diff --git a/.gitignore b/.gitignore --- a/.gitignore +++ b/.gitignore @@ -56,7 +56,9 @@ bench/p30 bench/candidates bench/gps +bench/gps2 bench/lowtests +bench/terpret proto/u-conjure test/expr test/conjurable diff --git a/Makefile b/Makefile --- a/Makefile +++ b/Makefile @@ -33,6 +33,8 @@ bench/p12 \ bench/p30 \ bench/gps \ + bench/gps2 \ + bench/terpret \ proto/u-conjure TESTS = \ diff --git a/README.md b/README.md --- a/README.md +++ b/README.md @@ -94,7 +94,7 @@ , pr (1::Int) , prim "+" ((+) :: Int -> Int -> Int) , prim "*" ((*) :: Int -> Int -> Int) - , prim "dec" (subtract 1 :: Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) ] running @@ -105,15 +105,16 @@ factorial :: Int -> Int -- testing 4 combinations of argument values - -- pruning with 22/42 rules + -- pruning with 27/65 rules -- looking through 3 candidates of size 1 - -- looking through 6 candidates of size 2 - -- looking through 16 candidates of size 3 - -- looking through 39 candidates of size 4 - -- looking through 78 candidates of size 5 - -- looking through 166 candidates of size 6 + -- looking through 4 candidates of size 2 + -- looking through 13 candidates of size 3 + -- looking through 34 candidates of size 4 + -- looking through 75 candidates of size 5 + -- looking through 183 candidates of size 6 + -- looking through 577 candidates of size 7 factorial 0 = 1 - factorial x = x * factorial (dec x) + factorial x = x * factorial (x - 1) in less than a second. diff --git a/TODO.md b/TODO.md --- a/TODO.md +++ b/TODO.md @@ -3,25 +3,27 @@ A non-exhaustive list of things TO DO for Conjure. -* pretty-print top-level ifs? +* consider not breaking in some cases (increased crossproduct of patterns) -* carry on implementing all GPS benches while taking notes on the paper +* reduce the number of `deconstructions` considered: -* consider memoizing `recs ep` in `candidateDefnsC` - and a sub function with `vs` arguments. + 1. place a `traceShowId` in `deconstructions :: [Expr]` + 2. run the GCD example -* remove `requireDescent=False` requirement from `gcd` - (add and use `isDeconstruction`) - this would also eliminate the requirement of providing `dec` + [_ `mod` y :: Int,x `mod` _ :: Int,_ `mod` x :: Int,y `mod` _ :: Int,0 `mod` _ :: Int,0 `mod` _ :: Int] -* consider not breaking in some cases (increased crossproduct of patterns) + 3. there's no need to go variable by variable. Just generating expressions + with a _single hole_ should be enough. + ### for later * allow specifying properties that need to be true * allow recursion under any lazy functions (discover them by testing!) + +* pretty-print top-level ifs? * exclude magic numbers? e.g.: `1+1`? diff --git a/bench/candidates.hs b/bench/candidates.hs --- a/bench/candidates.hs +++ b/bench/candidates.hs @@ -38,7 +38,7 @@ , pr (1 :: Int) , prim "+" ((+) :: Int -> Int -> Int) , prim "*" ((+) :: Int -> Int -> Int) - , prim "dec" (subtract 1 :: Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) ] printCandidates 9 6 "?" (undefined :: Int -> Int -> Int) diff --git a/bench/candidates.out b/bench/candidates.out --- a/bench/candidates.out +++ b/bench/candidates.out @@ -1,7487 +1,9727 @@ Candidates for: foo :: Int -> Int - pruning with 13/26 rules - [3,2,4,3,12,7,27,25,58] direct candidates, 0 duplicates - [3,6,15,34,61,111,174,344,529] pattern candidates, 0 duplicates - -rules: -dec 1 == 0 -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 -1 + dec x == x -dec x + 1 == 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: - -foo x = x - -foo x = 0 - -foo x = 1 - -foo x = dec x - -foo x = dec 0 - -foo x = x + x - -foo x = x + 1 - -foo x = 1 + x - -foo x = 1 + 1 - -foo x = dec 0 + x - -foo x = x + dec x - -foo x = x + dec 0 - -foo x = x + (x + x) - -foo x = x + (x + 1) - -foo x = x + (1 + x) - -foo x = x + (1 + 1) - -foo x = 1 + (x + x) - -foo x = 1 + (x + 1) - -foo x = 1 + (1 + x) - -foo x = 1 + (1 + 1) - -foo x = dec x + dec x - -foo x = dec x + dec 0 - -foo x = dec 0 + dec x - -foo x = dec 0 + dec 0 - -foo x = x + (dec 0 + x) - -foo x = x + (x + dec x) - -foo x = x + (x + dec 0) - -foo x = 1 + (dec 0 + x) - -foo x = 1 + (x + dec x) - -foo x = 1 + (x + dec 0) - -foo x = dec 0 + (x + x) - - -pattern candidates: - -foo x = x - -foo x = 0 - -foo x = 1 - -foo x = dec x - -foo x = dec 0 - -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 0 = 0 -foo x = dec x - -foo 0 = 0 -foo x = dec 0 - -foo 0 = 1 -foo x = dec x - -foo 0 = 1 -foo x = dec 0 - -foo 0 = dec 0 -foo x = x - -foo 0 = dec 0 -foo x = 0 - -foo 0 = dec 0 -foo x = 1 - -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 = foo (dec x) - -foo 0 = 1 -foo x = foo (dec x) - -foo x = dec 0 + x - -foo x = x + dec x - -foo x = x + dec 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 = 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 = dec 0 -foo x = dec x - -foo 0 = 1 + 1 -foo x = x - -foo 0 = 1 + 1 -foo x = 0 - -foo 0 = 1 + 1 -foo x = 1 - -foo 1 = 0 -foo x = dec x - -foo 1 = 0 -foo x = dec 0 - -foo 1 = 1 -foo x = dec x - -foo 1 = 1 -foo x = dec 0 - -foo 1 = dec 0 -foo x = x - -foo 1 = dec 0 -foo x = 0 - -foo 1 = dec 0 -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 = dec (foo (dec x)) - -foo 0 = 1 -foo x = dec (foo (dec x)) - -foo 0 = dec 0 -foo x = foo (dec x) - -foo 1 = 0 -foo x = foo (dec x) - -foo 1 = 1 -foo x = foo (dec x) - -foo x = x + (x + x) - -foo x = x + (x + 1) - -foo x = x + (1 + x) - -foo x = x + (1 + 1) - -foo x = 1 + (x + x) - -foo x = 1 + (x + 1) - -foo x = 1 + (1 + x) - -foo x = 1 + (1 + 1) - -foo x = dec x + dec x - -foo x = dec x + dec 0 - -foo x = dec 0 + dec x - -foo x = dec 0 + dec 0 - -foo 0 = 0 -foo x = dec 0 + x - -foo 0 = 0 -foo x = x + dec x - -foo 0 = 0 -foo x = x + dec 0 - -foo 0 = 1 -foo x = dec 0 + x - -foo 0 = 1 -foo x = x + dec x - -foo 0 = 1 -foo x = x + dec 0 - -foo 0 = dec 0 -foo x = x + x - -foo 0 = dec 0 -foo x = x + 1 - -foo 0 = dec 0 -foo x = 1 + x - -foo 0 = dec 0 -foo x = 1 + 1 - -foo 0 = 1 + 1 -foo x = dec x - -foo 0 = 1 + 1 -foo x = dec 0 - -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 = 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 = dec 0 -foo x = dec x - -foo 1 = 1 + 1 -foo x = x - -foo 1 = 1 + 1 -foo x = 0 - -foo 1 = 1 + 1 -foo x = 1 - -foo 0 = 0 -foo 1 = 0 -foo x = dec x - -foo 0 = 0 -foo 1 = 0 -foo x = dec 0 - -foo 0 = 0 -foo 1 = 1 -foo x = dec x - -foo 0 = 0 -foo 1 = 1 -foo x = dec 0 - -foo 0 = 0 -foo 1 = dec 0 -foo x = x - -foo 0 = 0 -foo 1 = dec 0 -foo x = 0 - -foo 0 = 0 -foo 1 = dec 0 -foo x = 1 - -foo 0 = 1 -foo 1 = 0 -foo x = dec x - -foo 0 = 1 -foo 1 = 0 -foo x = dec 0 - -foo 0 = 1 -foo 1 = 1 -foo x = dec x - -foo 0 = 1 -foo 1 = 1 -foo x = dec 0 - -foo 0 = 1 -foo 1 = dec 0 -foo x = x - -foo 0 = 1 -foo 1 = dec 0 -foo x = 0 - -foo 0 = 1 -foo 1 = dec 0 -foo x = 1 - -foo 0 = dec 0 -foo 1 = 0 -foo x = x - -foo 0 = dec 0 -foo 1 = 0 -foo x = 0 - -foo 0 = dec 0 -foo 1 = 0 -foo x = 1 - -foo 0 = dec 0 -foo 1 = 1 -foo x = x - -foo 0 = dec 0 -foo 1 = 1 -foo x = 0 - -foo 0 = dec 0 -foo 1 = 1 -foo x = 1 - -foo 0 = 0 -foo x = x + foo (dec x) - -foo 0 = 0 -foo x = foo (dec x) + x - -foo 0 = 0 -foo x = foo (dec x) + 1 - -foo 0 = 0 -foo x = 1 + foo (dec x) - -foo 0 = 1 -foo x = x + foo (dec x) - -foo 0 = 1 -foo x = foo (dec x) + x - -foo 0 = 1 -foo x = foo (dec x) + 1 - -foo 0 = 1 -foo x = 1 + foo (dec x) - -foo 0 = dec 0 -foo x = dec (foo (dec x)) - -foo 0 = 1 + 1 -foo x = foo (dec x) - -foo 1 = 0 -foo x = dec (foo (dec x)) - -foo 1 = 1 -foo x = dec (foo (dec x)) - -foo 1 = dec 0 -foo x = foo (dec x) - -foo 0 = 0 -foo 1 = 0 -foo x = foo (dec x) - -foo 0 = 0 -foo 1 = 1 -foo x = foo (dec x) - -foo 0 = 1 -foo 1 = 0 -foo x = foo (dec x) - -foo 0 = 1 -foo 1 = 1 -foo x = foo (dec x) - -foo x = x + (dec 0 + x) - -foo x = x + (x + dec x) - -foo x = x + (x + dec 0) - -foo x = 1 + (dec 0 + x) - -foo x = 1 + (x + dec x) - -foo x = 1 + (x + dec 0) - -foo x = dec 0 + (x + 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 = 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 = dec x + dec x - -foo 0 = 0 -foo x = dec x + dec 0 - -foo 0 = 0 -foo x = dec 0 + dec x - -foo 0 = 0 -foo x = dec 0 + dec 0 - -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 = 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 = dec x + dec x - -foo 0 = 1 -foo x = dec x + dec 0 - -foo 0 = 1 -foo x = dec 0 + dec x - -foo 0 = 1 -foo x = dec 0 + dec 0 - -foo 0 = dec 0 -foo x = dec 0 + x - -foo 0 = dec 0 -foo x = x + dec x - -foo 0 = dec 0 -foo x = x + dec 0 - -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 + 1) -foo x = x - -foo 0 = 1 + (1 + 1) -foo x = 0 - -foo 0 = 1 + (1 + 1) -foo x = 1 - -foo 0 = dec 0 + dec 0 -foo x = x - -foo 0 = dec 0 + dec 0 -foo x = 0 - -foo 0 = dec 0 + dec 0 -foo x = 1 - -foo 1 = 0 -foo x = dec 0 + x - -foo 1 = 0 -foo x = x + dec x - -foo 1 = 0 -foo x = x + dec 0 - -foo 1 = 1 -foo x = dec 0 + x - -foo 1 = 1 -foo x = x + dec x - -foo 1 = 1 -foo x = x + dec 0 - -foo 1 = dec 0 -foo x = x + x - -foo 1 = dec 0 -foo x = x + 1 - -foo 1 = dec 0 -foo x = 1 + x - -foo 1 = dec 0 -foo x = 1 + 1 - -foo 1 = 1 + 1 -foo x = dec x - -foo 1 = 1 + 1 -foo x = dec 0 - -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 = 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 = dec 0 -foo x = dec x - -foo 0 = 0 -foo 1 = dec 0 -foo x = dec 0 - -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 = 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 = 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 = dec 0 -foo x = dec x - -foo 0 = 1 -foo 1 = dec 0 -foo x = dec 0 - -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 = dec 0 -foo 1 = 0 -foo x = dec x - -foo 0 = dec 0 -foo 1 = 0 -foo x = dec 0 - -foo 0 = dec 0 -foo 1 = 1 -foo x = dec x - -foo 0 = dec 0 -foo 1 = 1 -foo x = dec 0 - -foo 0 = dec 0 -foo 1 = dec 0 -foo x = x - -foo 0 = dec 0 -foo 1 = dec 0 -foo x = 0 - -foo 0 = dec 0 -foo 1 = dec 0 -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 - - -Candidates for: ? :: Int -> Int -> Int - pruning with 10/23 rules - [3,3,4,10,17,26,54,93,162] direct candidates, 0 duplicates - [3,11,39,88,245,596,1542,3881,9466] 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 = 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 ? 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 = y - -0 ? 0 = 0 -0 ? x = x -x ? 0 = 0 -x ? y = 0 - -0 ? 0 = 0 -0 ? x = 0 -x ? 0 = x -x ? y = x - -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 + 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 ? 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 = 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 = 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 - -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 = 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 = dec x -x ? y = dec y - -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 ? 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 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 = x -x ? y = 0 ? dec x - -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 = x -x ? y = dec y ? x - -x ? 0 = x -x ? y = dec y ? y - -x ? 0 = x -x ? y = dec y ? 0 - -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 = 0 -x ? y = 0 ? dec x - -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 - -x ? 0 = 0 -x ? y = dec y ? x - -x ? 0 = 0 -x ? y = dec y ? y - -x ? 0 = 0 -x ? y = dec y ? 0 - -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 = x -x ? y = 0 ? dec x - -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 = x -x ? y = dec y ? x - -0 ? x = x -x ? y = dec y ? y - -0 ? x = x -x ? y = dec y ? 0 - -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 = 0 -x ? y = 0 ? dec x - -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 - -0 ? x = 0 -x ? y = dec y ? x - -0 ? x = 0 -x ? y = dec y ? y - -0 ? x = 0 -x ? y = dec y ? 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 ? 0 = x -x ? y = dec x + y - -x ? 0 = x -x ? y = dec y + x - -x ? 0 = x -x ? y = dec 0 + x - -x ? 0 = x -x ? y = dec 0 + y - -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 + y - -x ? 0 = 0 -x ? y = dec y + x - -x ? 0 = 0 -x ? y = dec 0 + x - -x ? 0 = 0 -x ? y = dec 0 + y - -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 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 = 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 0 + x -x ? y = x - -x ? 0 = dec 0 + x -x ? y = y - -x ? 0 = dec 0 + 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 - -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 + y - -0 ? x = x -x ? y = dec y + x - -0 ? x = x -x ? y = dec 0 + x - -0 ? x = x -x ? y = dec 0 + y - -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 + y - -0 ? x = 0 -x ? y = dec y + x - -0 ? x = 0 -x ? y = dec 0 + x - -0 ? x = 0 -x ? y = dec 0 + y - -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 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 = 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 0 + x -x ? y = x - -0 ? x = dec 0 + x -x ? y = y - -0 ? x = dec 0 + 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 + 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 = 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 = 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 = 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 = 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 = dec x -x ? y = dec x - -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 = dec 0 -x ? y = dec 0 - -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 = 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 y - -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 = 0 -x ? y = dec 0 - -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 = 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 = y - -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 = 0 -x ? 0 = dec 0 -x ? y = 0 - -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 = x - -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 0 -0 ? x = dec 0 -x ? 0 = 0 -x ? y = 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 = 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 - -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 = 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 - -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 x) - -x ? 0 = x -x ? y = dec (y ? dec y) - -x ? 0 = x -x ? y = dec (0 ? dec x) - -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 = x -x ? y = dec (dec y ? x) - -x ? 0 = x -x ? y = dec (dec y ? y) - -x ? 0 = x -x ? y = dec (dec y ? 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 (y ? dec x) - -x ? 0 = 0 -x ? y = dec (y ? dec y) - -x ? 0 = 0 -x ? y = dec (0 ? dec x) - -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 = 0 -x ? y = dec (dec y ? x) - -x ? 0 = 0 -x ? y = dec (dec y ? y) - -x ? 0 = 0 -x ? y = dec (dec y ? 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 = y ? dec x - -x ? 0 = dec x -x ? y = y ? dec y - -x ? 0 = dec x -x ? y = 0 ? dec x - -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 x -x ? y = dec y ? x - -x ? 0 = dec x -x ? y = dec y ? y - -x ? 0 = dec x -x ? y = dec y ? 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 = y ? dec x - -x ? 0 = dec 0 -x ? y = y ? dec y - -x ? 0 = dec 0 -x ? y = 0 ? dec x - -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 - -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 y ? 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 (y ? dec x) - -0 ? x = x -x ? y = dec (y ? dec y) - -0 ? x = x -x ? y = dec (0 ? dec x) - -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 = x -x ? y = dec (dec y ? x) - -0 ? x = x -x ? y = dec (dec y ? y) - -0 ? x = x -x ? y = dec (dec y ? 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 (y ? dec x) - -0 ? x = 0 -x ? y = dec (y ? dec y) - -0 ? x = 0 -x ? y = dec (0 ? dec x) - -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 = 0 -x ? y = dec (dec y ? x) - -0 ? x = 0 -x ? y = dec (dec y ? y) - -0 ? x = 0 -x ? y = dec (dec y ? 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 = y ? dec x - -0 ? x = dec x -x ? y = y ? dec y - -0 ? x = dec x -x ? y = 0 ? dec x - -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 x -x ? y = dec y ? x - -0 ? x = dec x -x ? y = dec y ? y - -0 ? x = dec x -x ? y = dec y ? 