == x + dec y -dec x + y == x + dec y -dec y + x == dec x + y -x + dec 0 == dec x -dec 0 + x == dec x -y + (x + z) == x + (y + z) -z + (x + y) == x + (y + z) -z + (y + x) == x + (y + z) -y + dec (dec x) == x + dec (dec y) -dec (dec x) + y == x + dec (dec y) -x + dec (dec 0) == dec (dec x) -dec (dec 0) + x == dec (dec x) - -class of 2 equivalent candidates: - - x ? y = x - - 0 ? x = 0 - x ? y = x - - -class of 2 equivalent candidates: - - x ? y = y - - x ? 0 = 0 - x ? y = y - - -class of 3 equivalent candidates: - - x ? y = 0 - - x ? 0 = 0 - x ? y = dec x ? 0 - - 0 ? x = 0 - x ? y = 0 ? dec y - - -class of 4 equivalent candidates: - - x ? y = dec x - - x ? y = dec 0 + x - - x ? y = x + dec 0 - - 0 ? x = dec 0 - x ? y = dec x - - -class of 4 equivalent candidates: - - x ? y = dec y - - x ? y = dec 0 + y - - x ? y = y + dec 0 - - x ? 0 = dec 0 - x ? y = dec y - - -class of 2 equivalent candidates: - - x ? 0 = x - x ? y = y - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = x - x ? y = y - - -class of 2 equivalent candidates: - - x ? 0 = x - x ? y = 0 - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = x - x ? y = 0 - - -class of 2 equivalent candidates: - - x ? 0 = 0 - x ? y = x - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = 0 - x ? y = x - - -class of 2 equivalent candidates: - - 0 ? x = x - x ? y = 0 - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = 0 - x ? y = 0 - - -class of 2 equivalent candidates: - - 0 ? x = 0 - x ? y = y - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = 0 - x ? y = y - - -class of 2 equivalent candidates: - - x ? y = x + x - - 0 ? x = 0 - x ? y = x + x - - -class of 8 equivalent candidates: - - x ? y = x + y - - x ? y = y + x - - x ? 0 = x - x ? y = x + y - - x ? 0 = x - x ? y = y + x - - 0 ? x = x - x ? y = x + y - - 0 ? x = x - x ? y = y + x - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = x - x ? y = x + y - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = x - x ? y = y + x - - -class of 2 equivalent candidates: - - x ? y = y + y - - x ? 0 = 0 - x ? y = y + y - - -class of 5 equivalent candidates: - - x ? y = dec (dec x) - - x ? y = dec (dec 0) + x - - x ? y = x + dec (dec 0) - - x ? y = dec x + dec 0 - - x ? y = dec 0 + dec x - - -class of 5 equivalent candidates: - - x ? y = dec (dec y) - - x ? y = dec (dec 0) + y - - x ? y = y + dec (dec 0) - - x ? y = dec y + dec 0 - - x ? y = dec 0 + dec y - - -class of 2 equivalent candidates: - - x ? y = dec (dec 0) - - x ? y = dec 0 + dec 0 - - -class of 5 equivalent candidates: - - x ? 0 = x - x ? y = dec x - - x ? 0 = x - x ? y = dec x ? 0 - - x ? 0 = x - x ? y = dec 0 + x - - x ? 0 = x - x ? y = x + dec 0 - - 0 ? 0 = 0 - 0 ? x = dec 0 - x ? 0 = x - x ? y = dec x - - -class of 4 equivalent candidates: - - x ? 0 = x - x ? y = dec y - - x ? 0 = x - x ? y = dec 0 + y - - x ? 0 = x - x ? y = y + dec 0 - - 0 ? 0 = 0 - 0 ? x = dec x - x ? 0 = x - x ? y = dec y - - -class of 2 equivalent candidates: - - x ? 0 = x - x ? y = dec 0 - - 0 ? 0 = 0 - 0 ? x = dec 0 - x ? 0 = x - x ? y = dec 0 - - -class of 4 equivalent candidates: - - x ? 0 = 0 - x ? y = dec x - - x ? 0 = 0 - x ? y = dec 0 + x - - x ? 0 = 0 - x ? y = x + dec 0 - - 0 ? 0 = 0 - 0 ? x = dec 0 - x ? 0 = 0 - x ? y = dec x - - -class of 3 equivalent candidates: - - x ? 0 = 0 - x ? y = dec y - - x ? 0 = 0 - x ? y = dec 0 + y - - x ? 0 = 0 - x ? y = y + dec 0 - - -class of 4 equivalent candidates: - - x ? 0 = dec x - x ? y = x - - x ? 0 = dec 0 + x - x ? y = x - - x ? 0 = x + dec 0 - x ? y = x - - 0 ? 0 = dec 0 - 0 ? x = 0 - x ? 0 = dec x - x ? y = x - - -class of 4 equivalent candidates: - - x ? 0 = dec x - x ? y = y - - x ? 0 = dec 0 + x - x ? y = y - - x ? 0 = x + dec 0 - x ? y = y - - 0 ? 0 = dec 0 - 0 ? x = x - x ? 0 = dec x - x ? y = y - - -class of 4 equivalent candidates: - - x ? 0 = dec x - x ? y = 0 - - x ? 0 = dec 0 + x - x ? y = 0 - - x ? 0 = x + dec 0 - x ? y = 0 - - 0 ? 0 = dec 0 - 0 ? x = 0 - x ? 0 = dec x - x ? y = 0 - - -class of 2 equivalent candidates: - - x ? 0 = dec 0 - x ? y = x - - 0 ? 0 = dec 0 - 0 ? x = 0 - x ? 0 = dec 0 - x ? y = x - - -class of 4 equivalent candidates: - - 0 ? x = x - x ? y = dec x - - 0 ? x = x - x ? y = dec 0 + x - - 0 ? x = x - x ? y = x + dec 0 - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = dec x - x ? y = dec x - - -class of 5 equivalent candidates: - - 0 ? x = x - x ? y = dec y - - 0 ? x = x - x ? y = 0 ? dec y - - 0 ? x = x - x ? y = dec 0 + y - - 0 ? x = x - x ? y = y + dec 0 - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = dec 0 - x ? y = dec y - - -class of 2 equivalent candidates: - - 0 ? x = x - x ? y = dec 0 - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = dec 0 - x ? y = dec 0 - - -class of 3 equivalent candidates: - - 0 ? x = 0 - x ? y = dec x - - 0 ? x = 0 - x ? y = dec 0 + x - - 0 ? x = 0 - x ? y = x + dec 0 - - -class of 4 equivalent candidates: - - 0 ? x = 0 - x ? y = dec y - - 0 ? x = 0 - x ? y = dec 0 + y - - 0 ? x = 0 - x ? y = y + dec 0 - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = dec 0 - x ? y = dec y - - -class of 4 equivalent candidates: - - 0 ? x = dec x - x ? y = x - - 0 ? x = dec 0 + x - x ? y = x - - 0 ? x = x + dec 0 - x ? y = x - - 0 ? 0 = dec 0 - 0 ? x = dec x - x ? 0 = x - x ? y = x - - -class of 4 equivalent candidates: - - 0 ? x = dec x - x ? y = y - - 0 ? x = dec 0 + x - x ? y = y - - 0 ? x = x + dec 0 - x ? y = y - - 0 ? 0 = dec 0 - 0 ? x = dec x - x ? 0 = 0 - x ? y = y - - -class of 4 equivalent candidates: - - 0 ? x = dec x - x ? y = 0 - - 0 ? x = dec 0 + x - x ? y = 0 - - 0 ? x = x + dec 0 - x ? y = 0 - - 0 ? 0 = dec 0 - 0 ? x = dec x - x ? 0 = 0 - x ? y = 0 - - -class of 2 equivalent candidates: - - 0 ? x = dec 0 - x ? y = y - - 0 ? 0 = dec 0 - 0 ? x = dec 0 - x ? 0 = 0 - x ? y = y - - -class of 2 equivalent candidates: - - x ? y = dec x + x - - x ? y = x + dec x - - -class of 4 equivalent candidates: - - x ? y = dec x + y - - x ? y = dec y + x - - x ? y = x + dec y - - x ? y = y + dec x - - -class of 2 equivalent candidates: - - x ? y = dec y + y - - x ? y = y + dec y - - -class of 2 equivalent candidates: - - x ? 0 = x - x ? y = x + x - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = x - x ? y = x + x - - -class of 2 equivalent candidates: - - x ? 0 = 0 - x ? y = x + x - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = 0 - x ? y = x + x - - -class of 4 equivalent candidates: - - x ? 0 = 0 - x ? y = x + y - - x ? 0 = 0 - x ? y = y + x - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = 0 - x ? y = x + y - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = 0 - x ? y = y + x - - -class of 2 equivalent candidates: - - x ? 0 = x + x - x ? y = x - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = x + x - x ? y = x - - -class of 2 equivalent candidates: - - x ? 0 = x + x - x ? y = y - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = x + x - x ? y = y - - -class of 2 equivalent candidates: - - x ? 0 = x + x - x ? y = 0 - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = x + x - x ? y = 0 - - -class of 2 equivalent candidates: - - 0 ? x = x - x ? y = y + y - - 0 ? 0 = 0 - 0 ? x = x - x ? 0 = 0 - x ? y = y + y - - -class of 4 equivalent candidates: - - 0 ? x = 0 - x ? y = x + y - - 0 ? x = 0 - x ? y = y + x - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = x - x ? y = x + y - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = x - x ? y = y + x - - -class of 2 equivalent candidates: - - 0 ? x = 0 - x ? y = y + y - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = 0 - x ? y = y + y - - -class of 2 equivalent candidates: - - 0 ? x = x + x - x ? y = x - - 0 ? 0 = 0 - 0 ? x = x + x - x ? 0 = x - x ? y = x - - -class of 2 equivalent candidates: - - 0 ? x = x + x - x ? y = y - - 0 ? 0 = 0 - 0 ? x = x + x - x ? 0 = 0 - x ? y = y - - -class of 2 equivalent candidates: - - 0 ? x = x + x - x ? y = 0 - - 0 ? 0 = 0 - 0 ? x = x + x - x ? 0 = 0 - x ? y = 0 - - -class of 3 equivalent candidates: - - x ? y = dec (dec x) + x - - x ? y = x + dec (dec x) - - x ? y = dec x + dec x - - -class of 6 equivalent candidates: - - x ? y = dec (dec x) + y - - x ? y = dec (dec y) + x - - x ? y = x + dec (dec y) - - x ? y = y + dec (dec x) - - x ? y = dec x + dec y - - x ? y = dec y + dec x - - -class of 3 equivalent candidates: - - x ? y = dec (dec y) + y - - x ? y = y + dec (dec y) - - x ? y = dec y + dec y - - -class of 3 equivalent candidates: - - x ? y = x + (x + y) - - x ? y = x + (y + x) - - x ? y = y + (x + x) - - -class of 3 equivalent candidates: - - x ? y = x + (y + y) - - x ? y = y + (x + y) - - x ? y = y + (y + x) - - -class of 2 equivalent candidates: - - x ? 0 = x - x ? y = dec x + x - - x ? 0 = x - x ? y = x + dec x - - -class of 4 equivalent candidates: - - x ? 0 = x - x ? y = dec x + y - - x ? 0 = x - x ? y = dec y + x - - x ? 0 = x - x ? y = x + dec y - - x ? 0 = x - x ? y = y + dec x - - -class of 2 equivalent candidates: - - x ? 0 = x - x ? y = dec y + y - - x ? 0 = x - x ? y = y + dec y - - -class of 2 equivalent candidates: - - x ? 0 = 0 - x ? y = dec x + x - - x ? 0 = 0 - x ? y = x + dec x - - -class of 4 equivalent candidates: - - x ? 0 = 0 - x ? y = dec x + y - - x ? 0 = 0 - x ? y = dec y + x - - x ? 0 = 0 - x ? y = x + dec y - - x ? 0 = 0 - x ? y = y + dec x - - -class of 2 equivalent candidates: - - x ? 0 = 0 - x ? y = dec y + y - - x ? 0 = 0 - x ? y = y + dec y - - -class of 2 equivalent candidates: - - x ? 0 = dec x - x ? y = x + y - - x ? 0 = dec x - x ? y = y + x - - -class of 2 equivalent candidates: - - x ? 0 = dec 0 - x ? y = x + y - - x ? 0 = dec 0 - x ? y = y + x - - -class of 2 equivalent candidates: - - x ? 0 = dec x + x - x ? y = x - - x ? 0 = x + dec x - x ? y = x - - -class of 2 equivalent candidates: - - x ? 0 = dec x + x - x ? y = y - - x ? 0 = x + dec x - x ? y = y - - -class of 2 equivalent candidates: - - x ? 0 = dec x + x - x ? y = 0 - - x ? 0 = x + dec x - x ? y = 0 - - -class of 2 equivalent candidates: - - 0 ? x = x - x ? y = dec x + x - - 0 ? x = x - x ? y = x + dec x - - -class of 4 equivalent candidates: - - 0 ? x = x - x ? y = dec x + y - - 0 ? x = x - x ? y = dec y + x - - 0 ? x = x - x ? y = x + dec y - - 0 ? x = x - x ? y = y + dec x - - -class of 2 equivalent candidates: - - 0 ? x = x - x ? y = dec y + y - - 0 ? x = x - x ? y = y + dec y - - -class of 2 equivalent candidates: - - 0 ? x = 0 - x ? y = dec x + x - - 0 ? x = 0 - x ? y = x + dec x - - -class of 4 equivalent candidates: - - 0 ? x = 0 - x ? y = dec x + y - - 0 ? x = 0 - x ? y = dec y + x - - 0 ? x = 0 - x ? y = x + dec y - - 0 ? x = 0 - x ? y = y + dec x - - -class of 2 equivalent candidates: - - 0 ? x = 0 - x ? y = dec y + y - - 0 ? x = 0 - x ? y = y + dec y - - -class of 2 equivalent candidates: - - 0 ? x = dec x - x ? y = x + y - - 0 ? x = dec x - x ? y = y + x - - -class of 2 equivalent candidates: - - 0 ? x = dec 0 - x ? y = x + y - - 0 ? x = dec 0 - x ? y = y + x - - -class of 2 equivalent candidates: - - 0 ? x = dec x + x - x ? y = x - - 0 ? x = x + dec x - x ? y = x - - -class of 2 equivalent candidates: - - 0 ? x = dec x + x - x ? y = y - - 0 ? x = x + dec x - x ? y = y - - -class of 2 equivalent candidates: - - 0 ? x = dec x + x - x ? y = 0 - - 0 ? x = x + dec x - x ? y = 0 - - -class of 2 equivalent candidates: - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = 0 - x ? y = x + y - - 0 ? 0 = 0 - 0 ? x = 0 - x ? 0 = 0 - x ? y = y + x - - - -Redundant candidates for: goo :: [Int] -> [Int] - pruning with 4/4 rules - [2,1,2,3,4] candidates - 9/12 unique candidates - 3/12 redundant candidates - -rules: -xs ++ [] == xs -[] ++ xs == xs -(xs ++ ys) ++ zs == xs ++ (ys ++ zs) -(x:xs) ++ ys == x:(xs ++ ys) - -class of 3 equivalent candidates: - - goo xs = xs - - goo [] = [] - goo (x:xs) = x:xs - - goo [] = [] - goo (x:xs) = x:goo xs - - -class of 2 equivalent candidates: - - goo xs = [] - - goo [] = [] - goo (x:xs) = goo xs - - - -Redundant candidates for: ?? :: [Int] -> [Int] -> [Int] - pruning with 4/4 rules - [3,7,13,66,152] candidates - 161/241 unique candidates - 80/241 redundant candidates - -rules: -xs ++ [] == xs -[] ++ xs == xs -(xs ++ ys) ++ zs == xs ++ (ys ++ zs) -(x:xs) ++ ys == x:(xs ++ ys) - -class of 4 equivalent candidates: - - xs ?? ys = xs - - xs ?? [] = xs - xs ?? (x:ys) = xs ?? ys - - xs ?? [] = xs - xs ?? (x:ys) = xs ?? [] - - [] ?? xs = [] - (x:xs) ?? ys = x:xs - - -class of 4 equivalent candidates: - - xs ?? ys = ys - - [] ?? xs = xs - (x:xs) ?? ys = xs ?? ys - - [] ?? xs = xs - (x:xs) ?? ys = [] ?? ys - - xs ?? [] = [] - xs ?? (x:ys) = x:ys - - -class of 29 equivalent candidates: - - xs ?? ys = [] - - xs ?? [] = [] - xs ?? (x:ys) = xs ?? ys - - xs ?? [] = [] - xs ?? (x:ys) = xs ?? [] - - xs ?? [] = [] - xs ?? (x:ys) = ys ?? xs - - xs ?? [] = [] - xs ?? (x:ys) = ys ?? ys - - xs ?? [] = [] - xs ?? (x:ys) = ys ?? [] - - xs ?? [] = [] - xs ?? (x:ys) = [] ?? xs - - xs ?? [] = [] - xs ?? (x:ys) = [] ?? ys - - [] ?? xs = [] - (x:xs) ?? ys = xs ?? xs - - [] ?? xs = [] - (x:xs) ?? ys = xs ?? ys - - [] ?? xs = [] - (x:xs) ?? ys = xs ?? [] - - [] ?? xs = [] - (x:xs) ?? ys = ys ?? xs - - [] ?? xs = [] - (x:xs) ?? ys = ys ?? [] - - [] ?? xs = [] - (x:xs) ?? ys = [] ?? xs - - [] ?? xs = [] - (x:xs) ?? ys = [] ?? ys - - [] ?? [] = [] - [] ?? (x:xs) = xs ?? xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = xs ?? [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = [] ?? xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = xs ?? xs - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = xs ?? [] - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] ?? xs - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = xs ?? xs - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = xs ?? ys - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = xs ?? [] - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = ys ?? xs - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = ys ?? ys - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = ys ?? [] - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = [] ?? xs - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = [] ?? ys - - -class of 3 equivalent candidates: - - xs ?? [] = xs - xs ?? (x:ys) = ys - - xs ?? [] = xs - xs ?? (x:ys) = ys ?? [] - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = x:xs - (x:xs) ?? (y:ys) = ys - - -class of 5 equivalent candidates: - - xs ?? [] = xs - xs ?? (x:ys) = [] - - xs ?? [] = xs - xs ?? (x:ys) = ys ?? ys - - xs ?? [] = xs - xs ?? (x:ys) = [] ?? xs - - xs ?? [] = xs - xs ?? (x:ys) = [] ?? ys - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = x:xs - (x:xs) ?? (y:ys) = [] - - -class of 2 equivalent candidates: - - xs ?? [] = [] - xs ?? (x:ys) = xs - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = x:xs - - -class of 2 equivalent candidates: - - xs ?? [] = [] - xs ?? (x:ys) = ys - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs ?? [] - (x:xs) ?? (y:ys) = ys - - -class of 5 equivalent candidates: - - [] ?? xs = xs - (x:xs) ?? ys = [] - - [] ?? xs = xs - (x:xs) ?? ys = xs ?? xs - - [] ?? xs = xs - (x:xs) ?? ys = xs ?? [] - - [] ?? xs = xs - (x:xs) ?? ys = ys ?? [] - - [] ?? [] = [] - [] ?? (x:xs) = x:xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = [] - - -class of 2 equivalent candidates: - - [] ?? xs = [] - (x:xs) ?? ys = xs - - [] ?? [] = [] - [] ?? (x:xs) = [] ?? xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = xs - - -class of 2 equivalent candidates: - - [] ?? xs = [] - (x:xs) ?? ys = ys - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = y:ys - - -class of 3 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = xs ?? xs - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = ys ?? ys - - -class of 2 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = xs - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs ?? [] - (x:xs) ?? (y:ys) = xs - - -class of 7 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs ?? xs - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs ?? [] - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = xs ?? xs - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = xs ?? [] - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = ys ?? ys - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = ys ?? [] - - -class of 2 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = ys - - [] ?? [] = [] - [] ?? (x:xs) = [] ?? xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = ys - - -class of 7 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = xs ?? xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = [] ?? xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = [] - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = xs ?? xs - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = ys ?? ys - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = [] ?? xs - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = [] ?? ys - - -class of 5 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = xs - - [] ?? [] = [] - [] ?? (x:xs) = xs ?? [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = xs - - [] ?? [] = [] - [] ?? (x:xs) = [] ?? xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = xs - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = xs ?? [] - (x:xs) ?? (y:ys) = xs - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] ?? xs - (x:xs) ?? (y:ys) = xs - - -class of 5 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = ys - - [] ?? [] = [] - [] ?? (x:xs) = xs ?? [] - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = ys - - [] ?? [] = [] - [] ?? (x:xs) = [] ?? xs - (x:xs) ?? [] = [] - (x:xs) ?? (y:ys) = ys - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = xs ?? [] - (x:xs) ?? (y:ys) = ys - - [] ?? [] = [] - [] ?? (x:xs) = [] - (x:xs) ?? [] = [] ?? xs - (x:xs) ?? (y:ys) = ys - - -class of 2 equivalent candidates: - - [] ?? xs = xs - (x:xs) ?? ys = [] ?? xs - - [] ?? [] = [] - [] ?? (x:xs) = x:xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = xs - - -class of 2 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = xs ?? ys - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = ys ?? xs - - -class of 2 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = xs ?? [] - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = [] ?? xs - - -class of 2 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = ys ?? [] - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = [] ?? ys - - -class of 2 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs ?? xs - (x:xs) ?? (y:ys) = xs - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = [] ?? xs - (x:xs) ?? (y:ys) = xs - - -class of 2 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = xs ?? xs - (x:xs) ?? (y:ys) = ys - - [] ?? [] = [] - [] ?? (x:xs) = xs - (x:xs) ?? [] = [] ?? xs - (x:xs) ?? (y:ys) = ys - - -class of 2 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = xs ?? xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = xs - - [] ?? [] = [] - [] ?? (x:xs) = xs ?? [] - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = xs - - -class of 2 equivalent candidates: - - [] ?? [] = [] - [] ?? (x:xs) = xs ?? xs - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = ys - - [] ?? [] = [] - [] ?? (x:xs) = xs ?? [] - (x:xs) ?? [] = xs - (x:xs) ?? (y:ys) = ys - - - -Redundant candidates for: ton :: Bool -> Bool - pruning with 39/49 rules - [3,3,0,0,0] candidates - 4/6 unique candidates - 2/6 redundant candidates - -rules: -not False == True -not True == False -p && p == p -p || p == p -not (not p) == p -p && False == False -p && True == p -False && p == False -True && p == p -p || False == p -p || True == True -False || p == p -True || p == True -not (p && q) == not p || not q -not (p && q) == not q || not p -not (p || q) == not p && not q -not (p || q) == not q && not p -p && not p == False -not p && p == False -p || not p == True -not p || p == True -(p && q) && r == p && (q && r) -(p && q) && r == q && (p && r) -(p || q) || r == p || (q || r) -(p || q) || r == q || (p || r) -p && (p && q) == p && q -p && (q && p) == p && q -p && (q && p) == q && p -p || (p || q) == p || q -p || (q || p) == p || q -p || (q || p) == q || p -p && (p || q) == p -p && (q || p) == p -(p || q) && p == p -(p || q) && q == q -p || p && q == p -p || q && p == p -p && q || p == p -p && q || q == q -equations: -q && p == p && q -q || p == p || q -q && (p && r) == p && (q && r) -r && (p && q) == p && (q && r) -r && (q && p) == p && (q && r) -q || (p || r) == p || (q || r) -r || (p || q) == p || (q || r) -r || (q || p) == p || (q || r) -(r || q) && p == p && (q || r) -r && q || p == p || q && r - -class of 2 equivalent candidates: - - ton p = p - - ton False = False - ton True = True - - -class of 2 equivalent candidates: - - ton p = not p - - ton False = True - ton True = False - - - -Redundant candidates for: &| :: Bool -> Bool -> Bool - pruning with 39/49 rules - [4,14,26,8,4] candidates - 16/56 unique candidates - 40/56 redundant candidates - -rules: -not False == True -not True == False -p && p == p -p || p == p -not (not p) == p -p && False == False -p && True == p -False && p == False -True && p == p -p || False == p -p || True == True -False || p == p -True || p == True -not (p && q) == not p || not q -not (p && q) == not q || not p -not (p || q) == not p && not q -not (p || q) == not q && not p -p && not p == False -not p && p == False -p || not p == True -not p || p == True -(p && q) && r == p && (q && r) -(p && q) && r == q && (p && r) -(p || q) || r == p || (q || r) -(p || q) || r == q || (p || r) -p && (p && q) == p && q -p && (q && p) == p && q -p && (q && p) == q && p -p || (p || q) == p || q -p || (q || p) == p || q -p || (q || p) == q || p -p && (p || q) == p -p && (q || p) == p -(p || q) && p == p -(p || q) && q == q -p || p && q == p -p || q && p == p -p && q || p == p -p && q || q == q -equations: -q && p == p && q -q || p == p || q -q && (p && r) == p && (q && r) -r && (p && q) == p && (q && r) -r && (q && p) == p && (q && r) -q || (p || r) == p || (q || r) -r || (p || q) == p || (q || r) -r || (q || p) == p || (q || r) -(r || q) && p == p && (q || r) -r && q || p == p || q && r - -class of 2 equivalent candidates: - - p &| q = p - - False &| p = False - True &| p = True - - -class of 2 equivalent candidates: - - p &| q = q - - p &| False = False - p &| True = True - - -class of 2 equivalent candidates: - - p &| q = not p - - False &| p = True - True &| p = False - - -class of 2 equivalent candidates: - - p &| q = not q - - p &| False = True - p &| True = False - - -class of 5 equivalent candidates: - - p &| False = p - p &| True = False - - False &| p = False - True &| p = not p - - False &| False = False - False &| True = False - True &| False = True - True &| True = False - - p &| q = not q && p - - p &| q = p && not q - - -class of 5 equivalent candidates: - - p &| False = p - p &| True = True - - False &| p = p - True &| p = True - - p &| q = p || q - - p &| q = q || p - - False &| False = False - False &| True = True - True &| False = True - True &| True = True - - -class of 5 equivalent candidates: - - p &| False = False - p &| True = p - - False &| p = False - True &| p = p - - p &| q = p && q - - p &| q = q && p - - False &| False = False - False &| True = False - True &| False = False - True &| True = True - - -class of 5 equivalent candidates: - - p &| False = True - p &| True = p - - False &| p = not p - True &| p = True - - False &| False = True - False &| True = False - True &| False = True - True &| True = True - - p &| q = not q || p - - p &| q = p || not q - - -class of 5 equivalent candidates: - - False &| p = p - True &| p = False - - p &| False = False - p &| True = not p - - False &| False = False - False &| True = True - True &| False = False - True &| True = False - - p &| q = not p && q - - p &| q = q && not p - - -class of 5 equivalent candidates: - - False &| p = True - True &| p = p - - p &| False = not p - p &| True = True - - False &| False = True - False &| True = True - True &| False = False - True &| True = True - - p &| q = not p || q - - p &| q = q || not p - - -class of 3 equivalent candidates: - - p &| False = p - p &| True = not p - - False &| p = p - True &| p = not p - - False &| False = False - False &| True = True - True &| False = True - True &| True = False - - -class of 5 equivalent candidates: - - p &| False = True - p &| True = not p - - False &| p = True - True &| p = not p - - False &| False = True - False &| True = True - True &| False = True - True &| True = False - - p &| q = not p || not q - - p &| q = not q || not p - - -class of 3 equivalent candidates: - - p &| False = not p - p &| True = p - - False &| p = not p - True &| p = p - - False &| False = True - False &| True = False - True &| False = False - True &| True = True - - -class of 5 equivalent candidates: - - p &| False = not p - p &| True = False - - False &| p = not p - True &| p = False - - False &| False = True - False &| True = False - True &| False = False - True &| True = False - - p &| q = not p && not q - - p &| q = not q && not p + [3,3,8,13,29] candidates + 46/56 unique candidates + 10/56 redundant candidates + +rules: +x * y == x + y +x * y == y + x +x - x == 0 +x + 0 == x +0 + x == x +x - 0 == x +(x + y) + z == x + (y + z) +(x + y) + z == y + (x + z) +x - (y - z) == z + (x - y) +(x - y) - z == x - (y + z) +(x - y) - z == x - (z + y) +(x + y) - z == x + (y - z) +(x + y) - z == y + (x - z) +x + (y - x) == y +(x - y) + y == x +equations: +y + x == x + y +y + (x + z) == x + (y + z) +z + (x + y) == x + (y + z) +z + (y + x) == x + (y + z) +y + (x - z) == x + (y - z) +(x - z) + y == x + (y - z) +(z - y) + x == (x - y) + z +y - (x + y) == 0 - x +y - (y + x) == 0 - x +z - (y + z) == x - (x + y) +z - (y + z) == x - (y + x) +z - (z + y) == x - (x + y) +z - (z + y) == x - (y + x) +x + (0 - y) == x - y +(0 - y) + x == x - y +x - (x + 1) == 0 - 1 +x - (1 + x) == 0 - 1 +y - (y + 1) == x - (x + 1) +y - (y + 1) == x - (1 + x) +y - (1 + y) == x - (1 + x) + +class of 2 equivalent candidates: + + foo x = x + x + + foo 0 = 0 + foo x = x + x + + +class of 2 equivalent candidates: + + foo x = x + 1 + + foo 0 = 1 + foo x = x + 1 + + +class of 2 equivalent candidates: + + foo x = x - 1 + + foo 1 = 0 + foo x = x - 1 + + +class of 4 equivalent candidates: + + foo x = 0 - x + + foo 0 = 0 + foo x = 0 - x + + foo x = x - (x + x) + + foo x = 1 - (x + 1) + + +class of 4 equivalent candidates: + + foo x = 1 - x + + foo 0 = 1 + foo x = 1 - x + + foo x = 1 + (0 - x) + + foo 1 = 0 + foo x = 1 - x + + +class of 2 equivalent candidates: + + foo x = x + (x + 1) + + foo x = 1 + (x + x) + + + +Redundant candidates for: ? :: Int -> Int -> Int + pruning with 10/23 rules + [3,7,22,53,129] candidates + 192/214 unique candidates + 22/214 redundant candidates + +rules: +x * y == x + y +x * y == y + x +x + 0 == x +0 + x == x +dec (x + y) == x + dec y +dec (x + y) == y + dec x +dec (x + y) == dec x + y +dec (x + y) == dec y + x +(x + y) + z == x + (y + z) +(x + y) + z == y + (x + z) +equations: +y + x == x + y +y + dec x == x + dec y +dec x + y == x + dec y +dec y + x == dec x + y +x + dec 0 == dec x +dec 0 + x == dec x +y + (x + z) == x + (y + z) +z + (x + y) == x + (y + z) +z + (y + x) == x + (y + z) +y + dec (dec x) == x + dec (dec y) +dec (dec x) + y == x + dec (dec y) +x + dec (dec 0) == dec (dec x) +dec (dec 0) + x == dec (dec x) + +class of 2 equivalent candidates: + + x ? y = x + x + + 0 ? x = 0 + x ? y = x + x + + +class of 4 equivalent candidates: + + x ? y = x + y + + x ? 0 = x + x ? y = x + y + + 0 ? x = x + x ? y = x + y + + 0 ? x = x + x ? 0 = x + x ? y = x + y + + +class of 2 equivalent candidates: + + x ? y = y + y + + x ? 0 = 0 + x ? y = y + y + + +class of 2 equivalent candidates: + + x ? y = x + dec y + + x ? y = y + dec x + + +class of 2 equivalent candidates: + + x ? 0 = x + x ? y = x + x + + 0 ? x = 0 + x ? 0 = x + x ? y = x + x + + +class of 2 equivalent candidates: + + x ? 0 = 0 + x ? y = x + x + + 0 ? x = 0 + x ? 0 = 0 + x ? y = x + x + + +class of 2 equivalent candidates: + + x ? 0 = 0 + x ? y = x + y + + 0 ? x = x + x ? 0 = 0 + x ? y = x + y + + +class of 2 equivalent candidates: + + 0 ? x = x + x ? y = y + y + + 0 ? x = x + x ? 0 = 0 + x ? y = y + y + + +class of 2 equivalent candidates: + + 0 ? x = 0 + x ? y = x + y + + 0 ? x = 0 + x ? 0 = x + x ? y = x + y + + +class of 2 equivalent candidates: + + 0 ? x = 0 + x ? y = y + y + + 0 ? x = 0 + x ? 0 = 0 + x ? y = y + y + + +class of 2 equivalent candidates: + + x ? y = x + (x + y) + + x ? y = y + (x + x) + + +class of 2 equivalent candidates: + + x ? y = x + (y + y) + + x ? y = y + (x + y) + + +class of 2 equivalent candidates: + + x ? y = x + dec (dec x) + + x ? y = dec x + dec x + + +class of 3 equivalent candidates: + + x ? y = x + dec (dec y) + + x ? y = y + dec (dec x) + + x ? y = dec x + dec y + + +class of 2 equivalent candidates: + + x ? y = y + dec (dec y) + + x ? y = dec y + dec y + + +class of 2 equivalent candidates: + + x ? 0 = x + x ? y = x + dec y + + x ? 0 = x + x ? y = y + dec x + + +class of 2 equivalent candidates: + + x ? 0 = 0 + x ? y = x + dec y + + x ? 0 = 0 + x ? y = y + dec x + + +class of 2 equivalent candidates: + + 0 ? x = x + x ? y = x + dec y + + 0 ? x = x + x ? y = y + dec x + + +class of 2 equivalent candidates: + + 0 ? x = 0 + x ? y = x + dec y + + 0 ? x = 0 + x ? y = y + dec x + + + +Redundant candidates for: goo :: [Int] -> [Int] + pruning with 4/4 rules + [2,1,2,2,4] candidates + 9/11 unique candidates + 2/11 redundant candidates + +rules: +xs ++ [] == xs +[] ++ xs == xs +(xs ++ ys) ++ zs == xs ++ (ys ++ zs) +(x:xs) ++ ys == x:(xs ++ ys) + +class of 2 equivalent candidates: + + goo xs = xs + + goo [] = [] + goo (x:xs) = x:goo xs + + +class of 2 equivalent candidates: + + goo xs = [] + + goo [] = [] + goo (x:xs) = goo xs + + + +Redundant candidates for: ?? :: [Int] -> [Int] -> [Int] + pruning with 4/4 rules + [3,7,15,57,134] candidates + 160/216 unique candidates + 56/216 redundant candidates + +rules: +xs ++ [] == xs +[] ++ xs == xs +(xs ++ ys) ++ zs == xs ++ (ys ++ zs) +(x:xs) ++ ys == x:(xs ++ ys) + +class of 2 equivalent candidates: + + xs ?? ys = xs + + xs ?? [] = xs + xs ?? (x:ys) = xs ?? ys + + +class of 2 equivalent candidates: + + xs ?? ys = ys + + [] ?? xs = xs + (x:xs) ?? ys = xs ?? ys + + +class of 22 equivalent candidates: + + xs ?? ys = [] + + xs ?? [] = [] + xs ?? (x:ys) = xs ?? ys + + xs ?? [] = [] + xs ?? (x:ys) = ys ?? xs + + xs ?? [] = [] + xs ?? (x:ys) = ys ?? ys + + xs ?? [] = [] + xs ?? (x:ys) = [] ?? xs + + xs ?? [] = [] + xs ?? (x:ys) = [] ?? ys + + [] ?? xs = [] + (x:xs) ?? ys = xs ?? xs + + [] ?? xs = [] + (x:xs) ?? ys = xs ?? ys + + [] ?? xs = [] + (x:xs) ?? ys = xs ?? [] + + [] ?? xs = [] + (x:xs) ?? ys = ys ?? xs + + [] ?? xs = [] + (x:xs) ?? ys = ys ?? [] + + [] ?? xs = [] + (x:xs) ?? [] = xs ?? xs + (x:xs) ?? (y:ys) = [] + + [] ?? xs = [] + (x:xs) ?? [] = xs ?? [] + (x:xs) ?? (y:ys) = [] + + [] ?? xs = [] + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = xs ?? xs + + [] ?? xs = [] + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = xs ?? ys + + [] ?? xs = [] + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = xs ?? [] + + [] ?? xs = [] + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = ys ?? xs + + [] ?? xs = [] + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = ys ?? ys + + [] ?? xs = [] + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = ys ?? [] + + [] ?? [] = [] + [] ?? (x:xs) = xs ?? xs + (x:xs) ?? ys = [] + + [] ?? [] = [] + [] ?? (x:xs) = xs ?? [] + (x:xs) ?? ys = [] + + [] ?? [] = [] + [] ?? (x:xs) = [] ?? xs + (x:xs) ?? ys = [] + + +class of 5 equivalent candidates: + + xs ?? [] = xs + xs ?? (x:ys) = [] + + xs ?? [] = xs + xs ?? (x:ys) = ys ?? ys + + xs ?? [] = xs + xs ?? (x:ys) = [] ?? xs + + xs ?? [] = xs + xs ?? (x:ys) = [] ?? ys + + [] ?? xs = [] + (x:xs) ?? [] = x:xs + (x:xs) ?? (y:ys) = [] + + +class of 2 equivalent candidates: + + xs ?? [] = [] + xs ?? (x:ys) = xs + + [] ?? xs = [] + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = x:xs + + +class of 3 equivalent candidates: + + xs ?? [] = [] + xs ?? (x:ys) = ys + + [] ?? [] = [] + [] ?? (x:xs) = xs + (x:xs) ?? ys = xs ?? ys + + [] ?? [] = [] + [] ?? (x:xs) = xs + (x:xs) ?? ys = [] ?? ys + + +class of 10 equivalent candidates: + + [] ?? xs = xs + (x:xs) ?? ys = [] + + [] ?? xs = xs + (x:xs) ?? ys = xs ?? xs + + [] ?? xs = xs + (x:xs) ?? ys = xs ?? [] + + [] ?? xs = xs + (x:xs) ?? ys = ys ?? [] + + [] ?? xs = xs + (x:xs) ?? [] = xs ?? xs + (x:xs) ?? (y:ys) = [] + + [] ?? xs = xs + (x:xs) ?? [] = xs ?? [] + (x:xs) ?? (y:ys) = [] + + [] ?? xs = xs + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = xs ?? xs + + [] ?? xs = xs + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = xs ?? [] + + [] ?? xs = xs + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = ys ?? ys + + [] ?? xs = xs + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = ys ?? [] + + +class of 2 equivalent candidates: + + [] ?? xs = [] + (x:xs) ?? ys = xs + + [] ?? [] = [] + [] ?? (x:xs) = [] ?? xs + (x:xs) ?? ys = xs + + +class of 3 equivalent candidates: + + [] ?? xs = [] + (x:xs) ?? ys = ys + + [] ?? [] = [] + [] ?? (x:xs) = xs ?? [] + (x:xs) ?? ys = ys + + [] ?? [] = [] + [] ?? (x:xs) = [] ?? xs + (x:xs) ?? ys = ys + + +class of 3 equivalent candidates: + + [] ?? xs = xs + (x:xs) ?? [] = xs + (x:xs) ?? (y:ys) = [] + + [] ?? xs = xs + (x:xs) ?? [] = xs + (x:xs) ?? (y:ys) = xs ?? xs + + [] ?? xs = xs + (x:xs) ?? [] = xs + (x:xs) ?? (y:ys) = ys ?? ys + + +class of 2 equivalent candidates: + + [] ?? xs = xs + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = xs + + [] ?? xs = xs + (x:xs) ?? [] = xs ?? [] + (x:xs) ?? (y:ys) = xs + + +class of 2 equivalent candidates: + + [] ?? xs = xs + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = ys + + [] ?? xs = xs + (x:xs) ?? [] = xs ?? [] + (x:xs) ?? (y:ys) = ys + + +class of 3 equivalent candidates: + + [] ?? xs = [] + (x:xs) ?? [] = xs + (x:xs) ?? (y:ys) = [] + + [] ?? xs = [] + (x:xs) ?? [] = xs + (x:xs) ?? (y:ys) = xs ?? xs + + [] ?? xs = [] + (x:xs) ?? [] = xs + (x:xs) ?? (y:ys) = ys ?? ys + + +class of 2 equivalent candidates: + + [] ?? xs = [] + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = xs + + [] ?? xs = [] + (x:xs) ?? [] = xs ?? [] + (x:xs) ?? (y:ys) = xs + + +class of 2 equivalent candidates: + + [] ?? xs = [] + (x:xs) ?? [] = [] + (x:xs) ?? (y:ys) = ys + + [] ?? xs = [] + (x:xs) ?? [] = xs ?? [] + (x:xs) ?? (y:ys) = ys + + +class of 2 equivalent candidates: + + [] ?? [] = [] + [] ?? (x:xs) = xs + (x:xs) ?? ys = ys + + [] ?? [] = [] + [] ?? (x:xs) = xs ?? xs + (x:xs) ?? ys = ys + + +class of 4 equivalent candidates: + + [] ?? [] = [] + [] ?? (x:xs) = xs + (x:xs) ?? ys = [] + + [] ?? [] = [] + [] ?? (x:xs) = xs + (x:xs) ?? ys = xs ?? xs + + [] ?? [] = [] + [] ?? (x:xs) = xs + (x:xs) ?? ys = xs ?? [] + + [] ?? [] = [] + [] ?? (x:xs) = xs + (x:xs) ?? ys = ys ?? [] + + +class of 2 equivalent candidates: + + [] ?? xs = xs + (x:xs) ?? ys = ys ?? xs + + [] ?? xs = xs + (x:xs) ?? [] = xs + (x:xs) ?? (y:ys) = xs ?? ys + + +class of 2 equivalent candidates: + + [] ?? [] = [] + [] ?? (x:xs) = xs ?? xs + (x:xs) ?? ys = xs + + [] ?? [] = [] + [] ?? (x:xs) = xs ?? [] + (x:xs) ?? ys = xs + + + +Redundant candidates for: ton :: Bool -> Bool + pruning with 39/49 rules + [3,2,0,0,0] candidates + 4/5 unique candidates + 1/5 redundant candidates + +rules: +not False == True +not True == False +p && p == p +p || p == p +not (not p) == p +p && False == False +p && True == p +False && p == False +True && p == p +p || False == p +p || True == True +False || p == p +True || p == True +not (p && q) == not p || not q +not (p && q) == not q || not p +not (p || q) == not p && not q +not (p || q) == not q && not p +p && not p == False +not p && p == False +p || not p == True +not p || p == True +(p && q) && r == p && (q && r) +(p && q) && r == q && (p && r) +(p || q) || r == p || (q || r) +(p || q) || r == q || (p || r) +p && (p && q) == p && q +p && (q && p) == p && q +p && (q && p) == q && p +p || (p || q) == p || q +p || (q || p) == p || q +p || (q || p) == q || p +p && (p || q) == p +p && (q || p) == p +(p || q) && p == p +(p || q) && q == q +p || p && q == p +p || q && p == p +p && q || p == p +p && q || q == q +equations: +q && p == p && q +q || p == p || q +q && (p && r) == p && (q && r) +r && (p && q) == p && (q && r) +r && (q && p) == p && (q && r) +q || (p || r) == p || (q || r) +r || (p || q) == p || (q || r) +r || (q || p) == p || (q || r) +(r || q) && p == p && (q || r) +r && q || p == p || q && r + +class of 2 equivalent candidates: + + ton p = not p + + ton False = True + ton True = False + + + +Redundant candidates for: &| :: Bool -> Bool -> Bool + pruning with 39/49 rules + [4,12,20,6,2] candidates + 16/44 unique candidates + 28/44 redundant candidates + +rules: +not False == True +not True == False +p && p == p +p || p == p +not (not p) == p +p && False == False +p && True == p +False && p == False +True && p == p +p || False == p +p || True == True +False || p == p +True || p == True +not (p && q) == not p || not q +not (p && q) == not q || not p +not (p || q) == not p && not q +not (p || q) == not q && not p +p && not p == False +not p && p == False +p || not p == True +not p || p == True +(p && q) && r == p && (q && r) +(p && q) && r == q && (p && r) +(p || q) || r == p || (q || r) +(p || q) || r == q || (p || r) +p && (p && q) == p && q +p && (q && p) == p && q +p && (q && p) == q && p +p || (p || q) == p || q +p || (q || p) == p || q +p || (q || p) == q || p +p && (p || q) == p +p && (q || p) == p +(p || q) && p == p +(p || q) && q == q +p || p && q == p +p || q && p == p +p && q || p == p +p && q || q == q +equations: +q && p == p && q +q || p == p || q +q && (p && r) == p && (q && r) +r && (p && q) == p && (q && r) +r && (q && p) == p && (q && r) +q || (p || r) == p || (q || r) +r || (p || q) == p || (q || r) +r || (q || p) == p || (q || r) +(r || q) && p == p && (q || r) +r && q || p == p || q && r + +class of 2 equivalent candidates: + + p &| q = not p + + False &| p = True + True &| p = False + + +class of 4 equivalent candidates: + + p &| q = not q + + p &| False = True + p &| True = False + + False &| p = not p + True &| False = True + True &| True = False + + False &| False = True + False &| True = False + True &| p = not p + + +class of 4 equivalent candidates: + + p &| False = p + p &| True = False + + False &| p = False + True &| p = not p + + False &| p = False + True &| False = True + True &| True = False + + p &| q = p && not q + + +class of 3 equivalent candidates: + + p &| False = p + p &| True = True + + False &| p = p + True &| p = True + + p &| q = p || q + + +class of 3 equivalent candidates: + + p &| False = False + p &| True = p + + False &| p = False + True &| p = p + + p &| q = p && q + + +class of 4 equivalent candidates: + + p &| False = True + p &| True = p + + False &| p = not p + True &| p = True + + False &| False = True + False &| True = False + True &| p = True + + p &| q = p || not q + + +class of 3 equivalent candidates: + + False &| p = p + True &| p = False + + p &| False = False + p &| True = not p + + p &| q = q && not p + + +class of 3 equivalent candidates: + + False &| p = True + True &| p = p + + p &| False = not p + p &| True = True + + p &| q = q || not p + + +class of 3 equivalent candidates: + + p &| False = p + p &| True = not p + + False &| p = p + True &| p = not p + + False &| p = p + True &| False = True + True &| True = False + + +class of 4 equivalent candidates: + + p &| False = True + p &| True = not p + + False &| p = True + True &| p = not p + + False &| p = True + True &| False = True + True &| True = False + + p &| q = not p || not q + + +class of 3 equivalent candidates: + + p &| False = not p + p &| True = p + + False &| p = not p + True &| p = p + + False &| False = True + False &| True = False + True &| p = p + + +class of 4 equivalent candidates: + + p &| False = not p + p &| True = False + + False &| p = not p + True &| p = False + + False &| False = True + False &| True = False + True &| p = False + + p &| q = not p && not q diff --git a/bench/runtime/lapmatrud/bench/candidates.runtime b/bench/runtime/lapmatrud/bench/candidates.