diff --git a/.github/workflows/build.yml b/.github/workflows/build.yml --- a/.github/workflows/build.yml +++ b/.github/workflows/build.yml @@ -1,8 +1,10 @@ # Builds and tests this Haskell project on "GitHub Actions" # -# 2021 Rudy Matela +# 2021-2023 Rudy Matela # # some docs: https://github.com/haskell/actions/tree/main/setup +# +# The official haskell docker image: https://hub.docker.com/_/haskell name: build on: [push] jobs: @@ -11,24 +13,26 @@ steps: - name: Cache ~/.cabal/packages - uses: actions/cache@v2 + uses: actions/cache@v3 with: path: ~/.cabal/packages - key: v1-${{ runner.os }}-cabal-packages-${{ hashFiles('hello.cabal') }} + key: v1-${{ runner.os }}-cabal-packages-${{ hashFiles('*.cabal') }} restore-keys: v1-${{ runner.os }}-cabal-packages- - name: Cache ~/.cabal and ~/.ghc - uses: actions/cache@v2 + uses: actions/cache@v3 with: path: | ~/.cabal !~/.cabal/packages ~/.ghc - key: v1-${{ runner.os }}-cabal-ghc-latest-${{ hashFiles('hello.cabal') }} + key: v1-${{ runner.os }}-cabal-ghc-latest-${{ hashFiles('*.cabal') }} restore-keys: v1-${{ runner.os }}-cabal-ghc-latest- - run: du -hd3 ~/.cabal ~/.ghc || true + - run: make --version + - run: haddock --version || sudo apt-get install ghc-haddock - run: ghc --version - run: cabal --version @@ -36,50 +40,54 @@ - run: ghc-pkg list - name: Check out repository - uses: actions/checkout@v2 + uses: actions/checkout@v3 - - run: cabal update + - run: git --version + - run: make install-dependencies - - run: make - - run: make test - - run: make haddock + # 2023-07: some projects were failing with missing base for GHC 9.6. + # Here we compile through cabal only provisionally. + # - run: make + # - run: make test + # - run: make haddock - run: make test-sdist - - run: make test-via-cabal + #- run: make test-via-cabal + - run: cabal configure --enable-tests --enable-benchmarks --ghc-options="-O0" + - run: cabal build + - run: cabal test + - run: cabal haddock test-with-ghc: strategy: matrix: ghc: + - '9.4' + - '9.2' - '9.0' - '8.10' - - '8.8' - '8.6' - - '8.4' - '8.2' - - '8.0' - - '7.10' - - '7.8' runs-on: ubuntu-latest needs: build-and-test container: haskell:${{ matrix.ghc }} steps: - name: Cache ~/.cabal/packages - uses: actions/cache@v2 + uses: actions/cache@v3 with: path: ~/.cabal/packages - key: v1-${{ runner.os }}-cabal-packages-${{ hashFiles('hello.cabal') }} + key: v1-${{ runner.os }}-cabal-packages-${{ hashFiles('*.cabal') }} restore-keys: v1-${{ runner.os }}-cabal-packages- - name: Cache ~/.cabal and ~/.ghc - uses: actions/cache@v2 + uses: actions/cache@v3 with: path: | ~/.cabal !~/.cabal/packages ~/.ghc - key: v1-${{ runner.os }}-cabal-ghc-${{ matrix.ghc }}-${{ hashFiles('hello.cabal') }} + key: v1-${{ runner.os }}-cabal-ghc-${{ matrix.ghc }}-${{ hashFiles('*.cabal') }} restore-keys: v1-${{ runner.os }}-cabal-ghc-${{ matrix.ghc }}- - run: du -hd3 ~/.cabal ~/.ghc || true @@ -94,9 +102,8 @@ - run: ghc-pkg list - name: Check out repository - uses: actions/checkout@v2 + uses: actions/checkout@v3 - - run: cabal update - run: make install-dependencies - run: make @@ -110,12 +117,12 @@ needs: build-and-test steps: - name: Setup Haskell's GHC and Cabal as required by current Stackage LTS - uses: haskell/actions/setup@v1 - with: # lts-18.14 - ghc-version: '8.10.7' - cabal-version: '3.2' + uses: haskell/actions/setup@v2 + with: # lts-19.19 + ghc-version: '9.0.2' + cabal-version: '3.4' - - uses: actions/cache@v2 + - uses: actions/cache@v3 with: path: ~/.stack key: v1-${{ runner.os }}-stack-${{ hashFiles('stack.yaml') }} @@ -124,5 +131,5 @@ - run: stack --version - name: Check out repository - uses: actions/checkout@v2 + uses: actions/checkout@v3 - run: make test-via-stack diff --git a/.gitignore b/.gitignore --- a/.gitignore +++ b/.gitignore @@ -42,6 +42,7 @@ eg/pow eg/tapps eg/tree +eg/bst eg/replicate eg/setelem eg/sort diff --git a/Makefile b/Makefile --- a/Makefile +++ b/Makefile @@ -2,11 +2,12 @@ GHCFLAGS = -O2 -v0 \ $(shell grep -q "Arch Linux" /etc/lsb-release && echo -dynamic) HADDOCKFLAGS = \ - -i $(shell find ~/.cabal -name express.haddock | tail -1) \ + -i $(shell find ~/.cabal -name leancheck.haddock | tail -1) \ + -i $(shell find ~/.cabal -name express.haddock | tail -1) \ -i $(shell find ~/.cabal -name speculate.haddock | tail -1) \ -i $(shell find /usr/share/doc/ghc/html/libraries -name template-haskell.haddock | tail -1) \ - $(shell grep -q "Arch Linux" /etc/lsb-release && echo --optghc=-dynamic) -HADDOCK_FILTER = grep -v "^Warning: Couldn't find .haddock for export [A-Z]$$" + $(shell grep -q "Arch Linux" /etc/lsb-release && echo --optghc=-dynamic) \ + | grep -v "^Warning: Couldn't find .haddock for export [A-Z]$$" INSTALL_DEPS = leancheck express speculate EG = \ @@ -28,6 +29,7 @@ eg/spec \ eg/tapps \ eg/tree \ + eg/bst \ bench/candidates \ bench/ill-hit \ bench/longshot \ diff --git a/TODO.md b/TODO.md --- a/TODO.md +++ b/TODO.md @@ -3,29 +3,14 @@ A non-exhaustive list of things TO DO for Conjure. -* test `conjureCases` from `deriveConjurable`; -* consider not breaking in some cases (increased crossproduct of patterns) - -* reduce the number of `deconstructions` considered: - - 1. place a `traceShowId` in `deconstructions :: [Expr]` - 2. run the GCD example - - [_ `mod` y :: Int,x `mod` _ :: Int,_ `mod` x :: Int,y `mod` _ :: Int,0 `mod` _ :: Int,0 `mod` _ :: Int] - - 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 if and case expressions? -* pretty-print top-level ifs? +* consider not breaking arguments in some cases + (increased crossproduct of patterns). + but which cases? * exclude magic numbers? e.g.: `1+1`? @@ -49,5 +34,5 @@ This file is part of Conjure, -(C) 2021 Rudy Matela, +(C) 2021-2022 Rudy Matela, Distribued under the 3-Clause BSD license. diff --git a/bench/candidates.hs b/bench/candidates.hs --- a/bench/candidates.hs +++ b/bench/candidates.hs @@ -75,3 +75,13 @@ , prim "||" (||) , prim "not" not ] + + -- Degenerate case: + -- lots of redundancy, since 0 `mod` 0 = undefined + -- Speculate is not able to discover that x `mod` x = 0 + -- nevertheless useful for observing candidate filtering + -- through other means + printCandidates 9 6 "gcd" (undefined :: Int -> Int -> Int) + [ pr (0::Int) + , prim "`mod`" (mod :: Int -> Int -> Int) + ] diff --git a/bench/candidates.out b/bench/candidates.out --- a/bench/candidates.out +++ b/bench/candidates.out @@ -869,8874 +869,11186 @@ 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 - -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 + [3,10,40,116,338,953,2702,7505,20728] pattern candidates, 0 duplicates + +rules: +x * y == x + y +x * y == y + x +x + 0 == x +0 + x == x +dec (x + y) == x + dec y +dec (x + y) == y + dec x +dec (x + y) == dec x + y +dec (x + y) == dec y + x +(x + y) + z == x + (y + z) +(x + y) + z == y + (x + z) +equations: +y + x == x + y +y + dec x == x + dec y +dec x + y == x + dec y +dec y + x == dec x + y +x + dec 0 == dec x +dec 0 + x == dec x +y + (x + z) == x + (y + z) +z + (x + y) == x + (y + z) +z + (y + x) == x + (y + z) +y + dec (dec x) == x + dec (dec y) +dec (dec x) + y == x + dec (dec y) +x + dec (dec 0) == dec (dec x) +dec (dec 0) + x == dec (dec x) + +direct candidates: + +x ? y = x + +x ? y = y + +x ? y = 0 + +x ? y = dec x + +x ? y = dec y + +x ? y = dec 0 + +x ? y = x + x + +x ? y = x + y + +x ? y = y + x + +x ? y = y + y + +x ? y = dec x + y + +x ? y = dec y + x + +x ? y = dec 0 + x + +x ? y = dec 0 + y + +x ? y = x + dec x + +x ? y = x + dec y + +x ? y = x + dec 0 + +x ? y = y + dec x + +x ? y = y + dec y + +x ? y = y + dec 0 + +x ? y = x + (x + x) + +x ? y = x + (x + y) + +x ? y = x + (y + x) + +x ? y = x + (y + y) + +x ? y = y + (x + x) + +x ? y = y + (x + y) + +x ? y = y + (y + x) + +x ? y = y + (y + y) + +x ? y = dec x + dec x + +x ? y = dec x + dec y + +x ? y = dec x + dec 0 + +x ? y = dec y + dec x + +x ? y = dec y + dec y + +x ? y = dec y + dec 0 + +x ? y = dec 0 + dec x + +x ? y = dec 0 + dec y + +x ? y = dec 0 + dec 0 + +x ? y = x + (dec x + y) + +x ? y = x + (dec y + x) + +x ? y = x + (dec 0 + x) + +x ? y = x + (dec 0 + y) + +x ? y = x + (x + dec x) + +x ? y = x + (x + dec y) + +x ? y = x + (x + dec 0) + +x ? y = x + (y + dec x) + +x ? y = x + (y + dec y) + +x ? y = x + (y + dec 0) + +x ? y = y + (dec x + y) + +x ? y = y + (dec y + x) + +x ? y = y + (dec 0 + x) + +x ? y = y + (dec 0 + y) + +x ? y = y + (x + dec x) + +x ? y = y + (x + dec y) + +x ? y = y + (x + dec 0) + +x ? y = y + (y + dec x) + +x ? y = y + (y + dec y) + +x ? y = y + (y + dec 0) + +x ? y = dec x + (y + y) + +x ? y = dec y + (x + x) + +x ? y = dec 0 + (x + x) + +x ? y = dec 0 + (x + y) + +x ? y = dec 0 + (y + x) + +x ? y = dec 0 + (y + y) + + +pattern candidates: + +x ? y = x + +x ? y = y + +x ? y = 0 + +x ? y = dec x + +x ? y = dec y + +x ? y = dec 0 + +x ? 0 = x +x ? y = y + +x ? 0 = x +x ? y = 0 + +x ? 0 = 0 +x ? y = x + +x ? 0 = 0 +x ? y = y + +0 ? x = x +x ? y = 0 + +0 ? x = 0 +x ? y = x + +0 ? x = 0 +x ? y = y + +x ? y = x + x + +x ? y = x + y + +x ? y = y + x + +x ? y = y + y + +x ? y = dec (dec x) + +x ? y = dec (dec y) + +x ? y = dec (dec 0) + +x ? 0 = x +x ? y = dec x + +x ? 0 = x +x ? y = dec y + +x ? 0 = x +x ? y = dec 0 + +x ? 0 = 0 +x ? y = dec x + +x ? 0 = 0 +x ? y = dec y + +x ? 0 = 0 +x ? y = dec 0 + +x ? 0 = dec x +x ? y = x + +x ? 0 = dec x +x ? y = y + +x ? 0 = dec x +x ? y = 0 + +x ? 0 = dec 0 +x ? y = x + +x ? 0 = dec 0 +x ? y = y + +x ? 0 = dec 0 +x ? y = 0 + +0 ? x = x +x ? y = dec x + +0 ? x = x +x ? y = dec y + +0 ? x = x +x ? y = dec 0 + +0 ? x = 0 +x ? y = dec x + +0 ? x = 0 +x ? y = dec y + +0 ? x = 0 +x ? y = dec 0 + +0 ? x = dec x +x ? y = x + +0 ? x = dec x +x ? y = y + +0 ? x = dec x +x ? y = 0 + +0 ? x = dec 0 +x ? y = x + +0 ? x = dec 0 +x ? y = y + +0 ? x = dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = y + +x ? y = dec x + x + +x ? y = dec x + y + +x ? y = dec y + x + +x ? y = dec y + y + +x ? y = dec 0 + x + +x ? y = dec 0 + y + +x ? y = dec (dec (dec x)) + +x ? y = dec (dec (dec y)) + +x ? y = dec (dec (dec 0)) + +x ? y = x + dec x + +x ? y = x + dec y + +x ? y = x + dec 0 + +x ? y = y + dec x + +x ? y = y + dec y + +x ? y = y + dec 0 + +x ? 0 = x +x ? y = x + x + +x ? 0 = x +x ? y = x + y + +x ? 0 = x +x ? y = y + x + +x ? 0 = x +x ? y = y + y + +x ? 0 = x +x ? y = dec (dec x) + +x ? 0 = x +x ? y = dec (dec y) + +x ? 0 = x +x ? y = dec (dec 0) + +x ? 0 = 0 +x ? y = x + x + +x ? 0 = 0 +x ? y = x + y + +x ? 0 = 0 +x ? y = y + x + +x ? 0 = 0 +x ? y = y + y + +x ? 0 = 0 +x ? y = dec (dec x) + +x ? 0 = 0 +x ? y = dec (dec y) + +x ? 0 = 0 +x ? y = dec (dec 0) + +x ? 0 = dec x +x ? y = dec y + +x ? 0 = dec x +x ? y = dec 0 + +x ? 0 = dec 0 +x ? y = dec x + +x ? 0 = dec 0 +x ? y = dec y + +x ? 0 = x + x +x ? y = x + +x ? 0 = x + x +x ? y = y + +x ? 0 = x + x +x ? y = 0 + +x ? 0 = dec (dec x) +x ? y = x + +x ? 0 = dec (dec x) +x ? y = y + +x ? 0 = dec (dec x) +x ? y = 0 + +x ? 0 = dec (dec 0) +x ? y = x + +x ? 0 = dec (dec 0) +x ? y = y + +x ? 0 = dec (dec 0) +x ? y = 0 + +0 ? x = x +x ? y = x + x + +0 ? x = x +x ? y = x + y + +0 ? x = x +x ? y = y + x + +0 ? x = x +x ? y = y + y + +0 ? x = x +x ? y = dec (dec x) + +0 ? x = x +x ? y = dec (dec y) + +0 ? x = x +x ? y = dec (dec 0) + +0 ? x = 0 +x ? y = x + x + +0 ? x = 0 +x ? y = x + y + +0 ? x = 0 +x ? y = y + x + +0 ? x = 0 +x ? y = y + y + +0 ? x = 0 +x ? y = dec (dec x) + +0 ? x = 0 +x ? y = dec (dec y) + +0 ? x = 0 +x ? y = dec (dec 0) + +0 ? x = dec x +x ? y = dec 0 + +0 ? x = dec 0 +x ? y = dec x + +0 ? x = dec 0 +x ? y = dec y + +0 ? x = x + x +x ? y = x + +0 ? x = x + x +x ? y = y + +0 ? x = x + x +x ? y = 0 + +0 ? x = dec (dec x) +x ? y = x + +0 ? x = dec (dec x) +x ? y = y + +0 ? x = dec (dec x) +x ? y = 0 + +0 ? x = dec (dec 0) +x ? y = x + +0 ? x = dec (dec 0) +x ? y = y + +0 ? x = dec (dec 0) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = 0 + +x ? 