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 = y ? dec x - -0 ? x = dec 0 -x ? y = y ? dec y - -0 ? x = dec 0 -x ? y = 0 ? dec x - -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 ? x = dec 0 -x ? y = dec y ? x - -0 ? x = dec 0 -x ? y = dec y ? y - -0 ? x = dec 0 -x ? y = dec y ? 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 = 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 = 0 ? dec x - -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 = 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 y ? 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 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 = 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 = 0 ? dec x - -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 -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 y ? 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 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 = 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 = 0 ? dec x - -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 = 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 y ? 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 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 = 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 = 0 ? dec x - -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 - -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 y ? 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) - -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 = 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 = 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 = 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 = 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 = 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 + y - -x ? 0 = dec x -x ? y = dec y + x - -x ? 0 = dec x -x ? y = dec 0 + x - -x ? 0 = dec x -x ? y = dec 0 + 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 = 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 + y - -x ? 0 = dec 0 -x ? y = dec y + x - -x ? 0 = dec 0 -x ? y = dec 0 + x - -x ? 0 = dec 0 -x ? y = dec 0 + y - -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 = 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 = 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 = x + (x + x) -x ? y = x - -x ? 0 = x + (x + x) -x ? y = y - -x ? 0 = x + (x + 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 - -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 = 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 = 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 = 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 = 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 = 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 = 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 + y - -0 ? x = dec x -x ? y = dec y + x - -0 ? x = dec x -x ? y = dec 0 + x - -0 ? x = dec x -x ? y = dec 0 + 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 = 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 + y - -0 ? x = dec 0 -x ? y = dec y + x - -0 ? x = dec 0 -x ? y = dec 0 + x - -0 ? x = dec 0 -x ? y = dec 0 + y - -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 = y + y - -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 = 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 = x + (x + x) -x ? y = x - -0 ? x = x + (x + x) -x ? y = y - -0 ? x = x + (x + 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 = 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 + 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 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 = 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 + 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 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 = 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 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 = 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 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 = 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 + 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 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 = 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 + 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 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 = 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 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 = 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 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 = 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 = 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 = 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 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 = 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 = 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 = 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 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 - - -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 - [3,8,15,66,152,362,1400,2084,11820] 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 = 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) = ys - -[] ?? [] = [] -[] ?? (x:xs) = xs -(x:xs) ?? [] = [] -(x:xs) ?? (y: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 -(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 -(x:xs) ?? ys = ys ++ ys - -[] ?? 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 - [3,1,0,0,0,0,0,0,0] 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 - - -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 - [4,2,4,8,4,32,36,112,264] direct candidates, 0 duplicates - [4,14,30,8,4,32,36,112,264] 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 && (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 - - -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 = False -True &| False = True -True &| True = True - -False &| False = False -False &| True = True -True &| False = False -True &| True = False - -False &| False = False -False &| True = True -True &| False = False -True &| True = True - -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 = False - -False &| False = True -False &| True = False -True &| False = True -True &| True = True - -False &| False = True -False &| True = True -True &| False = False -True &| True = False - -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 = 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 + 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 + [3,11,42,117,346,955,2714,7510,20754] 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 = 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 = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = x + +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 y + +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 x + +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 = dec 0 +x ? y = dec 0 + +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 y + +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 = 0 +x ? y = dec 0 + +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 = y + +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 = 0 +x ? 0 = dec 0 +x ? y = 0 + +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 = x + +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 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = 0 + +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 = y + y + +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 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 + [3,8,15,66,152,362,1400,2084,11820] 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 = 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) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y: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 +(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 +(x:xs) ?? ys = ys ++ ys + +[] ?? 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 + [4,14,30,8,4,48,60,280,592] 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 = False +True &| False = True +True &| True = True + +False &| False = False +False &| True = True +True &| False = False +True &| True = False + +False &| False = False +False &| True = True +True &| False = False +True &| True = True + +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 = False + +False &| False = True +False &| True = False +True &| False = True +True &| True = True + +False &| False = True +False &| True = True +True &| False = False +True &| True = False + +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 diff --git a/bench/gps.hs b/bench/gps.hs --- a/bench/gps.hs +++ b/bench/gps.hs @@ -2,14 +2,31 @@ -- -- Copyright (C) 2021 Rudy Matela -- Distributed under the 3-Clause BSD licence (see the file LICENSE). +{-# LANGUAGE CPP #-} import Conjure import System.Environment (getArgs) import Data.Char (isLetter) -- GPS bench #5 import Data.Char (isSpace) -- GPS bench #7 import Data.Ratio ((%), numerator, denominator) -- GPS bench #10 +import Data.List (findIndex) -- GPS bench #12 +import Data.Maybe (fromJust) -- GPS bench #12 +import Data.List (sort) -- GPS bench #16 +import Data.Char (chr,ord) -- GPS bench #24 +-- GPS bench #16: +#if __GLASGOW_HASKELL__ >= 710 +import Data.List (isSubsequenceOf) +#else +isSubsequenceOf :: Eq a => [a] -> [a] -> Bool +isSubsequenceOf [] _ = True +isSubsequenceOf (_:_) [] = False +isSubsequenceOf (x:xs) (y:ys) + | x == y = xs `isSubsequenceOf` ys + | otherwise = (x:xs) `isSubsequenceOf` ys +#endif + gps1p :: Int -> Float -> Float gps1p 0 1.0 = 1.0 gps1p 1 0.0 = 1.0 @@ -315,6 +332,571 @@ , prim "wallisNext" wallisNext ] + +-- GPS Benchmark #11 -- Lengths Backwards + +gps11p :: [String] -> [Int] +gps11p ["a"] = [1] +gps11p ["aa","a"] = [1,2] +gps11p ["a","aa"] = [2,1] +gps11p ["a","aa","aaa"] = [3,2,1] + +gps11g :: [String] -> [Int] +gps11g = reverse . map length + +gps11g2 :: [String] -> [Int] +gps11g2 [] = [] +gps11g2 (s:ss) = gps11g2 ss ++ [length s] + +gps11c :: IO () +gps11c = do + conjure "gps11" gps11p + [ prim "reverse" (reverse :: [Int] -> [Int]) + , prim "length" (length :: String -> Int) + , prim "map" (map :: (String -> Int) -> [String] -> [Int]) + ] + + conjure "gps11" gps11p + [ pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "++" ((++) :: [Int] -> [Int] -> [Int]) + , prim "length" (length :: String -> Int) + ] + + +-- GPS Benchmark #12 -- Last index or zero + +gps12p :: [Int] -> Int +gps12p [0] = 0 +gps12p [0,0] = 1 +gps12p [0,1,0] = 2 +gps12p [1,0,1] = 1 +gps12p [0,1,0,1] = 2 +gps12p [1,0,1,0] = 3 + +gps12g :: [Int] -> Int +gps12g xs = length xs - fromJust (findIndex (0==) (reverse xs)) - 1 + +gps12c :: IO () +gps12c = do + conjureWith args{maxSize = 11} "gps12" gps12p + [ prim "length" (length :: [Int] -> Int) + , prim "reverse" (reverse :: [Int] -> [Int]) + , prim "findIndex" (findIndex :: (Int -> Bool) -> [Int] -> Maybe Int) + , prim "fromJust" (fromJust :: Maybe Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) + , prim "==" ((==) :: Int -> Int -> Bool) + , pr (0 :: Int) + , pr (1 :: Int) + ] + + +-- GPS Benchmark #13 -- Vector Average -- + +gps13p :: [Rational] -> Rational +gps13p [0,2] = 1 +gps13p [1,2,3] = 2 +gps13p [0,0,2,2] = 1 + +gps13g :: [Rational] -> Rational +gps13g qs = foldr (+) 0 qs / fromIntegral (length qs) + +gps13c :: IO () +gps13c = do + conjure "gps13" gps13p + [ prim "0" (0 :: Rational) + , prim "+" ((+) :: Rational -> Rational -> Rational) + , prim "/" ((/) :: Rational -> Rational -> Rational) + , prim "foldr" (foldr :: (Rational -> Rational -> Rational) -> Rational -> [Rational] -> Rational) + , prim "length" (length :: [Rational] -> Int) + , prim "fromIntegral" (fromIntegral :: Int -> Rational) + ] + + +-- GPS Benchmark #14 -- Count Odds -- + +gps14p :: [Int] -> Int +gps14p [0,0] = 0 +gps14p [1,3] = 2 +gps14p [0,1,2] = 1 + +gps14g :: [Int] -> Int +gps14g xs = length (filter odd xs) + where + odd x = x `mod` 2 /= 0 + +gps14g2 :: [Int] -> Int +gps14g2 [] = 0 +gps14g2 (x:xs) = if x `mod` 2 == 0 + then gps14g2 xs + else 1 + gps14g2 xs + +odd' :: Int -> Bool +odd' 0 = False +odd' 1 = True +odd' 2 = False +odd' 3 = True +odd' 4 = False +odd' 5 = True + +gps14c :: IO () +gps14c = do + conjure "odd" odd' + [ pr (0 :: Int) + , pr (1 :: Int) + , pr (2 :: Int) + , prim "`mod`" (mod :: Int -> Int -> Int) + , prim "/=" ((/=) :: Int -> Int -> Bool) + ] + + conjure "gps14" gps14p + [ prim "odd" (odd :: Int -> Bool) + , prim "filter" (filter :: (Int -> Bool) -> [Int] -> [Int]) + , prim "length" (length :: [Int] -> Int) + ] + + -- hah! I was expecting Conjure to use an if like above, but it was smarter: + -- gps14 [] = 0 + -- gps14 (x:xs) = x `mod` 2 + gps14 xs + conjureWith args{maxSize=13} "gps14" gps14p + [ pr (0 :: Int) + , pr (1 :: Int) + , pr (2 :: Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "`mod`" (mod :: Int -> Int -> Int) + , prim "==" ((==) :: Int -> Int -> Bool) + , prif (undefined :: Int) + ] + + +-- GPS Benchmark #15 -- Mirror Image -- + +gps15p :: [Int] -> [Int] -> Bool +gps15p [0] [0] = True +gps15p [0,1] [1,0] = True +gps15p [0,1] [0,1] = False +gps15p [0,0,1] [1,0,0] = True +gps15p [0,0,1] [0,0,1] = False + +gps15g :: [Int] -> [Int] -> Bool +gps15g xs ys = reverse xs == ys + +gps15c :: IO () +gps15c = do + conjure "gps15" gps15p + [ prim "==" ((==) :: [Int] -> [Int] -> Bool) + , prim "reverse" (reverse :: [Int] -> [Int]) + ] + + +-- GPS Benchmark #16 -- Super Anagrams -- + +gps16p :: String -> String -> Bool +gps16p "a" "aa" = True +gps16p "aa" "a" = False +gps16p "ab" "ba" = True +gps16p "ba" "ab" = True +gps16p "ab" "c" = False +gps16p "ab" "aba" = True + +gps16g :: String -> String -> Bool +gps16g cs ds = sort cs `isSubsequenceOf` sort ds + +gps16c :: IO () +gps16c = do + conjure "gps16" gps16p + [ prim "`isSubsequenceOf`" (isSubsequenceOf :: String -> String -> Bool) + , prim "sort" (sort :: String -> String) + ] + + +-- GPS Benchmark #17 -- Sum of Squares -- +-- Given integer _n_, return the sum of squaring each integer in 1 to n +gps17p :: Int -> Int +gps17p 1 = 1 +gps17p 2 = 5 +gps17p 3 = 14 +gps17p 4 = 30 + +gps17g :: Int -> Int +gps17g n = n * n + gps17g (n - 1) + +gps17c :: IO () +gps17c = conjure "gps17" gps17p + [ pr (0 :: Int) + , pr (1 :: Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "*" ((*) :: Int -> Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) + ] + + +-- GPS Benchmark #18 -- Vectors Summed -- +-- zip with sum +gps18p :: [Int] -> [Int] -> [Int] +gps18p [0] [0] = [0] +gps18p [0,1] [0,1] = [0,2] +gps18p [1,0] [0,1] = [1,1] + +gps18g :: [Int] -> [Int] -> [Int] +gps18g xs ys = zipWith (+) xs ys + +gps18g' :: [Int] -> [Int] -> [Int] +gps18g' [] [] = [] +gps18g' (x:xs) (y:ys) = x + y : gps18g' xs ys + +gps18c :: IO () +gps18c = do + conjure "gps18" gps18p + [ pr ([] :: [Int]) + , prim "+" ((+) :: Int -> Int -> Int) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + ] + + conjure "gps18" gps18p + [ prim "+" ((+) :: Int -> Int -> Int) + , prim "zipWith" (zipWith :: (Int -> Int -> Int) -> [Int] -> [Int] -> [Int]) + ] + + +-- GPS Benchmark #19 -- X-Word Lines -- + +gps19p :: Int -> String -> String +gps19p 1 "a a a" = "a\na\na\n" +gps19p 1 "a\na\na" = "a\na\na\n" +gps19p 2 "a a a" = "a a\na\n" + +gps19g :: Int -> String -> String +gps19g x xs = unlines (map unwords (chunk x (words xs))) + +chunk :: Int -> [String] -> [[String]] +chunk x [] = [] +chunk x ss = take x ss : chunk x (drop x ss) + +gps19c :: IO () +gps19c = do + -- speculate takes too long + conjure "gps19" gps19p $ take 0 + [ prim "words" (words :: String -> [String]) +-- , prim "words" (lines :: String -> [String]) + , prim "unwords" (unwords :: [String] -> String) + , prim "unlines" (unlines :: [String] -> String) +-- , prim "chunk" (chunk :: Int -> [String] -> [[String]]) +-- , prim "map" (map :: ([String] -> String) -> [[String]] -> [String]) + ] + + +-- GPS Benchmark #20 -- Pig Latin -- +gps20p :: String -> String +gps20p "hello world" = "ellohay orldway" +gps20p "a string" = "aay tringsay" + +gps20g :: String -> String +gps20g = pig + +pig :: String -> String +pig = unwords . map pig1 . words + +pig1 :: String -> String +pig1 (c:cs) = if isVowel c + then (c:cs) ++ "ay" + else cs ++ (c:"ay") + +pig1' :: String -> String +pig1' "hello" = "ellohay" +pig1' "world" = "orldway" +pig1' "string" = "tringsay" +pig1' "east" = "eastay" + +isVowel :: Char -> Bool +isVowel 'a' = True +isVowel 'e' = True +isVowel 'i' = True +isVowel 'o' = True +isVowel 'u' = True +isVowel 'y' = True +isVowel _ = False + +isVowel' :: Char -> Bool +isVowel' 'a' = True +isVowel' 'e' = True +isVowel' 'i' = True +isVowel' 'o' = True +isVowel' 'u' = True +isVowel' 'y' = True +isVowel' ' ' = False +isVowel' 'b' = False +isVowel' 'c' = False +isVowel' 'd' = False +isVowel' 'f' = False +isVowel' 'g' = False + +gps20c :: IO () +gps20c = do + conjureWith args{maxSize=22} "isVowel" isVowel' + [ pr 'a' + , pr 'e' + , pr 'i' + , pr 'o' + , pr 'u' + , pr 'y' + , pr True + , pr False + ] + + let force = [ [val "hello"] + , [val "world"] + , [val "string"] + , [val "east"] + ] + conjureWith args{forceTests=force, maxSize=14} "pig1" pig1' + [ pr "ay" + , prif (undefined :: String) + , prim "isVowel" isVowel + , prim "++" ((++) :: String -> String -> String) + , prim ":" ((:) :: Char -> String -> String) + ] + + let force = [ [val "hello world"] + , [val "a string"] + ] + conjureWith args{forceTests=force} "gps20c" gps20p + [ prim "words" (words :: String -> [String]) + , prim "unwords" (unwords :: [String] -> String) + , prim "map" (map :: (String -> String) -> [String] -> [String]) + , prim "pig1" pig1 + ] + + + +-- GPS Benchmark #21 -- Negative To Zero -- + +gps21p :: [Int] -> [Int] +gps21p [1] = [1] +gps21p [-1] = [0] +gps21p [1,-1] = [1,0] +gps21p [-1,1] = [0,1] + +gps21g :: [Int] -> [Int] +gps21g [] = [] +gps21g (x:xs) = (if x < 0 then 0 else x) : gps21g xs + +gps21c :: IO () +gps21c = conjure "gps21" gps21p + [ pr ([] :: [Int]) + , pr (0 :: Int) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "<" ((<) :: Int -> Int -> Bool) + , prif (undefined :: Int) + ] + + +-- GPS Benchmark #22 -- Scrabble Score -- +gps22p :: String -> Int +gps22p "a" = 1 +gps22p "hello" = 8 +gps22p "world" = 9 +gps22p "scrabble" = 14 + +gps22g :: String -> Int +gps22g = scrabble + +scrabble :: String -> Int +scrabble s = foldr (+) 0 (map scrabble1 s) + +scrabble1 :: Char -> Int +scrabble1 'd' = 2 +scrabble1 'g' = 2 +scrabble1 'b' = 3 +scrabble1 'c' = 3 +scrabble1 'm' = 3 +scrabble1 'p' = 3 +scrabble1 'f' = 4 +scrabble1 'h' = 4 +scrabble1 'v' = 4 +scrabble1 'w' = 4 +scrabble1 'y' = 4 +scrabble1 'k' = 5 +scrabble1 'j' = 8 +scrabble1 'x' = 8 +scrabble1 'q' = 10 +scrabble1 'z' = 10 +scrabble1 _ = 1 -- aeilnorstu + +gps22c :: IO () +gps22c = do + let force = [ [val "a"] + , [val "hello"] + , [val "world"] + , [val "scrabble"] + ] + + conjureWith args{forceTests=force} "gps22" gps22p + [ pr (0 :: Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "map" (map :: (Int -> Int) -> [Int] -> [Int]) + , prim "scrabble1" scrabble1 + ] + + +-- GPS Benchmark #23 -- Word Stats -- +gps23p :: String -> ([(Int,Int)], Int, Double) +gps23p "" = ([], 0, 1.0/0.0) + +gps23c :: IO () +gps23c = do + conjure "gps23" gps23p [] + + +-- GPS Benchmark #24 -- Checksum -- +gps24p :: String -> Char +gps24p "a" = 'A' +gps24p "aa" = '"' +gps24p "a a" = 'B' +gps24p "b" = 'B' + +gps24g :: String -> Char +gps24g s = chr (sum (map ord s) `mod` 64 + ord ' ') + +gps24c :: IO () +gps24c = conjure "gps24" gps24p + [ pr ' ' + , pr (64 :: Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "`mod`" (mod :: Int -> Int -> Int) + , prim "sum" (sum :: [Int] -> Int) + , prim "ord" ord + , prim "chr" chr + , prim "map" (map :: (Char -> Int) -> String -> [Int]) + ] + + +-- GPS Benchmark #25 -- Digits -- +gps25p :: Int -> [Int] +gps25p 0 = [0] +gps25p 1 = [1] +gps25p 12 = [2,1] +gps25p (-21) = [1,-2] + +gps25g :: Int -> [Int] +gps25g 0 = [0] -- 3 +gps25g n = if abs n < 10 -- 8 + then [n] -- 11 + else abs (n `rem` 10) : gps25g (n `quot` 10) -- 20 + +-- out of reach performance-wise +gps25c :: IO () +gps25c = conjureWith args{maxSize=20} "gps25" gps25p $ take 0 + [ pr (0 :: Int) + , pr (10 :: Int) + , pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prif (undefined :: [Int]) + , prim "abs" (abs :: Int -> Int) + , prim "<" ((<) :: Int -> Int -> Bool) + , prim "rem" (rem :: Int -> Int -> Int) + , prim "quot" (quot :: Int -> Int -> Int) + ] + + +-- GPS Benchmark #26 -- Grade -- +gps26p :: Int -> Int -> Int -> Int -> Int -> Char +gps26p 4 3 2 1 5 = 'A' +gps26p 4 3 2 1 4 = 'A' +gps26p 4 3 2 1 3 = 'B' +gps26p 4 3 2 1 2 = 'C' +gps26p 4 3 2 1 1 = 'D' +gps26p 4 3 2 1 0 = 'F' + +gps26g :: Int -> Int -> Int -> Int -> Int -> Char +gps26g a b c d x + | x >= a = 'A' + | x >= b = 'B' + | x >= c = 'C' + | x >= d = 'D' + | otherwise = 'F' + +-- out of reach performance-wise +gps26c :: IO () +gps26c = conjureWith args{maxSize=2} "gps26" gps26p + [ pr 'A' + , pr 'B' + , pr 'C' + , pr 'D' + , pr 'F' + , prif (undefined :: Char) + , prim ">=" ((>=) :: Int -> Int -> Bool) + ] + + +-- GPS Benchmark #27 -- Median -- +gps27p :: Int -> Int -> Int -> Int +gps27p 0 1 2 = 1 +gps27p 1 0 2 = 1 +gps27p (-1) 1 0 = 0 + +gps27g :: Int -> Int -> Int -> Int +gps27g x y z + | y < x && x < z = x -- 8 + | x < y && y < z = y -- 16 + | otherwise = z -- 17 +-- Conjure found a smaller implementation! + +gps27c :: IO () +gps27c = conjure "gps27" gps27p + [ prim "<" ((<) :: Int -> Int -> Bool) + , prim "&&" (&&) + , prif (undefined :: Int) + ] + + +-- GPS Benchmark #28 -- Smallest -- +gps28p :: Int -> Int -> Int -> Int -> Int +gps28p 0 1 2 3 = 0 +gps28p 3 2 1 0 = 0 +gps28p 1 0 2 3 = 0 +gps28p 3 2 0 1 = 0 +gps28p 1 1 1 2 = 1 + +gps28c :: IO () +gps28c = conjure "gps28" gps28p + [ prim "`min`" (min :: Int -> Int -> Int) + ] + + +-- GPS Benchmark #29 -- Syllables -- +gps29p :: String -> Int +gps29p "pub" = 1 +gps29p "hello" = 2 +gps29p "world" = 1 +gps29p "string" = 1 +gps29p "haskell" = 2 +gps29p "photography" = 4 + +gps29g :: String -> Int +gps29g "" = 0 +gps29g (c:cs) = if isVowel c + then 1 + gps29g cs + else gps29g cs + +gps29c :: IO () +gps29c = conjureWith args{forceTests=force} "gps29" gps29p + [ pr (0 :: Int) + , pr (1 :: Int) + , prim "+" ((+) :: Int->Int->Int) + , prif (undefined :: Int) + , prim "isVowel" isVowel + ] + where + force = [ [val "pub"] + , [val "hello"] + , [val "world"] + , [val "string"] + , [val "haskell"] + , [val "photography"] + ] + + main :: IO () main = do as <- getArgs @@ -334,4 +916,23 @@ , gps8c , gps9c , gps10c + , gps11c + , gps12c + , gps13c + , gps14c + , gps15c + , gps16c + , gps17c + , gps18c + , gps19c + , gps20c + , gps21c + , gps22c + , gps23c + , gps24c + , gps25c + , gps26c + , gps27c + , gps28c + , gps29c ] diff --git a/bench/gps.out b/bench/gps.out --- a/bench/gps.out +++ b/bench/gps.out @@ -35,7 +35,7 @@ -- looking through 0 candidates of size 5 -- looking through 1536 candidates of size 6 -- looking through 0 candidates of size 7 --- looking through 26127 candidates of size 8 +-- looking through 26991 candidates of size 8 gps3 x y z = enumFromThenTo x (x + z) (y - 1) gps3 :: Int -> Int -> Int -> [Int] @@ -83,10 +83,10 @@ -- pruning with 16/18 rules -- looking through 4 candidates of size 1 -- looking through 9 candidates of size 2 --- looking through 30 candidates of size 3 --- looking through 125 candidates of size 4 --- looking through 415 candidates of size 5 --- looking through 1602 candidates of size 6 +-- looking through 33 candidates of size 3 +-- looking through 146 candidates of size 4 +-- looking through 624 candidates of size 5 +-- looking through 2576 candidates of size 6 cannot conjure gps7 :: [Char] -> ([Char],Int) @@ -100,9 +100,9 @@ -- looking through 7 candidates of size 6 -- looking through 16 candidates of size 7 -- looking through 39 candidates of size 8 --- looking through 86 candidates of size 9 --- looking through 193 candidates of size 10 --- looking through 414 candidates of size 11 +-- looking through 88 candidates of size 9 +-- looking through 201 candidates of size 10 +-- looking through 442 candidates of size 11 gps7 cs = (init (unlines (words cs)),length (filter (not . isSpace) cs)) gps8 :: [Char] -> [Char] -> [(Int,Char,Char)] @@ -121,11 +121,11 @@ -- looking through 4 candidates of size 3 -- looking through 4 candidates of size 4 -- looking through 10 candidates of size 5 --- looking through 25 candidates of size 6 --- looking through 35 candidates of size 7 --- looking through 87 candidates of size 8 --- looking through 150 candidates of size 9 --- looking through 272 candidates of size 10 +-- looking through 24 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 gps9 x = filter even (filter (x >) (map sq [1..x])) wallisNext :: Ratio Integer -> Ratio Integer @@ -138,7 +138,7 @@ -- looking through 4 candidates of size 5 -- looking through 118 candidates of size 6 -- looking through 5 candidates of size 7 --- looking through 825 candidates of size 8 +-- looking through 1001 candidates of size 8 wallisNext (x % y) = (y + 1) % (x + 1) gps10 :: Int -> Ratio Integer @@ -153,4 +153,324 @@ -- looking through 8 candidates of size 7 -- looking through 13 candidates of size 8 gps10 x = product (take x (iterate wallisNext (2 % 3))) + +gps11 :: [[Char]] -> [Int] +-- testing 4 combinations of argument values +-- pruning with 1/1 rules +-- looking through 0 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 +gps11 css = reverse (map length css) + +gps11 :: [[Char]] -> [Int] +-- testing 4 combinations of argument values +-- pruning with 4/4 rules +-- looking through 1 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 1 candidates of size 3 +-- looking through 0 candidates of size 4 +-- looking through 1 candidates of size 5 +-- looking through 2 candidates of size 6 +-- looking through 0 candidates of size 7 +-- looking through 2 candidates of size 8 +gps11 [] = [] +gps11 (cs:css) = gps11 css ++ [length cs] + +gps12 :: [Int] -> Int +-- testing 6 combinations of argument values +-- 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 +gps12 xs = (length xs - fromJust (findIndex (0 ==) (reverse xs))) - 1 + +gps13 :: [Ratio Integer] -> Ratio Integer +-- testing 3 combinations of argument values +-- pruning with 4/8 rules +-- looking through 1 candidates of size 1 +-- 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 +gps13 qs = foldr (+) 0 qs / fromIntegral (length qs) + +odd :: Int -> Bool +-- testing 6 combinations of argument values +-- 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 0 candidates of size 4 +-- looking through 71 candidates of size 5 +odd x = x `mod` 2 /= 0 + +gps14 :: [Int] -> Int +-- testing 3 combinations of argument values +-- pruning with 1/1 rules +-- looking through 0 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 1 candidates of size 4 +gps14 xs = length (filter odd xs) + +gps14 :: [Int] -> Int +-- testing 3 combinations of argument values +-- 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 +gps14 [] = 0 +gps14 (x:xs) = x `mod` 2 + gps14 xs + +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 +gps15 xs ys = reverse xs == ys + +gps16 :: [Char] -> [Char] -> Bool +-- testing 6 combinations of argument values +-- pruning with 3/3 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 2 candidates of size 3 +-- looking through 6 candidates of size 4 +-- looking through 2 candidates of size 5 +gps16 cs ds = sort cs `isSubsequenceOf` sort ds + +gps17 :: Int -> Int +-- 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 13 candidates of size 3 +-- looking through 34 candidates of size 4 +-- looking through 75 candidates of size 5 +-- looking through 183 candidates of size 6 +-- looking through 577 candidates of size 7 +-- looking through 1311 candidates of size 8 +-- looking through 5282 candidates of size 9 +gps17 0 = 0 +gps17 x = x * x + gps17 (x - 1) + +gps18 :: [Int] -> [Int] -> [Int] +-- testing 3 combinations of argument values +-- pruning with 2/6 rules +-- looking through 3 candidates of size 1 +-- looking through 8 candidates of size 2 +-- looking through 11 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 791 candidates of size 7 +-- looking through 220 candidates of size 8 +-- looking through 4632 candidates of size 9 +gps18 [] [] = [] +gps18 [] (x:xs) = xs +gps18 (x:xs) [] = xs +gps18 (x:xs) (y:ys) = x + y:gps18 xs ys + +gps18 :: [Int] -> [Int] -> [Int] +-- testing 3 combinations of argument values +-- pruning with 2/7 rules +-- looking through 2 candidates of size 1 +-- looking through 2 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 12 candidates of size 4 +gps18 xs ys = zipWith (+) xs ys + +gps19 :: Int -> [Char] -> [Char] +-- testing 2 combinations of argument values +-- pruning with 0/0 rules +-- looking through 1 candidates of size 1 +-- looking through 0 candidates of size 2 +cannot conjure + +isVowel :: Char -> Bool +-- testing 12 combinations of argument values +-- pruning with 0/0 rules +-- looking through 2 candidates of size 1 +-- looking through 2 candidates of size 2 +-- looking through 2 candidates of size 3 +-- looking through 8 candidates of size 4 +-- looking through 8 candidates of size 5 +-- looking through 14 candidates of size 6 +-- looking through 28 candidates of size 7 +-- looking through 32 candidates of size 8 +-- looking through 40 candidates of size 9 +-- looking through 54 candidates of size 10 +-- looking through 78 candidates of size 11 +-- looking through 78 candidates of size 12 +-- looking through 102 candidates of size 13 +-- looking through 88 candidates of size 14 +-- looking through 104 candidates of size 15 +-- looking through 136 candidates of size 16 +-- looking through 122 candidates of size 17 +-- looking through 92 candidates of size 18 +-- looking through 92 candidates of size 19 +-- looking through 62 candidates of size 20 +-- looking through 62 candidates of size 21 +-- looking through 126 candidates of size 22 +isVowel 'a' = True +isVowel 'e' = True +isVowel 'i' = True +isVowel 'o' = True +isVowel 'u' = True +isVowel 'y' = True +isVowel c = False + +pig1 :: [Char] -> [Char] +-- testing 4 combinations of argument values +-- pruning with 5/5 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 5 candidates of size 3 +-- looking through 8 candidates of size 4 +-- looking through 14 candidates of size 5 +-- looking through 28 candidates of size 6 +-- looking through 47 candidates of size 7 +-- looking through 126 candidates of size 8 +-- looking through 201 candidates of size 9 +-- looking through 631 candidates of size 10 +-- looking through 1069 candidates of size 11 +-- looking through 3495 candidates of size 12 +-- looking through 6382 candidates of size 13 +-- looking through 20409 candidates of size 14 +pig1 "" = "ay" +pig1 (c:cs) = if isVowel c then (c:cs) ++ "ay" else cs ++ (c:"ay") + +gps20c :: [Char] -> [Char] +-- testing 2 combinations of argument values +-- pruning with 1/1 rules +-- looking through 1 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 2 candidates of size 3 +-- looking through 3 candidates of size 4 +-- looking through 5 candidates of size 5 +gps20c cs = unwords (map pig1 (words cs)) + +gps21 :: [Int] -> [Int] +-- testing 4 combinations of argument values +-- pruning with 4/4 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 3 candidates of size 3 +-- looking through 6 candidates of size 4 +-- looking through 5 candidates of size 5 +-- looking through 13 candidates of size 6 +-- looking through 9 candidates of size 7 +-- looking through 30 candidates of size 8 +-- looking through 25 candidates of size 9 +-- looking through 65 candidates of size 10 +gps21 [] = [] +gps21 (x:xs) = (if x < 0 then 0 else x):gps21 xs + +gps22 :: [Char] -> Int +-- testing 4 combinations of argument values +-- pruning with 5/9 rules +-- looking through 1 candidates of size 1 +-- looking through 0 candidates of size 2 +-- 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 +gps22 "" = 0 +gps22 (c:cs) = scrabble1 c + gps22 cs + +gps23 :: [Char] -> ([(Int,Int)],Int,Double) +-- testing 1 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +cannot conjure + +gps24 :: [Char] -> Char +-- testing 4 combinations of argument values +-- pruning with 15/19 rules +-- looking through 1 candidates of size 1 +-- 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 +gps24 cs = chr (ord ' ' + sum (map ord cs) `mod` 64) + +gps25 :: Int -> [Int] +-- testing 4 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +cannot conjure + +gps26 :: Int -> Int -> Int -> Int -> Int -> Char +-- testing 5 combinations of argument values +-- pruning with 4/4 rules +-- looking through 5 candidates of size 1 +-- looking through 0 