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/candidates.runtime @@ -0,0 +1,1 @@ +8.0 diff --git a/bench/runtime/lapmatrud/bench/gps.runtime b/bench/runtime/lapmatrud/bench/gps.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/gps.runtime @@ -0,0 +1,1 @@ +15.1 diff --git a/bench/runtime/lapmatrud/bench/gps2.runtime b/bench/runtime/lapmatrud/bench/gps2.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/gps2.runtime @@ -0,0 +1,1 @@ +12.3 diff --git a/bench/runtime/lapmatrud/bench/ill-hit.runtime b/bench/runtime/lapmatrud/bench/ill-hit.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/ill-hit.runtime @@ -0,0 +1,1 @@ +0.5 diff --git a/bench/runtime/lapmatrud/bench/longshot.runtime b/bench/runtime/lapmatrud/bench/longshot.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/longshot.runtime @@ -0,0 +1,1 @@ +0.0 diff --git a/bench/runtime/lapmatrud/bench/lowtests.runtime b/bench/runtime/lapmatrud/bench/lowtests.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/lowtests.runtime @@ -0,0 +1,1 @@ +0.2 diff --git a/bench/runtime/lapmatrud/bench/p12.runtime b/bench/runtime/lapmatrud/bench/p12.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/p12.runtime @@ -0,0 +1,1 @@ +1.7 diff --git a/bench/runtime/lapmatrud/bench/p30.runtime b/bench/runtime/lapmatrud/bench/p30.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/p30.runtime @@ -0,0 +1,1 @@ +0.0 diff --git a/bench/runtime/lapmatrud/bench/redundants.runtime b/bench/runtime/lapmatrud/bench/redundants.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/redundants.runtime @@ -0,0 +1,1 @@ +1.9 diff --git a/bench/runtime/lapmatrud/bench/self.runtime b/bench/runtime/lapmatrud/bench/self.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/self.runtime @@ -0,0 +1,1 @@ +0.0 diff --git a/bench/runtime/lapmatrud/bench/take-drop.runtime b/bench/runtime/lapmatrud/bench/take-drop.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/take-drop.runtime @@ -0,0 +1,1 @@ +0.3 diff --git a/bench/runtime/lapmatrud/bench/terpret.runtime b/bench/runtime/lapmatrud/bench/terpret.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/terpret.runtime @@ -0,0 +1,1 @@ +7.2 diff --git a/bench/runtime/lapmatrud/bench/weird.runtime b/bench/runtime/lapmatrud/bench/weird.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/bench/weird.runtime @@ -0,0 +1,1 @@ +2.8 diff --git a/bench/runtime/lapmatrud/eg/arith.runtime b/bench/runtime/lapmatrud/eg/arith.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/arith.runtime @@ -0,0 +1,1 @@ +0.9 diff --git a/bench/runtime/lapmatrud/eg/bools.runtime b/bench/runtime/lapmatrud/eg/bools.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/bools.runtime @@ -0,0 +1,1 @@ +1.8 diff --git a/bench/runtime/lapmatrud/eg/bst.runtime b/bench/runtime/lapmatrud/eg/bst.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/bst.runtime @@ -0,0 +1,1 @@ +7.9 diff --git a/bench/runtime/lapmatrud/eg/count.runtime b/bench/runtime/lapmatrud/eg/count.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/count.runtime @@ -0,0 +1,1 @@ +0.5 diff --git a/bench/runtime/lapmatrud/eg/dupos.runtime b/bench/runtime/lapmatrud/eg/dupos.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/dupos.runtime @@ -0,0 +1,1 @@ +3.6 diff --git a/bench/runtime/lapmatrud/eg/factorial.runtime b/bench/runtime/lapmatrud/eg/factorial.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/factorial.runtime @@ -0,0 +1,1 @@ +1.3 diff --git a/bench/runtime/lapmatrud/eg/fib01.runtime b/bench/runtime/lapmatrud/eg/fib01.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/fib01.runtime @@ -0,0 +1,1 @@ +2.4 diff --git a/bench/runtime/lapmatrud/eg/fibonacci.runtime b/bench/runtime/lapmatrud/eg/fibonacci.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/fibonacci.runtime @@ -0,0 +1,1 @@ +3.7 diff --git a/bench/runtime/lapmatrud/eg/gcd.runtime b/bench/runtime/lapmatrud/eg/gcd.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/gcd.runtime @@ -0,0 +1,1 @@ +0.1 diff --git a/bench/runtime/lapmatrud/eg/higher.runtime b/bench/runtime/lapmatrud/eg/higher.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/higher.runtime @@ -0,0 +1,1 @@ +0.6 diff --git a/bench/runtime/lapmatrud/eg/id.runtime b/bench/runtime/lapmatrud/eg/id.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/id.runtime @@ -0,0 +1,1 @@ +0.0 diff --git a/bench/runtime/lapmatrud/eg/ints.runtime b/bench/runtime/lapmatrud/eg/ints.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/ints.runtime @@ -0,0 +1,1 @@ +0.5 diff --git a/bench/runtime/lapmatrud/eg/list.runtime b/bench/runtime/lapmatrud/eg/list.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/list.runtime @@ -0,0 +1,1 @@ +1.0 diff --git a/bench/runtime/lapmatrud/eg/pow.runtime b/bench/runtime/lapmatrud/eg/pow.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/pow.runtime @@ -0,0 +1,1 @@ +4.7 diff --git a/bench/runtime/lapmatrud/eg/replicate.runtime b/bench/runtime/lapmatrud/eg/replicate.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/replicate.runtime @@ -0,0 +1,1 @@ +0.3 diff --git a/bench/runtime/lapmatrud/eg/setelem.runtime b/bench/runtime/lapmatrud/eg/setelem.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/setelem.runtime @@ -0,0 +1,1 @@ +1.1 diff --git a/bench/runtime/lapmatrud/eg/sort.runtime b/bench/runtime/lapmatrud/eg/sort.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/sort.runtime @@ -0,0 +1,1 @@ +1.1 diff --git a/bench/runtime/lapmatrud/eg/spec.runtime b/bench/runtime/lapmatrud/eg/spec.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/spec.runtime @@ -0,0 +1,1 @@ +0.6 diff --git a/bench/runtime/lapmatrud/eg/subset.runtime b/bench/runtime/lapmatrud/eg/subset.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/subset.runtime @@ -0,0 +1,1 @@ +0.5 diff --git a/bench/runtime/lapmatrud/eg/tapps.runtime b/bench/runtime/lapmatrud/eg/tapps.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/tapps.runtime @@ -0,0 +1,1 @@ +0.4 diff --git a/bench/runtime/lapmatrud/eg/tree.runtime b/bench/runtime/lapmatrud/eg/tree.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/eg/tree.runtime @@ -0,0 +1,1 @@ +1.2 diff --git a/bench/runtime/lapmatrud/proto/u-conjure.runtime b/bench/runtime/lapmatrud/proto/u-conjure.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/proto/u-conjure.runtime @@ -0,0 +1,1 @@ +0.2 diff --git a/bench/runtime/lapmatrud/versions b/bench/runtime/lapmatrud/versions new file mode 100644 --- /dev/null +++ b/bench/runtime/lapmatrud/versions @@ -0,0 +1,4 @@ +GHC 9.0.2 +leancheck-1.0.0 +express-1.0.12 +speculate-0.4.14 diff --git a/bench/runtime/zero/bench/candidates.runtime b/bench/runtime/zero/bench/candidates.runtime --- a/bench/runtime/zero/bench/candidates.runtime +++ b/bench/runtime/zero/bench/candidates.runtime @@ -1,1 +1,1 @@ -10.9 +10.0 diff --git a/bench/runtime/zero/bench/gps.runtime b/bench/runtime/zero/bench/gps.runtime --- a/bench/runtime/zero/bench/gps.runtime +++ b/bench/runtime/zero/bench/gps.runtime @@ -1,1 +1,1 @@ -26.2 +28.4 diff --git a/bench/runtime/zero/bench/gps2.runtime b/bench/runtime/zero/bench/gps2.runtime --- a/bench/runtime/zero/bench/gps2.runtime +++ b/bench/runtime/zero/bench/gps2.runtime @@ -1,1 +1,1 @@ -9.3 +22.6 diff --git a/bench/runtime/zero/bench/p12.runtime b/bench/runtime/zero/bench/p12.runtime --- a/bench/runtime/zero/bench/p12.runtime +++ b/bench/runtime/zero/bench/p12.runtime @@ -1,1 +1,1 @@ -3.0 +3.1 diff --git a/bench/runtime/zero/bench/redundants.runtime b/bench/runtime/zero/bench/redundants.runtime --- a/bench/runtime/zero/bench/redundants.runtime +++ b/bench/runtime/zero/bench/redundants.runtime @@ -1,1 +1,1 @@ -3.7 +2.7 diff --git a/bench/runtime/zero/bench/self.runtime b/bench/runtime/zero/bench/self.runtime --- a/bench/runtime/zero/bench/self.runtime +++ b/bench/runtime/zero/bench/self.runtime @@ -1,1 +1,1 @@ -0.0 +0.1 diff --git a/bench/runtime/zero/bench/terpret.runtime b/bench/runtime/zero/bench/terpret.runtime --- a/bench/runtime/zero/bench/terpret.runtime +++ b/bench/runtime/zero/bench/terpret.runtime @@ -1,1 +1,1 @@ -12.0 +15.1 diff --git a/bench/runtime/zero/bench/weird.runtime b/bench/runtime/zero/bench/weird.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/zero/bench/weird.runtime @@ -0,0 +1,1 @@ +5.1 diff --git a/bench/runtime/zero/eg/bst.runtime b/bench/runtime/zero/eg/bst.runtime --- a/bench/runtime/zero/eg/bst.runtime +++ b/bench/runtime/zero/eg/bst.runtime @@ -1,1 +1,1 @@ -6.6 +13.9 diff --git a/bench/runtime/zero/eg/count.runtime b/bench/runtime/zero/eg/count.runtime --- a/bench/runtime/zero/eg/count.runtime +++ b/bench/runtime/zero/eg/count.runtime @@ -1,1 +1,1 @@ -0.6 +0.7 diff --git a/bench/runtime/zero/eg/dupos.runtime b/bench/runtime/zero/eg/dupos.runtime --- a/bench/runtime/zero/eg/dupos.runtime +++ b/bench/runtime/zero/eg/dupos.runtime @@ -1,1 +1,1 @@ -4.0 +7.4 diff --git a/bench/runtime/zero/eg/fib01.runtime b/bench/runtime/zero/eg/fib01.runtime --- a/bench/runtime/zero/eg/fib01.runtime +++ b/bench/runtime/zero/eg/fib01.runtime @@ -1,1 +1,1 @@ -5.7 +2.3 diff --git a/bench/runtime/zero/eg/fibonacci.runtime b/bench/runtime/zero/eg/fibonacci.runtime --- a/bench/runtime/zero/eg/fibonacci.runtime +++ b/bench/runtime/zero/eg/fibonacci.runtime @@ -1,1 +1,1 @@ -4.7 +8.2 diff --git a/bench/runtime/zero/eg/gcd.runtime b/bench/runtime/zero/eg/gcd.runtime --- a/bench/runtime/zero/eg/gcd.runtime +++ b/bench/runtime/zero/eg/gcd.runtime @@ -1,1 +1,1 @@ -0.1 +0.2 diff --git a/bench/runtime/zero/eg/higher.runtime b/bench/runtime/zero/eg/higher.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/zero/eg/higher.runtime @@ -0,0 +1,1 @@ +1.2 diff --git a/bench/runtime/zero/eg/id.runtime b/bench/runtime/zero/eg/id.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/zero/eg/id.runtime @@ -0,0 +1,1 @@ +0.0 diff --git a/bench/runtime/zero/eg/list.runtime b/bench/runtime/zero/eg/list.runtime --- a/bench/runtime/zero/eg/list.runtime +++ b/bench/runtime/zero/eg/list.runtime @@ -1,1 +1,1 @@ -1.2 +2.0 diff --git a/bench/runtime/zero/eg/pow.runtime b/bench/runtime/zero/eg/pow.runtime --- a/bench/runtime/zero/eg/pow.runtime +++ b/bench/runtime/zero/eg/pow.runtime @@ -1,1 +1,1 @@ -3.7 +6.9 diff --git a/bench/runtime/zero/eg/replicate.runtime b/bench/runtime/zero/eg/replicate.runtime --- a/bench/runtime/zero/eg/replicate.runtime +++ b/bench/runtime/zero/eg/replicate.runtime @@ -1,1 +1,1 @@ -0.4 +0.5 diff --git a/bench/runtime/zero/eg/setelem.runtime b/bench/runtime/zero/eg/setelem.runtime --- a/bench/runtime/zero/eg/setelem.runtime +++ b/bench/runtime/zero/eg/setelem.runtime @@ -1,1 +1,1 @@ -1.9 +2.1 diff --git a/bench/runtime/zero/eg/sort.runtime b/bench/runtime/zero/eg/sort.runtime --- a/bench/runtime/zero/eg/sort.runtime +++ b/bench/runtime/zero/eg/sort.runtime @@ -1,1 +1,1 @@ -1.2 +2.5 diff --git a/bench/runtime/zero/eg/spec.runtime b/bench/runtime/zero/eg/spec.runtime --- a/bench/runtime/zero/eg/spec.runtime +++ b/bench/runtime/zero/eg/spec.runtime @@ -1,1 +1,1 @@ -1.0 +1.2 diff --git a/bench/runtime/zero/eg/subset.runtime b/bench/runtime/zero/eg/subset.runtime --- a/bench/runtime/zero/eg/subset.runtime +++ b/bench/runtime/zero/eg/subset.runtime @@ -1,1 +1,1 @@ -0.8 +0.9 diff --git a/bench/runtime/zero/eg/tree.runtime b/bench/runtime/zero/eg/tree.runtime --- a/bench/runtime/zero/eg/tree.runtime +++ b/bench/runtime/zero/eg/tree.runtime @@ -1,1 +1,1 @@ -1.9 +2.2 diff --git a/bench/self.out b/bench/self.out --- a/bench/self.out +++ b/bench/self.out @@ -2,18 +2,18 @@ -- testing 360 combinations of argument values -- pruning with 0/0 rules -- looking through 4 candidates of size 1 --- looking through 17 candidates of size 2 --- looking through 80 candidates of size 3 --- tested 23 candidates +-- looking through 15 candidates of size 2 +-- looking through 55 candidates of size 3 +-- tested 21 candidates x ? y = x + y (?) :: Int -> Int -> Int -- testing 360 combinations of argument values -- pruning with 0/0 rules -- looking through 4 candidates of size 1 --- looking through 17 candidates of size 2 --- looking through 80 candidates of size 3 --- tested 39 candidates +-- looking through 15 candidates of size 2 +-- looking through 55 candidates of size 3 +-- tested 33 candidates x ? y = x * y i :: Int -> Int @@ -21,7 +21,7 @@ -- pruning with 0/0 rules -- looking through 3 candidates of size 1 -- looking through 2 candidates of size 2 --- looking through 21 candidates of size 3 +-- looking through 13 candidates of size 3 -- tested 8 candidates i x = x + 1 @@ -30,7 +30,7 @@ -- pruning with 0/0 rules -- looking through 3 candidates of size 1 -- looking through 3 candidates of size 2 --- looking through 20 candidates of size 3 --- tested 26 candidates +-- looking through 12 candidates of size 3 +-- tested 18 candidates cannot conjure diff --git a/bench/take-drop.out b/bench/take-drop.out --- a/bench/take-drop.out +++ b/bench/take-drop.out @@ -4,15 +4,12 @@ -- looking through 2 candidates of size 1 -- looking through 3 candidates of size 2 -- looking through 4 candidates of size 3 --- looking through 14 candidates of size 4 --- looking through 26 candidates of size 5 --- looking through 41 candidates of size 6 --- looking through 130 candidates of size 7 --- looking through 125 candidates of size 8 --- looking through 458 candidates of size 9 --- tested 488 candidates -drop 0 [] = [] -drop 0 (x:xs) = x:xs +-- looking through 9 candidates of size 4 +-- looking through 10 candidates of size 5 +-- looking through 28 candidates of size 6 +-- looking through 50 candidates of size 7 +-- tested 63 candidates +drop 0 xs = xs drop x [] = [] drop x (y:xs) = drop (x - 1) xs @@ -22,15 +19,14 @@ -- looking through 2 candidates of size 1 -- looking through 3 candidates of size 2 -- looking through 4 candidates of size 3 --- looking through 14 candidates of size 4 --- looking through 26 candidates of size 5 --- looking through 41 candidates of size 6 --- looking through 130 candidates of size 7 --- looking through 125 candidates of size 8 --- looking through 458 candidates of size 9 --- tested 477 candidates -take 0 [] = [] -take 0 (x:xs) = [] +-- looking through 9 candidates of size 4 +-- looking through 10 candidates of size 5 +-- looking through 28 candidates of size 6 +-- looking through 50 candidates of size 7 +-- looking through 83 candidates of size 8 +-- looking through 133 candidates of size 9 +-- tested 249 candidates +take 0 xs = [] take x [] = [] take x (y:xs) = y:take (x - 1) xs diff --git a/bench/terpret.out b/bench/terpret.out --- a/bench/terpret.out +++ b/bench/terpret.out @@ -54,13 +54,15 @@ -- looking through 10 candidates of size 4 -- looking through 125 candidates of size 5 -- looking through 225 candidates of size 6 --- looking through 1075 candidates of size 7 --- looking through 2457 candidates of size 8 --- looking through 8154 candidates of size 9 --- looking through 27432 candidates of size 10 --- looking through 174839 candidates of size 11 --- tested 137576 candidates -cshift (p,q,r) = if p then (p,r,q) else (p,q,r) +-- looking through 625 candidates of size 7 +-- looking through 1242 candidates of size 8 +-- looking through 3087 candidates of size 9 +-- looking through 8937 candidates of size 10 +-- looking through 102875 candidates of size 11 +-- tested 40385 candidates +cshift (p,q,r) = if p + then (p,r,q) + else (p,q,r) cshift :: Bool -> Bool -> Bool -> (Bool,Bool,Bool) -- testing 4 combinations of argument values @@ -71,10 +73,10 @@ -- looking through 0 candidates of size 3 -- looking through 125 candidates of size 4 -- looking through 225 candidates of size 5 --- looking through 1035 candidates of size 6 --- looking through 2457 candidates of size 7 --- looking through 20250 candidates of size 8 --- tested 22191 candidates +-- looking through 585 candidates of size 6 +-- looking through 1242 candidates of size 7 +-- looking through 15183 candidates of size 8 +-- tested 15459 candidates cshift False p q = (False,p,q) cshift True p q = (True,q,p) @@ -86,8 +88,12 @@ -- looking through 0 candidates of size 1 -- looking through 0 candidates of size 2 -- looking through 9 candidates of size 3 --- looking through 19 candidates of size 4 --- tested 28 candidates +-- looking through 18 candidates of size 4 +-- looking through 27 candidates of size 5 +-- looking through 72 candidates of size 6 +-- looking through 245 candidates of size 7 +-- looking through 663 candidates of size 8 +-- tested 1034 candidates fadder False False False = (False,False) fadder False False True = (False,True) fadder False True False = (False,True) @@ -128,13 +134,12 @@ -- pruning with 4/7 rules -- looking through 1 candidates of size 1 -- looking through 1 candidates of size 2 --- looking through 2 candidates of size 3 --- looking through 6 candidates of size 4 --- looking through 18 candidates of size 5 --- looking through 23 candidates of size 6 --- looking through 68 candidates of size 7 --- tested 64 candidates -[] `access` 0 = undefined +-- looking through 1 candidates of size 3 +-- looking through 4 candidates of size 4 +-- looking through 9 candidates of size 5 +-- looking through 17 candidates of size 6 +-- looking through 46 candidates of size 7 +-- tested 46 candidates [] `access` x = undefined (x:xs) `access` 0 = x (x:xs) `access` y = xs `access` (y - 1) diff --git a/bench/weird.hs b/bench/weird.hs new file mode 100644 --- /dev/null +++ b/bench/weird.hs @@ -0,0 +1,33 @@ +-- weird.hs: conjures some weird or unintuitive functions +-- +-- Copyright (C) 2024 Rudy Matela +-- Distributed under the 3-Clause BSD licence (see the file LICENSE). +-- +-- This file conjures some weird or unintuitive functions. +-- It is here to make sure we do not overprune! +import Conjure + +-- | xor for integers +-- +-- returns the sum but only when one of the arguments is 0 +(^^^) :: Int -> Int -> Int +0 ^^^ y = y +x ^^^ 0 = x +_ ^^^ _ = 0 + +main :: IO () +main = do + conjure "^^^" (^^^) primitives + conjureWith args{usePatterns = False} "^^^" (^^^) primitives + +primitives :: [Prim] +primitives = + [ pr (0::Int) + , pr (1::Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "*" ((*) :: Int -> Int -> Int) + , prim "==" ((==) :: Int -> Int -> Bool) + , prim "&&" (&&) + , prim "||" (||) + , prif (undefined :: Int) + ] diff --git a/bench/weird.out b/bench/weird.out new file mode 100644 --- /dev/null +++ b/bench/weird.out @@ -0,0 +1,29 @@ +(^^^) :: Int -> Int -> Int +-- testing 360 combinations of argument values +-- pruning with 56/91 rules +-- looking through 4 candidates of size 1 +-- looking through 15 candidates of size 2 +-- looking through 39 candidates of size 3 +-- tested 36 candidates +0 ^^^ x = x +x ^^^ 0 = x +x ^^^ y = 0 + +(^^^) :: Int -> Int -> Int +-- testing 360 combinations of argument values +-- pruning with 56/91 rules +-- looking through 4 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 9 candidates of size 3 +-- looking through 0 candidates of size 4 +-- looking through 39 candidates of size 5 +-- looking through 60 candidates of size 6 +-- looking through 225 candidates of size 7 +-- looking through 1056 candidates of size 8 +-- looking through 1488 candidates of size 9 +-- looking through 13020 candidates of size 10 +-- tested 7148 candidates +x ^^^ y = if 0 == x * y + then x + y + else 0 + diff --git a/changelog.md b/changelog.md --- a/changelog.md +++ b/changelog.md @@ -2,6 +2,18 @@ ============================ +v0.5.8 +------ + +* prune redundant mutants using a few new rules +* rework numeric recursion criteria +* improve pretty-printing +* improve error handling +* refactor and lint throughout +* Conjurable tuples up to 12-tuples +* bump Express requirement to v1.0.14 (bugfix) + + v0.5.6 ------ diff --git a/code-conjure.cabal b/code-conjure.cabal --- a/code-conjure.cabal +++ b/code-conjure.cabal @@ -3,7 +3,7 @@ -- Copyright (C) 2021 Rudy Matela -- Distributed under the 3-Clause BSD licence (see the file LICENSE). name: code-conjure -version: 0.5.6 +version: 0.5.8 synopsis: synthesize Haskell functions out of partial definitions description: Conjure is a tool that synthesizes Haskell functions out of partial definitions. @@ -43,6 +43,10 @@ , bench/runtime/zero/eg/*.runtime , bench/runtime/zero/proto/*.runtime , bench/runtime/zero/versions + , bench/runtime/lapmatrud/bench/*.runtime + , bench/runtime/lapmatrud/eg/*.runtime + , bench/runtime/lapmatrud/proto/*.runtime + , bench/runtime/lapmatrud/versions , bench/versions , bench/time , test/sdist @@ -60,7 +64,7 @@ source-repository this type: git location: https://github.com/rudymatela/conjure - tag: v0.5.6 + tag: v0.5.8 library exposed-modules: Conjure @@ -76,7 +80,7 @@ , leancheck >= 1.0.0 , template-haskell , speculate >= 0.4.14 - , express >= 1.0.8 + , express >= 1.0.14 hs-source-dirs: src default-language: Haskell2010 diff --git a/eg/arith.out b/eg/arith.out --- a/eg/arith.out +++ b/eg/arith.out @@ -2,49 +2,49 @@ -- testing 4 combinations of argument values -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 2 candidates of size 2 --- looking through 8 candidates of size 3 --- tested 6 candidates +-- looking through 1 candidates of size 2 +-- looking through 6 candidates of size 3 +-- tested 5 candidates double x = x + x triple :: Int -> Int -- testing 4 combinations of argument values -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 2 candidates of size 2 --- looking through 8 candidates of size 3 --- looking through 8 candidates of size 4 --- looking through 21 candidates of size 5 --- tested 25 candidates +-- looking through 1 candidates of size 2 +-- looking through 6 candidates of size 3 +-- looking through 4 candidates of size 4 +-- looking through 11 candidates of size 5 +-- tested 15 candidates triple x = x + (x + x) add :: Int -> Int -> Int -- testing 4 combinations of argument values -- pruning with 14/25 rules -- looking through 4 candidates of size 1 --- looking through 17 candidates of size 2 --- looking through 61 candidates of size 3 --- tested 23 candidates +-- looking through 15 candidates of size 2 +-- looking through 39 candidates of size 3 +-- tested 21 candidates add x y = x + y square :: Int -> Int -- testing 3 combinations of argument values -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 2 candidates of size 2 --- looking through 7 candidates of size 3 --- tested 10 candidates +-- looking through 1 candidates of size 2 +-- looking through 4 candidates of size 3 +-- tested 7 candidates square x = x * x cube :: Int -> Int -- testing 3 combinations of argument values -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 2 candidates of size 2 --- looking through 7 candidates of size 3 --- looking through 8 candidates of size 4 --- looking through 21 candidates of size 5 --- tested 36 candidates +-- looking through 1 candidates of size 2 +-- looking through 4 candidates of size 3 +-- looking through 3 candidates of size 4 +-- looking through 11 candidates of size 5 +-- tested 19 candidates cube x = x * (x * x) tnpo :: Int -> Int @@ -52,11 +52,11 @@ -- pruning with 14/25 rules -- looking through 3 candidates of size 1 -- looking through 2 candidates of size 2 --- looking through 8 candidates of size 3 --- looking through 8 candidates of size 4 --- looking through 21 candidates of size 5 --- looking through 21 candidates of size 6 --- looking through 90 candidates of size 7 --- tested 87 candidates +-- looking through 6 candidates of size 3 +-- looking through 5 candidates of size 4 +-- looking through 11 candidates of size 5 +-- looking through 11 candidates of size 6 +-- looking through 35 candidates of size 7 +-- tested 40 candidates tnpo x = x + (x + (x + 1)) diff --git a/eg/bools.out b/eg/bools.out --- a/eg/bools.out +++ b/eg/bools.out @@ -5,8 +5,8 @@ -- looking through 6 candidates of size 2 -- looking through 12 candidates of size 3 -- looking through 16 candidates of size 4 --- looking through 42 candidates of size 5 --- tested 45 candidates +-- looking through 28 candidates of size 5 +-- tested 42 candidates and [] = True and (p:ps) = p && and ps @@ -17,7 +17,7 @@ -- looking through 6 candidates of size 2 -- looking through 12 candidates of size 3 -- looking through 16 candidates of size 4 --- looking through 42 candidates of size 5 +-- looking through 28 candidates of size 5 -- tested 39 candidates or [] = False or (p:ps) = p || or ps diff --git a/eg/bst.hs b/eg/bst.hs --- a/eg/bst.hs +++ b/eg/bst.hs @@ -3,27 +3,14 @@ -- Copyright (C) 2021 Rudy Matela -- Distributed under the 3-Clause BSD licence (see the file LICENSE). {-# LANGUAGE CPP, TemplateHaskell #-} -#if __GLASGOW_HASKELL__ == 708 -{-# LANGUAGE DeriveDataTypeable, StandaloneDeriving #-} -import Data.Typeable (Typeable) -#endif import Conjure import Test.LeanCheck import Data.Express hiding (height,size) --- TODO: remove the following import --- and fix build on GHC 7.10 and 7.8 --- the generation of -: and ->>>: somehow fails. -import Test.LeanCheck.Utils - data Tree = Leaf | Node Tree Int Tree deriving (Eq, Ord, Show, Read) - -#if __GLASGOW_HASKELL__ == 708 -deriving instance Typeable Tree -#endif deriveExpress ''Tree diff --git a/eg/bst.out b/eg/bst.out --- a/eg/bst.out +++ b/eg/bst.out @@ -4,18 +4,18 @@ -- looking through 1 candidates of size 1 -- looking through 0 candidates of size 2 -- looking through 0 candidates of size 3 --- looking through 8 candidates of size 4 +-- looking through 7 candidates of size 4 -- looking through 0 candidates of size 5 -- looking through 0 candidates of size 6 -- looking through 0 candidates of size 7 --- looking through 60 candidates of size 8 --- looking through 120 candidates of size 9 +-- looking through 31 candidates of size 8 +-- looking through 92 candidates of size 9 -- looking through 0 candidates of size 10 --- looking through 452 candidates of size 11 --- looking through 424 candidates of size 12 --- tested 707 candidates +-- looking through 304 candidates of size 11 +-- looking through 199 candidates of size 12 +-- tested 501 candidates mem x Leaf = False -mem x (Node t1 y t2) = mem x t1 || (x == y || mem x t2) +mem x (Node t1 y t2) = mem x t1 || (mem x t2 || x == y) mem :: Int -> Tree -> Bool -- testing 360 combinations of argument values @@ -42,16 +42,16 @@ -- looking through 2 candidates of size 1 -- looking through 2 candidates of size 2 -- looking through 0 candidates of size 3 --- looking through 10 candidates of size 4 +-- looking through 8 candidates of size 4 -- looking through 21 candidates of size 5 -- looking through 0 candidates of size 6 --- looking through 118 candidates of size 7 +-- looking through 86 candidates of size 7 -- looking through 239 candidates of size 8 --- looking through 216 candidates of size 9 --- looking through 2204 candidates of size 10 --- looking through 3651 candidates of size 11 --- looking through 8280 candidates of size 12 --- tested 14743 candidates +-- looking through 104 candidates of size 9 +-- looking through 1558 candidates of size 10 +-- looking through 3555 candidates of size 11 +-- looking through 4028 candidates of size 12 +-- tested 9603 candidates cannot conjure before :: Int -> Tree -> Tree @@ -59,16 +59,16 @@ -- looking through 2 candidates of size 1 -- looking through 2 candidates of size 2 -- looking through 0 candidates of size 3 --- looking through 10 candidates of size 4 +-- looking through 8 candidates of size 4 -- looking through 21 candidates of size 5 -- looking through 0 candidates of size 6 --- looking through 118 candidates of size 7 +-- looking through 86 candidates of size 7 -- looking through 239 candidates of size 8 --- looking through 216 candidates of size 9 --- looking through 2204 candidates of size 10 --- looking through 3651 candidates of size 11 --- looking through 8280 candidates of size 12 --- tested 14743 candidates +-- looking through 104 candidates of size 9 +-- looking through 1558 candidates of size 10 +-- looking through 3555 candidates of size 11 +-- looking through 4028 candidates of size 12 +-- tested 9603 candidates cannot conjure before :: Int -> Tree -> Tree @@ -76,16 +76,16 @@ -- looking through 2 candidates of size 1 -- looking through 2 candidates of size 2 -- looking through 0 candidates of size 3 --- looking through 10 candidates of size 4 +-- looking through 8 candidates of size 4 -- looking through 21 candidates of size 5 -- looking through 0 candidates of size 6 --- looking through 94 candidates of size 7 +-- looking through 68 candidates of size 7 -- looking through 287 candidates of size 8 --- looking through 72 candidates of size 9 --- looking through 1784 candidates of size 10 +-- looking through 32 candidates of size 9 +-- looking through 1216 candidates of size 10 -- looking through 5103 candidates of size 11 --- looking through 3312 candidates of size 12 --- tested 10687 candidates +-- looking through 1472 candidates of size 12 +-- tested 8211 candidates cannot conjure beyond :: Int -> Tree -> Tree @@ -93,16 +93,16 @@ -- looking through 2 candidates of size 1 -- looking through 2 candidates of size 2 -- looking through 0 candidates of size 3 --- looking through 10 candidates of size 4 +-- looking through 8 candidates of size 4 -- looking through 21 candidates of size 5 -- looking through 0 candidates of size 6 --- looking through 94 candidates of size 7 +-- looking through 68 candidates of size 7 -- looking through 287 candidates of size 8 --- looking through 72 candidates of size 9 --- looking through 1784 candidates of size 10 +-- looking through 32 candidates of size 9 +-- looking through 1216 candidates of size 10 -- looking through 5103 candidates of size 11 --- looking through 3312 candidates of size 12 --- tested 10687 candidates +-- looking through 1472 candidates of size 12 +-- tested 8211 candidates cannot conjure union :: Tree -> Tree -> Tree @@ -110,13 +110,13 @@ -- pruning with 6/8 rules -- looking through 3 candidates of size 1 -- looking through 11 candidates of size 2 --- looking through 40 candidates of size 3 +-- looking through 32 candidates of size 3 -- looking through 84 candidates of size 4 --- looking through 784 candidates of size 5 --- looking through 1032 candidates of size 6 --- looking through 8368 candidates of size 7 --- looking through 24890 candidates of size 8 --- looking through 75535 candidates of size 9 --- tested 110747 candidates +-- looking through 470 candidates of size 5 +-- looking through 1074 candidates of size 6 +-- looking through 4088 candidates of size 7 +-- looking through 16373 candidates of size 8 +-- looking through 41366 candidates of size 9 +-- tested 63501 candidates cannot conjure diff --git a/eg/count.out b/eg/count.out --- a/eg/count.out +++ b/eg/count.out @@ -14,16 +14,16 @@ -- pruning with 8/13 rules -- looking through 2 candidates of size 1 -- looking through 2 candidates of size 2 --- looking through 1 candidates of size 3 --- looking through 8 candidates of size 4 --- looking through 9 candidates of size 5 --- looking through 24 candidates of size 6 --- looking through 46 candidates of size 7 --- looking through 114 candidates of size 8 --- looking through 272 candidates of size 9 --- looking through 648 candidates of size 10 --- looking through 1549 candidates of size 11 --- tested 1751 candidates +-- looking through 0 candidates of size 3 +-- looking through 4 candidates of size 4 +-- looking through 8 candidates of size 5 +-- looking through 14 candidates of size 6 +-- looking through 24 candidates of size 7 +-- looking through 58 candidates of size 8 +-- looking through 134 candidates of size 9 +-- looking through 302 candidates of size 10 +-- looking through 633 candidates of size 11 +-- tested 909 candidates count x [] = 0 -count x (y:xs) = (if x == y then 1 else 0) + count x xs +count x (y:xs) = count x xs + (if x == y then 1 else 0) diff --git a/eg/dupos.out b/eg/dupos.out --- a/eg/dupos.out +++ b/eg/dupos.out @@ -12,15 +12,17 @@ -- looking through 16 candidates of size 9 -- looking through 25 candidates of size 10 -- looking through 36 candidates of size 11 --- looking through 65 candidates of size 12 --- looking through 141 candidates of size 13 --- looking through 322 candidates of size 14 --- looking through 644 candidates of size 15 --- looking through 1189 candidates of size 16 --- looking through 2189 candidates of size 17 --- tested 2599 candidates +-- looking through 63 candidates of size 12 +-- looking through 133 candidates of size 13 +-- looking through 299 candidates of size 14 +-- looking through 602 candidates of size 15 +-- looking through 1116 candidates of size 16 +-- looking through 2031 candidates of size 17 +-- tested 2451 candidates duplicates [] = [] -duplicates (x:xs) = if elem x xs && not (elem x (duplicates xs)) then x:duplicates xs else duplicates xs +duplicates (x:xs) = if elem x xs && not (elem x (duplicates xs)) + then x:duplicates xs + else duplicates xs duplicates :: [Int] -> [Int] -- pruning with 21/26 rules @@ -35,15 +37,17 @@ -- looking through 16 candidates of size 9 -- looking through 25 candidates of size 10 -- looking through 36 candidates of size 11 --- looking through 65 candidates of size 12 --- looking through 141 candidates of size 13 --- looking through 322 candidates of size 14 --- looking through 644 candidates of size 15 --- looking through 1189 candidates of size 16 --- looking through 2189 candidates of size 17 --- tested 2599 candidates +-- looking through 63 candidates of size 12 +-- looking through 133 candidates of size 13 +-- looking through 299 candidates of size 14 +-- looking through 602 candidates of size 15 +-- looking through 1116 candidates of size 16 +-- looking through 2031 candidates of size 17 +-- tested 2451 candidates duplicates [] = [] -duplicates (x:xs) = if elem x xs && not (elem x (duplicates xs)) then x:duplicates xs else duplicates xs +duplicates (x:xs) = if elem x xs && not (elem x (duplicates xs)) + then x:duplicates xs + else duplicates xs positionsFrom :: Int -> A -> [A] -> [Int] -- testing 360 combinations of argument values @@ -52,17 +56,17 @@ -- looking through 0 candidates of size 2 -- looking through 2 candidates of size 3 -- looking through 5 candidates of size 4 --- looking through 15 candidates of size 5 --- looking through 25 candidates of size 6 --- looking through 59 candidates of size 7 --- looking through 108 candidates of size 8 --- looking through 256 candidates of size 9 --- looking through 458 candidates of size 10 --- looking through 1064 candidates of size 11 --- looking through 1948 candidates of size 12 --- looking through 4516 candidates of size 13 --- looking through 8304 candidates of size 14 --- tested 8844 candidates +-- looking through 13 candidates of size 5 +-- looking through 18 candidates of size 6 +-- looking through 47 candidates of size 7 +-- looking through 77 candidates of size 8 +-- looking through 193 candidates of size 9 +-- looking through 293 candidates of size 10 +-- looking through 738 candidates of size 11 +-- looking through 1173 candidates of size 12 +-- looking through 2879 candidates of size 13 +-- looking through 4537 candidates of size 14 +-- tested 5610 candidates positionsFrom x y [] = [] positionsFrom x y (z:xs) = (if y == z then (x :) else id) (positionsFrom (x + 1) y xs) diff --git a/eg/factorial.out b/eg/factorial.out --- a/eg/factorial.out +++ b/eg/factorial.out @@ -2,13 +2,13 @@ -- testing 4 combinations of argument values -- pruning with 27/65 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 11 candidates of size 3 --- looking through 29 candidates of size 4 --- looking through 60 candidates of size 5 --- looking through 145 candidates of size 6 --- looking through 501 candidates of size 7 --- tested 283 candidates +-- looking through 3 candidates of size 2 +-- looking through 7 candidates of size 3 +-- looking through 13 candidates of size 4 +-- looking through 35 candidates of size 5 +-- looking through 61 candidates of size 6 +-- looking through 219 candidates of size 7 +-- tested 153 candidates factorial 0 = 1 factorial x = x * factorial (x - 1) @@ -16,11 +16,11 @@ -- testing 4 combinations of argument values -- pruning with 32/72 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 11 candidates of size 3 --- looking through 29 candidates of size 4 --- looking through 60 candidates of size 5 --- looking through 209 candidates of size 6 --- tested 148 candidates +-- looking through 3 candidates of size 2 +-- looking through 7 candidates of size 3 +-- looking through 13 candidates of size 4 +-- looking through 35 candidates of size 5 +-- looking through 101 candidates of size 6 +-- tested 89 candidates factorial x = foldr (*) 1 [1..x] diff --git a/eg/fib01.out b/eg/fib01.out --- a/eg/fib01.out +++ b/eg/fib01.out @@ -3,10 +3,10 @@ -- pruning with 6/10 rules -- looking through 4 candidates of size 1 -- looking through 24 candidates of size 2 --- looking through 241 candidates of size 3 --- looking through 1958 candidates of size 4 --- looking through 10540 candidates of size 5 --- tested 12767 candidates +-- looking through 115 candidates of size 3 +-- looking through 470 candidates of size 4 +-- looking through 1657 candidates of size 5 +-- tested 2270 candidates cannot conjure fib01 :: Int -> Int -> Int -> Int @@ -14,14 +14,14 @@ -- pruning with 18/37 rules -- looking through 4 candidates of size 1 -- looking through 4 candidates of size 2 --- looking through 9 candidates of size 3 --- looking through 21 candidates of size 4 --- looking through 43 candidates of size 5 --- looking through 84 candidates of size 6 --- looking through 192 candidates of size 7 --- looking through 391 candidates of size 8 --- looking through 840 candidates of size 9 --- looking through 6414 candidates of size 10 --- tested 8002 candidates +-- looking through 10 candidates of size 3 +-- looking through 16 candidates of size 4 +-- looking through 44 candidates of size 5 +-- looking through 92 candidates of size 6 +-- looking through 234 candidates of size 7 +-- looking through 544 candidates of size 8 +-- looking through 1332 candidates of size 9 +-- looking through 7796 candidates of size 10 +-- tested 10076 candidates cannot conjure diff --git a/eg/fibonacci.out b/eg/fibonacci.out --- a/eg/fibonacci.out +++ b/eg/fibonacci.out @@ -3,17 +3,17 @@ -- pruning with 21/44 rules -- looking through 4 candidates of size 1 -- looking through 3 candidates of size 2 --- looking through 14 candidates of size 3 --- looking through 17 candidates of size 4 --- looking through 58 candidates of size 5 --- looking through 69 candidates of size 6 --- looking through 357 candidates of size 7 --- looking through 390 candidates of size 8 --- looking through 2265 candidates of size 9 --- looking through 2502 candidates of size 10 --- looking through 15355 candidates of size 11 --- looking through 17083 candidates of size 12 --- tested 21324 candidates +-- looking through 9 candidates of size 3 +-- looking through 11 candidates of size 4 +-- looking through 34 candidates of size 5 +-- looking through 41 candidates of size 6 +-- looking through 187 candidates of size 7 +-- looking through 204 candidates of size 8 +-- looking through 957 candidates of size 9 +-- looking through 1059 candidates of size 10 +-- looking through 5295 candidates of size 11 +-- looking through 6052 candidates of size 12 +-- tested 8126 candidates fibonacci 0 = 1 fibonacci 1 = 1 fibonacci x = fibonacci (x - 1) + fibonacci (x - 2) diff --git a/eg/gcd.out b/eg/gcd.out --- a/eg/gcd.out +++ b/eg/gcd.out @@ -2,12 +2,12 @@ -- testing 11 combinations of argument values -- pruning with 0/0 rules -- looking through 3 candidates of size 1 --- looking through 7 candidates of size 2 --- looking through 18 candidates of size 3 --- looking through 60 candidates of size 4 --- looking through 150 candidates of size 5 --- looking through 469 candidates of size 6 --- tested 243 candidates +-- looking through 5 candidates of size 2 +-- looking through 11 candidates of size 3 +-- looking through 50 candidates of size 4 +-- looking through 98 candidates of size 5 +-- looking