0 = x +x ? y = x ? dec y + +x ? 0 = x +x ? y = y ? dec y + +x ? 0 = x +x ? y = 0 ? dec y + +x ? 0 = x +x ? y = dec x ? x + +x ? 0 = x +x ? y = dec x ? y + +x ? 0 = x +x ? y = dec x ? 0 + +x ? 0 = 0 +x ? y = x ? dec y + +x ? 0 = 0 +x ? y = y ? dec y + +x ? 0 = 0 +x ? y = 0 ? dec y + +x ? 0 = 0 +x ? y = dec x ? x + +x ? 0 = 0 +x ? y = dec x ? y + +x ? 0 = 0 +x ? y = dec x ? 0 + +0 ? x = x +x ? y = x ? dec y + +0 ? x = x +x ? y = y ? dec y + +0 ? x = x +x ? y = 0 ? dec y + +0 ? x = x +x ? y = dec x ? x + +0 ? x = x +x ? y = dec x ? y + +0 ? x = x +x ? y = dec x ? 0 + +0 ? x = 0 +x ? y = x ? dec y + +0 ? x = 0 +x ? y = y ? dec y + +0 ? x = 0 +x ? y = 0 ? dec y + +0 ? x = 0 +x ? y = dec x ? x + +0 ? x = 0 +x ? y = dec x ? y + +0 ? x = 0 +x ? y = dec x ? 0 + +x ? y = dec (dec x) + x + +x ? y = dec (dec x) + y + +x ? y = dec (dec y) + x + +x ? y = dec (dec y) + y + +x ? y = dec (dec 0) + x + +x ? y = dec (dec 0) + y + +x ? y = dec (dec (dec (dec x))) + +x ? y = dec (dec (dec (dec y))) + +x ? y = dec (dec (dec (dec 0))) + +x ? y = x + (x + x) + +x ? y = x + (x + y) + +x ? y = x + (y + x) + +x ? y = x + (y + y) + +x ? y = x + dec (dec x) + +x ? y = x + dec (dec y) + +x ? y = x + dec (dec 0) + +x ? y = y + (x + x) + +x ? y = y + (x + y) + +x ? y = y + (y + x) + +x ? y = y + (y + y) + +x ? y = y + dec (dec x) + +x ? y = y + dec (dec y) + +x ? y = y + dec (dec 0) + +x ? y = dec x + dec x + +x ? y = dec x + dec y + +x ? y = dec x + dec 0 + +x ? y = dec y + dec x + +x ? y = dec y + dec y + +x ? y = dec y + dec 0 + +x ? y = dec 0 + dec x + +x ? y = dec 0 + dec y + +x ? y = dec 0 + dec 0 + +x ? 0 = x +x ? y = dec x + x + +x ? 0 = x +x ? y = dec x + y + +x ? 0 = x +x ? y = dec y + x + +x ? 0 = x +x ? y = dec y + y + +x ? 0 = x +x ? y = dec 0 + x + +x ? 0 = x +x ? y = dec 0 + y + +x ? 0 = x +x ? y = dec (dec (dec x)) + +x ? 0 = x +x ? y = dec (dec (dec y)) + +x ? 0 = x +x ? y = dec (dec (dec 0)) + +x ? 0 = x +x ? y = x + dec x + +x ? 0 = x +x ? y = x + dec y + +x ? 0 = x +x ? y = x + dec 0 + +x ? 0 = x +x ? y = y + dec x + +x ? 0 = x +x ? y = y + dec y + +x ? 0 = x +x ? y = y + dec 0 + +x ? 0 = 0 +x ? y = dec x + x + +x ? 0 = 0 +x ? y = dec x + y + +x ? 0 = 0 +x ? y = dec y + x + +x ? 0 = 0 +x ? y = dec y + y + +x ? 0 = 0 +x ? y = dec 0 + x + +x ? 0 = 0 +x ? y = dec 0 + y + +x ? 0 = 0 +x ? y = dec (dec (dec x)) + +x ? 0 = 0 +x ? y = dec (dec (dec y)) + +x ? 0 = 0 +x ? y = dec (dec (dec 0)) + +x ? 0 = 0 +x ? y = x + dec x + +x ? 0 = 0 +x ? y = x + dec y + +x ? 0 = 0 +x ? y = x + dec 0 + +x ? 0 = 0 +x ? y = y + dec x + +x ? 0 = 0 +x ? y = y + dec y + +x ? 0 = 0 +x ? y = y + dec 0 + +x ? 0 = dec x +x ? y = x + x + +x ? 0 = dec x +x ? y = x + y + +x ? 0 = dec x +x ? y = y + x + +x ? 0 = dec x +x ? y = y + y + +x ? 0 = dec x +x ? y = dec (dec x) + +x ? 0 = dec x +x ? y = dec (dec y) + +x ? 0 = dec x +x ? y = dec (dec 0) + +x ? 0 = dec 0 +x ? y = x + x + +x ? 0 = dec 0 +x ? y = x + y + +x ? 0 = dec 0 +x ? y = y + x + +x ? 0 = dec 0 +x ? y = y + y + +x ? 0 = dec 0 +x ? y = dec (dec x) + +x ? 0 = dec 0 +x ? y = dec (dec y) + +x ? 0 = dec 0 +x ? y = dec (dec 0) + +x ? 0 = x + x +x ? y = dec x + +x ? 0 = x + x +x ? y = dec y + +x ? 0 = x + x +x ? y = dec 0 + +x ? 0 = dec (dec x) +x ? y = dec x + +x ? 0 = dec (dec x) +x ? y = dec y + +x ? 0 = dec (dec x) +x ? y = dec 0 + +x ? 0 = dec (dec 0) +x ? y = dec x + +x ? 0 = dec (dec 0) +x ? y = dec y + +x ? 0 = dec (dec 0) +x ? y = dec 0 + +x ? 0 = dec x + x +x ? y = x + +x ? 0 = dec x + x +x ? y = y + +x ? 0 = dec x + x +x ? y = 0 + +x ? 0 = dec 0 + x +x ? y = x + +x ? 0 = dec 0 + x +x ? y = y + +x ? 0 = dec 0 + x +x ? y = 0 + +x ? 0 = dec (dec (dec x)) +x ? y = x + +x ? 0 = dec (dec (dec x)) +x ? y = y + +x ? 0 = dec (dec (dec x)) +x ? y = 0 + +x ? 0 = dec (dec (dec 0)) +x ? y = x + +x ? 0 = dec (dec (dec 0)) +x ? y = y + +x ? 0 = dec (dec (dec 0)) +x ? y = 0 + +x ? 0 = x + dec x +x ? y = x + +x ? 0 = x + dec x +x ? y = y + +x ? 0 = x + dec x +x ? y = 0 + +x ? 0 = x + dec 0 +x ? y = x + +x ? 0 = x + dec 0 +x ? y = y + +x ? 0 = x + dec 0 +x ? y = 0 + +0 ? x = x +x ? y = dec x + x + +0 ? x = x +x ? y = dec x + y + +0 ? x = x +x ? y = dec y + x + +0 ? x = x +x ? y = dec y + y + +0 ? x = x +x ? y = dec 0 + x + +0 ? x = x +x ? y = dec 0 + y + +0 ? x = x +x ? y = dec (dec (dec x)) + +0 ? x = x +x ? y = dec (dec (dec y)) + +0 ? x = x +x ? y = dec (dec (dec 0)) + +0 ? x = x +x ? y = x + dec x + +0 ? x = x +x ? y = x + dec y + +0 ? x = x +x ? y = x + dec 0 + +0 ? x = x +x ? y = y + dec x + +0 ? x = x +x ? y = y + dec y + +0 ? x = x +x ? y = y + dec 0 + +0 ? x = 0 +x ? y = dec x + x + +0 ? x = 0 +x ? y = dec x + y + +0 ? x = 0 +x ? y = dec y + x + +0 ? x = 0 +x ? y = dec y + y + +0 ? x = 0 +x ? y = dec 0 + x + +0 ? x = 0 +x ? y = dec 0 + y + +0 ? x = 0 +x ? y = dec (dec (dec x)) + +0 ? x = 0 +x ? y = dec (dec (dec y)) + +0 ? x = 0 +x ? y = dec (dec (dec 0)) + +0 ? x = 0 +x ? y = x + dec x + +0 ? x = 0 +x ? y = x + dec y + +0 ? x = 0 +x ? y = x + dec 0 + +0 ? x = 0 +x ? y = y + dec x + +0 ? x = 0 +x ? y = y + dec y + +0 ? x = 0 +x ? y = y + dec 0 + +0 ? x = dec x +x ? y = x + x + +0 ? x = dec x +x ? y = x + y + +0 ? x = dec x +x ? y = y + x + +0 ? x = dec x +x ? y = y + y + +0 ? x = dec x +x ? y = dec (dec x) + +0 ? x = dec x +x ? y = dec (dec y) + +0 ? x = dec x +x ? y = dec (dec 0) + +0 ? x = dec 0 +x ? y = x + x + +0 ? x = dec 0 +x ? y = x + y + +0 ? x = dec 0 +x ? y = y + x + +0 ? x = dec 0 +x ? y = y + y + +0 ? x = dec 0 +x ? y = dec (dec x) + +0 ? x = dec 0 +x ? y = dec (dec y) + +0 ? x = dec 0 +x ? y = dec (dec 0) + +0 ? x = x + x +x ? y = dec x + +0 ? x = x + x +x ? y = dec y + +0 ? x = x + x +x ? y = dec 0 + +0 ? x = dec (dec x) +x ? y = dec x + +0 ? x = dec (dec x) +x ? y = dec y + +0 ? x = dec (dec x) +x ? y = dec 0 + +0 ? x = dec (dec 0) +x ? y = dec x + +0 ? x = dec (dec 0) +x ? y = dec y + +0 ? x = dec (dec 0) +x ? y = dec 0 + +0 ? x = dec x + x +x ? y = x + +0 ? x = dec x + x +x ? y = y + +0 ? x = dec x + x +x ? y = 0 + +0 ? x = dec 0 + x +x ? y = x + +0 ? x = dec 0 + x +x ? y = y + +0 ? x = dec 0 + x +x ? y = 0 + +0 ? x = dec (dec (dec x)) +x ? y = x + +0 ? x = dec (dec (dec x)) +x ? y = y + +0 ? x = dec (dec (dec x)) +x ? y = 0 + +0 ? x = dec (dec (dec 0)) +x ? y = x + +0 ? x = dec (dec (dec 0)) +x ? y = y + +0 ? x = dec (dec (dec 0)) +x ? y = 0 + +0 ? x = x + dec x +x ? y = x + +0 ? x = x + dec x +x ? y = y + +0 ? x = x + dec x +x ? y = 0 + +0 ? x = x + dec 0 +x ? y = x + +0 ? x = x + dec 0 +x ? y = y + +0 ? x = x + dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = x + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = x + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = y + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = y + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = x + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = x + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = y + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = y + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + x +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + x +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec x) +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec x) +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec x) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec 0) +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec 0) +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec 0) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = x + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = x + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = y + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = y + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = x + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = x + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = y + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = y + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + x +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + x +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec 0) +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec 0) +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec 0) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = x +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = x +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = x +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = x +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = 0 +x ? y = 0 + +x ? 0 = x +x ? y = x ? dec (dec y) + +x ? 0 = x +x ? y = y ? dec (dec y) + +x ? 0 = x +x ? y = 0 ? dec (dec y) + +x ? 0 = x +x ? y = dec x ? dec x + +x ? 0 = x +x ? y = dec x ? dec y + +x ? 0 = x +x ? y = dec x ? dec 0 + +x ? 0 = x +x ? y = dec y ? dec y + +x ? 0 = x +x ? y = dec 0 ? dec y + +x ? 0 = x +x ? y = dec (dec x) ? x + +x ? 0 = x +x ? y = dec (dec x) ? y + +x ? 0 = x +x ? y = dec (dec x) ? 0 + +x ? 0 = 0 +x ? y = x ? dec (dec y) + +x ? 0 = 0 +x ? y = y ? dec (dec y) + +x ? 0 = 0 +x ? y = 0 ? dec (dec y) + +x ? 0 = 0 +x ? y = dec x ? dec x + +x ? 0 = 0 +x ? y = dec x ? dec y + +x ? 0 = 0 +x ? y = dec x ? dec 0 + +x ? 0 = 0 +x ? y = dec y ? dec y + +x ? 0 = 0 +x ? y = dec 0 ? dec y + +x ? 0 = 0 +x ? y = dec (dec x) ? x + +x ? 0 = 0 +x ? y = dec (dec x) ? y + +x ? 0 = 0 +x ? y = dec (dec x) ? 0 + +0 ? x = x +x ? y = x ? dec (dec y) + +0 ? x = x +x ? y = y ? dec (dec y) + +0 ? x = x +x ? y = 0 ? dec (dec y) + +0 ? x = x +x ? y = dec x ? dec x + +0 ? x = x +x ? y = dec x ? dec y + +0 ? x = x +x ? y = dec x ? dec 0 + +0 ? x = x +x ? y = dec y ? dec y + +0 ? x = x +x ? y = dec 0 ? dec y + +0 ? x = x +x ? y = dec (dec x) ? x + +0 ? x = x +x ? y = dec (dec x) ? y + +0 ? x = x +x ? y = dec (dec x) ? 0 + +0 ? x = 0 +x ? y = x ? dec (dec y) + +0 ? x = 0 +x ? y = y ? dec (dec y) + +0 ? x = 0 +x ? y = 0 ? dec (dec y) + +0 ? x = 0 +x ? y = dec x ? dec x + +0 ? x = 0 +x ? y = dec x ? dec y + +0 ? x = 0 +x ? y = dec x ? dec 0 + +0 ? x = 0 +x ? y = dec y ? dec y + +0 ? x = 0 +x ? y = dec 0 ? dec y + +0 ? x = 0 +x ? y = dec (dec x) ? x + +0 ? x = 0 +x ? y = dec (dec x) ? y + +0 ? x = 0 +x ? y = dec (dec x) ? 0 + +x ? 0 = x +x ? y = dec (x ? dec y) + +x ? 0 = x +x ? y = dec (y ? dec y) + +x ? 0 = x +x ? y = dec (0 ? dec y) + +x ? 0 = x +x ? y = dec (dec x ? x) + +x ? 0 = x +x ? y = dec (dec x ? y) + +x ? 0 = x +x ? y = dec (dec x ? 0) + +x ? 0 = 0 +x ? y = dec (x ? dec y) + +x ? 0 = 0 +x ? y = dec (y ? dec y) + +x ? 0 = 0 +x ? y = dec (0 ? dec y) + +x ? 0 = 0 +x ? y = dec (dec x ? x) + +x ? 0 = 0 +x ? y = dec (dec x ? y) + +x ? 0 = 0 +x ? y = dec (dec x ? 0) + +x ? 0 = dec x +x ? y = x ? dec y + +x ? 0 = dec x +x ? y = y ? dec y + +x ? 0 = dec x +x ? y = 0 ? dec y + +x ? 0 = dec x +x ? y = dec x ? x + +x ? 0 = dec x +x ? y = dec x ? y + +x ? 0 = dec x +x ? y = dec x ? 0 + +x ? 0 = dec 0 +x ? y = x ? dec y + +x ? 0 = dec 0 +x ? y = y ? dec y + +x ? 0 = dec 0 +x ? y = 0 ? dec y + +x ? 0 = dec 0 +x ? y = dec x ? x + +x ? 0 = dec 0 +x ? y = dec x ? y + +x ? 0 = dec 0 +x ? y = dec x ? 0 + +0 ? x = x +x ? y = dec (x ? dec y) + +0 ? x = x +x ? y = dec (y ? dec y) + +0 ? x = x +x ? y = dec (0 ? dec y) + +0 ? x = x +x ? y = dec (dec x ? x) + +0 ? x = x +x ? y = dec (dec x ? y) + +0 ? x = x +x ? y = dec (dec x ? 0) + +0 ? x = 0 +x ? y = dec (x ? dec y) + +0 ? x = 0 +x ? y = dec (y ? dec y) + +0 ? x = 0 +x ? y = dec (0 ? dec y) + +0 ? x = 0 +x ? y = dec (dec x ? x) + +0 ? x = 0 +x ? y = dec (dec x ? y) + +0 ? x = 0 +x ? y = dec (dec x ? 