candidates of size 2 +cannot conjure + +gps27 :: Int -> Int -> Int -> Int +-- testing 3 combinations of argument values +-- pruning with 14/18 rules +-- looking through 3 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 +-- looking through 0 candidates of size 5 +-- looking through 36 candidates of size 6 +-- 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 2592 candidates of size 11 +gps27 x y z = if x < y then (if y < z then y else z) else x + +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 0 candidates of size 4 +-- looking through 24 candidates of size 5 +-- looking through 0 candidates of size 6 +-- looking through 72 candidates of size 7 +gps28 x y z x' = x `min` (y `min` (z `min` x')) + +gps29 :: [Char] -> Int +-- testing 6 combinations of argument values +-- 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 +gps29 "" = 0 +gps29 (c:cs) = (if isVowel c then 1 else 0) + gps29 cs diff --git a/bench/gps2.hs b/bench/gps2.hs new file mode 100644 --- /dev/null +++ b/bench/gps2.hs @@ -0,0 +1,797 @@ +-- gps2.hs: General Program Synthesis Benchmark Suite II +-- +-- Copyright (C) 2021 Rudy Matela +-- Distributed under the 3-Clause BSD licence (see the file LICENSE). +{-# LANGUAGE CPP, TemplateHaskell #-} +#if __GLASGOW_HASKELL__ <= 710 +{-# LANGUAGE DeriveDataTypeable, StandaloneDeriving #-} +import Data.Typeable +#endif +import Conjure +import System.Environment (getArgs) + +import Data.List (findIndex, inits) -- for #1 +import Data.Char (toUpper) -- for #4 +import Data.Ratio ((%)) -- for #7 +import Data.List (findIndices, tails, isPrefixOf) -- for #12 +import Data.Maybe (fromJust) -- for #23 +import Test.LeanCheck -- for #24 +import Data.Express -- for #24 + + +gps1p :: [Int] -> Maybe Int +gps1p [0,-1,2] = Just 1 +gps1p [-1,0,1] = Just 0 +gps1p [0,-1] = Just 1 +gps1p [1,-1,-1] = Just 2 + +gps1g :: [Int] -> Maybe Int +gps1g xs = findIndex (0 >) (map sum (tail (inits xs))) + +gps1p2 :: Int -> [Int] -> Int +gps1p2 0 [0,-1,2] = 1 +gps1p2 0 [-1,0,1] = 0 +gps1p2 0 [0,-1] = 1 +gps1p2 0 [1,-1,-1] = 2 +gps1p2 0 [0,0,0,-1] = 3 +gps1p2 0 [1,0,0,-2] = 3 + +-- efficient gps1, unreachable performance-wise +gps1g2 :: Int -> [Int] -> Int +gps1g2 t [] = undefined -- 1 +gps1g2 t (x:xs) = if t + x < 0 -- 7 + then 0 -- 8 + else 1 + gps1g2 (t + x) xs -- 15 + +gps1c :: IO () +gps1c = do + conjure "gps1" gps1p + [ pr (0 :: Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim ">" ((>) :: Int -> Int -> Bool) + , prim "foldr" (foldr :: (Int -> Int -> Int) -> Int -> [Int] -> Int) + , prim "sum" (sum :: [Int] -> Int) + , prim "findIndex" (findIndex :: (Int -> Bool) -> [Int] -> Maybe Int) + , prim "map" (map :: ([Int] -> Int) -> [[Int]] -> [Int]) + , prim "inits" (inits :: [Int] -> [[Int]]) + , prim "tail" (tail :: [[Int]] -> [[Int]]) + ] + + conjure "gps1" gps1p + [ pr (0 :: Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim ">" ((>) :: Int -> Int -> Bool) + , prim "sum" (sum :: [Int] -> Int) + , prim "foldr" (foldr :: (Int -> Int -> Int) -> Int -> [Int] -> Int) + , prim "findIndex" (findIndex :: (Int -> Bool) -> [Int] -> Maybe Int) + , prim "map" (map :: ([Int] -> Int) -> [[Int]] -> [Int]) + , prim "inits" (inits :: [Int] -> [[Int]]) + , prim "tail" (tail :: [[Int]] -> [[Int]]) + ] + + conjureWithMaxSize 4 "gps1" gps1p2 + [ pr (0 :: Int) + , pr (1 :: Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "<" ((<) :: Int -> Int -> Bool) + , prif (undefined :: Int) + , prim "undefined" (undefined :: Int) + ] + + +gps2p :: Double -> Double -> Int -> Double +gps2p 2 1 1 = 2 + 1 +gps2p 2 1 2 = 2 + 1 + 1 + 0.5 +gps2p 2 1 3 = 2 + 1 + 1 + 0.5 + 0.5 + 0.25 +gps2p 3 1 1 = 3 + 1 +gps2p 3 1 2 = 3 + 1 + 1/3 + +-- apex to apex +gps2g :: Double -> Double -> Int -> Double +gps2g h0 h1 0 = h0 + h1 +gps2g h0 h1 n = h0 + h1 + gps2g h1 (h1 * (h1 / h0)) (n - 1) +-- size 17, out of reach performance-wise + +gps2c :: IO () +gps2c = do + conjureWithMaxSize 6 "gps2" gps2p + [ pr (0 :: Int) + , pr (1 :: Int) + , prim "*" ((*) :: Double -> Double -> Double) + , prim "/" ((/) :: Double -> Double -> Double) + , prim "+" ((+) :: Double -> Double -> Double) + , prim "-" ((-) :: Double -> Double -> Double) + ] + + +gps3p :: String -> Int +gps3p "X X X X X X X X X XXX" = 300 +gps3p "11 11 11 11 11 11 11 11 11 11" = 20 + +-- unreachable performance wise +-- I presume a correct solution would need around 30 symbols +gps3c :: IO () +gps3c = conjure "gps3" gps3p [] + + + +-- GPS #4: https://www.codewars.com/kata/517abf86da9663f1d2000003 +-- Here we're using the problem description as in the GPS2 paper +-- considering we're getting input as kebab-case. +-- Nevertheless, this one is out of reach performance-wise (and OOM-wise) + +gps4p :: String -> String +gps4p "the-stealth-warrior" = "theStealthWarrior" +--gps4p "The_Stealth_Warrior" = "TheStealthWarrior" +gps4p "camel-case" = "camelCase" +--gps4p "snake_case" = "snakeCase" +gps4p "Kebab-Case" = "KebabCase" + +gps4g :: String -> String +gps4g "" = "" +-- gps4g ('-':c:cs) = toUpper c : gps4g cs +-- gps4g ('_':c:cs) = toUpper c : gps4g cs +gps4g (c:cs) = if c == '-' -- 5 + then if null cs -- 8 + then "" -- 9 + else toUpper (head cs) : gps4g (tail cs) -- 16 + else c : gps4g cs -- 19 + +gps4c :: IO () +gps4c = do + let force = [ [val "the-stealth-warrior"] +-- , [val "The_Stealth_Warrior"] + , [val "camel-case"] +-- , [val "snake_case"] + , [val "Kebab-Case"] + ] + conjureWith args{forceTests = force, maxSize=6} "gps4" gps4p + [ pr '-' +-- , pr '_' + , pr ("" :: String) + , prim ":" ((:) :: Char -> String -> String) + , prim "==" ((==) :: Char -> Char -> Bool) + , prim "head" (head :: String -> Char) + , prim "tail" (tail :: String -> String) + , prif (undefined :: Char) + , prif (undefined :: String) + , prim "toUpper" (toUpper :: Char -> Char) + ] + + +-- GPS #5: Coin Sums +-- https://projecteuler.net/problem=31 +-- +-- The problem description is inconsistent with the one given in the paper. +-- +-- Paper: change-making problem in USD +-- Euler: How many different ways can £2 be made using any number of coins? +-- +-- I'm considering the Paper to be canonical, as the one in Project Euler can be solved by a constant function: +-- +-- solution :: Int +-- solution = 6378216738 -- <-- arbitrary example here, I didn't run the numbers +gps5p :: Int -> [Int] +gps5p 100 = [4, 0, 0, 0] +gps5p 50 = [2, 0, 0, 0] +gps5p 25 = [1, 0, 0, 0] +gps5p 30 = [1, 0, 1, 0] +gps5p 20 = [0, 2, 0, 0] +gps5p 3 = [0, 0, 0, 3] + +gps5p2 :: [Int] -> Int -> [Int] +gps5p2 [25, 10, 5, 1] 100 = [4, 0, 0, 0] +gps5p2 [25, 10, 5, 1] 50 = [2, 0, 0, 0] +gps5p2 [25, 10, 5, 1] 25 = [1, 0, 0, 0] +gps5p2 [25, 10, 5, 1] 30 = [1, 0, 1, 0] +gps5p2 [25, 10, 5, 1] 20 = [0, 2, 0, 0] +gps5p2 [25, 10, 5, 1] 3 = [0, 0, 0, 3] + +coins :: [Int] +coins = [25, 10, 5, 1] + +gps5g :: Int -> [Int] +gps5g = tell coins + +tell :: [Int] -> Int -> [Int] +tell [] a = [] +tell (n:ns) a = a `div` n : tell ns (a `mod` n) + +gps5c :: IO () +gps5c = do + -- cannot conjure directly due to needing to introduce a local definition + conjure "gps5" gps5p + [ + ] + + let force = [ [val coins, val (100::Int)] + , [val coins, val ( 50::Int)] + , [val coins, val ( 25::Int)] + , [val coins, val ( 30::Int)] + , [val coins, val ( 20::Int)] + , [val coins, val ( 3::Int)] + ] + conjureWith args{forceTests=force} "tell" gps5p2 + [ pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "`div`" (div :: Int -> Int -> Int) + , prim "`mod`" (mod :: Int -> Int -> Int) + ] + + conjureWith args{forceTests=force} "gps5" gps5p + [ pr coins + , prim "tell" tell + ] + + conjureWith args{forceTests=force} "gps5" gps5p + [ pr (1 :: Int) + , pr (5 :: Int) + , pr (10 :: Int) + , pr (25 :: Int) + , pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "tell" tell + ] + + +-- GPS2#6: Cut Vector +gps6g :: [Int] -> Int +gps6g xs = snd + $ minimum + [ (abs (sum xs0 - sum xs1), i) + | i <- [0 .. length xs] + , let (xs0, xs1) = splitAt i xs + ] + +-- not Conjurable due to needing local definition +-- I conjecture it _perhaps-maybe_ this could be done in 3 steps, +-- but I'll leave this as-is for now. +gps6c :: IO () +gps6c = + conjure "gps6" gps6g [] + + +-- GPS2#7: Dice Game +-- https://projecteuler.net/problem=31 +-- https://github.com/thelmuth/Clojush/blob/psb2-v1.0/src/clojush/problems/psb2/dice_game.clj +-- again, paper and problem are inconsistent, going with paper +gps7p :: Integer -> Integer -> Rational +gps7p 4 6 = 1 % 4 +gps7p 6 4 = 7 % 12 +gps7p 2 4 = 1 % 8 +gps7p 4 2 = 5 % 8 +gps7p 2 6 = 1 % 12 +gps7p 6 2 = 3 % 4 + +gps7g :: Integer -> Integer -> Rational +gps7g peter colin = sum (map (min colin) [0 .. (peter-1)]) % (colin*peter) + +-- out of reach performance-wise +gps7c :: IO () +gps7c = do + conjureWith args{maxSize=6} "gps7" gps7p $ take 0 + [ pr (0 :: Integer) + , pr (1 :: Integer) + , prim "%" ((%) :: Integer -> Integer -> Rational) + , prim "+" ((+) :: Integer -> Integer -> Integer) + , prim "-" ((-) :: Integer -> Integer -> Integer) + , prim "*" ((*) :: Integer -> Integer -> Integer) + , prim "min" (min :: Integer -> Integer -> Integer) + , prim ".." (enumFromTo :: Integer -> Integer -> [Integer]) + , prim "map" (map :: (Integer -> Integer) -> [Integer] -> [Integer]) + , prim "sum" (sum :: [Integer] -> Integer) + ] + + +gps8p :: Int -> [Int] -> (Int,Int) +gps8p 2 [1,1,2] = (1,1) +gps8p 3 [1,1,2] = (1,2) +gps8p 2 [0,1,0,2] = (0,2) + +gps8g :: Int -> [Int] -> (Int,Int) +gps8g x xs = head [(y,z) | y <- xs, z <- xs, y + z == x] +-- this one could be generated but is a bit of a stretch... +-- it is unintuitive to provide the given symbols +-- gps8g :: Int -> [Int] -> (Int,Int) +-- gps8g x xs = head $ filter ((x ==) . uncurry (+)) $ liftA2 (,) xs xs + +gps8c :: IO () +gps8c = conjure "gps" gps8p [] + + +-- GPSB#9: Fizz Buzz (CW) +gps9p :: Int -> String +gps9p 3 = "Fizz" +gps9p 4 = "4" +gps9p 5 = "Buzz" +gps9p 6 = "Fizz" +gps9p 10 = "Buzz" +gps9p 15 = "FizzBuzz" +gps9p 17 = "17" + +gps9g :: Int -> String +gps9g x + | x `div` 3 == 0 = "Fizz" -- 7 + | x `div` 5 == 0 = "Buzz" -- 14 + | x `div` 3 == 0 && x `div` 5 == 0 = "FizzBuzz" -- 27 + | otherwise = show x -- 29 + +-- probably unreachable performance-wise +gps9c :: IO () +gps9c = conjure "gps" gps9p [] + + +gps10p :: [Int] -> Int +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 +-- ?? + +-- unreachable due to lambda +gps10c :: IO () +gps10c = conjure "gps" gps9p [] + + +gps11p :: Int -> Int -> Int +gps11p = gcd' + where + gcd' :: Int -> Int -> Int + gcd' 1 1 = 1 + gcd' 1 2 = 1 + gcd' 2 1 = 1 + gcd' 2 2 = 2 + gcd' 2 6 = 2 + gcd' 6 2 = 2 + gcd' 3 6 = 3 + gcd' 6 3 = 3 + gcd' 6 9 = 3 + gcd' 9 6 = 3 + gcd' 12 18 = 6 + +gps11c :: IO () +gps11c = conjure "gcd a b" gps11p + [ pr (0::Int) + , prim "`mod`" (mod :: Int -> Int -> Int) + ] + -- generated function: + -- gcd x 0 = x + -- gcd x y = gcd y (x `mod` y) + + +gps12p :: String -> String -> [Int] +gps12p "a" "a" = [0] +gps12p "aa" "a" = [0,1] +gps12p "aa" "aa" = [0] +gps12p "a a" "a" = [0,2] +gps12p "b" "a" = [] + +gps12g :: String -> String -> [Int] +gps12g s s' = findIndices (s' `isPrefixOf`) (tails s) + +gps12c :: IO () +gps12c = conjure "gps12" gps12p + [ prim "findIndices" (findIndices :: (String -> Bool) -> [String] -> [Int]) + , prim "`isPrefixOf`" (isPrefixOf :: String -> String -> Bool) + , prim "tails" (tails :: String -> [String]) + ] + + +gps13p :: [Int] -> [Int] +gps13p [0,1] = [1] +gps13p [1,0] = [1,0] +gps13p [2,0,1] = [2,1] +gps13p [1,0,1] = [1] + +gps13g :: [Int] -> [Int] +gps13g = leaders + where + leaders [] = [] + leaders (x:xs) = if all (x >) xs + then x : leaders xs + else leaders xs + +gps13c :: IO () +gps13c = conjure "gps13_leaders" gps13p + [ pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim ">" ((>) :: Int -> Int -> Bool) + , prim "all" (all :: (Int -> Bool) -> [Int] -> Bool) + , prif (undefined :: [Int]) + ] + + +gps14p :: [Int] -> Int +gps14p = undefined + +-- 30 symbols +gps14g :: [Int] -> Int +gps14g = luhn + where + luhn :: [Int] -> Int + luhn xs = sum (firsts xs ++ map double9 (seconds xs)) + where + double9 :: Int -> Int + double9 x = if xx > 9 + then xx - 9 + else xx + where + xx = x * 2 + seconds :: [Int] -> [Int] + seconds [] = [] + seconds (x:y:xs) = y : seconds xs + firsts :: [Int] -> [Int] + firsts [] = [] + firsts (x:y:xs) = x : seconds xs + +-- cannot Conjure directly +gps14c :: IO () +gps14c = conjure "gps14_luhn" gps14p + [ + ] + + +gps15p :: () -> () +gps15p = undefined + +-- skipped +gps15c :: IO () +gps15c = conjure "gps15_mastermind" gps15p [] + + +gps16p :: String -> String +gps16p "a" = "a" +gps16p "aaa" = "a" +gps16p "aa" = "aa" +gps16p "a a" = " " +gps16p " a " = "a" +gps16p "a a " = " a" +gps16p " a a " = " " + +gps16g1 :: String -> String +gps16g1 s = if odd len + then take 1 (drop (len `div` 2) s) + else take 2 (drop (len `div` 2 - 1) s) + where + len = length s + +gps16g2 :: String -> String +gps16g2 "" = "" +gps16g2 (c:cs) = if length cs <= 1 + then c : cs + else gps16g2 (init cs) + +gps16c :: IO () +gps16c = conjure "gps16_middle" gps16p + [ pr "" + , pr (1 :: Int) + , prim "<=" ((<=) :: Int -> Int -> Bool) + , prim ":" ((:) :: Char -> String -> String) + , prim "length" (length :: String -> Int) + , prim "init" (init :: String -> String) + , prif (undefined :: String) + ] + + +gps17p :: [Int] -> Int +gps17p [0,1,0] = 0 +gps17p [1,1] = 1 +gps17p [1,1,0] = 1 +gps17p [0,1,1] = 1 +gps17p [1,1,1] = 2 + +gps17g :: [Int] -> Int +gps17g xs = pds xs + where + pds [] = 0 + pds (x:xs) = if not (null xs) && x == head xs + then x + pds xs + else pds xs +-- surprise: +-- gps17_pds [] = 0 +-- gps17_pds (x:xs) = (if not (null xs) && head xs == x then x else 0) + gps17_pds xs + + +-- can generate at size 15 in 18 seconds +-- setting limit of 5 for faster output +gps17c :: IO () +gps17c = conjureWith args{maxSize=5} "gps17_pds" gps17p + [ pr (0 :: Int) + , prif (undefined :: Int) + , prim "not" not + , prim "null" (null :: [Int] -> Bool) + , prim "==" ((==) :: Int -> Int -> Bool) + , prim "head" (head :: [Int] -> Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "&&" (&&) + ] + + +gps18p :: [Double] -> [Double] -> Double +gps18p [1.0] [0.5] = 0.5 +gps18p [2.0] [0.5] = 1.0 +gps18p [1.0,1.0] [0.5,0.0] = 1.5 +gps18p [1.0,1.0] [0.0,0.5] = 1.5 + +gps18g :: [Double] -> [Double] -> Double +gps18g prices discounts = foldr (+) 0 (zipWith (*) prices (map (1-) discounts)) + +-- this was OOM'd +gps18c :: IO () +gps18c = conjureWithMaxSize 6 "gps18_price" gps18p + [ pr (0 :: Double) + , pr (1 :: Double) + , prim "+" ((+) :: Double -> Double -> Double) + , prim "*" ((*) :: Double -> Double -> Double) + , prim "-" ((-) :: Double -> Double -> Double) +-- , prim "foldr" (foldr :: (Double -> Double -> Double) -> Double -> [Double] -> Double) +-- , prim "zipWith" (zipWith :: (Double -> Double -> Double) -> [Double] -> [Double] -> [Double]) +-- , prim "map" (map :: (Double -> Double) -> [Double] -> [Double]) + ] + + +gps19p :: Int -> Double -> Double -> Double -> Double +gps19p 0 0 1.0 1.0 = 0 +gps19p 1 0 1.0 1.0 = 0 +gps19p 2 0 1.0 1.0 = 0 +gps19p 1 0 1.0 2.0 = 1.0 +gps19p 2 0 1.0 2.0 = 2.0 +gps19p 1 0 0.5 1.0 = 0.5 +gps19p 2 0 0.5 1.0 = 1.0 + +gps19g :: Int -> Double -> Double -> Double -> Double +gps19g 0 total melt fall = total +gps19g n total melt fall = gps19p (n-1) (max 0 (total-melt+fall)) melt fall +-- size 14, out of reach performance wise. + +gps19c :: IO () +gps19c = conjureWithMaxSize 6 "gps19_snowday" gps19p + [ pr (0 :: Int) + , pr (1 :: Int) + , prim "max" (max :: Double -> Double -> Double) + , prim "+" ((+) :: Double -> Double -> Double) + , prim "-" ((-) :: Double -> Double -> Double) + ] + + +gps20p :: String -> Bool +gps20p = undefined + +gps20c :: IO () +gps20c = conjure "gps20" gps20p + [ + ] + + +gps21p :: String -> String +gps21p "word" = "word" +gps21p "words" = "sdrow" +gps21p "word words" = "word sdrow" +gps21p "words word" = "sdrow word" + +spin' :: String -> String +spin' "abc" = "abc" +spin' "abcd" = "abcd" +spin' "word" = "word" +spin' "abcde" = "edcba" +spin' "words" = "sdrow" +spin' "hello" = "olleh" +spin' "world" = "dlrow" + +spin :: String -> String +spin w = if length w >= 5 + then reverse w + else w + +gps21g :: String -> String +gps21g s = unwords (map spin (words s)) + +gps21c :: IO () +gps21c = do + let force = [ [ val "abc" ] + , [ val "abcd" ] + , [ val "word" ] + , [ val "abcde" ] + , [ val "words" ] + , [ val "hello" ] + , [ val "world" ] + ] + conjureWith args{maxSize=12, forceTests=force} "spin" spin' + [ prim "length" (length :: String -> Int) + , prim "reverse" (reverse :: String -> String) + , prif (undefined :: String) + , prim ">=" ((>=) :: Int -> Int -> Bool) + , pr (5 :: Int) + ] + + let force = [ [ val "word" ] + , [ val "words" ] + , [ val "word words" ] + , [ val "words word" ] + ] + conjureWith args{maxSize=12, forceTests=force} "gps21_spinwords" gps21p + [ prim "words" words + , prim "unwords" unwords + , prim "spin" (spin :: String -> String) + , prim "map" (map :: (String -> String) -> [String] -> [String]) + , prim "length" (length :: String -> Int) + , prim "reverse" (reverse :: String -> String) + , prif (undefined :: String) + , prim ">=" ((>=) :: Int -> Int -> Bool) + , pr (5 :: Int) + ] + + +digits :: Int -> [Int] +digits 0 = [] +digits n = n `mod` 10 : digits (n `div` 10) + +digits' :: Int -> [Int] +digits' 1 = [1] +digits' 12 = [2,1] +digits' 21 = [1,2] +digits' 123 = [3,2,1] +digits' 321 = [1,2,3] + +gps22p :: Int -> String +gps22p = undefined + +gps22c :: IO () +gps22c = do + conjure "digits" digits' + [ pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "`div`" (div :: Int -> Int -> Int) + , prim "`mod`" (mod :: Int -> Int -> Int) + , prim "div10" ((`div` 10) :: Int -> Int) + , pr (10 :: Int) + ] + + conjure "gps22" gps22p + [ + ] + + +gps23p :: String -> String -> String -> String +gps23p "abcd" "abcd" "abcd" = "abcd" +gps23p "abcd" "dcba" "abcd" = "dcba" +gps23p "abcd" "1234" "abcd" = "1234" +gps23p "abcd" "1234" "bacd" = "2134" + +gps23g :: String -> String -> String -> String +-- gps23g f t s = map (\c -> fromJust $ c `lookup` zipWith (,) f t) s +gps23g f t s = map (fromJust . (`lookup` zipWith (,) f t)) s + +gps23c :: IO () +gps23c = do + let force = [ [ val "abcd", val "abcd", val "abcd" ] + , [ val "abcd", val "dcba", val "abcd" ] + , [ val "abcd", val "1234", val "abcd" ] + , [ val "abcd", val "1234", val "bacd" ] + ] + -- cannot conjure, needs lambda + conjureWith args{forceTests=force} "gps23" gps23p + [ + ] + + +data Twitter = Tweet Int + | TooMany + | Empty + deriving (Eq, Show) + +#if __GLASGOW_HASKELL__ <= 710 +deriving instance Typeable Twitter + +instance Express Twitter where + expr (Tweet n) = value "Tweet" Tweet :$ expr n + expr t = val t +#else +deriveExpress ''Twitter +#endif +deriveListable ''Twitter +deriveName ''Twitter + +instance Conjurable Twitter where + conjureExpress = reifyExpress + conjureEquality = reifyEquality + conjureTiers = reifyTiers + +gps24g_twitter :: String -> Twitter +gps24g_twitter "" = Empty +gps24g_twitter s = if length s > 140 + then TooMany + else Tweet (length s) + +gps24p_twitter :: String -> Twitter +gps24p_twitter "" = Empty +gps24p_twitter "abcd" = Tweet 4 +gps24p_twitter "0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij" = Tweet 140 +gps24p_twitter "0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghijX" = TooMany + +gps24c :: IO () +gps24c = do + let force = [ [ val "" ] + , [ val "abcd" ] + , [ val "0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij" ] + , [ val "0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghij0123456789abcdefghijX" ] + ] + conjureWith args{maxTests=360, forceTests=force} "gps24" gps24p_twitter + [ pr Empty + , pr TooMany + , prim "Tweet" Tweet + , prif (undefined :: Twitter) + , pr "" + , prim ":" ((:) :: Char -> String -> String) + , prim "length" (length :: String -> Int) + , pr (140 :: Int) + , prim ">" ((>) :: Int -> Int -> Bool) + ] + + +gps25p :: [Double] -> [Double] -> Double +gps25p [0] [1] = 1 +gps25p [1] [0] = 1 +gps25p [-1] [1] = 2 +gps25p [0,1] [1,0] = sqrt 2 +gps25p [1,0] [0,1] = sqrt 2 +gps25p [0,0,1] [1,0,0] = sqrt 2 +gps25p [0,1,2] [1,2,3] = sqrt 3 + +gps25g :: [Double] -> [Double] -> Double +gps25g v1 v2 = sqrt (foldr (+) 0 (map (^2) (zipWith (-) v1 v2))) + +-- out of reach performance-wise +gps25c :: IO () +gps25c = conjureWith args{maxSize=6} "gps25" gps25p + [ pr (0 :: Double) + , pr (2 :: Double) + , prim "+" ((+) :: Double -> Double -> Double) + , prim "-" ((-) :: Double -> Double -> Double) + , prim "**" ((**) :: Double -> Double -> Double) + , prim "zipWith" (zipWith :: (Double -> Double -> Double) -> [Double] -> [Double] -> [Double]) + , prim "map" (map :: (Double -> Double) -> [Double] -> [Double]) + , prim "sqrt" (sqrt :: Double -> Double) + ] + + +main :: IO () +main = do + as <- getArgs + case as of + [] -> sequence_ gpss + (n:_) -> gpss !! (read n - 1) + + +gpss :: [IO ()] +gpss = [ gps1c + , gps2c + , gps3c + , gps4c + , gps5c + , gps6c + , gps7c + , gps8c + , gps9c + , gps10c + , gps11c + , gps12c + , gps13c + , gps14c + , gps15c + , gps16c + , gps17c + , gps18c + , gps19c + , gps20c + , gps21c + , gps22c + , gps23c + , gps24c + , gps25c + ] diff --git a/bench/gps2.out b/bench/gps2.out new file mode 100644 --- /dev/null +++ b/bench/gps2.out @@ -0,0 +1,349 @@ +gps1 :: [Int] -> Maybe Int +-- testing 4 combinations of argument values +-- pruning with 11/21 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 1 candidates of size 4 +-- looking through 0 candidates of size 5 +-- looking through 0 candidates of size 6 +-- looking through 1 candidates of size 7 +-- looking through 0 candidates of size 8 +-- looking through 1 candidates of size 9 +-- looking through 2 candidates of size 10 +gps1 xs = findIndex (0 >) (map (foldr (+) 0) (tail (inits xs))) + +gps1 :: [Int] -> Maybe Int +-- testing 4 combinations of argument values +-- pruning with 11/21 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 1 candidates of size 4 +-- looking through 1 candidates of size 5 +-- looking through 0 candidates of size 6 +-- looking through 1 candidates of size 7 +-- looking through 4 candidates of size 8 +gps1 xs = findIndex (0 >) (map sum (tail (inits xs))) + +gps1 :: Int -> [Int] -> Int +-- 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 255 candidates of size 3 +-- looking through 451 candidates of size 4 +cannot conjure + +gps2 :: Double -> Double -> Int -> Double +-- testing 5 combinations of argument values +-- pruning with 3/3 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 314 candidates of size 5 +-- looking through 1432 candidates of size 6 +cannot conjure + +gps3 :: [Char] -> Int +-- testing 0 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +cannot conjure + +gps4 :: [Char] -> [Char] +-- testing 3 combinations of argument values +-- pruning with 13/20 rules +-- looking through 2 candidates of size 1 +-- looking through 3 candidates of size 2 +-- looking through 9 candidates of size 3 +-- looking through 22 candidates of size 4 +-- looking through 54 candidates of size 5 +-- looking through 124 candidates of size 6 +cannot conjure + +gps5 :: Int -> [Int] +-- testing 6 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +cannot conjure + +tell :: [Int] -> Int -> [Int] +-- testing 6 combinations of argument values +-- pruning with 0/0 rules +-- 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 6 candidates of size 5 +-- looking through 61 candidates of size 6 +-- looking through 26 candidates of size 7 +-- looking through 538 candidates of size 8 +-- looking through 134 candidates of size 9 +-- looking through 5299 candidates of size 10 +tell [] x = [] +tell (x:xs) y = y `div` x:tell xs (y `mod` x) + +gps5 :: Int -> [Int] +-- testing 6 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 1 candidates of size 3 +gps5 x = tell [25,10,5,1] x + +gps5 :: Int -> [Int] +-- testing 6 combinations of argument values +-- pruning with 12/12 rules +-- looking through 1 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 5 candidates of size 3 +-- looking through 9 candidates of size 4 +-- looking through 44 candidates of size 5 +-- looking through 91 candidates of size 6 +-- looking through 441 candidates of size 7 +-- looking through 946 candidates of size 8 +-- looking through 4450 candidates of size 9 +-- looking through 9710 candidates of size 10 +-- looking through 44767 candidates of size 11 +gps5 x = tell [25,10,5,1] x + +gps6 :: [Int] -> Int +-- testing 360 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +cannot conjure + +gps7 :: Integer -> Integer -> Ratio Integer +-- testing 6 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +cannot conjure + +gps :: Int -> [Int] -> (Int,Int) +-- testing 3 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +cannot conjure + +gps :: Int -> [Char] +-- testing 7 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +cannot conjure + +gps :: Int -> [Char] +-- testing 7 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +cannot conjure + +gcd :: Int -> Int -> Int +-- testing 11 combinations of argument values +-- pruning with 0/0 rules +-- looking through 3 candidates of size 1 +-- looking through 8 candidates of size 2 +-- looking through 20 candidates of size 3 +-- looking through 60 candidates of size 4 +-- looking through 150 candidates of size 5 +-- looking through 472 candidates of size 6 +gcd x 0 = x +gcd x y = gcd y (x `mod` y) + +gps12 :: [Char] -> [Char] -> [Int] +-- testing 5 combinations of argument values +-- pruning with 1/2 rules +-- 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 +-- looking through 4 candidates of size 5 +gps12 cs ds = findIndices (ds `isPrefixOf`) (tails cs) + +gps13_leaders :: [Int] -> [Int] +-- testing 4 combinations of argument values +-- pruning with 5/5 rules +-- looking through 2 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 5 +-- looking through 2 candidates of size 6 +-- looking through 1 candidates of size 7 +-- looking through 4 candidates of size 8 +-- looking through 7 candidates of size 9 +-- looking through 21 candidates of size 10 +-- looking through 34 candidates of size 11 +-- looking through 67 candidates of size 12 +gps13_leaders [] = [] +gps13_leaders (x:xs) = if all (x >) xs then x:gps13_leaders xs else gps13_leaders xs + +gps14_luhn :: [Int] -> Int +-- testing 0 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +cannot conjure + +gps15_mastermind :: () -> () +-- testing 0 combinations of argument values +-- pruning with 0/1 rules +-- looking through 1 candidates of size 1 +gps15_mastermind u = u + +gps16_middle :: [Char] -> [Char] +-- testing 7 combinations of argument values +-- pruning with 10/11 rules +-- looking through 2 candidates of size 1 +-- looking through 3 candidates of size 2 +-- looking through 7 candidates of size 3 +-- looking through 12 candidates of size 4 +-- looking through 21 candidates of size 5 +-- looking through 34 candidates of size 6 +-- looking through 61 candidates of size 7 +-- looking through 126 candidates of size 8 +-- looking through 307 candidates of size 9 +-- looking through 824 candidates of size 10 +-- looking through 2282 candidates of size 11 +-- looking through 6293 candidates of size 12 +gps16_middle "" = "" +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 +-- pruning with 29/40 rules +-- looking through 1 candidates of size 1 +-- 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 +cannot conjure + +gps18_price :: [Double] -> [Double] -> Double +-- testing 4 combinations of argument values +-- pruning with 28/34 rules +-- looking through 2 candidates of size 1 +-- looking through 8 candidates of size 2 +-- looking through 72 candidates of size 3 +-- looking through 72 candidates of size 4 +-- looking through 1192 candidates of size 5 +-- looking through 696 candidates of size 6 +cannot conjure + +gps19_snowday :: Int -> Double -> Double -> Double -> Double +-- testing 7 combinations of argument values +-- pruning with 7/12 rules +-- looking through 3 candidates of size 1 +-- looking through 7 candidates of size 2 +-- looking through 31 candidates of size 3 +-- looking through 169 candidates of size 4 +-- looking through 543 candidates of size 5 +-- looking through 3608 candidates of size 6 +cannot conjure + +gps20 :: [Char] -> Bool +-- testing 0 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +cannot conjure + +spin :: [Char] -> [Char] +-- testing 7 combinations of argument values +-- pruning with 6/6 rules +-- reasoning produced 1 incorrect properties, please re-run with more tests for faster results +-- looking through 1 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 0 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 4 candidates of size 8 +spin cs = if length cs >= 5 then reverse cs else cs + +gps21_spinwords :: [Char] -> [Char] +-- testing 4 combinations of argument values +-- pruning with 16/16 rules +-- reasoning produced 2 incorrect properties, please re-run with more tests for faster results +-- looking through 1 candidates of size 1 +-- looking through 2 candidates of size 2 +-- looking through 2 candidates of size 3 +-- looking through 3 candidates of size 4 +-- looking through 5 candidates of size 5 +gps21_spinwords cs = unwords (map spin (words cs)) + +digits :: Int -> [Int] +-- testing 5 combinations of argument values +-- pruning with 7/7 rules +-- looking through 1 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 2 candidates of size 3 +-- looking through 3 candidates of size 4 +-- looking through 15 candidates of size 5 +-- looking through 14 candidates of size 6 +-- looking through 97 candidates of size 7 +-- looking through 104 candidates of size 8 +-- looking through 871 candidates of size 9 +-- looking through 935 candidates of size 10 +-- looking through 9018 candidates of size 11 +-- looking through 9686 candidates of size 12 +cannot conjure + +gps22 :: Int -> [Char] +-- testing 0 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +cannot conjure + +gps23 :: [Char] -> [Char] -> [Char] -> [Char] +-- testing 4 combinations of argument values +-- pruning with 0/0 rules +-- looking through 3 candidates of size 1 +-- looking through 12 candidates of size 2 +-- looking through 33 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 +cannot conjure + +gps24 :: [Char] -> Twitter +-- testing 4 combinations of argument values +-- pruning with 4/8 rules +-- reasoning produced 4 incorrect properties, please re-run with more tests for faster results +-- looking through 2 candidates of size 1 +-- looking through 3 candidates of size 2 +-- looking through 8 candidates of size 3 +-- looking through 7 candidates of size 4 +-- looking through 4 candidates of size 5 +-- looking through 5 candidates of size 6 +-- looking through 10 candidates of size 7 +-- looking through 48 candidates of size 8 +-- looking through 138 candidates of size 9 +-- looking through 232 candidates of size 10 +-- looking through 322 candidates of size 11 +-- looking through 483 candidates of size 12 +gps24 "" = Empty +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 33/58 rules +-- looking through 2 candidates of size 1 +-- looking through 9 candidates of size 2 +-- looking through 89 candidates of size 3 +-- looking through 312 candidates of size 4 +-- looking through 2262 candidates of size 5 +-- looking through 9400 candidates of size 6 +cannot conjure + diff --git a/bench/ill-hit.out b/bench/ill-hit.out --- a/bench/ill-hit.out +++ b/bench/ill-hit.out @@ -4,8 +4,8 @@ -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 -- looking through 6 candidates of size 3 --- looking through 19 candidates of size 4 --- looking through 31 candidates of size 5 +-- looking through 20 candidates of size 4 +-- looking through 34 candidates of size 5 sum [] = 0 sum (x:xs) = x + sum xs @@ -15,8 +15,8 @@ -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 -- looking through 6 candidates of size 3 --- looking through 19 candidates of size 4 --- looking through 31 candidates of size 5 +-- looking through 20 candidates of size 4 +-- looking through 34 candidates of size 5 sum [] = 0 sum (x:xs) = x + sum xs @@ -26,8 +26,8 @@ -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 -- looking through 6 candidates of size 3 --- looking through 19 candidates of size 4 --- looking through 31 candidates of size 5 +-- looking through 20 candidates of size 4 +-- looking through 34 candidates of size 5 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 @@ -8,7 +8,7 @@ sort xs `isSubsequenceOf` ys == xs `isSubsequenceOf` sort ys -} --- reasoning produced incorrect properties, please re-run with more tests for faster results +-- reasoning produced 1 incorrect properties, please re-run with more tests for faster results -- looking through 0 candidates of size 1 -- looking through 0 candidates of size 2 -- looking through 2 candidates of size 3 diff --git a/bench/p12.out b/bench/p12.out --- a/bench/p12.out +++ b/bench/p12.out @@ -4,10 +4,10 @@ -- pruning with 67/100 rules -- looking through 3 candidates of size 1 -- looking through 7 candidates of size 2 --- looking through 22 candidates of size 3 --- looking through 62 candidates of size 4 --- looking through 175 candidates of size 5 --- looking through 542 candidates of size 6 +-- looking through 25 candidates of size 3 +-- looking through 82 candidates of size 4 +-- looking through 263 candidates of size 5 +-- looking through 960 candidates of size 6 factorial 0 = 1 factorial x = x * factorial (dec x) 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 @@ -10.0 +9.1 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 @@ -15.2 +23.8 diff --git a/bench/runtime/zero/bench/gps2.runtime b/bench/runtime/zero/bench/gps2.