through 359 candidates of size 6 +-- tested 171 candidates gcd x 0 = x gcd x y = gcd y (x `mod` y) diff --git a/eg/higher.hs b/eg/higher.hs new file mode 100644 --- /dev/null +++ b/eg/higher.hs @@ -0,0 +1,55 @@ +-- higher.hs: conjuring higher order functions +-- +-- Copyright (C) 2024 Rudy Matela +-- Distributed under the 3-Clause BSD licence (see the file LICENSE). +import Conjure + +appSpec :: ((Int -> Int) -> Int -> Int) -> Bool +appSpec ($) = ((+1) $ 2) == 3 + && (abs $ (-2)) == 2 + && (negate $ 1) == -1 + +-- composeSpec :: ((Int->Int) -> (Int->Int) -> Int->Int) -> + +composeSpec :: ((Int->Int) -> (Int->Int) -> Int->Int) -> Bool +composeSpec (.) = ((*2) . (*3)) 1 == 6 + && (abs . negate) 7 == 7 + && (negate . abs) 8 == -8 + +flipSpec :: ((Int -> Int -> Int) -> Int -> Int -> Int) -> Bool +flipSpec flip = flip const 1 2 == 2 + && flip div 2 10 == 5 + && flip mod 3 10 == 1 + +mapSpec :: ((Int -> Int) -> [Int] -> [Int]) -> Bool +mapSpec map = map (*2) [1,2,3] == [2,4,6] + && map abs [0,-1,-2] == [0,1,2] + +foldSpec :: ((Int -> Int -> Int) -> Int -> [Int] -> Int) -> Bool +foldSpec fold = fold (+) 0 [1,2,3] == 6 + && fold (*) 1 [1,2,3] == 6 + && fold (+) 0 [1,2,3,4] == 10 + && fold (*) 1 [1,2,3,4] == 24 + +filterSpec :: ((Int -> Bool) -> [Int] -> [Int]) -> Bool +filterSpec filter = filter (<0) [0,1,-1,2,-2] == [-1,-2] + && filter (>0) [0,1,-1,2,-2] == [1,2] + && filter odd [0,1,2,3,4,5,6] == [1,3,5] + && filter even [0,1,2,3,4,5,6] == [0,2,4,6] + +main :: IO () +main = do + conjureFromSpec "$" appSpec primitives + conjureFromSpec "." composeSpec primitives + conjureFromSpec "flip" flipSpec primitives + conjureFromSpec "map" mapSpec primitives + conjureFromSpec "fold" foldSpec primitives + conjureFromSpec "filter" filterSpec primitives + + +primitives :: [Prim] +primitives = + [ pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prif (undefined :: [Int]) + ] diff --git a/eg/higher.out b/eg/higher.out new file mode 100644 --- /dev/null +++ b/eg/higher.out @@ -0,0 +1,69 @@ +($) :: (Int -> Int) -> Int -> Int +-- pruning with 3/3 rules +-- looking through 1 candidates of size 1 +-- looking through 1 candidates of size 2 +-- tested 2 candidates +f $ x = f x + +(.) :: (Int -> Int) -> (Int -> Int) -> Int -> Int +-- pruning with 3/3 rules +-- looking through 1 candidates of size 1 +-- looking through 2 candidates of size 2 +-- looking through 4 candidates of size 3 +-- tested 5 candidates +(f . g) x = f (g x) + +flip :: (Int -> Int -> Int) -> Int -> Int -> Int +-- pruning with 3/3 rules +-- looking through 2 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 4 candidates of size 3 +-- tested 5 candidates +flip f x y = f y x + +map :: (Int -> Int) -> [Int] -> [Int] +-- pruning with 3/3 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 3 candidates of size 4 +-- looking through 2 candidates of size 5 +-- looking through 6 candidates of size 6 +-- looking through 9 candidates of size 7 +-- tested 17 candidates +map f [] = [] +map f (x:xs) = f x:map f xs + +fold :: (Int -> Int -> Int) -> Int -> [Int] -> Int +-- pruning with 3/3 rules +-- looking through 1 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 1 candidates of size 3 +-- looking through 6 candidates of size 4 +-- looking through 4 candidates of size 5 +-- looking through 23 candidates of size 6 +-- looking through 26 candidates of size 7 +-- tested 45 candidates +fold f x [] = x +fold f x (y:xs) = fold f (f x y) xs + +filter :: (Int -> Bool) -> [Int] -> [Int] +-- pruning with 3/3 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 3 candidates of size 4 +-- looking through 0 candidates of size 5 +-- looking through 6 candidates of size 6 +-- looking through 0 candidates of size 7 +-- looking through 22 candidates of size 8 +-- looking through 0 candidates of size 9 +-- looking through 65 candidates of size 10 +-- looking through 0 candidates of size 11 +-- looking through 219 candidates of size 12 +-- tested 166 candidates +filter f [] = [] +filter f (x:xs) = if f x + then x:filter f xs + else filter f xs + diff --git a/eg/id.hs b/eg/id.hs new file mode 100644 --- /dev/null +++ b/eg/id.hs @@ -0,0 +1,12 @@ +-- id.hs: conjuring the identity and const functions +-- +-- Copyright (C) 2024 Rudy Matela +-- Distributed under the 3-Clause BSD licence (see the file LICENSE). +import Conjure + +main :: IO () +main = do + -- conjure needs no primitives + -- to figure out the implementation of id and const + conjure "id" (id :: Int -> Int) [] + conjure "const" (const :: Int -> Int -> Int) [] diff --git a/eg/id.out b/eg/id.out new file mode 100644 --- /dev/null +++ b/eg/id.out @@ -0,0 +1,14 @@ +id :: Int -> Int +-- testing 360 combinations of argument values +-- pruning with 0/0 rules +-- looking through 1 candidates of size 1 +-- tested 1 candidates +id x = x + +const :: Int -> Int -> Int +-- testing 360 combinations of argument values +-- pruning with 0/0 rules +-- looking through 2 candidates of size 1 +-- tested 1 candidates +const x y = x + diff --git a/eg/ints.out b/eg/ints.out --- a/eg/ints.out +++ b/eg/ints.out @@ -22,10 +22,10 @@ -- pruning with 14/25 rules -- looking through 2 candidates of size 1 -- looking through 4 candidates of size 2 --- looking through 3 candidates of size 3 --- looking through 13 candidates of size 4 --- looking through 14 candidates of size 5 --- tested 23 candidates +-- looking through 2 candidates of size 3 +-- looking through 6 candidates of size 4 +-- looking through 6 candidates of size 5 +-- tested 15 candidates sum [] = 0 sum (x:xs) = x + sum xs @@ -34,10 +34,10 @@ -- pruning with 14/25 rules -- looking through 2 candidates of size 1 -- looking through 4 candidates of size 2 --- looking through 3 candidates of size 3 --- looking through 13 candidates of size 4 --- looking through 14 candidates of size 5 --- tested 33 candidates +-- looking through 2 candidates of size 3 +-- looking through 6 candidates of size 4 +-- looking through 6 candidates of size 5 +-- tested 20 candidates product [] = 1 product (x:xs) = x * product xs @@ -46,9 +46,9 @@ -- pruning with 15/26 rules -- looking through 2 candidates of size 1 -- looking through 4 candidates of size 2 --- looking through 3 candidates of size 3 --- looking through 16 candidates of size 4 --- tested 10 candidates +-- looking through 2 candidates of size 3 +-- looking through 9 candidates of size 4 +-- tested 9 candidates sum xs = foldr (+) 0 xs product :: [Int] -> Int @@ -56,8 +56,8 @@ -- pruning with 15/26 rules -- looking through 2 candidates of size 1 -- looking through 4 candidates of size 2 --- looking through 3 candidates of size 3 --- looking through 16 candidates of size 4 --- tested 12 candidates +-- looking through 2 candidates of size 3 +-- looking through 9 candidates of size 4 +-- tested 11 candidates product xs = foldr (*) 1 xs diff --git a/eg/list.hs b/eg/list.hs --- a/eg/list.hs +++ b/eg/list.hs @@ -52,6 +52,12 @@ ordered' [x,y] = x <= y ordered' [x,y,z] = x <= y && y <= z +zip' :: [Int] -> [Int] -> [(Int,Int)] +zip' [] [] = [] +zip' [x] [a] = [(x,a)] +zip' [x,y] [a,b] = [(x,a),(y,b)] +zip' [x,y,z] [a,b,c] = [(x,a),(y,b),(z,c)] + main :: IO () main = do conjure "length" length' @@ -75,6 +81,12 @@ [ pr ([] :: [Int]) , prim ":" ((:) :: Int -> [Int] -> [Int]) , prim "foldr" (foldr :: (Int -> [Int] -> [Int]) -> [Int] -> [Int] -> [Int]) + ] + + conjure "zip" (zip') + [ pr ([] :: [(Int,Int)]) + , prim ":" ((:) :: (Int,Int) -> [(Int,Int)] -> [(Int,Int)]) + , prim "," ((,) :: Int -> Int -> (Int,Int)) ] conjure "\\/" (\/) diff --git a/eg/list.out b/eg/list.out --- a/eg/list.out +++ b/eg/list.out @@ -3,10 +3,10 @@ -- pruning with 4/8 rules -- looking through 2 candidates of size 1 -- looking through 2 candidates of size 2 --- looking through 2 candidates of size 3 --- looking through 4 candidates of size 4 --- looking through 5 candidates of size 5 --- tested 13 candidates +-- looking through 1 candidates of size 3 +-- looking through 2 candidates of size 4 +-- looking through 2 candidates of size 5 +-- tested 9 candidates length [] = 0 length (x:xs) = length xs + 1 @@ -29,11 +29,11 @@ -- pruning with 0/0 rules -- looking through 3 candidates of size 1 -- looking through 7 candidates of size 2 --- looking through 9 candidates of size 3 --- looking through 44 candidates of size 4 --- looking through 116 candidates of size 5 --- looking through 80 candidates of size 6 --- tested 233 candidates +-- looking through 11 candidates of size 3 +-- looking through 37 candidates of size 4 +-- looking through 98 candidates of size 5 +-- looking through 94 candidates of size 6 +-- tested 194 candidates [] ++ xs = xs (x:xs) ++ ys = x:(xs ++ ys) @@ -42,21 +42,38 @@ -- pruning with 2/2 rules -- looking through 3 candidates of size 1 -- looking through 7 candidates of size 2 --- looking through 9 candidates of size 3 --- looking through 48 candidates of size 4 --- tested 54 candidates +-- looking through 11 candidates of size 3 +-- looking through 41 candidates of size 4 +-- tested 48 candidates xs ++ ys = foldr (:) ys xs +zip :: [Int] -> [Int] -> [(Int,Int)] +-- testing 360 combinations of argument values +-- pruning with 0/0 rules +-- looking through 1 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 8 candidates of size 4 +-- looking through 6 candidates of size 5 +-- looking through 2 candidates of size 6 +-- looking through 14 candidates of size 7 +-- looking through 9 candidates of size 8 +-- looking through 31 candidates of size 9 +-- tested 50 candidates +zip [] xs = [] +zip (x:xs) [] = [] +zip (x:xs) (y:ys) = (x,y):zip xs ys + (\/) :: [Int] -> [Int] -> [Int] -- testing 360 combinations of argument values -- pruning with 0/0 rules -- looking through 3 candidates of size 1 -- looking through 7 candidates of size 2 --- looking through 9 candidates of size 3 --- looking through 44 candidates of size 4 --- looking through 116 candidates of size 5 --- looking through 80 candidates of size 6 --- tested 235 candidates +-- looking through 11 candidates of size 3 +-- looking through 37 candidates of size 4 +-- looking through 98 candidates of size 5 +-- looking through 94 candidates of size 6 +-- tested 196 candidates [] \/ xs = xs (x:xs) \/ ys = x:ys \/ xs @@ -68,31 +85,31 @@ -- looking through 3 candidates of size 3 -- looking through 3 candidates of size 4 -- looking through 4 candidates of size 5 --- looking through 16 candidates of size 6 --- looking through 28 candidates of size 7 --- looking through 48 candidates of size 8 --- looking through 100 candidates of size 9 --- looking through 226 candidates of size 10 --- looking through 482 candidates of size 11 --- tested 616 candidates +-- looking through 12 candidates of size 6 +-- looking through 20 candidates of size 7 +-- looking through 30 candidates of size 8 +-- looking through 56 candidates of size 9 +-- looking through 114 candidates of size 10 +-- looking through 216 candidates of size 11 +-- tested 333 candidates ordered [] = True -ordered (x:xs) = (null xs || x <= head xs) && ordered xs +ordered (x:xs) = ordered xs && (null xs || x <= head xs) merge :: [Int] -> [Int] -> [Int] -- testing 360 combinations of argument values -- pruning with 1/2 rules -- looking through 2 candidates of size 1 -- looking through 1 candidates of size 2 --- looking through 2 candidates of size 3 +-- looking through 3 candidates of size 3 -- looking through 12 candidates of size 4 --- looking through 16 candidates of size 5 --- looking through 20 candidates of size 6 --- looking through 84 candidates of size 7 --- looking through 40 candidates of size 8 --- looking through 368 candidates of size 9 --- looking through 216 candidates of size 10 --- looking through 1838 candidates of size 11 --- looking through 1016 candidates of size 12 --- tested 3615 candidates +-- looking through 22 candidates of size 5 +-- looking through 32 candidates of size 6 +-- looking through 75 candidates of size 7 +-- looking through 108 candidates of size 8 +-- looking through 256 candidates of size 9 +-- looking through 520 candidates of size 10 +-- looking through 1329 candidates of size 11 +-- looking through 2632 candidates of size 12 +-- tested 4992 candidates cannot conjure diff --git a/eg/pow.out b/eg/pow.out --- a/eg/pow.out +++ b/eg/pow.out @@ -2,14 +2,14 @@ -- testing 5 combinations of argument values -- pruning with 14/30 rules -- looking through 4 candidates of size 1 --- looking through 17 candidates of size 2 --- looking through 92 candidates of size 3 --- looking through 307 candidates of size 4 --- looking through 1632 candidates of size 5 --- looking through 3406 candidates of size 6 --- looking through 21046 candidates of size 7 --- looking through 34728 candidates of size 8 --- tested 27145 candidates +-- looking through 15 candidates of size 2 +-- looking through 45 candidates of size 3 +-- looking through 168 candidates of size 4 +-- looking through 491 candidates of size 5 +-- looking through 1729 candidates of size 6 +-- looking through 5420 candidates of size 7 +-- looking through 17952 candidates of size 8 +-- tested 8593 candidates pow x 0 = 1 pow x y = x * pow x (y - 1) @@ -17,11 +17,11 @@ -- testing 5 combinations of argument values -- pruning with 15/19 rules -- looking through 4 candidates of size 1 --- looking through 19 candidates of size 2 --- looking through 105 candidates of size 3 --- looking through 360 candidates of size 4 --- looking through 1649 candidates of size 5 --- looking through 4534 candidates of size 6 --- tested 6671 candidates +-- looking through 17 candidates of size 2 +-- looking through 59 candidates of size 3 +-- looking through 190 candidates of size 4 +-- looking through 594 candidates of size 5 +-- looking through 1784 candidates of size 6 +-- tested 2648 candidates cannot conjure diff --git a/eg/replicate.out b/eg/replicate.out --- a/eg/replicate.out +++ b/eg/replicate.out @@ -4,12 +4,12 @@ -- looking through 1 candidates of size 1 -- looking through 0 candidates of size 2 -- looking through 1 candidates of size 3 --- looking through 1 candidates of size 4 --- looking through 3 candidates of size 5 --- looking through 5 candidates of size 6 --- looking through 5 candidates of size 7 --- looking through 8 candidates of size 8 --- tested 17 candidates +-- looking through 2 candidates of size 4 +-- looking through 4 candidates of size 5 +-- looking through 6 candidates of size 6 +-- looking through 8 candidates of size 7 +-- looking through 19 candidates of size 8 +-- tested 33 candidates replicate 0 c = "" replicate x c = c:replicate (x - 1) c @@ -41,12 +41,12 @@ -- looking through 2 candidates of size 1 -- looking through 3 candidates of size 2 -- looking through 7 candidates of size 3 --- looking through 24 candidates of size 4 --- looking through 64 candidates of size 5 --- looking through 161 candidates of size 6 --- looking through 469 candidates of size 7 --- looking through 1303 candidates of size 8 --- tested 791 candidates +-- looking through 23 candidates of size 4 +-- looking through 62 candidates of size 5 +-- looking through 155 candidates of size 6 +-- looking through 446 candidates of size 7 +-- looking through 1231 candidates of size 8 +-- tested 751 candidates replicates "" x = "" replicates (c:cs) x = replicate x c ++ replicates cs x diff --git a/eg/setelem.out b/eg/setelem.out --- a/eg/setelem.out +++ b/eg/setelem.out @@ -4,14 +4,14 @@ -- looking through 2 candidates of size 1 -- looking through 3 candidates of size 2 -- looking through 5 candidates of size 3 --- looking through 28 candidates of size 4 --- looking through 73 candidates of size 5 --- looking through 124 candidates of size 6 --- looking through 305 candidates of size 7 --- looking through 999 candidates of size 8 --- tested 826 candidates +-- looking through 20 candidates of size 4 +-- looking through 48 candidates of size 5 +-- looking through 75 candidates of size 6 +-- looking through 149 candidates of size 7 +-- looking through 368 candidates of size 8 +-- tested 396 candidates elem x [] = False -elem x (y:xs) = x == y || elem x xs +elem x (y:xs) = elem x xs || x == y set :: [Int] -> Bool -- testing 360 combinations of argument values @@ -21,10 +21,10 @@ -- looking through 5 candidates of size 3 -- looking through 11 candidates of size 4 -- looking through 20 candidates of size 5 --- looking through 47 candidates of size 6 --- looking through 136 candidates of size 7 --- looking through 348 candidates of size 8 --- tested 256 candidates +-- looking through 41 candidates of size 6 +-- looking through 100 candidates of size 7 +-- looking through 234 candidates of size 8 +-- tested 200 candidates set [] = True -set (x:xs) = not (elem x xs) && set xs +set (x:xs) = set xs && not (elem x xs) diff --git a/eg/sort.out b/eg/sort.out --- a/eg/sort.out +++ b/eg/sort.out @@ -46,16 +46,16 @@ -- pruning with 4/4 rules -- looking through 3 candidates of size 1 -- looking through 7 candidates of size 2 --- looking through 9 candidates of size 3 --- looking through 44 candidates of size 4 --- looking through 116 candidates of size 5 --- looking through 80 candidates of size 6 --- looking through 715 candidates of size 7 --- looking through 164 candidates of size 8 --- looking through 3360 candidates of size 9 --- looking through 1448 candidates of size 10 --- looking through 12901 candidates of size 11 --- looking through 15208 candidates of size 12 --- tested 34055 candidates +-- looking through 11 candidates of size 3 +-- looking through 37 candidates of size 4 +-- looking through 98 candidates of size 5 +-- looking through 94 candidates of size 6 +-- looking through 383 candidates of size 7 +-- looking through 375 candidates of size 8 +-- looking through 1263 candidates of size 9 +-- looking through 2252 candidates of size 10 +-- looking through 4090 candidates of size 11 +-- looking through 13481 candidates of size 12 +-- tested 22094 candidates cannot conjure diff --git a/eg/spec.out b/eg/spec.out --- a/eg/spec.out +++ b/eg/spec.out @@ -1,17 +1,17 @@ square :: Int -> Int -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 9 candidates of size 3 --- tested 12 candidates +-- looking through 3 candidates of size 2 +-- looking through 6 candidates of size 3 +-- tested 9 candidates square x = x * x square :: Int -> Int -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 9 candidates of size 3 --- tested 12 candidates +-- looking through 3 candidates of size 2 +-- looking through 6 candidates of size 3 +-- tested 9 candidates square x = x * x sum :: [Int] -> Int @@ -20,7 +20,7 @@ -- looking through 2 candidates of size 2 -- looking through 3 candidates of size 3 -- looking through 4 candidates of size 4 --- looking through 7 candidates of size 5 +-- looking through 5 candidates of size 5 -- tested 11 candidates sum [] = 0 sum (x:xs) = x + sum xs @@ -29,11 +29,11 @@ -- pruning with 3/3 rules -- looking through 2 candidates of size 1 -- looking through 3 candidates of size 2 --- looking through 10 candidates of size 3 --- looking through 27 candidates of size 4 --- looking through 78 candidates of size 5 --- looking through 171 candidates of size 6 --- tested 194 candidates +-- looking through 11 candidates of size 3 +-- looking through 32 candidates of size 4 +-- looking through 100 candidates of size 5 +-- looking through 244 candidates of size 6 +-- tested 234 candidates [] ++ xs = xs (x:xs) ++ ys = x:(xs ++ ys) diff --git a/eg/subset.out b/eg/subset.out --- a/eg/subset.out +++ b/eg/subset.out @@ -3,15 +3,15 @@ -- pruning with 29/39 rules -- looking through 2 candidates of size 1 -- looking through 4 candidates of size 2 --- looking through 10 candidates of size 3 --- looking through 40 candidates of size 4 --- looking through 160 candidates of size 5 --- looking through 0 candidates of size 6 --- looking through 444 candidates of size 7 --- looking through 784 candidates of size 8 --- tested 1310 candidates +-- looking through 8 candidates of size 3 +-- looking through 34 candidates of size 4 +-- looking through 88 candidates of size 5 +-- looking through 40 candidates of size 6 +-- looking through 138 candidates of size 7 +-- looking through 644 candidates of size 8 +-- tested 790 candidates subset [] xs = True -subset (x:xs) ys = elem x ys && subset xs ys +subset (x:xs) ys = subset xs ys && elem x ys subset :: [Int] -> [Int] -> Bool -- testing 44 combinations of argument values diff --git a/eg/tapps.out b/eg/tapps.out --- a/eg/tapps.out +++ b/eg/tapps.out @@ -3,9 +3,9 @@ -- pruning with 14/25 rules -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 --- looking through 6 candidates of size 3 --- looking through 20 candidates of size 4 --- tested 18 candidates +-- looking through 5 candidates of size 3 +-- looking through 12 candidates of size 4 +-- tested 16 candidates third xs = head (tail (tail xs)) product :: [Int] -> Int @@ -13,10 +13,10 @@ -- pruning with 14/25 rules -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 --- looking through 6 candidates of size 3 --- looking through 20 candidates of size 4 --- looking through 34 candidates of size 5 --- tested 44 candidates +-- looking through 5 candidates of size 3 +-- looking through 12 candidates of size 4 +-- looking through 18 candidates of size 5 +-- tested 30 candidates product [] = 1 product (x:xs) = x * product xs @@ -25,8 +25,8 @@ -- pruning with 15/26 rules -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 --- looking through 6 candidates of size 3 --- looking through 23 candidates of size 4 --- tested 20 candidates +-- looking through 5 candidates of size 3 +-- looking through 15 candidates of size 4 +-- tested 18 candidates product xs = foldr (*) 1 xs diff --git a/eg/tree.hs b/eg/tree.hs --- a/eg/tree.hs +++ b/eg/tree.hs @@ -3,27 +3,14 @@ -- Copyright (C) 2021 Rudy Matela -- Distributed under the 3-Clause BSD licence (see the file LICENSE). {-# LANGUAGE CPP, TemplateHaskell #-} -#if __GLASGOW_HASKELL__ == 708 -{-# LANGUAGE DeriveDataTypeable, StandaloneDeriving #-} -import Data.Typeable (Typeable) -#endif import Conjure import Test.LeanCheck import Data.Express hiding (height,size) --- TODO: remove the following import --- and fix build on GHC 7.10 and 7.8 --- the generation of -: and ->>>: somehow fails. -import Test.LeanCheck.Utils - data Tree = Leaf | Node Tree Int Tree deriving (Eq, Ord, Show, Read) - -#if __GLASGOW_HASKELL__ == 708 -deriving instance Typeable Tree -#endif deriveExpress ''Tree diff --git a/eg/tree.out b/eg/tree.out --- a/eg/tree.out +++ b/eg/tree.out @@ -10,7 +10,9 @@ -- looking through 16 candidates of size 7 -- tested 9 candidates leftmost Leaf = undefined -leftmost (Node t1 x t2) = if nil t1 then x else leftmost t1 +leftmost (Node t1 x t2) = if nil t1 + then x + else leftmost t1 rightmost :: Tree -> Int -- testing 360 combinations of argument values @@ -24,20 +26,22 @@ -- looking through 16 candidates of size 7 -- tested 18 candidates rightmost Leaf = undefined -rightmost (Node t1 x t2) = if nil t2 then x else rightmost t2 +rightmost (Node t1 x t2) = if nil t2 + then x + else rightmost t2 size :: Tree -> Int -- testing 360 combinations of argument values -- pruning with 4/8 rules -- looking through 2 candidates of size 1 -- looking through 2 candidates of size 2 --- looking through 3 candidates of size 3 --- looking through 4 candidates of size 4 --- looking through 9 candidates of size 5 --- looking through 12 candidates of size 6 --- looking through 25 candidates of size 7 --- looking through 40 candidates of size 8 --- tested 71 candidates +-- looking through 2 candidates of size 3 +-- looking through 2 candidates of size 4 +-- looking through 4 candidates of size 5 +-- looking through 8 candidates of size 6 +-- looking through 12 candidates of size 7 +-- looking through 24 candidates of size 8 +-- tested 42 candidates size Leaf = 0 size (Node t1 x t2) = size t1 + (size t2 + 1) @@ -46,15 +50,15 @@ -- pruning with 49/65 rules -- looking through 3 candidates of size 1 -- looking through 3 candidates of size 2 --- looking through 4 candidates of size 3 --- looking through 13 candidates of size 4 --- looking through 30 candidates of size 5 --- looking through 88 candidates of size 6 --- looking through 320 candidates of size 7 --- looking through 1093 candidates of size 8 --- tested 475 candidates +-- looking through 2 candidates of size 3 +-- looking through 6 candidates of size 4 +-- looking through 14 candidates of size 5 +-- looking through 34 candidates of size 6 +-- looking through 98 candidates of size 7 +-- looking through 283 candidates of size 8 +-- tested 202 candidates height Leaf = -1 -height (Node t1 x t2) = max (height t1) (height t2) + 1 +height (Node t1 x t2) = 1 + max (height t1) (height t2) mem :: Int -> Tree -> Bool -- testing 360 combinations of argument values @@ -62,18 +66,18 @@ -- looking through 1 candidates of size 1 -- looking through 0 candidates of size 2 -- looking through 0 candidates of size 3 --- looking through 6 candidates of size 4 +-- looking through 5 candidates of size 4 -- looking through 0 candidates of size 5 -- looking through 0 candidates of size 6 -- looking through 0 candidates of size 7 --- looking through 34 candidates of size 8 +-- looking through 20 candidates of size 8 -- looking through 0 candidates of size 9 -- looking through 0 candidates of size 10 -- looking through 0 candidates of size 11 --- looking through 184 candidates of size 12 --- tested 107 candidates +-- looking through 96 candidates of size 12 +-- tested 92 candidates mem x Leaf = False -mem x (Node t1 y t2) = mem x t1 || (x == y || mem x t2) +mem x (Node t1 y t2) = mem x t1 || (mem x t2 || x == y) ordered :: Tree -> Bool -- testing 360 combinations of argument values @@ -85,12 +89,12 @@ -- looking through 10 candidates of size 5 -- looking through 20 candidates of size 6 -- looking through 0 candidates of size 7 --- looking through 64 candidates of size 8 --- looking through 144 candidates of size 9 --- looking through 112 candidates of size 10 --- looking through 964 candidates of size 11 --- looking through 1480 candidates of size 12 --- tested 2799 candidates +-- looking through 32 candidates of size 8 +-- looking through 80 candidates of size 9 +-- looking through 56 candidates of size 10 +-- looking through 450 candidates of size 11 +-- looking through 708 candidates of size 12 +-- tested 1361 candidates cannot conjure ordered :: Tree -> Bool diff --git a/mk/depend.mk b/mk/depend.mk --- a/mk/depend.mk +++ b/mk/depend.mk @@ -159,6 +159,19 @@ src/Conjure/Conjurable.hs \ src/Conjure/Conjurable/Derive.hs \ bench/terpret.hs +bench/weird: \ + bench/weird.hs \ + mk/toplibs +bench/weird.o: \ + src/Conjure/Utils.hs \ + src/Conjure/Prim.hs \ + src/Conjure.hs \ + src/Conjure/Expr.hs \ + src/Conjure/Engine.hs \ + src/Conjure/Defn.hs \ + src/Conjure/Conjurable.hs \ + src/Conjure/Conjurable/Derive.hs \ + bench/weird.hs eg/arith: \ eg/arith.hs \ mk/toplibs @@ -276,6 +289,32 @@ src/Conjure/Conjurable.hs \ src/Conjure/Conjurable/Derive.hs \ eg/gcd.hs +eg/higher: \ + eg/higher.hs \ + mk/toplibs +eg/higher.o: \ + src/Conjure/Utils.hs \ + src/Conjure/Prim.hs \ + src/Conjure.hs \ + src/Conjure/Expr.hs \ + src/Conjure/Engine.hs \ + src/Conjure/Defn.hs \ + src/Conjure/Conjurable.hs \ + src/Conjure/Conjurable/Derive.hs \ + eg/higher.hs +eg/id: \ + eg/id.hs \ + mk/toplibs +eg/id.o: \ + src/Conjure/Utils.hs \ + src/Conjure/Prim.hs \ + src/Conjure.hs \ + src/Conjure/Expr.hs \ + src/Conjure/Engine.hs \ + src/Conjure/Defn.hs \ + src/Conjure/Conjurable.hs \ + src/Conjure/Conjurable/Derive.hs \ + eg/id.hs eg/ints: \ eg/ints.hs \ mk/toplibs diff --git a/src/Conjure/Conjurable.hs b/src/Conjure/Conjurable.hs --- a/src/Conjure/Conjurable.hs +++ b/src/Conjure/Conjurable.hs @@ -26,7 +26,6 @@ , conjureAreEqual , conjureMkEquation , A, B, C, D, E, F - , conjureIsDeconstructor , conjureIsDeconstruction , candidateDeconstructionsFrom , candidateDeconstructionsFromHoled @@ -34,11 +33,14 @@ , conjureReification , conjureReification1 , conjureDynamicEq + , conjureIsNumeric , cevaluate , ceval , cevl , Name (..) , Express (..) + , conjureArgumentPats + , conjureMostGeneralCanonicalVariation ) where @@ -259,9 +261,11 @@ -- > conjureExpress = reifyExpress -- > ... reifyExpress :: (Express a, Show a) => a -> Expr -> Expr -reifyExpress a e = case value "expr" (expr -:> a) $$ e of - Nothing -> e -- TODO: consider throwing an error - Just e' -> eval e e' -- TODO: consider throwing an error +reifyExpress a e = case exprE $$ e of + Nothing -> e -- identity needed for types such as functions + Just e' -> eval (error $ "reifyExpress: cannot eval " ++ show e') e' + where + exprE = value "expr" (expr -:> a) mkExprTiers :: (Listable a, Show a, Typeable a) => a -> [[Expr]] mkExprTiers a = mapT val (tiers -: [[a]]) @@ -307,6 +311,13 @@ ((e':_):_) | typ e' == typ e -> etiers _ -> tf etc +conjureIsNumeric :: Conjurable f => f -> Expr -> Bool +conjureIsNumeric f e = case concat $ conjureTiersFor f e of + -- We assume tiers of numeric values start with 0 + -- not so unfair... + (Value "0" _):_ -> True + _ -> False + -- | Compute variable names for the given 'Expr' type. conjureNamesFor :: Conjurable f => f -> Expr -> [String] conjureNamesFor f e = head @@ -317,45 +328,27 @@ conjureMostGeneralCanonicalVariation f = canonicalizeWith (conjureNamesFor f) . fastMostGeneralVariation --- | Checks if an unary function encoded as an 'Expr' is a deconstructor. --- --- (cf. 'conjureIsDeconstruction') -conjureIsDeconstructor :: Conjurable f => f -> Int -> Expr -> Bool -conjureIsDeconstructor f maxTests e = case as of - [] -> False - (h:_) -> isDec h - where - as = [h | h <- hs, isWellTyped (e:$h), typ (e:$h) == typ h] - hs = conjureArgumentHoles f - isDec h = count is gs >= length gs `div` 2 - where - gs = take maxTests $ grounds (conjureTiersFor f) h - sz = head [sz | (_, _, _, _, _, sz) <- conjureReification f - , isWellTyped (sz :$ h)] - esz e = eval (0::Int) (sz :$ e) - is e' = errorToFalse $ esz (e :$ e') < esz e' - -- | Checks if an expression is a deconstruction. -- -- There should be a single 'hole' in the expression. -- --- 1. The result does not increase the size for at least half the time. --- 2. The result decreases in size for at least a third of the time. +-- It should decrease the size of all arguments that have +-- a size greater than 0. -- -- (cf. 'conjureIsDeconstructor') conjureIsDeconstruction :: Conjurable f => f -> Int -> Expr -> Bool conjureIsDeconstruction f maxTests ed = length (holes ed) == 1 && typ h == typ ed - && count is gs >= length gs `div` 2 - && count iz gs >= length gs `div` 3 + && all is gs where gs = take maxTests $ grounds (conjureTiersFor f) ed [h] = holes ed sz = head [sz | (_, _, _, _, _, sz) <- conjureReification f , isWellTyped (sz :$ h)] esz e = eval (0::Int) (sz :$ e) - is e = errorToFalse $ esz e <= esz (holeValue e) - iz e = errorToFalse $ esz e < esz (holeValue e) + x << 0 = True + x << y = x < y + is e = errorToFalse $ esz e << esz (holeValue e) holeValue e = fromMaybe err . lookup h . fromMaybe err @@ -429,13 +422,20 @@ conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = abs + conjureSize = size where size x | x < 0 = 0 + | otherwise = x +-- allows easy modification of the global size function for integer values +-- duplicated above in the Int instance for performance reasons +integralSize :: Integral a => a -> Int +integralSize = fromIntegral . size where size x | x < 0 = 0 + | otherwise = x + instance Conjurable Integer where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable Char where conjureExpress = reifyExpress @@ -525,7 +525,7 @@ , value "Just" (Just ->: mx) :$ hole x ] where - x = Just undefined -: mx + Just x = undefined -: mx conjureEquality mx = from <$> conjureEquality x where x = fromJust mx @@ -649,23 +649,80 @@ where (nf:nas) = words nm ++ repeat "" + -- | Computes tiers of sets of patterns for the given function. -- -- > > conjurePats [zero] "f" (undefined :: Int -> Int) -- > [[[f x :: Int]],[[f 0 :: Int,f x :: Int]]] conjurePats :: Conjurable f => [Expr] -> String -> f -> [[ [Expr] ]] -conjurePats es nm f = mapT (map mkApp . prods) $ cs +conjurePats es nm f = mapT (map mkApp) + $ combinePatternOptions + $ conjureArgumentPats es f where mkApp = foldApp . (ef:) . unfold . conjureMostGeneralCanonicalVariation f . fold ef = var (head $ words nm) f -- TODO: take the tail into account - cs = products $ zipWith mk (conjureArgumentHoles f) (conjureArgumentCases f) + + -- What a horrible enumeration hack below... + -- Good luck to anyone who plans to refactor this. + + -- after application of mkApp, we end up w/ [[ [Pat ] ]] + combinePatternOptions :: [ [[ [Expr] ]] ] -> [[ [[Expr]] ]] + combinePatternOptions [] = [[[[]]]] + combinePatternOptions (esss:essss) = concatPrefixesWithT esss + $ combinePatternOptions essss + + -- The three functions below are all transformations over the same type + -- [[ [[Expr]] ]] + -- That's tiers of complete LHS of function definitions. + -- We proceed right to left building all possible patterns. + + -- concatenates all of the possibilities of prefixing + -- from tiers of possibilities + concatPrefixesWithT :: [[[Expr]]] -> [[ [[Expr]] ]] -> [[ [[Expr]] ]] + concatPrefixesWithT esss r = concatMapT (`concatPrefixesWith` r) esss + + -- concatenates the possibilities of prefixing from a list of prefixes + concatPrefixesWith :: [Expr] -> [[ [[Expr]] ]] -> [[ [[Expr]] ]] + concatPrefixesWith es r = mapT concat $ products [prefixWith e r | e <- es] + + -- prefixes with the given expression + prefixWith :: Expr -> [[ [[Expr]] ]] -> [[ [[Expr]] ]] + prefixWith e = mapT (map (e:)) + + -- this was useful in figuring out the implementations + -- of the local functions above + -- + -- > prefixWith2 :: Expr -> Expr -> [[Bs]] -> [[Bs]] + -- > prefixWith2 e1 e2 r = productWith (++) (mapT (map (e1:)) r) + -- > (mapT (map (e2:)) r) + -- + -- A prefixWith3 would follow similarly. + + -- To debug the above function use: + -- > import Test.LeanCheck.Tiers (printTiers) + -- > printTiers 100 $ newConjurePats [zero] "f" (undefined :: Int -> Int -> Int) + + +-- | Returns a list of tiers of possible patterns for each argument. +-- +-- The outer list has the same number of elements as the number of arguments +-- of the given function. +-- +-- This function is internal and only used in the implementation of 'conjurePats'. +-- It may be removed from the API without further notice. +-- It has been temporarily promoted to public to help refactor 'conjurePats'. +conjureArgumentPats :: Conjurable f => [Expr] -> f -> [ [[ [Expr] ]] ] +conjureArgumentPats es f = zipWith mk (conjureArgumentHoles f) (conjureArgumentCases f) + where + -- deal with types that have no cases such as ints. mk h [] = mapT (++ [h]) $ setsOf [[e] | e <- es, typ e == typ h] + -- deal with types that have cases, such as lists, maybes, etc. mk h cs = [[[h]], [cs]] - tiersFor = conjureTiersFor f + prods :: [[a]] -> [[a]] prods = foldr (productWith (:)) [[]] where @@ -695,55 +752,55 @@ conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable Int16 where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable Int32 where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable Int64 where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable Word where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable Word8 where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable Word16 where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable Word32 where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable Word64 where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance (Integral a, Conjurable a, Listable a, Show a, Eq a, Express a) => Conjurable (Ratio a) where conjureExpress = reifyExpress @@ -769,37 +826,37 @@ conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable B where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable C where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable D where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable E where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize instance Conjurable F where conjureExpress = reifyExpress conjureEquality = reifyEquality conjureTiers = reifyTiers - conjureSize = fromIntegral . abs + conjureSize = integralSize -- Conjurable tuples -- @@ -926,7 +983,7 @@ . conjureType v . conjureType u . conjureType t - where (x,y,z,w,v,u,t) = xyzwvut + where (x,y,z,w,v,u,t) = xyzwvut conjureEquality xyzwvut = from <$> conjureEquality x <*> conjureEquality y @@ -954,7 +1011,308 @@ && u1 .....==. u2 && t1 ......== t2 --- TODO: go up to 12-tuples +instance ( Conjurable a, Listable a, Show a, Express a + , Conjurable b, Listable b, Show b, Express b + , Conjurable c, Listable c, Show c, Express c + , Conjurable d, Listable d, Show d, Express d + , Conjurable e, Listable e, Show e, Express e + , Conjurable f, Listable f, Show f, Express f + , Conjurable g, Listable g, Show g, Express g + , Conjurable h, Listable h, Show h, Express h + ) => Conjurable (a,b,c,d,e,f,g,h) where + conjureExpress = reifyExpress + conjureTiers = reifyTiers + conjureSubTypes xyzwvuts = conjureType x + . conjureType y + . conjureType z + . conjureType w + . conjureType v + . conjureType u + . conjureType t + . conjureType s + where (x,y,z,w,v,u,t,s) = xyzwvuts + conjureEquality xyzwvuts = from + <$> conjureEquality x + <*> conjureEquality y + <*> conjureEquality z + <*> conjureEquality w + <*> conjureEquality v + <*> conjureEquality u + <*> conjureEquality t + <*> conjureEquality s + where + (x,y,z,w,v,u,t,s) = xyzwvuts + from e1 e2 e3 e4 e5 e6 e7 e8 = value "==" (==) + where + (==.......) = evl e1 ==: x + (.==......) = evl e2 ==: y + (..==.....) = evl e3 ==: z + (...==....) = evl e4 ==: w + (....==...) = evl e5 ==: v + (.....==..) = evl e6 ==: u + (......==.) = evl e7 ==: t + (.......==) = evl e8 ==: s + (x1,y1,z1,w1,v1,u1,t1,s1) == (x2,y2,z2,w2,v2,u2,t2,s2) = x1 ==....... x2 + && y1 .==...... y2 + && z1 ..==..... z2 + && w1 ...==.... w2 + && v1 ....==... v2 + && u1 .....==.. u2 + && t1 ......==. t2 + && s1 .......== s2 + +instance ( Conjurable a, Listable a, Show a, Express a + , Conjurable b, Listable b, Show b, Express b + , Conjurable c, Listable c, Show c, Express c + , Conjurable d, Listable d, Show d, Express d + , Conjurable e, Listable e, Show e, Express e + , Conjurable f, Listable f, Show f, Express f + , Conjurable g, Listable g, Show g, Express g + , Conjurable h, Listable h, Show h, Express h + , Conjurable i, Listable i, Show i, Express i + ) => Conjurable (a,b,c,d,e,f,g,h,i) where + conjureExpress = reifyExpress + conjureTiers = reifyTiers + conjureSubTypes xyzwvutsr = conjureType x + . conjureType y + . conjureType z + . conjureType w + . conjureType v + . conjureType u + . conjureType t + . conjureType s + . conjureType r + where (x,y,z,w,v,u,t,s,r) = xyzwvutsr + conjureEquality xyzwvutsr = from + <$> conjureEquality x + <*> conjureEquality y + <*> conjureEquality z + <*> conjureEquality w + <*> conjureEquality v + <*> conjureEquality u + <*> conjureEquality t + <*> conjureEquality s + <*> conjureEquality r + where + (x,y,z,w,v,u,t,s,r) = xyzwvutsr + from e1 e2 e3 e4 e5 e6 e7 e8 e9 = value "==" (==) + where + (==........) = evl e1 ==: x + (.==.......) = evl e2 ==: y + (..==......) = evl e3 ==: z + (...==.....) = evl e4 ==: w + (....==....) = evl e5 ==: v + (.....==...) = evl e6 ==: u + (......==..) = evl e7 ==: t + (.......==.) = evl e8 ==: s + (........==) = evl e9 ==: r + (x1,y1,z1,w1,v1,u1,t1,s1,r1) == (x2,y2,z2,w2,v2,u2,t2,s2,r2) = x1 ==........ x2 + && y1 .==....... y2 + && z1 ..==...... z2 + && w1 ...==..... w2 + && v1 ....==.... v2 + && u1 .....==... u2 + && t1 ......==.. t2 + && s1 .......==. s2 + && r1 ........