0) + +0 ? x = dec x +x ? y = x ? dec y + +0 ? x = dec x +x ? y = y ? dec y + +0 ? x = dec x +x ? y = 0 ? dec y + +0 ? x = dec x +x ? y = dec x ? x + +0 ? x = dec x +x ? y = dec x ? y + +0 ? x = dec x +x ? y = dec x ? 0 + +0 ? x = dec 0 +x ? y = x ? dec y + +0 ? x = dec 0 +x ? y = y ? dec y + +0 ? x = dec 0 +x ? y = 0 ? dec y + +0 ? x = dec 0 +x ? y = dec x ? x + +0 ? x = dec 0 +x ? y = dec x ? y + +0 ? x = dec 0 +x ? y = dec x ? 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = x ? dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = y ? dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = 0 ? dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec x ? x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec x ? y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec x ? 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x ? x +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x ? 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x ? x +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x ? 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x ? x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x ? 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = x ? dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = y ? dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = 0 ? dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec x ? x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec x ? y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec x ? 0 + +0 ? 0 = 0 +0 ? x = x ? dec x +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 ? dec x +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = x ? dec x +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 ? dec x +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = x ? dec x +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 ? dec x +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x ? dec x +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 ? dec x +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = x ? dec x +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 ? dec x +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = x ? dec x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 ? dec x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = x ? dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = y ? dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = 0 ? dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec x ? x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec x ? y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec x ? 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x ? x +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x ? 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x ? x +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x ? 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x ? x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x ? 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = x ? dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = y ? dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = 0 ? dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x ? x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x ? y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x ? 0 + +x ? y = dec (dec (dec x)) + x + +x ? y = dec (dec (dec x)) + y + +x ? y = dec (dec (dec y)) + x + +x ? y = dec (dec (dec y)) + y + +x ? y = dec (dec (dec 0)) + x + +x ? y = dec (dec (dec 0)) + y + +x ? y = dec (dec (dec (dec (dec x)))) + +x ? y = dec (dec (dec (dec (dec y)))) + +x ? y = dec (dec (dec (dec (dec 0)))) + +x ? y = x + (dec x + x) + +x ? y = x + (dec x + y) + +x ? y = x + (dec y + x) + +x ? y = x + (dec y + y) + +x ? y = x + (dec 0 + x) + +x ? y = x + (dec 0 + y) + +x ? y = x + dec (dec (dec x)) + +x ? y = x + dec (dec (dec y)) + +x ? y = x + dec (dec (dec 0)) + +x ? y = x + (x + dec x) + +x ? y = x + (x + dec y) + +x ? y = x + (x + dec 0) + +x ? y = x + (y + dec x) + +x ? y = x + (y + dec y) + +x ? y = x + (y + dec 0) + +x ? y = y + (dec x + x) + +x ? y = y + (dec x + y) + +x ? y = y + (dec y + x) + +x ? y = y + (dec y + y) + +x ? y = y + (dec 0 + x) + +x ? y = y + (dec 0 + y) + +x ? y = y + dec (dec (dec x)) + +x ? y = y + dec (dec (dec y)) + +x ? y = y + dec (dec (dec 0)) + +x ? y = y + (x + dec x) + +x ? y = y + (x + dec y) + +x ? y = y + (x + dec 0) + +x ? y = y + (y + dec x) + +x ? y = y + (y + dec y) + +x ? y = y + (y + dec 0) + +x ? y = dec x + (x + x) + +x ? y = dec x + (x + y) + +x ? y = dec x + (y + x) + +x ? y = dec x + (y + y) + +x ? y = dec x + dec (dec x) + +x ? y = dec x + dec (dec y) + +x ? y = dec x + dec (dec 0) + +x ? y = dec y + (x + x) + +x ? y = dec y + (x + y) + +x ? y = dec y + (y + x) + +x ? y = dec y + (y + y) + +x ? y = dec y + dec (dec x) + +x ? y = dec y + dec (dec y) + +x ? y = dec y + dec (dec 0) + +x ? y = dec 0 + (x + x) + +x ? y = dec 0 + (x + y) + +x ? y = dec 0 + (y + x) + +x ? y = dec 0 + (y + y) + +x ? y = dec 0 + dec (dec x) + +x ? y = dec 0 + dec (dec y) + +x ? y = dec 0 + dec (dec 0) + +x ? y = dec (dec x) + dec x + +x ? y = dec (dec x) + dec y + +x ? y = dec (dec x) + dec 0 + +x ? y = dec (dec y) + dec x + +x ? y = dec (dec y) + dec y + +x ? y = dec (dec y) + dec 0 + +x ? y = dec (dec 0) + dec x + +x ? y = dec (dec 0) + dec y + +x ? y = dec (dec 0) + dec 0 + +x ? 0 = x +x ? y = dec (dec x) + x + +x ? 0 = x +x ? y = dec (dec x) + y + +x ? 0 = x +x ? y = dec (dec y) + x + +x ? 0 = x +x ? y = dec (dec y) + y + +x ? 0 = x +x ? y = dec (dec 0) + x + +x ? 0 = x +x ? y = dec (dec 0) + y + +x ? 0 = x +x ? y = dec (dec (dec (dec x))) + +x ? 0 = x +x ? y = dec (dec (dec (dec y))) + +x ? 0 = x +x ? y = dec (dec (dec (dec 0))) + +x ? 0 = x +x ? y = x + (x + x) + +x ? 0 = x +x ? y = x + (x + y) + +x ? 0 = x +x ? y = x + (y + x) + +x ? 0 = x +x ? y = x + (y + y) + +x ? 0 = x +x ? y = x + dec (dec x) + +x ? 0 = x +x ? y = x + dec (dec y) + +x ? 0 = x +x ? y = x + dec (dec 0) + +x ? 0 = x +x ? y = y + (x + x) + +x ? 0 = x +x ? y = y + (x + y) + +x ? 0 = x +x ? y = y + (y + x) + +x ? 0 = x +x ? y = y + (y + y) + +x ? 0 = x +x ? y = y + dec (dec x) + +x ? 0 = x +x ? y = y + dec (dec y) + +x ? 0 = x +x ? y = y + dec (dec 0) + +x ? 0 = x +x ? y = dec x + dec x + +x ? 0 = x +x ? y = dec x + dec y + +x ? 0 = x +x ? y = dec x + dec 0 + +x ? 0 = x +x ? y = dec y + dec x + +x ? 0 = x +x ? y = dec y + dec y + +x ? 0 = x +x ? y = dec y + dec 0 + +x ? 0 = x +x ? y = dec 0 + dec x + +x ? 0 = x +x ? y = dec 0 + dec y + +x ? 0 = x +x ? y = dec 0 + dec 0 + +x ? 0 = 0 +x ? y = dec (dec x) + x + +x ? 0 = 0 +x ? y = dec (dec x) + y + +x ? 0 = 0 +x ? y = dec (dec y) + x + +x ? 0 = 0 +x ? y = dec (dec y) + y + +x ? 0 = 0 +x ? y = dec (dec 0) + x + +x ? 0 = 0 +x ? y = dec (dec 0) + y + +x ? 0 = 0 +x ? y = dec (dec (dec (dec x))) + +x ? 0 = 0 +x ? y = dec (dec (dec (dec y))) + +x ? 0 = 0 +x ? y = dec (dec (dec (dec 0))) + +x ? 0 = 0 +x ? y = x + (x + x) + +x ? 0 = 0 +x ? y = x + (x + y) + +x ? 0 = 0 +x ? y = x + (y + x) + +x ? 0 = 0 +x ? y = x + (y + y) + +x ? 0 = 0 +x ? y = x + dec (dec x) + +x ? 0 = 0 +x ? y = x + dec (dec y) + +x ? 0 = 0 +x ? y = x + dec (dec 0) + +x ? 0 = 0 +x ? y = y + (x + x) + +x ? 0 = 0 +x ? y = y + (x + y) + +x ? 0 = 0 +x ? y = y + (y + x) + +x ? 0 = 0 +x ? y = y + (y + y) + +x ? 0 = 0 +x ? y = y + dec (dec x) + +x ? 0 = 0 +x ? y = y + dec (dec y) + +x ? 0 = 0 +x ? y = y + dec (dec 0) + +x ? 0 = 0 +x ? y = dec x + dec x + +x ? 0 = 0 +x ? y = dec x + dec y + +x ? 0 = 0 +x ? y = dec x + dec 0 + +x ? 0 = 0 +x ? y = dec y + dec x + +x ? 0 = 0 +x ? y = dec y + dec y + +x ? 0 = 0 +x ? y = dec y + dec 0 + +x ? 0 = 0 +x ? y = dec 0 + dec x + +x ? 0 = 0 +x ? y = dec 0 + dec y + +x ? 0 = 0 +x ? y = dec 0 + dec 0 + +x ? 0 = dec x +x ? y = dec x + x + +x ? 0 = dec x +x ? y = dec x + y + +x ? 0 = dec x +x ? y = dec y + x + +x ? 0 = dec x +x ? y = dec y + y + +x ? 0 = dec x +x ? y = dec 0 + x + +x ? 0 = dec x +x ? y = dec 0 + y + +x ? 0 = dec x +x ? y = dec (dec (dec x)) + +x ? 0 = dec x +x ? y = dec (dec (dec y)) + +x ? 0 = dec x +x ? y = dec (dec (dec 0)) + +x ? 0 = dec x +x ? y = x + dec x + +x ? 0 = dec x +x ? y = x + dec y + +x ? 0 = dec x +x ? y = x + dec 0 + +x ? 0 = dec x +x ? y = y + dec x + +x ? 0 = dec x +x ? y = y + dec y + +x ? 0 = dec x +x ? y = y + dec 0 + +x ? 0 = dec 0 +x ? y = dec x + x + +x ? 0 = dec 0 +x ? y = dec x + y + +x ? 0 = dec 0 +x ? y = dec y + x + +x ? 0 = dec 0 +x ? y = dec y + y + +x ? 0 = dec 0 +x ? y = dec 0 + x + +x ? 0 = dec 0 +x ? y = dec 0 + y + +x ? 0 = dec 0 +x ? y = dec (dec (dec x)) + +x ? 0 = dec 0 +x ? y = dec (dec (dec y)) + +x ? 0 = dec 0 +x ? y = dec (dec (dec 0)) + +x ? 0 = dec 0 +x ? y = x + dec x + +x ? 0 = dec 0 +x ? y = x + dec y + +x ? 0 = dec 0 +x ? y = x + dec 0 + +x ? 0 = dec 0 +x ? y = y + dec x + +x ? 0 = dec 0 +x ? y = y + dec y + +x ? 0 = dec 0 +x ? y = y + dec 0 + +x ? 0 = x + x +x ? y = x + y + +x ? 0 = x + x +x ? y = y + x + +x ? 0 = x + x +x ? y = y + y + +x ? 0 = x + x +x ? y = dec (dec x) + +x ? 0 = x + x +x ? y = dec (dec y) + +x ? 0 = x + x +x ? y = dec (dec 0) + +x ? 0 = dec (dec x) +x ? y = x + x + +x ? 0 = dec (dec x) +x ? y = x + y + +x ? 0 = dec (dec x) +x ? y = y + x + +x ? 0 = dec (dec x) +x ? y = y + y + +x ? 0 = dec (dec x) +x ? y = dec (dec y) + +x ? 0 = dec (dec x) +x ? y = dec (dec 0) + +x ? 0 = dec (dec 0) +x ? y = x + x + +x ? 0 = dec (dec 0) +x ? y = x + y + +x ? 0 = dec (dec 0) +x ? y = y + x + +x ? 0 = dec (dec 0) +x ? y = y + y + +x ? 0 = dec (dec 0) +x ? y = dec (dec x) + +x ? 0 = dec (dec 0) +x ? y = dec (dec y) + +x ? 0 = dec x + x +x ? y = dec x + +x ? 0 = dec x + x +x ? y = dec y + +x ? 0 = dec x + x +x ? y = dec 0 + +x ? 0 = dec 0 + x +x ? y = dec x + +x ? 0 = dec 0 + x +x ? y = dec y + +x ? 0 = dec 0 + x +x ? y = dec 0 + +x ? 0 = dec (dec (dec x)) +x ? y = dec x + +x ? 0 = dec (dec (dec x)) +x ? y = dec y + +x ? 0 = dec (dec (dec x)) +x ? y = dec 0 + +x ? 0 = dec (dec (dec 0)) +x ? y = dec x + +x ? 0 = dec (dec (dec 0)) +x ? y = dec y + +x ? 0 = dec (dec (dec 0)) +x ? y = dec 0 + +x ? 0 = x + dec x +x ? y = dec x + +x ? 0 = x + dec x +x ? y = dec y + +x ? 0 = x + dec x +x ? y = dec 0 + +x ? 0 = x + dec 0 +x ? y = dec x + +x ? 0 = x + dec 0 +x ? y = dec y + +x ? 0 = x + dec 0 +x ? y = dec 0 + +x ? 0 = dec (dec x) + x +x ? y = x + +x ? 0 = dec (dec x) + x +x ? y = y + +x ? 0 = dec (dec x) + x +x ? y = 0 + +x ? 0 = dec (dec 0) + x +x ? y = x + +x ? 0 = dec (dec 0) + x +x ? y = y + +x ? 0 = dec (dec 0) + x +x ? y = 0 + +x ? 0 = dec (dec (dec (dec x))) +x ? y = x + +x ? 0 = dec (dec (dec (dec x))) +x ? y = y + +x ? 0 = dec (dec (dec (dec x))) +x ? y = 0 + +x ? 0 = dec (dec (dec (dec 0))) +x ? y = x + +x ? 0 = dec (dec (dec (dec 0))) +x ? y = y + +x ? 0 = dec (dec (dec (dec 0))) +x ? y = 0 + +x ? 0 = x + (x + x) +x ? y = x + +x ? 0 = x + (x + x) +x ? y = y + +x ? 0 = x + (x + x) +x ? y = 0 + +x ? 0 = x + dec (dec x) +x ? y = x + +x ? 0 = x + dec (dec x) +x ? y = y + +x ? 0 = x + dec (dec x) +x ? y = 0 + +x ? 0 = x + dec (dec 0) +x ? y = x + +x ? 0 = x + dec (dec 0) +x ? y = y + +x ? 0 = x + dec (dec 0) +x ? y = 0 + +x ? 0 = dec x + dec x +x ? y = x + +x ? 0 = dec x + dec x +x ? y = y + +x ? 0 = dec x + dec x +x ? y = 0 + +x ? 0 = dec x + dec 0 +x ? y = x + +x ? 0 = dec x + dec 0 +x ? y = y + +x ? 0 = dec x + dec 0 +x ? y = 0 + +x ? 0 = dec 0 + dec x +x ? y = x + +x ? 0 = dec 0 + dec x +x ? y = y + +x ? 0 = dec 0 + dec x +x ? y = 0 + +x ? 0 = dec 0 + dec 0 +x ? y = x + +x ? 0 = dec 0 + dec 0 +x ? y = y + +x ? 