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/zero/bench/gps2.runtime @@ -0,0 +1,1 @@ +9.5 diff --git a/bench/runtime/zero/bench/lowtests.runtime b/bench/runtime/zero/bench/lowtests.runtime --- a/bench/runtime/zero/bench/lowtests.runtime +++ b/bench/runtime/zero/bench/lowtests.runtime @@ -1,1 +1,1 @@ -0.4 +0.3 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 @@ -3.5 +3.2 diff --git a/bench/runtime/zero/bench/take-drop.runtime b/bench/runtime/zero/bench/take-drop.runtime --- a/bench/runtime/zero/bench/take-drop.runtime +++ b/bench/runtime/zero/bench/take-drop.runtime @@ -1,1 +1,1 @@ -0.6 +0.5 diff --git a/bench/runtime/zero/bench/terpret.runtime b/bench/runtime/zero/bench/terpret.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/zero/bench/terpret.runtime @@ -0,0 +1,1 @@ +13.0 diff --git a/bench/runtime/zero/eg/bools.runtime b/bench/runtime/zero/eg/bools.runtime --- a/bench/runtime/zero/eg/bools.runtime +++ b/bench/runtime/zero/eg/bools.runtime @@ -1,1 +1,1 @@ -3.5 +3.3 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 @@ -0.7 +0.5 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 @@ -5.4 +3.1 diff --git a/bench/runtime/zero/eg/factorial.runtime b/bench/runtime/zero/eg/factorial.runtime --- a/bench/runtime/zero/eg/factorial.runtime +++ b/bench/runtime/zero/eg/factorial.runtime @@ -1,1 +1,1 @@ -0.8 +2.4 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 @@ -5.8 +4.4 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 @@ -1.4 +8.8 diff --git a/bench/runtime/zero/eg/ints.runtime b/bench/runtime/zero/eg/ints.runtime --- a/bench/runtime/zero/eg/ints.runtime +++ b/bench/runtime/zero/eg/ints.runtime @@ -1,1 +1,1 @@ -1.6 +1.0 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 @@ -0.5 +0.6 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 @@ -2.3 +3.5 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 @@ -2.5 +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 @@ -1.3 +1.0 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 @@ -2.2 +1.5 diff --git a/bench/runtime/zero/versions b/bench/runtime/zero/versions --- a/bench/runtime/zero/versions +++ b/bench/runtime/zero/versions @@ -1,4 +1,4 @@ GHC 8.10.4 leancheck-0.9.10 -express-1.0.4 -speculate-0.4.12 +express-1.0.6 +speculate-0.4.14 diff --git a/bench/take-drop.hs b/bench/take-drop.hs --- a/bench/take-drop.hs +++ b/bench/take-drop.hs @@ -8,6 +8,10 @@ drop' 1 [x,y] = [y] drop' 2 [x,y] = [] drop' 3 [x,y] = [] +drop' 0 [x,y,z] = [x,y,z] +drop' 1 [x,y,z] = [y,z] +drop' 2 [x,y,z] = [z] +drop' 3 [x,y,z] = [] take' :: Int -> [a] -> [a] take' 0 [] = [] @@ -19,28 +23,27 @@ main :: IO () main = do - -- drop n xs = if n==0 || null xs then xs else drop (dec n) (tail xs) - -- needs size 13 -- drop 0 [] = [] -- 1 -- drop 0 (x:xs) = x : xs -- 4 -- drop n [] = [] -- 5 - -- drop n (x:xs) = drop (dec n) xs -- 9 + -- drop n (x:xs) = drop (x - 1) xs -- 10 conjure "drop" (drop' :: Int -> [A] -> [A]) [ pr (0 :: Int) + , pr (1 :: Int) , pr ([] :: [A]) , prim ":" ((:) :: A -> [A] -> [A]) - , prim "dec" (subtract 1 :: Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) ] - -- take n xs = if n==0 || null xs then [] else head xs : take (dec n) (tail xs) - -- needs size 16 + -- take 0 [] = [] -- 1 + -- take 0 (x:xs) = [] -- 2 + -- take x [] = [] -- 3 + -- take x (y:xs) = y:take (x - 1) xs -- 10 conjureWithMaxSize 16 "take" (take' :: Int -> [A] -> [A]) [ pr (0 :: Int) + , pr (1 :: Int) , pr ([] :: [A]) - , prim "null" (null :: [A] -> Bool) , prim "||" ((||) :: Bool -> Bool -> Bool) - , prim "dec" ((\n -> n-1) :: Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) , prim ":" ((:) :: A -> [A] -> [A]) - , prim "head" (head :: [A] -> A) - , prim "tail" (tail :: [A] -> [A]) ] diff --git a/bench/take-drop.out b/bench/take-drop.out --- a/bench/take-drop.out +++ b/bench/take-drop.out @@ -1,32 +1,34 @@ drop :: Int -> [A] -> [A] --- testing 143 combinations of argument values --- pruning with 0/0 rules +-- testing 360 combinations of argument values +-- pruning with 4/7 rules -- looking through 2 candidates of size 1 -- looking through 3 candidates of size 2 --- looking through 3 candidates of size 3 --- looking through 5 candidates of size 4 --- looking through 23 candidates of size 5 --- looking through 24 candidates of size 6 --- looking through 55 candidates of size 7 --- looking through 71 candidates of size 8 +-- looking through 5 candidates of size 3 +-- looking through 15 candidates of size 4 +-- looking through 27 candidates of size 5 +-- looking through 41 candidates of size 6 +-- looking through 132 candidates of size 7 +-- looking through 127 candidates of size 8 +-- looking through 458 candidates of size 9 drop 0 [] = [] drop 0 (x:xs) = x:xs drop x [] = [] -drop x (y:xs) = drop (dec x) xs +drop x (y:xs) = drop (x - 1) xs take :: Int -> [A] -> [A] -- testing 143 combinations of argument values --- pruning with 14/18 rules +-- pruning with 14/21 rules -- looking through 2 candidates of size 1 --- looking through 5 candidates of size 2 --- looking through 17 candidates of size 3 --- looking through 42 candidates of size 4 --- looking through 136 candidates of size 5 --- looking through 363 candidates of size 6 --- looking through 921 candidates of size 7 --- looking through 2354 candidates of size 8 +-- looking through 3 candidates of size 2 +-- looking through 5 candidates of size 3 +-- looking through 15 candidates of size 4 +-- looking through 27 candidates of size 5 +-- looking through 41 candidates of size 6 +-- looking through 132 candidates of size 7 +-- looking through 127 candidates of size 8 +-- looking through 458 candidates of size 9 take 0 [] = [] take 0 (x:xs) = [] take x [] = [] -take x (y:xs) = y:take (dec x) xs +take x (y:xs) = y:take (x - 1) xs diff --git a/bench/terpret.hs b/bench/terpret.hs new file mode 100644 --- /dev/null +++ b/bench/terpret.hs @@ -0,0 +1,249 @@ +-- terpret.hs: Benchmark execution-model problems from TerpreT +-- +-- Copyright (C) 2021 Rudy Matela +-- Distributed under the 3-Clause BSD licence (see the file LICENSE). +-- +-- You can find the description of these exercises on: +-- +-- 1. https://dl.acm.org/doi/pdf/10.1145/3067695.3082533 +-- 2. https://www.microsoft.com/en-us/research/uploads/prod/2017/03/1608.04428.pdf +-- +-- Here, we prefer the simplified definitions found in 1. +import Conjure +import System.Environment (getArgs) + + +-- TerpreT #1 -- invert -- + +t1p :: [Bool] -> [Bool] +t1p [False] = [True] +t1p [True] = [False] +t1p [False,True] = [True,False] +t1p [True,False] = [False,True] + +t1g :: [Bool] -> [Bool] +t1g ps = map not ps + +t1c :: IO () +t1c = do + putStrLn "TerpreT benchmark #1: invert\n" + + conjure "invert" t1p primitives123 + +-- the same primitives are used for TerpreT #1, #2 and #3 +primitives123 :: [Prim] +primitives123 = + [ pr False + , pr True + , prim "not" not + , prim "&&" (&&) + , prim "||" (||) + , pr ([] :: [Bool]) + , prim ":" ((:) :: Bool -> [Bool] -> [Bool]) +--, prim "map" (map :: (Bool -> Bool) -> [Bool] -> [Bool]) + , prif (undefined :: [Bool]) + ] + + +-- TerpreT #2 -- prepend zero -- +t2p :: [Bool] -> [Bool] +t2p [True] = [False,True] +t2p [False] = [False,False] +t2p [True,True] = [False,True,True] + +t2g :: [Bool] -> [Bool] +t2g ps = False:ps + +t2c :: IO () +t2c = do + putStrLn "TerpreT benchmark #2: prepend zero\n" + + conjure "prependZero" t2p primitives123 + + +-- TerpreT #3 -- binary decrement -- +-- here we choose to represent with little-endian notation + +t3p :: [Bool] -> [Bool] +-- t3p [True] = [False] +t3p [True,True] = [False,True] -- 11 - 1 = 10 +t3p [False,True] = [True,False] -- 10 - 1 = 1 +t3p [False,True,True] = [True,False,True] -- 110 - 1 = 101 + +t3g :: [Bool] -> [Bool] +t3g [] = [] +t3g (p:ps) = if p + then False : ps + else True : t3g ps +-- hah! Conjure surprised me again: +-- decrement [] = [] +-- decrement (p:ps) = not p:(if p then ps else decrement ps) + +t3c :: IO () +t3c = do + putStrLn "TerpreT benchmark #3: binary decrement\n" + + conjure "decrement" t3p primitives123 + + +-- TerpreT #4 -- 2-bit controlled shift register -- + +t4p1 :: (Bool,Bool,Bool) -> (Bool,Bool,Bool) +t4p1 (False, True, False) = (False, True, False) +t4p1 (False, False, True) = (False, False, True) +t4p1 (True, True, False) = (True, False, True) +t4p1 (True, False, True) = (True, True, False) + +t4p2 :: Bool -> Bool -> Bool -> (Bool,Bool,Bool) +t4p2 False True False = (False, True, False) +t4p2 False False True = (False, False, True) +t4p2 True True False = (True, False, True) +t4p2 True False True = (True, True, False) + +t4g :: (Bool,Bool,Bool) -> (Bool,Bool,Bool) +t4g (False,p,q) = (False,p,q) +t4g (True,p,q) = (True,q,p) + +t4c :: IO () +t4c = do + putStrLn "TerpreT benchmark #4: controlled shift\n" + + conjure "cshift" t4p1 $ primitives123 ++ + [ prim ",," ((,,) :: Bool -> Bool -> Bool -> (Bool,Bool,Bool)) + , prif (undefined :: (Bool,Bool,Bool)) + ] + + conjure "cshift" t4p2 $ primitives123 ++ + [ prim ",," ((,,) :: Bool -> Bool -> Bool -> (Bool,Bool,Bool)) + , prif (undefined :: (Bool,Bool,Bool)) + ] + + +-- TerpreT #5 -- Full adder -- + +t5p :: Bool -> Bool -> Bool -> (Bool,Bool) +t5p False False False = (False,False) +t5p False False True = (False,True ) +t5p False True False = (False,True ) +t5p True False False = (False,True ) +t5p False True True = (True ,False) +t5p True False True = (True ,False) +t5p True True False = (True ,False) +t5p True True True = (True ,True ) + +t5c :: IO () +t5c = do + putStrLn "TerpreT benchmark #5: full adder\n" + + -- using primitives123 below works, but increases the runtime to 18 seconds + -- let's leave it commented out so runtime is faster when running automated tests + conjure "fadder" t5p $ -- primitives123 ++ + [ prim "not" not + , prim "," ((,) :: Bool -> Bool -> (Bool,Bool)) + , prim "==" ((==) :: Bool -> Bool -> Bool) +-- , prim "^^" ((/=) :: Bool -> Bool -> Bool) -- poor man's xor + , prif (undefined :: (Bool,Bool)) + ] +-- the printed function is weird, but correct +-- fadder p q r = if p == q then (p,r) else (r,not r) + + +-- TerpreT #6 -- 2-bit adder -- +t6p :: (Bool,Bool) -> (Bool,Bool) -> (Bool,Bool,Bool) +t6p (False,False) (False,False) = (False,False,False) +t6p (True ,False) (False,True ) = (False,True ,True ) +t6p (True ,True ) (False,True ) = (True ,False,False) +t6p (True ,True ) (True ,False) = (True ,False,True ) +t6p (True ,True ) (True ,True ) = (True ,True ,False) + +t6c :: IO () +t6c = do + putStrLn "TerpreT benchmark #6: 2-bit adder\n" + conjureWith args{maxSize=6} "adder2" t6p $ primitives123 ++ + [ prim ",," ((,,) :: Bool -> Bool -> Bool -> (Bool,Bool,Bool)) + , prim "==" ((==) :: Bool -> Bool -> Bool) + , prim "^^" ((/=) :: Bool -> Bool -> Bool) -- poor man's xor + , prif (undefined :: (Bool,Bool,Bool)) + ] + + +-- TerpreT #7 -- Access -- +t7p :: [A] -> Int -> A +t7p [1,0] 0 = 1 +t7p [1,0] 1 = 0 +t7p [0,2,1] 0 = 0 +t7p [0,2,1] 1 = 2 +t7p [0,2,1] 2 = 1 + +t7g :: [A] -> Int -> A +t7g [] _ = undefined +t7g (x:xs) 0 = x +t7g (x:xs) i = t7g xs (i-1) + +t7c :: IO () +t7c = do + putStrLn "TerpreT benchmark #7: access\n" + -- yes, one can implement index with index... + conjure "`access`" t7p + [ prim "!!" ((!!) :: [A] -> Int -> A) + ] + + conjure "`access`" t7p + [ pr (0 :: Int) + , pr (1 :: Int) + , pr ([] :: [A]) + , prim ":" ((:) :: A -> [A] -> [A]) + , prim "-" ((-) :: Int -> Int -> Int) + , prim "undefined" (undefined :: A) + ] + + +-- TerpreT #8 -- Decrement Elements -- +t8p :: [Int] -> [Int] +t8p [2] = [1] +t8p [1,0] = [0,-1] +t8p [0,1,2] = [-1,0,1] + +t8c :: IO () +t8c = do + putStrLn "TerpreT benchmark #8: decrement elements\n" + + conjure "decrelements" t8p + [ pr (1 :: Int) + , pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "-" ((-) :: Int -> Int -> Int) + , prim "map" (map :: (Int -> Int) -> [Int] -> [Int]) + ] + -- above, even though map is provided, Conjure cannot use it as it cannot + -- introduce lambdas + + conjure "decrelements" t8p + [ pr (1 :: Int) + , pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "-" ((-) :: Int -> Int -> Int) + , prim "map" (map :: (Int -> Int) -> [Int] -> [Int]) + , prim "subtract" (subtract :: Int -> Int -> Int) + ] + -- now above, the story changes because of subtract, map is used + + +main :: IO () +main = do + as <- getArgs + case as of + [] -> sequence_ ts + (n:_) -> ts !! (read n - 1) + + +ts :: [IO ()] +ts = [ t1c + , t2c + , t3c + , t4c + , t5c + , t6c + , t7c + , t8c + ] diff --git a/bench/terpret.out b/bench/terpret.out new file mode 100644 --- /dev/null +++ b/bench/terpret.out @@ -0,0 +1,156 @@ +TerpreT benchmark #1: invert + +invert :: [Bool] -> [Bool] +-- testing 4 combinations of argument values +-- pruning with 41/51 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 5 candidates of size 3 +-- looking through 10 candidates of size 4 +-- looking through 17 candidates of size 5 +-- looking through 41 candidates of size 6 +invert [] = [] +invert (p:ps) = not p:invert ps + +TerpreT benchmark #2: prepend zero + +prependZero :: [Bool] -> [Bool] +-- testing 3 combinations of argument values +-- pruning with 41/51 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 5 candidates of size 3 +prependZero ps = False:ps + +TerpreT benchmark #3: binary decrement + +decrement :: [Bool] -> [Bool] +-- testing 3 combinations of argument values +-- pruning with 41/51 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 5 candidates of size 3 +-- looking through 10 candidates of size 4 +-- looking through 17 candidates of size 5 +-- looking through 41 candidates of size 6 +-- looking through 86 candidates of size 7 +-- looking through 202 candidates of size 8 +-- looking through 487 candidates of size 9 +decrement [] = [] +decrement (p:ps) = not p:(if p then ps else decrement ps) + +TerpreT benchmark #4: controlled shift + +cshift :: (Bool,Bool,Bool) -> (Bool,Bool,Bool) +-- testing 4 combinations of argument values +-- pruning with 41/51 rules +-- reasoning produced 2 incorrect properties, please re-run with more tests for faster results +-- looking through 1 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 0 candidates of size 3 +-- 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 +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 +-- pruning with 41/51 rules +-- reasoning produced 2 incorrect properties, please re-run with more tests for faster results +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +-- 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 +cshift False p q = (False,p,q) +cshift True p q = (True,q,p) + +TerpreT benchmark #5: full adder + +fadder :: Bool -> Bool -> Bool -> (Bool,Bool) +-- testing 8 combinations of argument values +-- pruning with 18/22 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 9 candidates of size 3 +-- looking through 18 candidates of size 4 +-- looking through 45 candidates of size 5 +-- looking through 108 candidates of size 6 +-- looking through 300 candidates of size 7 +-- looking through 816 candidates of size 8 +-- looking through 2148 candidates of size 9 +-- looking through 5520 candidates of size 10 +-- looking through 13656 candidates of size 11 +fadder p q r = if p == q then (p,r) else (r,not r) + +TerpreT benchmark #6: 2-bit adder + +adder2 :: (Bool,Bool) -> (Bool,Bool) -> (Bool,Bool,Bool) +-- testing 5 combinations of argument values +-- pruning with 74/92 rules +-- reasoning produced 2 incorrect properties, please re-run with more tests for faster results +-- looking through 0 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 128 candidates of size 5 +-- looking through 408 candidates of size 6 +cannot conjure + +TerpreT benchmark #7: access + +access :: [A] -> Int -> A +-- testing 5 combinations of argument values +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 1 candidates of size 3 +xs `access` x = xs !! x + +access :: [A] -> Int -> A +-- testing 5 combinations of argument values +-- pruning with 4/7 rules +-- looking through 1 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 3 candidates of size 3 +-- looking through 7 candidates of size 4 +-- looking through 19 candidates of size 5 +-- looking through 23 candidates of size 6 +-- looking through 68 candidates of size 7 +[] `access` 0 = undefined +[] `access` x = undefined +(x:xs) `access` 0 = x +(x:xs) `access` y = xs `access` (y - 1) + +TerpreT benchmark #8: decrement elements + +decrelements :: [Int] -> [Int] +-- testing 3 combinations of argument values +-- pruning with 3/23 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 3 candidates of size 3 +-- looking through 7 candidates of size 4 +-- looking through 9 candidates of size 5 +-- looking through 29 candidates of size 6 +-- looking through 41 candidates of size 7 +decrelements [] = [] +decrelements (x:xs) = x - 1:decrelements xs + +decrelements :: [Int] -> [Int] +-- testing 3 combinations of argument values +-- pruning with 5/26 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 3 candidates of size 3 +-- looking through 8 candidates of size 4 +decrelements xs = map (subtract 1) xs + diff --git a/changelog.md b/changelog.md --- a/changelog.md +++ b/changelog.md @@ -2,6 +2,18 @@ ============================ +v0.4.4 +------ + +* remove need for explicit deconstructions: + - use `-` and `1` instead of `dec`; + - allow `mod` and `div` as deconstructions; +* bump Express requirement to v1.0.6 (bugfix); +* complete the GPS1 benchmark; +* add GPS2 and TerpreT benchmarks; +* minor fixes in the README. + + v0.4.2 ------ 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: 0.4.2 +version: 0.4.4 synopsis: conjure Haskell functions out of partial definitions description: Conjure is a tool that produces Haskell functions out of partial definitions. @@ -66,7 +66,7 @@ source-repository this type: git location: https://github.com/rudymatela/conjure - tag: v0.4.2 + tag: v0.4.4 library exposed-modules: Conjure @@ -81,8 +81,8 @@ build-depends: base >= 4 && < 5 , leancheck >= 0.9.10 , template-haskell - , speculate >= 0.4.12 - , express >= 1.0.4 + , speculate >= 0.4.14 + , express >= 1.0.6 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 @@ -29,8 +29,8 @@ -- looking through 4 candidates of size 2 -- looking through 9 candidates of size 3 -- looking through 23 candidates of size 4 --- looking through 26 candidates of size 5 --- looking through 65 candidates of size 6 --- looking through 75 candidates of size 7 +-- looking through 29 candidates of size 5 +-- looking through 71 candidates of size 6 +-- looking through 113 candidates of size 7 tnpo x = x + (x + (x + 1)) diff --git a/eg/bools.out b/eg/bools.out --- a/eg/bools.out +++ b/eg/bools.out @@ -5,7 +5,7 @@ -- looking through 6 candidates of size 2 -- looking through 12 candidates of size 3 -- looking through 16 candidates of size 4 --- looking through 40 candidates of size 5 +-- looking through 42 candidates of size 5 and [] = True and (p:ps) = p && and ps @@ -16,7 +16,7 @@ -- looking through 6 candidates of size 2 -- looking through 12 candidates of size 3 -- looking through 16 candidates of size 4 --- looking through 40 candidates of size 5 +-- looking through 42 candidates of size 5 or [] = False or (p:ps) = p || or ps diff --git a/eg/count.out b/eg/count.out --- a/eg/count.out +++ b/eg/count.out @@ -17,11 +17,11 @@ -- looking through 8 candidates of size 4 -- looking through 9 candidates of size 5 -- looking through 24 candidates of size 6 --- looking through 44 candidates of size 7 --- looking through 108 candidates of size 8 --- looking through 244 candidates of size 9 --- looking through 574 candidates of size 10 --- looking through 1320 candidates of size 11 +-- 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 count x [] = 0 count x (y:xs) = (if x == y then 1 else 0) + count x xs diff --git a/eg/dupos.out b/eg/dupos.out --- a/eg/dupos.out +++ b/eg/dupos.out @@ -16,8 +16,8 @@ -- looking through 141 candidates of size 13 -- looking through 322 candidates of size 14 -- looking through 644 candidates of size 15 --- looking through 1185 candidates of size 16 --- looking through 2153 candidates of size 17 +-- looking through 1189 candidates of size 16 +-- looking through 2189 candidates of size 17 duplicates [] = [] duplicates (x:xs) = if elem x xs && not (elem x (duplicates xs)) then x:duplicates xs else duplicates xs diff --git a/eg/factorial.hs b/eg/factorial.hs --- a/eg/factorial.hs +++ b/eg/factorial.hs @@ -13,77 +13,27 @@ main :: IO () main = do - -- using enumFromTo - conjure "factorial n" factorial - [ pr (1::Int) - , prim ".." (enumFromTo :: Int -> Int -> [Int]) - , prim "*" ((*) :: Int -> Int -> Int) - , prim "foldr" (foldr :: (Int -> Int -> Int) -> Int -> [Int] -> Int) - ] - -- explicit recursion conjure "factorial n" factorial [ pr (0::Int) , pr (1::Int) , prim "+" ((+) :: Int -> Int -> Int) , prim "*" ((*) :: Int -> Int -> Int) - , prim "dec" (subtract 1 :: Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) ] - --- the actual factorial function: --- factorial n = if n == 0 then 1 else n * factorial (n - 1) --- 1 2 3 4 5 6 7 8 9 10 11 symbols --- --- OR --- --- factorial n = if n == 0 then 1 else n * factorial (dec n) --- 1 2 3 4 5 6 7 8 9 10 symbols --- --- OR --- --- factorial n = if (isZero n) then 1 else (n * factorial (dec n)) --- 1 2 3 4 5 6 7 8 9 symbols - - -{- --- Paramorphism of Naturals encoded as integers -para :: (Int -> b -> b) -> b -> Int -> b -para (?) z = p - where - p n | n < 0 = z -- no negatives for you :-) - p 0 = z - p n = n ? p (n-1) - -The following works with a maxSize of 4, but not with a maxSize of 5. + -- should produce: + -- factorial 0 = 1 + -- factorial x = x * factorial (x - 1) - -- using a paramorphism + -- using foldr and enumFromTo conjure "factorial n" factorial - [ pr (1::Int) - , prim "para" (para :: (Int->Int->Int) -> Int -> Int -> Int) + [ pr (0::Int) + , pr (1::Int) + , prim "+" ((+) :: Int -> Int -> Int) , prim "*" ((*) :: Int -> Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) + , prim ".." (enumFromTo :: Int -> Int -> [Int]) + , prim "foldr" (foldr :: (Int -> Int -> Int) -> Int -> [Int] -> Int) ] - - -the factorial function is the following: - -fact = para (*) 1 - - -now consider the following grow_fast function: - -grow_fast = para (para (*)) 1 :: Integer -> Integer - - -> growFast 1 -1 -> growFast 2 -2 -> growFast 3 -6 -> growFast 4 -2880 -> growFast 5 -7148302174930174893017438921... 8000 digits! -> growFast 6 -stack overflow --} + -- should produce: + -- factorial x = foldr (*) 1 [1..x] diff --git a/eg/factorial.out b/eg/factorial.out --- a/eg/factorial.out +++ b/eg/factorial.out @@ -1,23 +1,24 @@ factorial :: Int -> Int -- testing 4 combinations of argument values --- pruning with 4/8 rules --- looking through 2 candidates of size 1 --- looking through 1 candidates of size 2 --- looking through 1 candidates of size 3 --- looking through 1 candidates of size 4 --- looking through 1 candidates of size 5 --- looking through 9 candidates of size 6 -factorial x = foldr (*) 1 [1..x] +-- pruning with 27/65 rules +-- looking through 3 candidates of size 1 +-- looking through 4 candidates of size 2 +-- looking through 13 candidates of size 3 +-- looking through 34 candidates of size 4 +-- looking through 75 candidates of size 5 +-- looking through 183 candidates of size 6 +-- looking through 577 candidates of size 7 +factorial 0 = 1 +factorial x = x * factorial (x - 1) factorial :: Int -> Int -- testing 4 combinations of argument values --- pruning with 22/42 rules +-- pruning with 32/72 rules -- looking through 3 candidates of size 1 --- looking through 6 candidates of size 2 --- looking through 16 candidates of size 3 --- looking through 39 candidates of size 4 --- looking through 78 candidates of size 5 --- looking through 166 candidates of size 6 -factorial 0 = 1 -factorial x = x * factorial (dec x) +-- looking through 4 candidates of size 2 +-- looking through 13 candidates of size 3 +-- looking through 34 candidates of size 4 +-- looking through 75 candidates of size 5 +-- looking through 247 candidates of size 6 +factorial x = foldr (*) 1 [1..x] diff --git a/eg/fib01.hs b/eg/fib01.hs --- a/eg/fib01.hs +++ b/eg/fib01.hs @@ -11,6 +11,12 @@ fib01 0 1 6 = 13 fib01 0 1 7 = 21 +fibonacci :: Int -> Int +fibonacci = f 0 1 + where + f x y 0 = y + f x y n = f y (x + y) (n - 1) + main :: IO () main = do conjureWithMaxSize 5 "fib01" fib01 diff --git a/eg/fib01.out b/eg/fib01.out --- a/eg/fib01.out +++ b/eg/fib01.out @@ -20,6 +20,6 @@ -- looking through 192 candidates of size 7 -- looking through 391 candidates of size 8 -- looking through 840 candidates of size 9 --- looking through 15630 candidates of size 10 +-- looking through 6414 candidates of size 10 cannot conjure diff --git a/eg/fibonacci.hs b/eg/fibonacci.hs --- a/eg/fibonacci.hs +++ b/eg/fibonacci.hs @@ -9,28 +9,17 @@ fibonacci 4 = 5 fibonacci 5 = 8 fibonacci 6 = 13 -fibonacci 7 = 21 main :: IO () main = do conjure "fibonacci n" fibonacci [ pr (0::Int) , pr (1::Int) - , prim "dec" (subtract 1 :: Int -> Int) + , pr (2::Int) , prim "+" ((+) :: Int -> Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) ] -- expected function: --- fibonacci n = if n <= 1 then 1 else fibonacci (dec n) + fibonacci (dec (dec n)) --- 1 2 3 4 5 6 7 8 9 10 11 12 13 - - -{- to note, if dec appears later than + in the primitives list: - -conjureWith ... = - -> print $ canReduceTo thy (xx -+- dec xx) (dec (dec xx)) -False -> print $ canReduceTo thy (dec (dec xx)) (xx -+- dec xx) -True - --} +-- fibonacci 0 = 1 +-- fibonacci 1 = 1 +-- fibonacci x = fibonacci (x - 1) + fibonacci (x - 2) diff --git a/eg/fibonacci.out b/eg/fibonacci.out --- a/eg/fibonacci.out +++ b/eg/fibonacci.out @@ -1,18 +1,19 @@ fibonacci :: Int -> Int --- testing 8 combinations of argument values --- pruning with 9/13 rules --- looking through 3 candidates of size 1 --- looking through 6 candidates of size 2 --- looking through 17 candidates of size 3 --- looking through 41 candidates of size 4 --- looking through 76 candidates of size 5 --- looking through 155 candidates of size 6 --- looking through 249 candidates of size 7 --- looking through 481 candidates of size 8 --- looking through 762 candidates of size 9 --- looking through 1413 candidates of size 10 --- looking through 2257 candidates of size 11 +-- testing 7 combinations of argument values +-- pruning with 21/44 rules +-- looking through 4 candidates of size 1 +-- looking through 9 candidates of size 2 +-- looking through 20 candidates of size 3 +-- looking through 87 candidates of size 4 +-- looking through 127 candidates of size 5 +-- looking through 420 candidates of size 6 +-- looking through 845 candidates of size 7 +-- looking through 2136 candidates of size 8 +-- looking through 5185 candidates of size 9 +-- looking through 13000 candidates of size 10 +-- looking through 33460 candidates of size 11 +-- looking through 85901 candidates of size 12 fibonacci 0 = 1 fibonacci 1 = 1 -fibonacci x = fibonacci (dec x) + fibonacci (dec (dec x)) +fibonacci x = fibonacci (x - 1) + fibonacci (x - 2) diff --git a/eg/gcd.hs b/eg/gcd.hs --- a/eg/gcd.hs +++ b/eg/gcd.hs @@ -18,7 +18,7 @@ gcd' 12 18 = 6 main :: IO () -main = conjureWith args{requireDescent=False} "gcd a b" gcd' +main = conjure "gcd a b" gcd' [ pr (0::Int) , prim "`mod`" (mod :: Int -> Int -> Int) ] diff --git a/eg/gcd.out b/eg/gcd.out --- a/eg/gcd.out +++ b/eg/gcd.out @@ -4,9 +4,9 @@ -- looking through 3 candidates of size 1 -- looking through 8 candidates of size 2 -- looking through 20 candidates of size 3 --- looking through 88 candidates of size 4 --- looking through 202 candidates of size 5 --- looking through 808 candidates of size 6 +-- looking through 60 candidates of size 4 +-- looking through 150 candidates of size 5 +-- looking through 472 candidates of size 6 gcd x 0 = x gcd x y = gcd y (x `mod` y) diff --git a/eg/ints.hs b/eg/ints.hs --- a/eg/ints.hs +++ b/eg/ints.hs @@ -25,12 +25,33 @@ main :: IO () main = do - conjure "second" (second :: [Int] -> Int) primitives - conjure "third" (third :: [Int] -> Int) primitives - conjure "sum" (sum' :: [Int] -> Int) primitives - conjure "product" (product' :: [Int] -> Int) primitives + conjure "second" (second :: [Int] -> Int) + [ prim "null" (null :: [Int] -> Bool) + , prim "head" (head :: [Int] -> Int) + , prim "tail" (tail :: [Int] -> [Int]) + ] - conjure "sum" (sum' :: [Int] -> Int) primitivesWithFold + conjure "third" (third :: [Int] -> Int) + [ prim "null" (null :: [Int] -> Bool) + , prim "head" (head :: [Int] -> Int) + , prim "tail" (tail :: [Int] -> [Int]) + ] + + conjure "sum" (sum' :: [Int] -> Int) + [ pr (0 :: Int) + , pr (1 :: Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "*" ((*) :: Int -> Int -> Int) + ] + + conjure "product" (product' :: [Int] -> Int) + [ pr (0 :: Int) + , pr (1 :: Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "*" ((*) :: Int -> Int -> Int) + ] + + conjure "sum" (sum' :: [Int] -> Int) primitivesWithFold conjure "product" (product' :: [Int] -> Int) primitivesWithFold primitives :: [Prim] @@ -39,9 +60,6 @@ , pr (1 :: Int) , prim "+" ((+) :: Int -> Int -> Int) , prim "*" ((*) :: Int -> Int -> Int) - , prim "null" (null :: [Int] -> Bool) - , prim "head" (head :: [Int] -> Int) - , prim "tail" (tail :: [Int] -> [Int]) ] primitivesWithFold :: [Prim] diff --git a/eg/ints.out b/eg/ints.out --- a/eg/ints.out +++ b/eg/ints.out @@ -1,28 +1,28 @@ second :: [Int] -> Int -- testing 360 combinations of argument values --- 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 +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 1 candidates of size 3 second xs = head (tail xs) third :: [Int] -> Int -- testing 360 combinations of argument values --- 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 19 candidates of size 4 +-- pruning with 0/0 rules +-- looking through 0 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 1 candidates of size 3 +-- looking through 1 candidates of size 4 third xs = head (tail (tail xs)) sum :: [Int] -> Int -- testing 360 combinations of argument values -- 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 19 candidates of size 4 --- looking through 31 candidates of size 5 +-- 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 sum [] = 0 sum (x:xs) = x + sum xs @@ -30,10 +30,10 @@ -- testing 360 combinations of argument values -- 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 19 candidates of size 4 --- looking through 31 candidates of size 5 +-- 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 product [] = 1 product (x:xs) = x * product xs @@ -41,17 +41,17 @@ -- testing 360 combinations of argument values -- 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 22 candidates of size 4 +-- looking through 4 candidates of size 2 +-- looking through 3 candidates of size 3 +-- looking through 16 candidates of size 4 sum xs = foldr (+) 0 xs product :: [Int] -> Int -- testing 360 combinations of argument values -- 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 22 candidates of size 4 +-- looking through 4 candidates of size 2 +-- looking through 3 candidates of size 3 +-- looking through 16 candidates of size 4 product xs = foldr (*) 1 xs diff --git a/eg/pow.hs b/eg/pow.hs --- a/eg/pow.hs +++ b/eg/pow.hs @@ -13,16 +13,13 @@ main :: IO () main = do - -- pow b e = if e == 0 then 1 else b * pow b (dec e) - -- 1 2 3 4 5 6 7 8 9 10 11 - -- somehow the above takes 30s to run, the two arguments - -- of the same type introduce the difficulty here. - -- with cases below, runtime is ok: + -- pow x 0 = 1 + -- pow x y = x * pow x (y - 1) conjureWithMaxSize 