== r2 + +instance ( Conjurable a, Listable a, Show a, Express a + , Conjurable b, Listable b, Show b, Express b + , Conjurable c, Listable c, Show c, Express c + , Conjurable d, Listable d, Show d, Express d + , Conjurable e, Listable e, Show e, Express e + , Conjurable f, Listable f, Show f, Express f + , Conjurable g, Listable g, Show g, Express g + , Conjurable h, Listable h, Show h, Express h + , Conjurable i, Listable i, Show i, Express i + , Conjurable j, Listable j, Show j, Express j + ) => Conjurable (a,b,c,d,e,f,g,h,i,j) where + conjureExpress = reifyExpress + conjureTiers = reifyTiers + conjureSubTypes xyzwvutsrq = conjureType x + . conjureType y + . conjureType z + . conjureType w + . conjureType v + . conjureType u + . conjureType t + . conjureType s + . conjureType r + . conjureType q + where (x,y,z,w,v,u,t,s,r,q) = xyzwvutsrq + conjureEquality xyzwvutsrq = from + <$> conjureEquality x + <*> conjureEquality y + <*> conjureEquality z + <*> conjureEquality w + <*> conjureEquality v + <*> conjureEquality u + <*> conjureEquality t + <*> conjureEquality s + <*> conjureEquality r + <*> conjureEquality q + where + (x,y,z,w,v,u,t,s,r,q) = xyzwvutsrq + from e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 = value "==" (==) + where + (==.........) = evl e1 ==: x + (.==........) = evl e2 ==: y + (..==.......) = evl e3 ==: z + (...==......) = evl e4 ==: w + (....==.....) = evl e5 ==: v + (.....==....) = evl e6 ==: u + (......==...) = evl e7 ==: t + (.......==..) = evl e8 ==: s + (........==.) = evl e9 ==: r + (.........==) = evl e10 ==: q + (x1,y1,z1,w1,v1,u1,t1,s1,r1,q1) == (x2,y2,z2,w2,v2,u2,t2,s2,r2,q2) = x1 ==......... x2 + && y1 .==........ y2 + && z1 ..==....... z2 + && w1 ...==...... w2 + && v1 ....==..... v2 + && u1 .....==.... u2 + && t1 ......==... t2 + && s1 .......==.. s2 + && r1 ........==. r2 + && q1 .........== q2 + + +instance ( Conjurable a, Listable a, Show a, Express a + , Conjurable b, Listable b, Show b, Express b + , Conjurable c, Listable c, Show c, Express c + , Conjurable d, Listable d, Show d, Express d + , Conjurable e, Listable e, Show e, Express e + , Conjurable f, Listable f, Show f, Express f + , Conjurable g, Listable g, Show g, Express g + , Conjurable h, Listable h, Show h, Express h + , Conjurable i, Listable i, Show i, Express i + , Conjurable j, Listable j, Show j, Express j + , Conjurable k, Listable k, Show k, Express k + ) => Conjurable (a,b,c,d,e,f,g,h,i,j,k) where + conjureExpress = reifyExpress + conjureTiers = reifyTiers + conjureSubTypes xyzwvutsrqp = conjureType x + . conjureType y + . conjureType z + . conjureType w + . conjureType v + . conjureType u + . conjureType t + . conjureType s + . conjureType r + . conjureType q + . conjureType p + where (x,y,z,w,v,u,t,s,r,q,p) = xyzwvutsrqp + conjureEquality xyzwvutsrqp = from + <$> conjureEquality x + <*> conjureEquality y + <*> conjureEquality z + <*> conjureEquality w + <*> conjureEquality v + <*> conjureEquality u + <*> conjureEquality t + <*> conjureEquality s + <*> conjureEquality r + <*> conjureEquality q + <*> conjureEquality p + where + (x,y,z,w,v,u,t,s,r,q,p) = xyzwvutsrqp + from e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 = value "==" (==) + where + (==..........) = evl e1 ==: x + (.==.........) = evl e2 ==: y + (..==........) = evl e3 ==: z + (...==.......) = evl e4 ==: w + (....==......) = evl e5 ==: v + (.....==.....) = evl e6 ==: u + (......==....) = evl e7 ==: t + (.......==...) = evl e8 ==: s + (........==..) = evl e9 ==: r + (.........==.) = evl e10 ==: q + (..........==) = evl e11 ==: p + (x1,y1,z1,w1,v1,u1,t1,s1,r1,q1,p1) == (x2,y2,z2,w2,v2,u2,t2,s2,r2,q2,p2) = x1 ==.......... x2 + && y1 .==......... y2 + && z1 ..==........ z2 + && w1 ...==....... w2 + && v1 ....==...... v2 + && u1 .....==..... u2 + && t1 ......==.... t2 + && s1 .......==... s2 + && r1 ........==.. r2 + && q1 .........==. q2 + && p1 ..........== p2 + + +instance ( Conjurable a, Listable a, Show a, Express a + , Conjurable b, Listable b, Show b, Express b + , Conjurable c, Listable c, Show c, Express c + , Conjurable d, Listable d, Show d, Express d + , Conjurable e, Listable e, Show e, Express e + , Conjurable f, Listable f, Show f, Express f + , Conjurable g, Listable g, Show g, Express g + , Conjurable h, Listable h, Show h, Express h + , Conjurable i, Listable i, Show i, Express i + , Conjurable j, Listable j, Show j, Express j + , Conjurable k, Listable k, Show k, Express k + , Conjurable l, Listable l, Show l, Express l + ) => Conjurable (a,b,c,d,e,f,g,h,i,j,k,l) where + conjureExpress = reifyExpress + conjureTiers = reifyTiers + conjureSubTypes xyzwvutsrqpo = conjureType x + . conjureType y + . conjureType z + . conjureType w + . conjureType v + . conjureType u + . conjureType t + . conjureType s + . conjureType r + . conjureType q + . conjureType p + . conjureType o + where (x,y,z,w,v,u,t,s,r,q,p,o) = xyzwvutsrqpo + conjureEquality xyzwvutsrqpo = from + <$> conjureEquality x + <*> conjureEquality y + <*> conjureEquality z + <*> conjureEquality w + <*> conjureEquality v + <*> conjureEquality u + <*> conjureEquality t + <*> conjureEquality s + <*> conjureEquality r + <*> conjureEquality q + <*> conjureEquality p + <*> conjureEquality o + where + (x,y,z,w,v,u,t,s,r,q,p,o) = xyzwvutsrqpo + from e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 = value "==" (==) + where + (==...........) = evl e1 ==: x + (.==..........) = evl e2 ==: y + (..==.........) = evl e3 ==: z + (...==........) = evl e4 ==: w + (....==.......) = evl e5 ==: v + (.....==......) = evl e6 ==: u + (......==.....) = evl e7 ==: t + (.......==....) = evl e8 ==: s + (........==...) = evl e9 ==: r + (.........==..) = evl e10 ==: q + (..........==.) = evl e11 ==: p + (...........==) = evl e12 ==: o + (x1,y1,z1,w1,v1,u1,t1,s1,r1,q1,p1,o1) == (x2,y2,z2,w2,v2,u2,t2,s2,r2,q2,p2,o2) = x1 ==........... x2 + && y1 .==.......... y2 + && z1 ..==......... z2 + && w1 ...==........ w2 + && v1 ....==....... v2 + && u1 .....==...... u2 + && t1 ......==..... t2 + && s1 .......==.... s2 + && r1 ........==... r2 + && q1 .........==.. q2 + && p1 ..........==. p2 + && o1 ...........== o2 + instance Name A instance Name B diff --git a/src/Conjure/Conjurable/Derive.hs b/src/Conjure/Conjurable/Derive.hs --- a/src/Conjure/Conjurable/Derive.hs +++ b/src/Conjure/Conjurable/Derive.hs @@ -127,10 +127,6 @@ conjureTiers = reifyTiers |] -- withTheReturnTypeOfs |++| (cxt |=>| inst) cxt |=>| inst `addFun` deriveSize t `mergeI` deriveSubTypes t `mergeI` deriveCases t --- TODO: derive conjureCases, e.g.: --- conjureCases mx = [ value "Nothing" (Nothing -: mx) --- , value "Just" (Just ->: mx) :$ hole x --- ] deriveCases :: Name -> DecsQ deriveCases t = do diff --git a/src/Conjure/Defn.hs b/src/Conjure/Defn.hs --- a/src/Conjure/Defn.hs +++ b/src/Conjure/Defn.hs @@ -19,8 +19,26 @@ , devl , devalFast , showDefn + , printDefn , defnApparentlyTerminates , isRedundantDefn + , isRedundantBySubsumption + , isRedundantByRepetition + , isRedundantByIntroduction + , hasRedundantRecursion + , isCompleteDefn + , isCompleteBndn + , simplifyDefn + , canonicalizeBndn + , canonicalizeBndnLast + , hasUnbound + , noUnbound + , isUndefined + , isDefined + , introduceVariableAt + , isBaseCase + , isRecursiveCase + , isRecursiveDefn , module Conjure.Expr ) where @@ -61,8 +79,22 @@ showDefn :: Defn -> String showDefn = unlines . map show1 where + show1 (lhs,Value "if" _ :$ c :$ t :$ e) = lhseqs ++ "if " ++ showExpr c + ++ "\n" ++ spaces ++ "then " ++ showExpr t + ++ "\n" ++ spaces ++ "else " ++ showExpr e + where + lhseqs = showExpr lhs ++ " = " + spaces = map (const ' ') lhseqs show1 (lhs,rhs) = showExpr lhs ++ " = " ++ showExpr rhs +-- | Pretty-prints a 'Defn' to the screen. +-- +-- > > printDefn sumDefn +-- > sum [] = 0 +-- > sum (x:xs) = x + sum xs +printDefn :: Defn -> IO () +printDefn = putStr . showDefn + type Memo = [(Expr, Maybe Dynamic)] -- | Evaluates an 'Expr' to a 'Dynamic' value @@ -198,38 +230,250 @@ defnApparentlyTerminates _ = True -- | Returns whether the given 'Defn' is redundant --- with regards to its patterns. +-- with regards to repetitions on RHSs. -- -- Here is an example of a redundant 'Defn': -- --- > 0 ? 0 = 0 --- > 0 ? x = 0 +-- > 0 ? 0 = 1 +-- > 0 ? x = 1 -- > x ? 0 = x -- > x ? y = x -- -- It is redundant because it is equivalent to: -- --- > 0 ? _ = 0 +-- > 0 ? _ = 1 -- > x ? _ = x -- --- If the given expression is incomplete ('hasHole') --- this function returns 'True' as nothing can be said. +-- This function safely handles holes on the RHSs +-- by being conservative in cases these are found: +-- nothing can be said before their fillings. isRedundantDefn :: Defn -> Bool -isRedundantDefn d = all isComplete (map snd d) - && any isRedundant1 (unfoldDefnArgs d) +isRedundantDefn d = isRedundantBySubsumption d + || isRedundantByRepetition d +-- || isRedundantByIntroduction d +-- we do not use isRedundantByIntroduction above +-- as it does not pay off in terms of runtime vs number of pruned candidates +-- +-- The number of candidates is reduced usually by less than 1% +-- and the runtime increases by 50% or sometimes 100%. + +-- | Returns whether the given 'Defn' is redundant +-- with regards to repetitions on RHSs. +-- +-- Here is an example of a redundant 'Defn': +-- +-- > 0 ? 0 = 1 +-- > 0 ? x = 1 +-- > x ? 0 = x +-- > x ? y = x +-- +-- It is redundant because it is equivalent to: +-- +-- > 0 ? _ = 1 +-- > x ? _ = x +-- +-- @1@ and @x@ are repeated in the results for when +-- the first arguments are @0@ and @x@. +isRedundantByRepetition :: Defn -> Bool +isRedundantByRepetition d = any anyAllEqual shovels where - isRedundant1 :: [(Expr,Expr)] -> Bool - isRedundant1 = all allEqual - . map (map snd) - . classifyOn fst - . map (unfoldPair . canonicalize . foldPair) + nArgs = length . tail . unfoldApp . fst $ head d + shovels :: [Expr -> Expr] + shovels = [digApp n | n <- [1..nArgs]] + anyAllEqual :: (Expr -> Expr) -> Bool + anyAllEqual shovel = any (\bs -> allEqual2 bs && isDefined bs) + . classifyOn fst + . map (canonicalizeBndn . first shovel) + $ d - -- Returns a list of degenerate 'Defn' - -- whose patterns have been "lensed" on each argument. - unfoldDefnArgs :: Defn -> [[(Expr,Expr)]] - unfoldDefnArgs = transpose . map unfoldBndnArgs - where - unfoldBndnArgs :: Bndn -> [(Expr,Expr)] - unfoldBndnArgs (p,r) = map (\a -> (a,r)) as - where - (_:as) = unfoldApp p +-- | Returns whether the given 'Defn' is redundant +-- with regards to case elimination +-- +-- The following is redundant according to this criterium: +-- +-- > foo [] = [] +-- > foo (x:xs) = x:xs +-- +-- It is equivalent to: +-- +-- > foo xs = xs +-- +-- The following is also redundant: +-- +-- > [] ?? xs = [] +-- > (x:xs) ?? ys = x:xs +-- +-- as it is equivalent to: +-- +-- > xs ?? ys == xs +-- +-- This function is not used as one of the criteria in 'isRedundantDefn' +-- because it does not pay-off +-- in terms of runtime vs number of pruned candidates. +isRedundantByIntroduction :: Defn -> Bool +isRedundantByIntroduction d = any anyAllEqual [1..nArgs] + where + nArgs = length . tail . unfoldApp . fst $ head d + anyAllEqual :: Int -> Bool + anyAllEqual i = any (\bs -> length bs >= 2 && isDefined bs && noConflicts i bs) + . classifyOn (digApp i . fst) + . map (canonicalizeBndnLast i) + $ d + noConflicts :: Int -> [Bndn] -> Bool + noConflicts i bs = case listConflicts (map snd bs) of + [] -> True + [es] -> es == [efxs $!! i | (efxs,_) <- bs] + _ -> False + +-- | Returns whether the given 'Defn' is redundant +-- with regards to recursions +-- +-- The following is redundant: +-- +-- > xs ?? [] = [] +-- > xs ?? (x:ys) = xs ?? [] +-- +-- The LHS of a base-case pattern, matches the RHS of a recursive pattern. +-- The second RHS may be replaced by simply @[]@ which makes it redundant. +hasRedundantRecursion :: Defn -> Bool +hasRedundantRecursion d = not (null rs) && any matchesRHS bs + where + (bs,rs) = partition isBaseCase d + matchesRHS (lhs,_) = any ((`hasAppInstanceOf` lhs) . snd) rs + +-- | Introduces a hole at a given position in the binding: +-- +-- > > introduceVariableAt 1 (xxs -?- (yy -:- yys), (yy -:- yys) -++- (yy -:- yys)) +-- > (xs ? (y:ys) :: [Int],(y:ys) ++ (y:ys) :: [Int]) +-- +-- > > introduceVariableAt 2 (xxs -?- (yy -:- yys), (yy -:- yys) -++- (yy -:- yys)) +-- > (xs ? x :: [Int],x ++ x :: [Int]) +-- +-- Relevant occurrences are replaced. +introduceVariableAt :: Int -> Bndn -> Bndn +introduceVariableAt i b@(l,r) + | isVar p = b -- already a variable +-- | isGround p = (newVar, r) -- enabling catches a different set of candidates + | otherwise = unfoldPair + $ foldPair b // [(p,newVar)] + where + p = l $!! i + newVar = newName `varAsTypeOf` p + newName = head $ variableNamesFromTemplate "x" \\ varnames l + varnames :: Expr -> [String] + varnames e = [n | Value ('_':n) _ <- vars e] + +-- | Returns whether the given 'Defn' is redundant +-- with regards to subsumption by latter patterns +-- +-- Here is an example of a redundant 'Defn' by this criterium: +-- +-- > foo 0 = 0 +-- > foo x = x +isRedundantBySubsumption :: Defn -> Bool +isRedundantBySubsumption = is . map foldPair . filter isCompleteBndn + -- above, we could have used noUnbound instead of isCompleteBndn + -- we use isCompleteBndn as it is faster + where + is [] = False + is (b:bs) = any (b `isInstanceOf`) bs || is bs + +-- | Returns whether the definition is complete, +-- i.e., whether it does not have any holes in the RHS. +isCompleteDefn :: Defn -> Bool +isCompleteDefn = all isCompleteBndn + +-- | Returns whether the binding is complete, +-- i.e., whether it does not have any holes in the RHS. +isCompleteBndn :: Bndn -> Bool +isCompleteBndn (_,rhs) = isComplete rhs + +-- | Simplifies a definition by removing redundant patterns +-- +-- This may be useful in the following case: +-- +-- > 0 ^^^ 0 = 0 +-- > 0 ^^^ x = x +-- > x ^^^ 0 = x +-- > _ ^^^ _ = 0 +-- +-- The first pattern is subsumed by the last pattern. +simplifyDefn :: Defn -> Defn +simplifyDefn [] = [] +simplifyDefn (b:bs) = [b | none (foldPair b `isInstanceOf`) $ map foldPair bs] + ++ simplifyDefn bs + +canonicalizeBndn :: Bndn -> Bndn +canonicalizeBndn = unfoldPair . canonicalize . foldPair + +canonicalizeBndnLast :: Int -> Bndn -> Bndn +canonicalizeBndnLast i (lhs,rhs) = (updateAppAt i (const ex') lhs', rhs') + where + (lhs_, ex) = extractApp i lhs + (lhs', ex', rhs') = unfoldTrio . canonicalize . foldTrio $ (lhs_, ex, rhs) + +-- | Returns whether a binding has undefined variables, +-- i.e., +-- there are variables in the RHS that are not declared in the LHS. +-- +-- > > hasUnbound (xx -:- xxs, xxs) +-- > False +-- +-- > > hasUnbound (xx -:- xxs, yys) +-- > True +-- +-- For 'Defn's, use 'isUndefined'. +hasUnbound :: Bndn -> Bool +hasUnbound = not . noUnbound + +noUnbound :: Bndn -> Bool +noUnbound (lhs,rhs) = all (`elem` nubVars lhs) (vars rhs) + +-- | Returns whether a 'Defn' has undefined variables, +-- i.e., +-- there are variables in the RHSs that are not declared in the LHSs. +-- +-- > > isUndefined [(nil, nil), (xx -:- xxs, xxs)] +-- > False +-- +-- > > isUndefined [(nil, xxs), (xx -:- xxs, yys)] +-- > True +-- +-- For single 'Bndn's, use 'hasUnbound'. +isUndefined :: Defn -> Bool +isUndefined = any hasUnbound + +isDefined :: Defn -> Bool +isDefined = not . isUndefined + +-- | Returns whether a binding is a base case. +-- +-- > > isBaseCase (ff (xx -:- nil), xx) +-- > True +-- +-- > > isBaseCase (ff (xx -:- xxs), ff xxs) +-- > False +-- +-- (cf. 'isRecursiveCase') +isBaseCase :: Bndn -> Bool +isBaseCase (lhs,rhs) = f `notElem` values rhs + where + (f:_) = unfoldApp lhs + +-- | Returns whether a binding is a base case. +-- +-- > > isRecursiveCase (ff (xx -:- nil), xx) +-- > False +-- +-- > > isRecursiveCase (ff (xx -:- xxs), ff xxs) +-- > True +-- +-- (cf. 'isBaseCase') +isRecursiveCase :: Bndn -> Bool +isRecursiveCase (lhs,rhs) = f `elem` values rhs + where + (f:_) = unfoldApp lhs + +-- | Returns whether a definition is recursive +isRecursiveDefn :: Defn -> Bool +isRecursiveDefn = any isRecursiveCase diff --git a/src/Conjure/Engine.hs b/src/Conjure/Engine.hs --- a/src/Conjure/Engine.hs +++ b/src/Conjure/Engine.hs @@ -166,6 +166,7 @@ , requireDescent :: Bool -- ^ require recursive calls to deconstruct arguments , usePatterns :: Bool -- ^ use pattern matching to create (recursive) candidates , copyBindings :: Bool -- ^ copy partial definition bindings in candidates + , atomicNumbers :: Bool -- ^ restrict constant/ground numeric expressions to atoms , showTheory :: Bool -- ^ show theory discovered by Speculate used in pruning , uniqueCandidates :: Bool -- ^ unique-modulo-testing candidates } @@ -193,6 +194,7 @@ , requireDescent = True , usePatterns = True , copyBindings = True + , atomicNumbers = True , showTheory = False , uniqueCandidates = False } @@ -358,9 +360,8 @@ -- | Return apparently unique candidate definitions -- where there is a single body. candidateDefns1 :: Conjurable f => Args -> String -> f -> [Prim] -> ([[Defn]], Thy) -candidateDefns1 args nm f ps = mapFst (mapT toDefn) $ candidateExprs args nm f ps +candidateDefns1 args nm f ps = first (mapT toDefn) $ candidateExprs args nm f ps where - mapFst f (x,y) = (f x, y) efxs = conjureVarApplication nm f toDefn e = [(efxs, e)] @@ -385,8 +386,7 @@ eh = holeAsTypeOf efxs efxs = conjureVarApplication nm f (ef:exs) = unfoldApp efxs - keep = isRootNormalE thy . fastMostGeneralVariation - ds = filter (conjureIsDeconstructor f maxTests) es + keep = isRootNormalC thy . fastMostGeneralVariation keepR | requireDescent = descends isDecOf efxs | otherwise = const True where @@ -414,7 +414,7 @@ -- | Return apparently unique candidate definitions -- using pattern matching. candidateDefnsC :: Conjurable f => Args -> String -> f -> [Prim] -> ([[Defn]], Thy) -candidateDefnsC Args{..} nm f ps = (concatMapT fillingsFor fss,thy) +candidateDefnsC Args{..} nm f ps = (discardT hasRedundantRecursion $ concatMapT fillingsFor fss,thy) where pats = conjurePats es nm f fss = concatMapT ps2fss pats @@ -424,14 +424,19 @@ efxs = conjureVarApplication nm f (ef:exs) = unfoldApp efxs - keep = isRootNormal thy . fastMostGeneralVariation + keep = isRootNormalC thy . fastMostGeneralVariation appsWith :: Expr -> [Expr] -> [[Expr]] - appsWith eh vs = enumerateAppsFor eh keep $ vs ++ es + appsWith eh vs = enumerateAppsFor eh k $ vs ++ es + where + k | atomicNumbers && conjureIsNumeric f eh = \e -> keepNumeric e && keep e + | otherwise = keep + -- discards non-atomic numeric ground expressions such as 1 + 1 + keepNumeric e = isFun e || isConst e || not (isGround e) ps2fss :: [Expr] -> [[Defn]] ps2fss pats = discardT isRedundantDefn - . discardT (allEqual . map snd) + . discardT (allEqual2 . map snd) . products $ map p2eess pats where @@ -638,6 +643,16 @@ where trie = T.fromList (rules thy) +-- the logic of this function is a bit twisted for performance +-- but nevertheless correct +isRootNormalC :: Thy -> Expr -> Bool +isRootNormalC thy e | not (isRootNormal thy e) = False +isRootNormalC thy (ef :$ ex :$ ey) + | ex <= ey = True + | not (isCommutative thy ef) = True + | isRootNormal thy (ef :$ ey :$ ex) = False +isRootNormalC _ _ = True + isRootNormalE :: Thy -> Expr -> Bool isRootNormalE thy e = isRootNormal thy e && null (filter (e ->-) [e2 //- bs | (_,bs,e2) <- T.lookup e trie]) @@ -645,16 +660,27 @@ trie = T.fromList $ equations thy ++ map swap (equations thy) (->-) = canReduceTo thy +isCommutative :: Thy -> Expr -> Bool +isCommutative thy eo = eo `elem` commutativeOperators thy +commutativeOperators :: Thy -> [Expr] +commutativeOperators thy = [ ef + | (ef :$ ex :$ ey, ef' :$ ey' :$ ex') <- equations thy + , ef == ef' + , ex == ex' + , ey == ey' + ] + + --- tiers utils --- productsWith :: ([a] -> a) -> [ [[a]] ] -> [[a]] productsWith f = mapT f . products --- TODO: move to LeanCheck? +-- TODO: move productsWith to LeanCheck? delayedProductsWith :: ([a] -> a) -> [ [[a]] ] -> [[a]] delayedProductsWith f xsss = productsWith f xsss `addWeight` length xsss --- TODO: move to LeanCheck? +-- TODO: move delayedProductsWith to LeanCheck? foldAppProducts :: Expr -> [ [[Expr]] ] -> [[Expr]] foldAppProducts ef = delayedProductsWith (foldApp . (ef:)) diff --git a/src/Conjure/Expr.hs b/src/Conjure/Expr.hs --- a/src/Conjure/Expr.hs +++ b/src/Conjure/Expr.hs @@ -12,7 +12,6 @@ , module Data.Express.Fixtures , rehole - , (>$$<) , funToVar , recursexpr , apparentlyTerminates @@ -30,10 +29,20 @@ , reval , useMatches , deholings + , varToConst + , hasAppInstanceOf , enumerateAppsFor , enumerateFillings + , digApp + , extractApp + , updateAppAt + , ($!!) + + , conflicts + , listConflicts + , module Conjure.Utils ) where @@ -94,6 +103,19 @@ funToVar (ef :$ ex) = funToVar ef :$ ex funToVar ef@(Value nm _) = nm `varAsTypeOf` ef +-- | Given a variable, returns a constant with the same name +varToConst :: Expr -> Expr +varToConst (Value ('_':nm) dyn) = Value nm dyn +varToConst _ = error "varToConst: can only be applied to variables" + +-- | Returns whether the first 'Expr' +-- has an application instance of the second 'Expr'. +hasAppInstanceOf :: Expr -> Expr -> Bool +e `hasAppInstanceOf` efxs = constApp e `hasInstanceOf` constApp efxs + where + constApp e = e //- [(ef,varToConst ef)] + (ef:_) = unfoldApp efxs + -- | Expands recursive calls on an expression -- until the given size limit is reached. -- @@ -187,18 +209,13 @@ -- | Creates a case 'Expr' of the type of the given proxy. -- -- > > caseForOrd (undefined :: Int) --- > case :: Bool -> Int -> Int -> Int -> Int +-- > case :: Ordering -> Int -> Int -> Int -> Int -- -- > > caseForOrd (undefined :: String) --- > case :: Bool -> [Char] -> [Char] -> [Char] -> [Char] +-- > case :: Ordering -> [Char] -> [Char] -> [Char] -> [Char] caseForOrd :: Typeable a => a -> Expr caseForOrd a = value "case" (\o x y z -> case o of LT -> x; EQ -> y; GT -> z `asTypeOf` a) --- | Application cross-product between lists of Exprs -(>$$<) :: [Expr] -> [Expr] -> [Expr] -exs >$$< eys = catMaybes [ex $$ ey | ex <- exs, ey <- eys] --- TODO: move >$$< into Data.Express? - -- | Lists terminal values in BFS order. -- -- (cf. 