0 = dec 0 + dec 0 +x ? y = 0 + +0 ? x = x +x ? y = dec (dec x) + x + +0 ? x = x +x ? y = dec (dec x) + y + +0 ? x = x +x ? y = dec (dec y) + x + +0 ? x = x +x ? y = dec (dec y) + y + +0 ? x = x +x ? y = dec (dec 0) + x + +0 ? x = x +x ? y = dec (dec 0) + y + +0 ? x = x +x ? y = dec (dec (dec (dec x))) + +0 ? x = x +x ? y = dec (dec (dec (dec y))) + +0 ? x = x +x ? y = dec (dec (dec (dec 0))) + +0 ? x = x +x ? y = x + (x + x) + +0 ? x = x +x ? y = x + (x + y) + +0 ? x = x +x ? y = x + (y + x) + +0 ? x = x +x ? y = x + (y + y) + +0 ? x = x +x ? y = x + dec (dec x) + +0 ? x = x +x ? y = x + dec (dec y) + +0 ? x = x +x ? y = x + dec (dec 0) + +0 ? x = x +x ? y = y + (x + x) + +0 ? x = x +x ? y = y + (x + y) + +0 ? x = x +x ? y = y + (y + x) + +0 ? x = x +x ? y = y + (y + y) + +0 ? x = x +x ? y = y + dec (dec x) + +0 ? x = x +x ? y = y + dec (dec y) + +0 ? x = x +x ? y = y + dec (dec 0) + +0 ? x = x +x ? y = dec x + dec x + +0 ? x = x +x ? y = dec x + dec y + +0 ? x = x +x ? y = dec x + dec 0 + +0 ? x = x +x ? y = dec y + dec x + +0 ? x = x +x ? y = dec y + dec y + +0 ? x = x +x ? y = dec y + dec 0 + +0 ? x = x +x ? y = dec 0 + dec x + +0 ? x = x +x ? y = dec 0 + dec y + +0 ? x = x +x ? y = dec 0 + dec 0 + +0 ? x = 0 +x ? y = dec (dec x) + x + +0 ? x = 0 +x ? y = dec (dec x) + y + +0 ? x = 0 +x ? y = dec (dec y) + x + +0 ? x = 0 +x ? y = dec (dec y) + y + +0 ? x = 0 +x ? y = dec (dec 0) + x + +0 ? x = 0 +x ? y = dec (dec 0) + y + +0 ? x = 0 +x ? y = dec (dec (dec (dec x))) + +0 ? x = 0 +x ? y = dec (dec (dec (dec y))) + +0 ? x = 0 +x ? y = dec (dec (dec (dec 0))) + +0 ? x = 0 +x ? y = x + (x + x) + +0 ? x = 0 +x ? y = x + (x + y) + +0 ? x = 0 +x ? y = x + (y + x) + +0 ? x = 0 +x ? y = x + (y + y) + +0 ? x = 0 +x ? y = x + dec (dec x) + +0 ? x = 0 +x ? y = x + dec (dec y) + +0 ? x = 0 +x ? y = x + dec (dec 0) + +0 ? x = 0 +x ? y = y + (x + x) + +0 ? x = 0 +x ? y = y + (x + y) + +0 ? x = 0 +x ? y = y + (y + x) + +0 ? x = 0 +x ? y = y + (y + y) + +0 ? x = 0 +x ? y = y + dec (dec x) + +0 ? x = 0 +x ? y = y + dec (dec y) + +0 ? x = 0 +x ? y = y + dec (dec 0) + +0 ? x = 0 +x ? y = dec x + dec x + +0 ? x = 0 +x ? y = dec x + dec y + +0 ? x = 0 +x ? y = dec x + dec 0 + +0 ? x = 0 +x ? y = dec y + dec x + +0 ? x = 0 +x ? y = dec y + dec y + +0 ? x = 0 +x ? y = dec y + dec 0 + +0 ? x = 0 +x ? y = dec 0 + dec x + +0 ? x = 0 +x ? y = dec 0 + dec y + +0 ? x = 0 +x ? y = dec 0 + dec 0 + +0 ? x = dec x +x ? y = dec x + x + +0 ? x = dec x +x ? y = dec x + y + +0 ? x = dec x +x ? y = dec y + x + +0 ? x = dec x +x ? y = dec y + y + +0 ? x = dec x +x ? y = dec 0 + x + +0 ? x = dec x +x ? y = dec 0 + y + +0 ? x = dec x +x ? y = dec (dec (dec x)) + +0 ? x = dec x +x ? y = dec (dec (dec y)) + +0 ? x = dec x +x ? y = dec (dec (dec 0)) + +0 ? x = dec x +x ? y = x + dec x + +0 ? x = dec x +x ? y = x + dec y + +0 ? x = dec x +x ? y = x + dec 0 + +0 ? x = dec x +x ? y = y + dec x + +0 ? x = dec x +x ? y = y + dec y + +0 ? x = dec x +x ? y = y + dec 0 + +0 ? x = dec 0 +x ? y = dec x + x + +0 ? x = dec 0 +x ? y = dec x + y + +0 ? x = dec 0 +x ? y = dec y + x + +0 ? x = dec 0 +x ? y = dec y + y + +0 ? x = dec 0 +x ? y = dec 0 + x + +0 ? x = dec 0 +x ? y = dec 0 + y + +0 ? x = dec 0 +x ? y = dec (dec (dec x)) + +0 ? x = dec 0 +x ? y = dec (dec (dec y)) + +0 ? x = dec 0 +x ? y = dec (dec (dec 0)) + +0 ? x = dec 0 +x ? y = x + dec x + +0 ? x = dec 0 +x ? y = x + dec y + +0 ? x = dec 0 +x ? y = x + dec 0 + +0 ? x = dec 0 +x ? y = y + dec x + +0 ? x = dec 0 +x ? y = y + dec y + +0 ? x = dec 0 +x ? y = y + dec 0 + +0 ? x = x + x +x ? y = x + y + +0 ? x = x + x +x ? y = y + x + +0 ? x = x + x +x ? y = dec (dec x) + +0 ? x = x + x +x ? y = dec (dec y) + +0 ? x = x + x +x ? y = dec (dec 0) + +0 ? x = dec (dec x) +x ? y = x + x + +0 ? x = dec (dec x) +x ? y = x + y + +0 ? x = dec (dec x) +x ? y = y + x + +0 ? x = dec (dec x) +x ? y = y + y + +0 ? x = dec (dec x) +x ? y = dec (dec 0) + +0 ? x = dec (dec 0) +x ? y = x + x + +0 ? x = dec (dec 0) +x ? y = x + y + +0 ? x = dec (dec 0) +x ? y = y + x + +0 ? x = dec (dec 0) +x ? y = y + y + +0 ? x = dec (dec 0) +x ? y = dec (dec x) + +0 ? x = dec (dec 0) +x ? y = dec (dec y) + +0 ? x = dec x + x +x ? y = dec x + +0 ? x = dec x + x +x ? y = dec y + +0 ? x = dec x + x +x ? y = dec 0 + +0 ? x = dec 0 + x +x ? y = dec x + +0 ? x = dec 0 + x +x ? y = dec y + +0 ? x = dec 0 + x +x ? y = dec 0 + +0 ? x = dec (dec (dec x)) +x ? y = dec x + +0 ? x = dec (dec (dec x)) +x ? y = dec y + +0 ? x = dec (dec (dec x)) +x ? y = dec 0 + +0 ? x = dec (dec (dec 0)) +x ? y = dec x + +0 ? x = dec (dec (dec 0)) +x ? y = dec y + +0 ? x = dec (dec (dec 0)) +x ? y = dec 0 + +0 ? x = x + dec x +x ? y = dec x + +0 ? x = x + dec x +x ? y = dec y + +0 ? x = x + dec x +x ? y = dec 0 + +0 ? x = x + dec 0 +x ? y = dec x + +0 ? x = x + dec 0 +x ? y = dec y + +0 ? x = x + dec 0 +x ? y = dec 0 + +0 ? x = dec (dec x) + x +x ? y = x + +0 ? x = dec (dec x) + x +x ? y = y + +0 ? x = dec (dec x) + x +x ? y = 0 + +0 ? x = dec (dec 0) + x +x ? y = x + +0 ? x = dec (dec 0) + x +x ? y = y + +0 ? x = dec (dec 0) + x +x ? y = 0 + +0 ? x = dec (dec (dec (dec x))) +x ? y = x + +0 ? x = dec (dec (dec (dec x))) +x ? y = y + +0 ? x = dec (dec (dec (dec x))) +x ? y = 0 + +0 ? x = dec (dec (dec (dec 0))) +x ? y = x + +0 ? x = dec (dec (dec (dec 0))) +x ? y = y + +0 ? x = dec (dec (dec (dec 0))) +x ? y = 0 + +0 ? x = x + (x + x) +x ? y = x + +0 ? x = x + (x + x) +x ? y = y + +0 ? x = x + (x + x) +x ? y = 0 + +0 ? x = x + dec (dec x) +x ? y = x + +0 ? x = x + dec (dec x) +x ? y = y + +0 ? x = x + dec (dec x) +x ? y = 0 + +0 ? x = x + dec (dec 0) +x ? y = x + +0 ? x = x + dec (dec 0) +x ? y = y + +0 ? x = x + dec (dec 0) +x ? y = 0 + +0 ? x = dec x + dec x +x ? y = x + +0 ? x = dec x + dec x +x ? y = y + +0 ? x = dec x + dec x +x ? y = 0 + +0 ? x = dec x + dec 0 +x ? y = x + +0 ? x = dec x + dec 0 +x ? y = y + +0 ? x = dec x + dec 0 +x ? y = 0 + +0 ? x = dec 0 + dec x +x ? y = x + +0 ? x = dec 0 + dec x +x ? y = y + +0 ? x = dec 0 + dec x +x ? y = 0 + +0 ? x = dec 0 + dec 0 +x ? y = x + +0 ? x = dec 0 + dec 0 +x ? y = y + +0 ? x = dec 0 + dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec x + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec x + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec y + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec y + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec 0 + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec 0 + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec (dec (dec x)) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec (dec (dec y)) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = dec (dec (dec 0)) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = x + dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = x + dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = x + dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = y + dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = y + dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x +x ? y = y + dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec x + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec x + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec y + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec y + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec 0 + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec 0 + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec (dec x)) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec (dec y)) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec (dec 0)) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = x + dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = x + dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = x + dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = y + dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = y + dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = 0 +x ? y = y + dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = x + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = x + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = y + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = y + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = x + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = x + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = y + x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = y + y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec x) +x ? y = dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec x) +x ? y = dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec x) +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec 0) +x ? y = dec x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec 0) +x ? y = dec y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec 0) +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x + x +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x + x +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec x + x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 + x +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 + x +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec 0 + x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec (dec x)) +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec (dec x)) +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec (dec x)) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec (dec 0)) +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec (dec 0)) +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = dec (dec (dec 0)) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + dec x +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + dec x +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + dec x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + dec 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + dec 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = x +x ? 