8 "pow" pow [ pr (0::Int) , pr (1::Int) , prim "*" ((*) :: Int -> Int -> Int) - , prim "dec" (subtract 1 :: Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) ] -- pow b e = if e == 0 then 1 else pow b (halve e) * pow b (halve e) * if odd e then b else 1 diff --git a/eg/pow.out b/eg/pow.out --- a/eg/pow.out +++ b/eg/pow.out @@ -1,15 +1,16 @@ pow :: Int -> Int -> Int -- testing 5 combinations of argument values --- pruning with 8/12 rules +-- pruning with 14/30 rules -- looking through 4 candidates of size 1 --- looking through 21 candidates of size 2 --- looking through 129 candidates of size 3 --- looking through 517 candidates of size 4 --- looking through 2346 candidates of size 5 --- looking through 8550 candidates of size 6 --- looking through 32487 candidates of size 7 +-- looking through 18 candidates of size 2 +-- looking through 101 candidates of size 3 +-- looking through 324 candidates of size 4 +-- looking through 1668 candidates of size 5 +-- looking through 3438 candidates of size 6 +-- looking through 21096 candidates of size 7 +-- looking through 34780 candidates of size 8 pow x 0 = 1 -pow x y = x * pow x (dec y) +pow x y = x * pow x (y - 1) pow :: Int -> Int -> Int -- testing 5 combinations of argument values @@ -17,8 +18,8 @@ -- looking through 4 candidates of size 1 -- looking through 20 candidates of size 2 -- looking through 114 candidates of size 3 --- looking through 376 candidates of size 4 --- looking through 1714 candidates of size 5 --- looking through 4706 candidates of size 6 +-- looking through 378 candidates of size 4 +-- looking through 1690 candidates of size 5 +-- looking through 4572 candidates of size 6 cannot conjure diff --git a/eg/replicate.hs b/eg/replicate.hs --- a/eg/replicate.hs +++ b/eg/replicate.hs @@ -27,7 +27,8 @@ main = do conjure "replicate" replicate' [ pr (0 :: Int) - , prim "dec" (subtract 1 :: Int -> Int) + , pr (1 :: Int) + , prim "-" ((-) :: Int -> Int -> Int) , pr "" , prim ":" ((:) :: Char -> String -> String) ] diff --git a/eg/replicate.out b/eg/replicate.out --- a/eg/replicate.out +++ b/eg/replicate.out @@ -1,15 +1,16 @@ replicate :: Int -> Char -> [Char] -- testing 360 combinations of argument values --- pruning with 0/0 rules +-- pruning with 4/7 rules -- looking through 1 candidates of size 1 -- looking through 0 candidates of size 2 -- looking through 1 candidates of size 3 -- looking through 2 candidates of size 4 --- looking through 2 candidates of size 5 --- looking through 3 candidates of size 6 --- looking through 4 candidates of size 7 +-- looking through 3 candidates of size 5 +-- looking through 7 candidates of size 6 +-- looking through 5 candidates of size 7 +-- looking through 16 candidates of size 8 replicate 0 c = "" -replicate x c = c:replicate (dec x) c +replicate x c = c:replicate (x - 1) c replicates :: [Char] -> Int -> [Char] -- testing 360 combinations of argument values diff --git a/eg/setelem.out b/eg/setelem.out --- a/eg/setelem.out +++ b/eg/setelem.out @@ -6,9 +6,9 @@ -- looking through 5 candidates of size 3 -- looking through 28 candidates of size 4 -- looking through 73 candidates of size 5 --- looking through 119 candidates of size 6 --- looking through 277 candidates of size 7 --- looking through 895 candidates of size 8 +-- looking through 124 candidates of size 6 +-- looking through 305 candidates of size 7 +-- looking through 999 candidates of size 8 elem x [] = False elem x (y:xs) = x == y || elem x xs @@ -20,9 +20,9 @@ -- looking through 7 candidates of size 3 -- looking through 19 candidates of size 4 -- looking through 35 candidates of size 5 --- looking through 81 candidates of size 6 --- looking through 229 candidates of size 7 --- looking through 546 candidates of size 8 +-- looking through 83 candidates of size 6 +-- looking through 245 candidates of size 7 +-- looking through 617 candidates of size 8 set [] = True set (x:xs) = not (elem x xs) && set xs diff --git a/eg/tapps.out b/eg/tapps.out --- a/eg/tapps.out +++ b/eg/tapps.out @@ -4,7 +4,7 @@ -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 -- looking through 6 candidates of size 3 --- looking through 19 candidates of size 4 +-- looking through 20 candidates of size 4 third xs = head (tail (tail xs)) product :: [Int] -> Int @@ -13,8 +13,8 @@ -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 -- looking through 6 candidates of size 3 --- looking through 19 candidates of size 4 --- looking through 31 candidates of size 5 +-- looking through 20 candidates of size 4 +-- looking through 34 candidates of size 5 product [] = 1 product (x:xs) = x * product xs @@ -24,6 +24,6 @@ -- looking through 2 candidates of size 1 -- looking through 5 candidates of size 2 -- looking through 6 candidates of size 3 --- looking through 22 candidates of size 4 +-- looking through 23 candidates of size 4 product xs = foldr (*) 1 xs diff --git a/eg/tree.out b/eg/tree.out --- a/eg/tree.out +++ b/eg/tree.out @@ -46,11 +46,11 @@ -- looking through 8 candidates of size 3 -- looking through 59 candidates of size 4 -- looking through 114 candidates of size 5 --- looking through 388 candidates of size 6 --- looking through 1255 candidates of size 7 --- looking through 3833 candidates of size 8 +-- looking through 472 candidates of size 6 +-- looking through 1440 candidates of size 7 +-- looking through 5781 candidates of size 8 height Leaf = -1 -height (Node t1 x t2) = 1 + max (height t1) (height t2) +height (Node t1 x t2) = max (height t1) (height t2) + 1 mem :: Int -> Tree -> Bool -- testing 360 combinations of argument values diff --git a/mk/depend.mk b/mk/depend.mk --- a/mk/depend.mk +++ b/mk/depend.mk @@ -16,6 +16,19 @@ src/Conjure/Defn.hs \ src/Conjure/Conjurable.hs \ bench/candidates.hs +bench/gps2: \ + bench/gps2.hs \ + mk/toplibs +bench/gps2.o: \ + src/Conjure/Utils.hs \ + src/Conjure/Spec.hs \ + src/Conjure/Prim.hs \ + src/Conjure.hs \ + src/Conjure/Expr.hs \ + src/Conjure/Engine.hs \ + src/Conjure/Defn.hs \ + src/Conjure/Conjurable.hs \ + bench/gps2.hs bench/gps: \ bench/gps.hs \ mk/toplibs @@ -120,6 +133,19 @@ src/Conjure/Defn.hs \ src/Conjure/Conjurable.hs \ bench/take-drop.hs +bench/terpret: \ + bench/terpret.hs \ + mk/toplibs +bench/terpret.o: \ + src/Conjure/Utils.hs \ + src/Conjure/Spec.hs \ + src/Conjure/Prim.hs \ + src/Conjure.hs \ + src/Conjure/Expr.hs \ + src/Conjure/Engine.hs \ + src/Conjure/Defn.hs \ + src/Conjure/Conjurable.hs \ + bench/terpret.hs eg/arith: \ eg/arith.hs \ mk/toplibs diff --git a/src/Conjure.hs b/src/Conjure.hs --- a/src/Conjure.hs +++ b/src/Conjure.hs @@ -76,6 +76,8 @@ , reifyEquality , reifyTiers , conjureType + , Name (..) + , Express (..) -- * Pure interfaces , conjpure diff --git a/src/Conjure/Conjurable.hs b/src/Conjure/Conjurable.hs --- a/src/Conjure/Conjurable.hs +++ b/src/Conjure/Conjurable.hs @@ -27,12 +27,16 @@ , conjureMkEquation , A, B, C, D, E, F , conjureIsDeconstructor + , conjureIsDeconstruction + , candidateDeconstructionsFrom , conjureIsUnbreakable , conjureReification , conjureReification1 , cevaluate , ceval , cevl + , Name (..) + , Express (..) ) where @@ -44,6 +48,7 @@ import Test.Speculate.Expr import Data.Functor ((<$>)) import Control.Applicative ((<*>)) +import Data.Express import Data.Int -- for instances import Data.Word -- for instances @@ -252,7 +257,37 @@ sz = head [sz | (_, _, _, _, _, sz) <- conjureReification f , isWellTyped (sz :$ h)] esz e = eval (0::Int) (sz :$ e) - is e = esz (h :$ e) < esz e + is e' = errorToFalse $ esz (e :$ e') < esz e' + +-- the result does not increase the size for at least half the time +-- the result decreases in size for at least a third of the time +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 + 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) + holeValue e = fromMaybe err + . lookup h + . fromMaybe err + $ e `match` ed + err = error "conjureIsDeconstructor: the impossible happened" + +candidateDeconstructionsFrom :: Expr -> [Expr] +candidateDeconstructionsFrom e = + [ e' + | v <- vars e + , typ v == typ e + , let e' = e //- [(v, holeAsTypeOf v)] + , length (holes e') == 1 + ] conjureIsUnbreakable :: Conjurable f => f -> Expr -> Bool conjureIsUnbreakable f e = head diff --git a/src/Conjure/Engine.hs b/src/Conjure/Engine.hs --- a/src/Conjure/Engine.hs +++ b/src/Conjure/Engine.hs @@ -23,6 +23,7 @@ , candidateDefnsC , conjureTheory , conjureTheoryWith + , deconstructions , module Data.Express , module Data.Express.Fixtures , module Test.Speculate.Engine @@ -40,7 +41,7 @@ import Test.LeanCheck.Tiers import Test.LeanCheck.Error (errorToTrue, errorToFalse, errorToNothing) -import Test.Speculate.Reason (Thy, rules, equations, canReduceTo, printThy, closureLimit) +import Test.Speculate.Reason (Thy, rules, equations, invalid, canReduceTo, printThy, closureLimit, doubleCheck) import Test.Speculate.Engine (theoryFromAtoms, groundBinds, boolTy) import Conjure.Expr @@ -99,15 +100,16 @@ -- | Arguments to be passed to 'conjureWith' or 'conjpureWith'. -- See 'args' for the defaults. data Args = Args - { maxTests :: Int -- ^ maximum number of tests to each candidate - , maxSize :: Int -- ^ maximum size of candidate bodies - , maxEvalRecursions :: Int -- ^ maximum number of recursive evaluations when testing candidates - , maxEquationSize :: Int -- ^ maximum size of equation operands - , maxSearchTests :: Int -- ^ maximum number of tests to search for defined values - , requireDescent :: Bool -- ^ require recursive calls to deconstruct arguments - , usePatterns :: Bool -- ^ use pattern matching to create (recursive) candidates - , showTheory :: Bool -- ^ show theory discovered by Speculate used in pruning - , forceTests :: [[Expr]] -- ^ force tests + { maxTests :: Int -- ^ maximum number of tests to each candidate + , maxSize :: Int -- ^ maximum size of candidate bodies + , maxEvalRecursions :: Int -- ^ maximum number of recursive evaluations when testing candidates + , maxEquationSize :: Int -- ^ maximum size of equation operands + , maxSearchTests :: Int -- ^ maximum number of tests to search for defined values + , maxDeconstructionSize :: Int -- ^ maximum size of deconstructions (e.g.: @_ - 1@) + , requireDescent :: Bool -- ^ require recursive calls to deconstruct arguments + , usePatterns :: Bool -- ^ use pattern matching to create (recursive) candidates + , showTheory :: Bool -- ^ show theory discovered by Speculate used in pruning + , forceTests :: [[Expr]] -- ^ force tests } @@ -123,15 +125,16 @@ -- * don't show the theory used in pruning args :: Args args = Args - { maxTests = 360 - , maxSize = 12 - , maxEvalRecursions = 60 - , maxEquationSize = 5 - , maxSearchTests = 100000 - , requireDescent = True - , usePatterns = True - , showTheory = False - , forceTests = [] + { maxTests = 360 + , maxSize = 12 + , maxEvalRecursions = 60 + , maxEquationSize = 5 + , maxSearchTests = 100000 + , maxDeconstructionSize = 4 + , requireDescent = True + , usePatterns = True + , showTheory = False + , forceTests = [] } @@ -147,8 +150,9 @@ putStrLn $ "{-" printThy thy putStrLn $ "-}" - when (filtered thy) $ - putStrLn $ "-- reasoning produced incorrect properties," -- TODO: add Num + when (not . null $ invalid thy) $ + putStrLn $ "-- reasoning produced " + ++ show (length (invalid thy)) ++ " incorrect properties," ++ " please re-run with more tests for faster results" pr 1 rs where @@ -263,11 +267,25 @@ (ef:exs) = unfoldApp efxs keep = isRootNormalE thy . fastMostGeneralVariation ds = filter (conjureIsDeconstructor f maxTests) es - keepR | requireDescent = descends (`elem` ds) efxs + keepR | requireDescent = descends isDecOf efxs | otherwise = const True + where + e `isDecOf` e' = not $ null + [ () + | d <- deconstructions + , m <- maybeToList (e `match` d) + , filter (uncurry (/=)) m == [(holeAsTypeOf e', e')] + ] + deconstructions :: [Expr] + deconstructions = filter (conjureIsDeconstruction f maxTests) + $ concatMap candidateDeconstructionsFrom + $ concat . take maxDeconstructionSize + $ concatMapT forN [hs] + where + hs = nub $ conjureArgumentHoles f recs = filterT keepR $ foldAppProducts ef [forN h | h <- conjureArgumentHoles f] - thy = filterTheory (===) + thy = doubleCheck (===) . theoryFromAtoms (===) maxEquationSize . (:[]) . nub $ cjHoles (prim nm f:ps) ++ [val False, val True] ++ es (===) = cjAreEqual (prim nm f:ps) maxTests @@ -285,7 +303,7 @@ efxs = conjureVarApplication nm f (ef:exs) = unfoldApp efxs - keep = isRootNormalE thy . fastMostGeneralVariation + keep = isRootNormal thy . fastMostGeneralVariation appsWith :: Expr -> [Expr] -> [[Expr]] appsWith eh vs = enumerateAppsFor eh keep $ vs ++ es @@ -314,8 +332,22 @@ fillingsFor = products . map fillingsFor1 ds = filter (conjureIsDeconstructor f maxTests) es - keepR ep | requireDescent = descends (`elem` ds) ep + keepR ep | requireDescent = descends isDecOf ep | otherwise = const True + where + e `isDecOf` e' = not $ null + [ () + | d <- deconstructions + , m <- maybeToList (e `match` d) + , filter (uncurry (/=)) m == [(holeAsTypeOf e', e')] + ] + deconstructions :: [Expr] + deconstructions = filter (conjureIsDeconstruction f maxTests) + $ concatMap candidateDeconstructionsFrom + $ concat . take maxDeconstructionSize + $ concatMapT (`appsWith` vars ep) [hs] + where + hs = nub $ conjureArgumentHoles f recs ep = filterT (keepR ep) . discardT (\e -> e == ep) $ recsV' (tail (vars ep)) @@ -331,7 +363,7 @@ where evrs = [(vs, recsV vs) | vs <- nubSort $ map (tail . vars) possiblePats] - thy = filterTheory (===) + thy = doubleCheck (===) . theoryFromAtoms (===) maxEquationSize . (:[]) . nub $ cjHoles (prim nm f:ps) ++ [val False, val True] ++ es (===) = cjAreEqual (prim nm f:ps) maxTests @@ -393,8 +425,8 @@ -- For all possible sets of arguments (2^n - 1 elements: 1 3 7 15 31), -- see if any projects the same variables while only using deconstructions -- and where there is at least a single deconstruction. -descends :: (Expr -> Bool) -> Expr -> Expr -> Bool -descends isDec e' e = any d1 ss +descends :: (Expr -> Expr -> Bool) -> Expr -> Expr -> Bool +descends isDecOf e' e = any d1 ss where desc = any d1 . uncurry useMatches . unzip d1 exys = all isNotConstruction exys @@ -403,12 +435,13 @@ exys = zip exs eys (_:exs) = unfoldApp e' (_:eys) = unfoldApp e - isDeconstruction (p,e) | isVar p = not (null cs) && all isDec cs + isDeconstruction (p,e) | isVar p = e `isDecOf` p | otherwise = size e < size p where cs = consts e - isNotConstruction (p,e) | isVar p = all isDec (consts e) + isNotConstruction (p,e) | isVar p = e == p || e `isDecOf` p | otherwise = size e <= size p -- TODO: allow filter and id somehow +-- TODO: improve this function with better isNotConstruction candidatesTD :: (Expr -> Bool) -> Expr -> [Expr] -> [[Expr]] @@ -466,29 +499,6 @@ where trie = T.fromList $ equations thy ++ map swap (equations thy) (->-) = canReduceTo thy - - ---- double checks --- - -filtered :: Thy -> Bool -filtered = (< 0) . closureLimit - -filterTheory :: (Expr -> Expr -> Bool) -> Thy -> Thy --- TODO: move filterTheory into Speculate, and add new Thy field "doubleChecked / invalid" --- or maybe have a third list of equations: --- invalid :: (Expr,Expr) --- that lists ones that were discarded -filterTheory (===) thy = thy - { rules = rs - , equations = es - , closureLimit = if r' && e' - then closureLimit thy - else -1 - } - where - correct = uncurry (===) - (rs,r') = filterAnd correct (rules thy) - (es,e') = filterAnd correct (equations thy) --- tiers utils --- diff --git a/stack.yaml b/stack.yaml --- a/stack.yaml +++ b/stack.yaml @@ -10,5 +10,5 @@ extra-deps: - leancheck-0.9.10 -- speculate-0.4.12 -- express-1.0.4 +- speculate-0.4.14 +- express-1.0.6 diff --git a/test/conjurable.hs b/test/conjurable.hs --- a/test/conjurable.hs +++ b/test/conjurable.hs @@ -211,7 +211,47 @@ ] ] ] + + , 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 + + , 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 + , isDecon (i_ -*- two) == False -- increases the size + + , candidateDeconstructionsFrom (div' xx yy) == [ div' i_ yy + , div' xx i_ + ] + , candidateDeconstructionsFrom (div' xx xx) == [] + , candidateDeconstructionsFrom ((xx -+- xx) -+- yy) == [(xx -+- xx) -+- i_] ] + +isDecon :: Expr -> Bool +isDecon = conjureIsDeconstruction (undefined :: [Int] -> [Char] -> [Bool]) 60 ffs :: Expr -> Expr ffs e = ffE :$ e