'values', 'holesBFS', 'fillBFS') @@ -452,6 +469,67 @@ deh e = if typ e == typ e' && isHole e then [e'] else [] + +-- | Dig a hole in a function application at the given position +-- +-- > > digApp 1 (one -+- two) +-- > _ + 2 :: Int +-- +-- > > digApp 2 (one -+- two) +-- > 1 + _ :: Int +digApp :: Int -> Expr -> Expr +digApp i = updateAppAt i holeAsTypeOf + +updateAppAt :: Int -> (Expr -> Expr) -> Expr -> Expr +updateAppAt i f = foldApp . updateAt i f . unfoldApp + +-- | Extracts the argument of a function application at the given position. +-- +-- > (one -+- two) $!! 1 +-- 1 :: Int +-- +-- > (one -+- two) $!! 2 +-- 2 :: Int +($!!) :: Expr -> Int -> Expr +e $!! i = unfoldApp e !! i + +-- | Extracts a value in a function application at the given position +-- +-- > > extractApp 1 (one -+- two) +-- > (_ + 2 :: Int, 1 :: Int) +-- +-- > > extractApp 2 (one -+- two) +-- > (1 + _ :: Int, 2 :: Int) +extractApp :: Int -> Expr -> (Expr,Expr) +extractApp i efxs = (foldApp $ updateAt i holeAsTypeOf es, es !! i) + where + es = unfoldApp efxs + + +-- | Lists conflicts between two expressions +-- +-- > > conflicts (one -+- two) (three -+- four) +-- > [(1 :: Int,3 :: Int), (2 :: Int,4 :: Int)] +-- +-- > > conflicts (xx -:- nil) (xx -:- yy -:- yys) +-- > [(nil, yy -:- yys)] +-- +-- > > conflicts (one -:- one -:- nil) (zero -:- zero -:- xx -:- xxs) +-- > [(1 :: Int,0 :: Int),([] :: [Int],x:xs :: [Int])] +conflicts :: Expr -> Expr -> [(Expr,Expr)] +conflicts e1 e2 | typ e1 /= typ e2 = [(e1,e2)] +conflicts (ef :$ ex) (eg :$ ey) = conflicts ef eg +++ conflicts ex ey +conflicts e1 e2 = [(e1,e2) | e1 /= e2] + +listConflicts :: [Expr] -> [[Expr]] +listConflicts es + | not (allEqualOn typ es) = [es] + | all isApp es = listConflicts [ef | ef :$ _ <- es] + +++ listConflicts [ex | _ :$ ex <- es] + | otherwise = [es | not (allEqual es)] + where + fun (ef :$ _) = ef + arg (_ :$ ex) = ex instance Express A where expr = val instance Express B where expr = val diff --git a/src/Conjure/Utils.hs b/src/Conjure/Utils.hs --- a/src/Conjure/Utils.hs +++ b/src/Conjure/Utils.hs @@ -29,6 +29,8 @@ , sets , headOr , allEqual + , allEqualOn + , allEqual2 , choices , choicesThat , foldr0 @@ -37,6 +39,12 @@ , classify -- from LeanCheck.Stats , classifyBy -- from LeanCheck.Stats , classifyOn -- from LeanCheck.Stats + , none + , updateAt + , first + , second + , both + , (+++) ) where @@ -51,14 +59,27 @@ import Test.LeanCheck.Stats (classify, classifyBy, classifyOn) + -- | Checks if all elements of a list are equal. +allEqual :: Eq a => [a] -> Bool +allEqual (x:y:etc) = x == y && allEqual (y:etc) +allEqual _ = True + + +allEqualOn :: Eq b => (a -> b) -> [a] -> Bool +allEqualOn f = allEqual . map f + + +-- | Checks if all elements of a list are equal. -- -- Exceptionally this function returns false for an empty or unit list. -allEqual :: Eq a => [a] -> Bool -allEqual [] = False -allEqual [x] = False -allEqual [x,y] = x == y -allEqual (x:y:xs) = x == y && allEqual (y:xs) +-- +-- This returns true when all elements are equal and the list +-- has a length greater than or equal to two. +allEqual2 :: Eq a => [a] -> Bool +allEqual2 [] = False +allEqual2 [_] = False +allEqual2 xs = allEqual xs -- | Counts the number of occurrences on a list. count :: (a -> Bool) -> [a] -> Int @@ -129,9 +150,9 @@ -- | Lists choices of values. choices :: [a] -> [(a,[a])] choices [] = [] -choices (x:xs) = (x,xs) : map (mapSnd (x:)) (choices xs) +choices (x:xs) = (x,xs) : map (second (x:)) (choices xs) where - mapSnd f (x,y) = (x,f y) + second f (x,y) = (x,f y) -- | Lists choices of values that follow a property. choicesThat :: (a -> [a] -> Bool) -> [a] -> [(a,[a])] @@ -149,3 +170,51 @@ -- | Indents a block of text with the provided amount of spaces indentBy :: Int -> String -> String indentBy n = unlines . map (replicate n ' ' ++) . lines + +none :: (a -> Bool) -> [a] -> Bool +none p = not . any p + +-- | Updates the value in a list at a given position. +-- +-- > > updateAt 2 (*10) [1,2,3,4] +-- > [1,2,30,4] +updateAt :: Int -> (a -> a) -> [a] -> [a] +updateAt _ _ [] = [] +updateAt 0 f (x:xs) = f x : xs +updateAt n f (x:xs) = x : updateAt (n-1) f xs + +-- | Applies a function to the first element of a pair. +-- Often known on the wild as @mapFst@. +-- +-- > > first (*10) (1,2) +-- > (10,2) +first :: (a -> a') -> (a,b) -> (a',b) +first f (x,y) = (f x, y) + +-- | Applies a function to the second element of a pair. +-- Often known on the wild as @mapSnd@. +-- +-- > > second (*100) (1,2) +-- > (1,200) +second :: (b -> b') -> (a,b) -> (a,b') +second f (x,y) = (x, f y) + +both :: (a -> b) -> (a,a) -> (b,b) +both f (x,y) = (f x, f y) + +nubMergeBy :: (a -> a -> Ordering) -> [a] -> [a] -> [a] +nubMergeBy cmp (x:xs) (y:ys) = case x `cmp` y of + LT -> x:nubMergeBy cmp xs (y:ys) + GT -> y:nubMergeBy cmp (x:xs) ys + EQ -> x:nubMergeBy cmp xs ys +nubMergeBy _ xs ys = xs ++ ys + +nubMergeOn :: Ord b => (a -> b) -> [a] -> [a] -> [a] +nubMergeOn f = nubMergeBy (compare `on` f) + +nubMerge :: Ord a => [a] -> [a] -> [a] +nubMerge = nubMergeBy compare + +(+++) :: Ord a => [a] -> [a] -> [a] +(+++) = nubMerge +infixr 5 +++ diff --git a/stack.yaml b/stack.yaml --- a/stack.yaml +++ b/stack.yaml @@ -9,6 +9,6 @@ - . extra-deps: -- leancheck-1.0.0 +- leancheck-1.0.2 - speculate-0.4.14 -- express-1.0.10 +- express-1.0.14 diff --git a/test/conjurable.hs b/test/conjurable.hs --- a/test/conjurable.hs +++ b/test/conjurable.hs @@ -114,6 +114,8 @@ , isNothing $ conjureEquality (undefined :: (Bool,Bool,Bool,Bool,Bool,Unit,Bool)) , isNothing $ conjureEquality (undefined :: (Bool,Bool,Bool,Bool,Bool,Bool,Unit)) + , conjureArgumentHoles (undefined :: Bool -> Int -> Bool -> ()) == [b_, i_, b_] + , length (conjureCases (undefined :: ())) == 1 , length (conjureCases (undefined :: Bool)) == 2 , length (conjureCases (undefined :: Int)) == 0 @@ -121,6 +123,15 @@ , length (conjureCases (undefined :: [Int])) == 2 , length (conjureCases (undefined :: [Bool])) == 2 + , conjureCases (undefined :: Bool) == [false, true] + , conjureCases (undefined :: Int) == [] + , conjureCases (undefined :: [Int]) == [nil, i_ -:- is_] + , conjureCases (undefined :: [Bool]) == [nilBool, b_ -:- bs_] + , conjureCases (undefined :: Maybe Int) == [nothing, just i_] + + , map length (conjureArgumentCases (undefined :: () -> Bool -> Int -> [Int] -> [Bool] -> ())) + == [1, 2, 0, 2, 2] + , conjurePats [zero, one] "f" (undefined :: Int -> Int) == [ [ [ ff xx ] @@ -150,7 +161,7 @@ ] ] - , take 4 (conjurePats [zero, one] "?" (undefined :: Int -> Int -> Int)) + , take 3 (conjurePats [zero, one] "?" (undefined :: Int -> Int -> Int)) == [ [ [ xx -?- yy ] ] @@ -164,106 +175,27 @@ , [ [ xx -?- one , xx -?- yy ] - , [ zero -?- zero - , zero -?- xx + , [ zero -?- xx , xx -?- zero , xx -?- yy ] - , [ one -?- xx - , xx -?- yy - ] - ] - , [ [ xx -?- zero - , xx -?- one - , xx -?- yy - ] - , [ zero -?- one + , [ zero -?- zero , zero -?- xx - , xx -?- one , xx -?- yy ] - , [ one -?- zero - , one -?- xx - , xx -?- zero - , xx -?- yy - ] - , [ zero -?- xx - , one -?- xx + , [ one -?- xx , xx -?- yy ] ] ] - , concat (conjurePats [zero, one] "foo" (undefined :: [Int] -> Int -> Int)) - == [ [ ffoo xxs xx - ] - , [ ffoo xxs zero - , ffoo xxs xx - ] - , [ ffoo nil xx - , ffoo (xx -:- xxs) yy - ] - , [ ffoo xxs one - , ffoo xxs xx - ] - , [ ffoo nil zero - , ffoo nil xx - , ffoo (xx -:- xxs) zero - , ffoo (xx -:- xxs) yy - ] - , [ ffoo xxs zero - , ffoo xxs one - , ffoo xxs xx - ] - , [ ffoo nil one - , ffoo nil xx - , ffoo (xx -:- xxs) one - , ffoo (xx -:- xxs) yy - ] - , [ ffoo nil zero - , ffoo nil one - , ffoo nil xx - , ffoo (xx -:- xxs) zero - , ffoo (xx -:- xxs) one - , ffoo (xx -:- xxs) yy - ] - ] + , mapT length (conjurePats [zero, one] "foo" (undefined :: [Int] -> Int -> Int)) + == [[1],[2,2],[2,3,3],[3,3,4,3],[4,4,4,4],[5,4,5],[5,5],[6]] - , concat (conjurePats [zero, one] "foo" (undefined :: Int -> [Int] -> Int)) - == [ [ ffoo xx xxs - ] - , [ ffoo xx nil - , ffoo xx (yy -:- xxs) - ] - , [ ffoo zero xxs - , ffoo xx xxs - ] - , [ ffoo zero nil - , ffoo zero (xx -:- xxs) - , ffoo xx nil - , ffoo xx (yy -:- xxs) - ] - , [ ffoo one xxs - , ffoo xx xxs - ] - , [ ffoo one nil - , ffoo one (xx -:- xxs) - , ffoo xx nil - , ffoo xx (yy -:- xxs) - ] - , [ ffoo zero xxs - , ffoo one xxs - , ffoo xx xxs - ] - , [ ffoo zero nil - , ffoo zero (xx -:- xxs) - , ffoo one nil - , ffoo one (xx -:- xxs) - , ffoo xx nil - , ffoo xx (yy -:- xxs) - ] - ] + , mapT length (conjurePats [zero, one] "foo" (undefined :: Int -> [Int] -> Int)) + == [[1],[2,2],[3,3,2],[4,3,3,3],[4,4,4,4],[5,5,5],[6]] + , conjurePats [] "foo" (undefined :: [Int] -> [Char] -> Int) == [ [ [ ffoo xxs ccs ] @@ -275,6 +207,15 @@ , ffoo (xx -:- xxs) ccs ] ] + , [ [ ffoo nilInt ccs + , ffoo (xx -:- xxs) emptyString + , ffoo (xx -:- xxs) (cc -:- ccs) + ] + , [ ffoo nilInt emptyString + , ffoo nilInt (cc -:- ccs) + , ffoo (xx -:- xxs) ccs + ] + ] , [ [ ffoo nilInt emptyString , ffoo nilInt (cc -:- ccs) , ffoo (xx -:- xxs) emptyString @@ -283,32 +224,18 @@ ] ] - , conjureIsDeconstructor (undefined :: Int -> Int) 60 (value "dec" (subtract 1 :: Int -> Int)) - == True - - , conjureIsDeconstructor (undefined :: Int -> Int) 60 (value "halve" ((`div` 2) :: Int -> Int)) - == True - - , conjureIsDeconstructor (undefined :: Int -> Int) 60 (value "double" ((*2) :: Int -> Int)) - == False - - , conjureIsDeconstructor (undefined :: Int -> Int) 60 (value "inc" ((+1) :: Int -> Int)) - == False -- _almost_ converges half the time - - , conjureIsDeconstructor (undefined :: [Int] -> Int) 60 (value "tail" (tail :: [Int] -> [Int])) - == True - - , conjureIsDeconstructor (undefined :: [Int] -> Int) 60 (value "prep" ((0:) :: [Int] -> [Int])) - == False - , isDecon (minus :$ i_ :$ one) == True , isDecon (minusOne -+- i_) == True , isDecon (div' i_ two) == True , isDecon (tail' is_) == True , isDecon (init' is_) == True - , isDecon (mod' i_ two) == True - , isDecon (mod' i_ xx) == True + -- counter-intuitive but true: x `mod` y is a deconstruction of y: + -- x `mod` y < y for y > 0 + , isDecon (mod' xx i_) == True + + , isDecon (mod' i_ two) == False -- does not deconstruct 1 + , isDecon (mod' i_ xx) == False -- may not deconstruct 1 , isDecon (div' xx yy) == False -- must have a hole to indicate the value being deconstructed , isDecon (div' i_ i_) == False -- two holes are not allowed , isDecon (head' is_) == False -- must deconstruct to the same type @@ -337,10 +264,19 @@ == Just False , fromDynamic (conjureDynamicEq ((+) :: Int -> Int -> Int) `dynApp` toDyn (1::Int) `dynApp` toDyn (1::Int)) == Just True + + , isNumeric one + , isNumeric (val (1 :: Integer)) + , isNumeric bee == False + , isNumeric (expr [0::Int]) == False + , isNumeric true == False ] isDecon :: Expr -> Bool isDecon = conjureIsDeconstruction (undefined :: [Int] -> [Char] -> [Bool]) 60 + +isNumeric :: Expr -> Bool +isNumeric = conjureIsNumeric (undefined :: [Int] -> [Char] -> [Bool] -> [Integer]) ffs :: Expr -> Expr ffs e = ffE :$ e diff --git a/test/defn.hs b/test/defn.hs --- a/test/defn.hs +++ b/test/defn.hs @@ -4,6 +4,7 @@ import Test import Conjure.Defn import Test.LeanCheck.Error (errorToLeft) +import Test.LeanCheck.Function () import Data.Express.Fixtures main :: IO () @@ -73,10 +74,85 @@ , holds n $ cevl 60 nullDefn === (null :: [Int] -> Bool) , holds n $ cevl 60 appendDefn ==== ((++) :: [Int] -> [Int] -> [Int]) + , holds n $ cevl 60 idDefn === (id :: Int -> Int) + , holds n $ cevl 60 constDefn ==== (const :: Int -> Int -> Int) + , holds n $ cevl 60 appDefn ==== (($) :: (Int->Int) -> Int -> Int) + , holds n $ \f g x -> cevl 60 composeDefn f g x == ((.) :: (Int->Int) -> (Int->Int) -> Int->Int) f g x + -- evaluating at the incorrect types should return Nothing , isNothing (cevaluate 60 sumDefn :: Maybe ([Bool] -> Bool)) , isNothing (cevaluate 60 andDefn :: Maybe ([Int] -> Int)) , isNothing (cevaluate 60 nullDefn :: Maybe ([Int] -> Int)) + + , noUnbound (xx -:- xxs, xxs) + , hasUnbound (xx -:- xxs, yys) + , isDefined sumDefn + , isDefined andDefn + , isDefined nullDefn + , isDefined [(nil, nil), (xx -:- xxs, xxs)] + , isUndefined [(nil, xxs), (xx -:- xxs, yys)] + + , isRedundantDefn sumDefn == False + , isRedundantDefn factDefn == False + , isRedundantDefn fact1Defn == False + , isRedundantDefn nullDefn == False + , isRedundantDefn isZeroDefn == False + , isRedundantDefn isOneDefn == False + , isRedundantDefn andDefn == False + , isRedundantDefn orDefn == False + , isRedundantDefn and1Defn == False + , isRedundantDefn or1Defn == False + , isRedundantDefn appendDefn == False + + , isRedundantDefn const0Defn == False + , isRedundantDefn const0RedundantDefn + + , isRedundantDefn constDefn == False + , isRedundantDefn constRedundantDefn + , isRedundantDefn constRedundantRedundantDefn + + , isRedundantDefn redundantDefn1 + , isRedundantDefn canonicalDefn1 == False + , isRedundantDefn redundantDefn2 + + , isRedundantDefn constBoolRedundantDefn + , isRedundantDefn idListRedundantDefn == False + -- NOTE: ^ ByIntroduction is not an active check + -- as it does not pay-off in terms of performance + + , isRedundantBySubsumption constRedundantDefn + , isRedundantByRepetition constRedundantDefn == False + , isRedundantByIntroduction constRedundantDefn + + , isRedundantBySubsumption constBoolRedundantDefn == False + , isRedundantByRepetition constBoolRedundantDefn + , isRedundantByIntroduction constBoolRedundantDefn + + , isRedundantBySubsumption idListRedundantDefn == False + , isRedundantByRepetition idListRedundantDefn == False + , isRedundantByIntroduction idListRedundantDefn + + , isRedundantBySubsumption constNilRedundantDefn == False + , isRedundantByRepetition constNilRedundantDefn + , isRedundantByIntroduction constNilRedundantDefn + + , isRedundantBySubsumption constFalseRedundantDefn == False + , isRedundantByRepetition constFalseRedundantDefn + , isRedundantByIntroduction constFalseRedundantDefn == True + + , canonicalizeBndn (introduceVariableAt 2 (const' xxs (yy -:- yys), (yy -:- yys) -++- (yy -:- yys))) + == (const' xxs yys, yys -++- yys) + + , canonicalizeBndnLast 1 (introduceVariableAt 2 (const' xxs (yy -:- yys), (yy -:- yys) -++- (yy -:- yys))) + == (const' zzs yys, yys -++- yys) + + , canonicalizeBndnLast 2 (introduceVariableAt 2 (const' xxs (yy -:- yys), (yy -:- yys) -++- (yy -:- yys))) + == (const' xxs zzs, zzs -++- zzs) + + , isBaseCase (ff (xx -:- nil), xx) + , not $ isBaseCase (ff (xx -:- xxs), ff xxs) + , isRecursiveCase (ff (xx -:- xxs), ff xxs) + , not $ isRecursiveCase (ff (xx -:- nil), xx) ] dvl :: Typeable a => Defn -> Expr -> a @@ -147,6 +223,91 @@ appendDefn = [ nil -++- xxs =- xxs , (xx -:- xxs) -++- yys =- xx -:- (xxs -++- yys) ] where exs -++- eys = appendV :$ exs :$ eys + +const0Defn :: Defn +const0Defn = [ xx =- zero ] + +const0RedundantDefn :: Defn +const0RedundantDefn = [ zero =- zero + , xx =- zero + ] + +idDefn :: Defn +idDefn = [ id' xx =- xx ] + +constDefn :: Defn +constDefn = [ const' xx yy =- xx ] + +appDefn :: Defn +appDefn = [ ffE -$- xx =- ff xx ] + +composeDefn :: Defn +composeDefn = [ (ffE -.- ggE) :$ xx =- ff (gg xx) ] + +constRedundantDefn :: Defn +constRedundantDefn = [ const' zero xx =- zero + , const' xx yy =- xx + ] + +constRedundantRedundantDefn :: Defn +constRedundantRedundantDefn = [ const' zero zero =- zero + , const' zero xx =- zero + , const' xx zero =- xx + , const' xx yy =- xx + ] + +constBoolRedundantDefn :: Defn +constBoolRedundantDefn = [ const' false false =- false + , const' false true =- false + , const' true false =- true + , const' true true =- true + ] + +idListRedundantDefn :: Defn +idListRedundantDefn = [ id' nil =- nil + , id' (xx -:- xxs) =- xx -:- xxs + ] + +constNilRedundantDefn :: Defn +constNilRedundantDefn = [ id' nil =- nil + , id' (xx -:- xxs) =- nil + ] + +constFalseRedundantDefn :: Defn +constFalseRedundantDefn = [ id' nil =- false + , id' (xx -:- xxs) =- false + ] + +-- Here is an example of a redundant 'Defn': +-- +-- > 0 ? 0 = 1 +-- > 0 ? x = 1 +-- > x ? 0 = x +-- > x ? y = x +redundantDefn1 :: Defn +redundantDefn1 = [ zero -?- zero =- one + , zero -?- xx =- one + , xx -?- zero =- xx + , xx -?- yy =- xx + ] + +-- The above is redundant because it is equivalent to: +-- +-- > 0 ? _ = 1 +-- > x ? _ = x +canonicalDefn1 :: Defn +canonicalDefn1 = [ zero -?- xx =- one + , xx -?- yy =- xx + ] + +-- same as redundantDefn1 but with arguments swapped +-- (mind the alpha-renaming) +redundantDefn2 :: Defn +redundantDefn2 = [ zero -?- zero =- one + , zero -?- xx =- xx + , xx -?- zero =- one + , xx -?- yy =- yy + ] (=-) = (,) infixr 0 =- diff --git a/test/expr.hs b/test/expr.hs --- a/test/expr.hs +++ b/test/expr.hs @@ -35,38 +35,6 @@ , apparentlyTerminates ffE (if' pp (ff xx) zero) == False , apparentlyTerminates ffE (if' (odd' (ff xx)) zero zero) == False - , [false, true, zero, one] >$$< [notE, andE, orE, plus, times] == [] - , [notE, andE, orE, plus, times] >$$< [false, true, zero, one] - == [ not' false - , not' true - , andE :$ false - , andE :$ true - , orE :$ false - , orE :$ true - , plus :$ zero - , plus :$ one - , times :$ zero - , times :$ one - ] - , [notE, andE, orE, plus, times] >$$< [false, true, zero, one] >$$< [false, true, zero, one] - == [ false -&&- false - , false -&&- true - , true -&&- false - , true -&&- true - , false -||- false - , false -||- true - , true -||- false - , true -||- true - , zero -+- zero - , zero -+- one - , one -+- zero - , one -+- one - , zero -*- zero - , zero -*- one - , one -*- zero - , one -*- one - ] - , holds n $ \e -> sort (valuesBFS e) == sort (values e) , holds n $ \e -> holesBFS e == filter isHole (valuesBFS e) , valuesBFS false == [false] @@ -174,4 +142,16 @@ , zero $$ zero == Nothing , zero $$|< zero == Nothing , zero $$** zero == Just (zero :$ zero) + + , conflicts (one -+- two) (three -+- four) == [(one,three), (two,four)] + , conflicts (xx -:- nil) (xx -:- yy -:- yys) == [(nil, yy -:- yys)] + , conflicts (one -:- one -:- nil) (zero -:- zero -:- xx -:- xxs) + == [(one,zero), (nil,xx -:- xxs)] + , listConflicts [(one -+- one), (two -+- two), (three -+- three)] == [[one,two,three]] + , listConflicts [(one -+- zz), (two -+- yy), (three -+- xx)] == [[zz,yy,xx],[one,two,three]] + + , holds n $ \e1 e2 -> map listPair (conflicts e1 e2) == listConflicts [e1,e2] ] + +listPair :: (a,a) -> [a] +listPair (x,y) = [x,y]