0 = x + dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec x + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec x + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec y + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec y + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec 0 + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec 0 + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec (dec x)) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec (dec y)) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec (dec 0)) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = x + dec x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = x + dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = x + dec 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = y + dec x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = y + dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x +x ? y = y + dec 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec y + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec y + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec 0 + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec 0 + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec (dec x)) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec (dec y)) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec (dec 0)) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = x + dec x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = x + dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = x + dec 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = y + dec x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = y + dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = y + dec 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = x + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = x + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = y + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = y + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = x + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = x + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = y + x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = y + y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = dec x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec 0) +x ? y = dec x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec 0) +x ? y = dec y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec 0) +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x + x +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x + x +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec x + x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 + x +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 + x +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec 0 + x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec (dec x)) +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec (dec x)) +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec (dec x)) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec (dec 0)) +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec (dec 0)) +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = dec (dec (dec 0)) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + dec x +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + dec x +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + dec x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + dec 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + dec 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = 0 +x ? 0 = x + dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = x + x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = x + y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = y + x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = y + y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = x + x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = x + y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = y + x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = y + y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec 0 +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec 0 +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x + x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x + x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = x + x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec (dec x) +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec (dec x) +x ? y = y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec (dec x) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec (dec 0) +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec (dec 0) +x ? y = y + +0 ? 0 = 0 +0 ? x = dec x +x ? 0 = dec (dec 0) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = x + x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = x + y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = y + x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = y + y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = x + x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = x + y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = y + x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = y + y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = dec (dec x) + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = dec (dec y) + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = dec (dec 0) + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec 0 +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec 0 +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x + x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x + x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = x + x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec (dec x) +x ? y = x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec (dec x) +x ? y = y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec (dec x) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec (dec 0) +x ? y = x + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec (dec 0) +x ? y = y + +0 ? 0 = 0 +0 ? x = dec 0 +x ? 0 = dec (dec 0) +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = dec x +x ? y = x + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = dec x +x ? y = y + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = x + x +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = dec x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = dec x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec x) +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = x +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = x +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = dec x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = dec x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec 0) +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec x + x +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x + x +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec x + x +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec x + x +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec x + x +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec x + x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec 0 + x +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec 0 + x +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec 0 + x +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec 0 + x +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec 0 + x +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec 0 + x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec (dec x)) +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec (dec x)) +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec (dec x)) +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec (dec x)) +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec (dec x)) +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec (dec x)) +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec (dec 0)) +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec (dec 0)) +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec (dec 0)) +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = dec (dec (dec 0)) +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = dec (dec (dec 0)) +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = dec (dec (dec 0)) +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x + dec x +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = x + dec x +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = x + dec x +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x + dec x +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = x + dec x +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = x + dec x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x + dec 0 +x ? 0 = x +x ? y = x + +0 ? 0 = 0 +0 ? x = x + dec 0 +x ? 0 = x +x ? y = y + +0 ? 0 = 0 +0 ? x = x + dec 0 +x ? 0 = x +x ? y = 0 + +0 ? 0 = 0 +0 ? x = x + dec 0 +x ? 0 = 0 +x ? y = x + +0 ? 0 = 0 +0 ? x = x + dec 0 +x ? 0 = 0 +x ? y = y + +0 ? 0 = 0 +0 ? x = x + dec 0 +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = x + x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = x + y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = y + x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = y + y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = dec (dec x) + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = dec (dec y) + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x +x ? y = dec (dec 0) + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = x + x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = x + y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = y + x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = y + y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec x) + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec y) + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = 0 +x ? y = dec (dec 0) + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec x +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec x +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec x +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec 0 +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x + x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x + x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = x + x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec (dec x) +x ? y = x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec (dec x) +x ? y = y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec (dec x) +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec (dec 0) +x ? y = x + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec (dec 0) +x ? y = y + +0 ? 0 = dec 0 +0 ? x = x +x ? 0 = dec (dec 0) +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = x + x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = x + y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = y + x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = y + y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec x) + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec y) + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x +x ? y = dec (dec 0) + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = x + x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = x + y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = y + x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = y + y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec x) + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec y) + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = 0 +x ? y = dec (dec 0) + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec x +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec 0 +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x + x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x + x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = x + x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec (dec x) +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec (dec 0) +x ? y = x + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec (dec 0) +x ? y = y + +0 ? 0 = dec 0 +0 ? x = 0 +x ? 0 = dec (dec 0) +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = x +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = x +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = dec x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = dec x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec x +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = dec x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = dec x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec 0 +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = x + x +x ? 0 = x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = x + x +x ? 0 = x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = x + x +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = x + x +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = x + x +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec 0 +0 ? x = x + x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec (dec x) +x ? 0 = x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec (dec x) +x ? 0 = x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec (dec x) +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec (dec x) +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec (dec 0) +x ? 0 = x +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec (dec 0) +x ? 0 = x +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec (dec 0) +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec 0 +0 ? x = dec (dec 0) +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec 0 +0 ? x = dec (dec 0) +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec 0 +0 ? x = dec (dec 0) +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = x +x ? y = dec x + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = x +x ? y = dec y + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = dec x +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = dec x +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = x +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = x +x ? y = dec x + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = x +x ? y = dec y + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = x +x ? y = dec 0 + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = 0 +x ? y = dec x + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = 0 +x ? y = dec y + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = 0 +x ? y = dec 0 + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = dec x +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = dec x +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = dec x +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = dec 0 +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = dec 0 +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = 0 +x ? 0 = dec 0 +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = dec x +x ? 0 = x +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = dec x +x ? 0 = x +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = dec x +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = dec x +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = dec x +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = dec x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = dec 0 +x ? 0 = x +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = dec 0 +x ? 0 = x +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = dec 0 +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec (dec 0) +0 ? x = dec 0 +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec (dec 0) +0 ? x = dec 0 +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec (dec 0) +0 ? x = dec 0 +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec (dec (dec 0)) +0 ? x = x +x ? 0 = x +x ? y = x + +0 ? 0 = dec (dec (dec 0)) +0 ? x = x +x ? 0 = x +x ? y = y + +0 ? 0 = dec (dec (dec 0)) +0 ? x = x +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec (dec (dec 0)) +0 ? x = x +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec (dec (dec 0)) +0 ? x = x +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec (dec (dec 0)) +0 ? x = x +x ? 0 = 0 +x ? y = 0 + +0 ? 0 = dec (dec (dec 0)) +0 ? x = 0 +x ? 0 = x +x ? y = x + +0 ? 0 = dec (dec (dec 0)) +0 ? x = 0 +x ? 0 = x +x ? y = y + +0 ? 0 = dec (dec (dec 0)) +0 ? x = 0 +x ? 0 = x +x ? y = 0 + +0 ? 0 = dec (dec (dec 0)) +0 ? x = 0 +x ? 0 = 0 +x ? y = x + +0 ? 0 = dec (dec (dec 0)) +0 ? x = 0 +x ? 0 = 0 +x ? y = y + +0 ? 0 = dec (dec (dec 0)) +0 ? x = 0 +x ? 0 = 0 +x ? y = 0 + + +Candidates for: goo :: [Int] -> [Int] + pruning with 4/4 rules + [2,0,1,0,1,0,1,0,1] direct candidates, 0 duplicates + [2,1,2,3,4,7,10,17,26] pattern candidates, 0 duplicates + +rules: +xs ++ [] == xs +[] ++ xs == xs +(xs ++ ys) ++ zs == xs ++ (ys ++ zs) +(x:xs) ++ ys == x:(xs ++ ys) + +direct candidates: + +goo xs = xs + +goo xs = [] + +goo xs = xs ++ xs + +goo xs = xs ++ (xs ++ xs) + + +pattern candidates: + +goo xs = xs + +goo xs = [] + +goo [] = [] +goo (x:xs) = xs + +goo [] = [] +goo (x:xs) = goo xs + +goo xs = xs ++ xs + +goo [] = [] +goo (x:xs) = x:xs + +goo [] = [] +goo (x:xs) = [x] + +goo [] = [] +goo (x:xs) = xs ++ xs + +goo [] = [] +goo (x:xs) = x:goo xs + +goo [] = [] +goo (x:xs) = xs ++ goo xs + +goo [] = [] +goo (x:xs) = goo xs ++ xs + +goo xs = xs ++ (xs ++ xs) + +goo [] = [] +goo (x:xs) = goo xs ++ goo xs + +goo [] = [] +goo (x:xs) = x:x:xs + +goo [] = [] +goo (x:xs) = [x,x] + +goo [] = [] +goo (x:xs) = x:(xs ++ xs) + +goo [] = [] +goo (x:xs) = xs ++ (x:xs) + +goo [] = [] +goo (x:xs) = xs ++ [x] + +goo [] = [] +goo (x:xs) = xs ++ (xs ++ xs) + + +Candidates for: ?? :: [Int] -> [Int] -> [Int] + pruning with 4/4 rules + [3,0,4,0,8,0,16,0,32] direct candidates, 0 duplicates + [3,7,13,66,152,361,1394,2084,11820] pattern candidates, 0 duplicates + +rules: +xs ++ [] == xs +[] ++ xs == xs +(xs ++ ys) ++ zs == xs ++ (ys ++ zs) +(x:xs) ++ ys == x:(xs ++ ys) + +direct candidates: + +xs ?? ys = xs + +xs ?? ys = ys + +xs ?? ys = [] + +xs ?? ys = xs ++ xs + +xs ?? ys = xs ++ ys + +xs ?? ys = ys ++ xs + +xs ?? ys = ys ++ ys + +xs ?? ys = xs ++ (xs ++ xs) + +xs ?? ys = xs ++ (xs ++ ys) + +xs ?? ys = xs ++ (ys ++ xs) + +xs ?? ys = xs ++ (ys ++ ys) + +xs ?? ys = ys ++ (xs ++ xs) + +xs ?? ys = ys ++ (xs ++ ys) + +xs ?? ys = ys ++ (ys ++ xs) + +xs ?? ys = ys ++ (ys ++ ys) + + +pattern candidates: + +xs ?? ys = xs + +xs ?? ys = ys + +xs ?? ys = [] + +xs ?? [] = xs +xs ?? (x:ys) = ys + +xs ?? [] = xs +xs ?? (x:ys) = [] + +xs ?? [] = [] +xs ?? (x:ys) = xs + +xs ?? [] = [] +xs ?? (x:ys) = ys + +[] ?? xs = xs +(x:xs) ?? ys = [] + +[] ?? xs = [] +(x:xs) ?? ys = xs + +[] ?? xs = [] +(x:xs) ?? ys = ys + +xs ?? ys = xs ++ xs + +xs ?? ys = xs ++ ys + +xs ?? ys = ys ++ xs + +xs ?? ys = ys ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = x:xs + +xs ?? [] = xs +xs ?? (x:ys) = x:ys + +xs ?? [] = xs +xs ?? (x:ys) = [x] + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = x:xs + +xs ?? [] = [] +xs ?? (x:ys) = x:ys + +xs ?? [] = [] +xs ?? (x:ys) = [x] + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = [] + +[] ?? xs = xs +(x:xs) ?? ys = x:xs + +[] ?? xs = xs +(x:xs) ?? ys = x:ys + +[] ?? xs = xs +(x:xs) ?? ys = [x] + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = x:xs + +[] ?? xs = [] +(x:xs) ?? ys = x:ys + +[] ?? xs = [] +(x:xs) ?? ys = [x] + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = xs + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] ?? xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] ?? xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] ?? xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs ?? [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] ?? xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] ?? xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] ?? xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs ?? xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs ?? [] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] ?? xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ?? [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] ?? xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] ?? ys + +xs ?? ys = xs ++ (xs ++ xs) + +xs ?? ys = xs ++ (xs ++ ys) + +xs ?? ys = xs ++ (ys ++ xs) + +xs ?? ys = xs ++ (ys ++ ys) + +xs ?? ys = ys ++ (xs ++ xs) + +xs ?? ys = ys ++ (xs ++ ys) + +xs ?? ys = ys ++ (ys ++ xs) + +xs ?? ys = ys ++ (ys ++ ys) + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = x:xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = x:ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [x] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = y:xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = y:ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [y] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = x:xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = x:ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [x] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = y:xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = y:ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [y] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = x:xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = x:ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [x] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = y:xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = y:ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [y] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = x:xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = x:ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [x] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = y:xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = y:ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [y] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ++ xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys ++ ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = x:xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = [x] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [] +(x:xs) ?? [] = xs ++ xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = x:xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = [x] +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? [] = xs +(x:xs) ?? (y:ys) = [] + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = xs + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = ys + +[] ?? [] = [] +[] ?? (x:xs) = xs ++ xs +(x:xs) ?? [] = [] +(x:xs) ?? (y:ys) = [] + +xs ?? [] = xs +xs ?? (x:ys) = (x:xs) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (x:xs) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (x:xs) ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = (x:ys) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (x:ys) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (x:ys) ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = [x] ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = [x] ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = [x] ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = (xs ++ xs) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (xs ++ xs) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (xs ++ xs) ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = (xs ++ ys) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (xs ++ ys) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (xs ++ ys) ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = (ys ++ xs) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (ys ++ xs) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (ys ++ xs) ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = (ys ++ ys) ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = (ys ++ ys) ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = (ys ++ ys) ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = (x:xs) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (x:xs) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (x:xs) ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = (x:ys) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (x:ys) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (x:ys) ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = [x] ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = [x] ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = [x] ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = (xs ++ xs) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (xs ++ xs) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (xs ++ xs) ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = (xs ++ ys) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (xs ++ ys) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (xs ++ ys) ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = (ys ++ xs) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (ys ++ xs) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (ys ++ xs) ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = (ys ++ ys) ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = (ys ++ ys) ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = (ys ++ ys) ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (x:xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (x:ys) + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? [x] + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? (ys ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (x:xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (x:ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? [x] + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? (ys ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? (x:xs) + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? (x:ys) + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? [x] + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? (xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? (xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? (ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? (ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (x:xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (x:ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? [x] + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? (ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (x:xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (x:ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? [x] + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? (ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? (x:xs) + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? (x:ys) + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? [x] + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? (xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? (xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? (ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? (ys ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = x:xs ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = x:xs ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = x:xs ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = x:ys ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = x:ys ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = x:ys ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = [x,] ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = [x,] ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ xs ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ xs ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ xs ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ ys ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ ys ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ ys ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ [] ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ [] ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ xs ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ xs ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ xs ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ ys ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ ys ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ ys ?? [] + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ [] ?? xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ [] ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? xs ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? ys ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? [] ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? xs ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? ys ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? [] ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? xs ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? ys ++ xs + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? xs ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? ys ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = xs ?? [] ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? xs ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? ys ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = ys ?? [] ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? xs ++ ys + +xs ?? [] = xs +xs ?? (x:ys) = [] ?? ys ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = x:xs ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = x:xs ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = x:xs ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = x:ys ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = x:ys ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = x:ys ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = [x,] ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = [x,] ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ xs ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ xs ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ xs ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ ys ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ ys ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ ys ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ [] ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ [] ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ xs ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ xs ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ xs ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ ys ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ ys ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ ys ?? [] + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ [] ?? xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ [] ?? ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? xs ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? ys ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? [] ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? xs ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? ys ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? [] ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? xs ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? ys ++ xs + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? xs ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? ys ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = xs ?? [] ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? xs ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? ys ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = ys ?? [] ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? xs ++ ys + +xs ?? [] = [] +xs ?? (x:ys) = [] ?? ys ++ ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = xs ?? xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = xs ?? ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = xs ?? [] + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys ?? xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys ?? ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys ?? [] + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = [] ?? xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = [] ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = x:xs ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = x:xs ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = x:xs ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = x:ys ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = x:ys ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = x:ys ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = [x,] ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = [x,] ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ xs ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ xs ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ xs ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ ys ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ ys ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ ys ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ [] ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ [] ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ xs ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ xs ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ xs ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ ys ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ ys ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ ys ?? [] + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ [] ?? xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ [] ?? ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? xs ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? ys ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? [] ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? xs ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? ys ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? [] ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? xs ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? ys ++ xs + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? xs ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? ys ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = xs ?? [] ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? xs ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? ys ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = ys ?? [] ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? xs ++ ys + +[] ?? xs = xs +(x:xs) ?? ys = [] ?? ys ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = x:xs ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = x:xs ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = x:xs ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = x:ys ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = x:ys ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = x:ys ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = [x,] ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = [x,] ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ xs ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ xs ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ xs ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ ys ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ ys ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ ys ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ [] ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ [] ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ xs ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ xs ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ xs ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ ys ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ ys ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ ys ?? [] + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ [] ?? xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ [] ?? ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? xs ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? ys ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? [] ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? xs ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? ys ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? [] ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? xs ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? ys ++ xs + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? xs ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? ys ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = xs ?? [] ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? xs ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? ys ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = ys ?? [] ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? xs ++ ys + +[] ?? xs = [] +(x:xs) ?? ys = [] ?? ys ++ ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = xs ?? xs + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = xs ?? ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = xs ?? [] + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = ys ?? xs + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = ys ?? ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = ys ?? [] + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = [] ?? xs + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = [] ?? ys + +xs ?? [] = xs +xs ?? (x:ys) = x:x:xs + +xs ?? [] = xs +xs ?? (x:ys) = x:x:ys + +xs ?? [] = xs +xs ?? (x:ys) = [x,x] + +xs ?? [] = xs +xs ?? (x:ys) = x:(xs ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = x:(xs ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = x:(ys ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = x:(ys ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (x:xs) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (x:ys) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ [x] + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (xs ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (xs ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (ys ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = xs ++ (ys ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (x:xs) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (x:ys) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ [x] + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (xs ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (xs ++ ys) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (ys ++ xs) + +xs ?? [] = xs +xs ?? (x:ys) = ys ++ (ys ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = x:x:xs + +xs ?? [] = [] +xs ?? (x:ys) = x:x:ys + +xs ?? [] = [] +xs ?? (x:ys) = [x,x] + +xs ?? [] = [] +xs ?? (x:ys) = x:(xs ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = x:(xs ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = x:(ys ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = x:(ys ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (x:xs) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (x:ys) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ [x] + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (xs ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (xs ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (ys ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = xs ++ (ys ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (x:xs) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (x:ys) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ [x] + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (xs ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (xs ++ ys) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (ys ++ xs) + +xs ?? [] = [] +xs ?? (x:ys) = ys ++ (ys ++ ys) + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = x:xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = x:ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = [x] + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = xs ++ ys + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys ++ xs + +xs ?? [] = xs ++ xs +xs ?? (x:ys) = ys ++ ys + +xs ?? [] = xs ++ (xs ++ xs) +xs ?? (x:ys) = xs + +xs ?? [] = xs ++ (xs ++ xs) +xs ?? (x:ys) = ys + +xs ?? [] = xs ++ (xs ++ xs) +xs ?? (x:ys) = [] + +[] ?? xs = xs +(x:xs) ?? ys = x:x:xs + +[] ?? xs = xs +(x:xs) ?? ys = x:x:ys + +[] ?? xs = xs +(x:xs) ?? ys = [x,x] + +[] ?? xs = xs +(x:xs) ?? ys = x:(xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = x:(xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = x:(ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = x:(ys ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (x:xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (x:ys) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ [x] + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = xs ++ (ys ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (x:xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (x:ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ [x] + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (xs ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (xs ++ ys) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (ys ++ xs) + +[] ?? xs = xs +(x:xs) ?? ys = ys ++ (ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = x:x:xs + +[] ?? xs = [] +(x:xs) ?? ys = x:x:ys + +[] ?? xs = [] +(x:xs) ?? ys = [x,x] + +[] ?? xs = [] +(x:xs) ?? ys = x:(xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = x:(xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = x:(ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = x:(ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (x:xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (x:ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ [x] + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = xs ++ (ys ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (x:xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (x:ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ [x] + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (xs ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (xs ++ ys) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (ys ++ xs) + +[] ?? xs = [] +(x:xs) ?? ys = ys ++ (ys ++ ys) + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = x:xs + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = x:ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = [x] + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = xs ++ ys + +[] ?? xs = xs ++ xs +(x:xs) ?? ys = ys ++ xs + +[] ?? xs = xs ++ (xs ++ xs) +(x:xs) ?? ys = xs + +[] ?? xs = xs ++ (xs ++ xs) +(x:xs) ?? ys = ys + +[] ?? xs = xs ++ (xs ++ xs) +(x:xs) ?? ys = [] + + +Candidates for: ton :: Bool -> Bool + pruning with 39/49 rules + [3,1,0,2,6,14,38,68,140] direct candidates, 0 duplicates + [3,3,0,0,0,0,0,0,0] pattern candidates, 0 duplicates + +rules: +not False == True +not True == False +p && p == p +p || p == p +not (not p) == p +p && False == False +p && True == p +False && p == False +True && p == p +p || False == p +p || True == True +False || p == p +True || p == True +not (p && q) == not p || not q +not (p && q) == not q || not p +not (p || q) == not p && not q +not (p || q) == not q && not p +p && not p == False +not p && p == False +p || not p == True +not p || p == True +(p && q) && r == p && (q && r) +(p && q) && r == q && (p && r) +(p || q) || r == p || (q || r) +(p || q) || r == q || (p || r) +p && (p && q) == p && q +p && (q && p) == p && q +p && (q && p) == q && p +p || (p || q) == p || q +p || (q || p) == p || q +p || (q || p) == q || p +p && (p || q) == p +p && (q || p) == p +(p || q) && p == p +(p || q) && q == q +p || p && q == p +p || q && p == p +p && q || p == p +p && q || q == q +equations: +q && p == p && q +q || p == p || q +q && (p && r) == p && (q && r) +r && (p && q) == p && (q && r) +r && (q && p) == p && (q && r) +q || (p || r) == p || (q || r) +r || (p || q) == p || (q || r) +r || (q || p) == p || (q || r) +(r || q) && p == p && (q || r) +r && q || p == p || q && r + +direct candidates: + +ton p = p + +ton p = False + +ton p = True + +ton p = not p + +ton p = p && ton p + +ton p = p || ton p + +ton p = p && ton (not p) + +ton p = p || ton (not p) + +ton p = p && not (ton p) + +ton p = not p && ton p + +ton p = p || not (ton p) + +ton p = not p || ton p + +ton p = p && not (ton (not p)) + +ton p = not p && ton (not p) + +ton p = p || not (ton (not p)) + +ton p = not p || ton (not p) + +ton p = p && (p && ton p) + +ton p = p && (ton p && p) + +ton p = p && (p || ton p) + +ton p = p && (ton p || p) + +ton p = not p && not (ton p) + +ton p = p || p && ton p + +ton p = p || ton p && p + +ton p = p || (p || ton p) + +ton p = p || (ton p || p) + +ton p = not p || not (ton p) + + +pattern candidates: + +ton p = p + +ton p = False + +ton p = True + +ton p = not p + +ton False = False +ton True = True + +ton False = True +ton True = False + + +Candidates for: &| :: Bool -> Bool -> Bool + pruning with 39/49 rules + [4,2,4,8,32,128,468,1576,4924] direct candidates, 0 duplicates + [4,14,26,8,4,48,60,280,592] pattern candidates, 0 duplicates + +rules: +not False == True +not True == False +p && p == p +p || p == p +not (not p) == p +p && False == False +p && True == p +False && p == False +True && p == p +p || False == p +p || True == True +False || p == p +True || p == True +not (p && q) == not p || not q +not (p && q) == not q || not p +not (p || q) == not p && not q +not (p || q) == not q && not p +p && not p == False +not p && p == False +p || not p == True +not p || p == True +(p && q) && r == p && (q && r) +(p && q) && r == q && (p && r) +(p || q) || r == p || (q || r) +(p || q) || r == q || (p || r) +p && (p && q) == p && q +p && (q && p) == p && q +p && (q && p) == q && p +p || (p || q) == p || q +p || (q || p) == p || q +p || (q || p) == q || p +p && (p || q) == p +p && (q || p) == p +(p || q) && p == p +(p || q) && q == q +p || p && q == p +p || q && p == p +p && q || p == p +p && q || q == q +equations: +q && p == p && q +q || p == p || q +q && (p && r) == p && (q && r) +r && (p && q) == p && (q && r) +r && (q && p) == p && (q && r) +q || (p || r) == p || (q || r) +r || (p || q) == p || (q || r) +r || (q || p) == p || (q || r) +(r || q) && p == p && (q || r) +r && q || p == p || q && r + +direct candidates: + +p &| q = p + +p &| q = q + +p &| q = False + +p &| q = True + +p &| q = not p + +p &| q = not q + +p &| q = p && q + +p &| q = q && p + +p &| q = p || q + +p &| q = q || p + +p &| q = not p && q + +p &| q = not q && p + +p &| q = not p || q + +p &| q = not q || p + +p &| q = p && not q + +p &| q = q && not p + +p &| q = p || not q + +p &| q = q || not p + +p &| q = not p && not q + +p &| q = not q && not p + +p &| q = not p || not q + +p &| q = not q || not p + +p &| q = p && p &| p + +p &| q = p && p &| q + +p &| q = p && p &| False + +p &| q = p && p &| True + +p &| q = p && q &| q + +p &| q = p && False &| q + +p &| q = p && True &| q + +p &| q = q && p &| p + +p &| q = q && p &| q + +p &| q = q && p &| False + +p &| q = q && p &| True + +p &| q = q && q &| q + +p &| q = q && False &| q + +p &| q = q && True &| q + +p &| q = p || p &| p + +p &| q = p || p &| q + +p &| q = p || p &| False + +p &| q = p || p &| True + +p &| q = p || q &| q + +p &| q = p || False &| q + +p &| q = p || True &| q + +p &| q = q || p &| p + +p &| q = q || p &| q + +p &| q = q || p &| False + +p &| q = q || p &| True + +p &| q = q || q &| q + +p &| q = q || False &| q + +p &| q = q || True &| q + +p &| q = p && (not p && q) + +p &| q = p && (not p || q) + +p &| q = p && (q && not p) + +p &| q = p && (q || not p) + +p &| q = q && (not q && p) + +p &| q = q && (not q || p) + +p &| q = q && (p && not q) + +p &| q = q && (p || not q) + +p &| q = p || not p && q + +p &| q = p || (not p || q) + +p &| q = p || q && not p + +p &| q = p || (q || not p) + +p &| q = q || not q && p + +p &| q = q || (not q || p) + +p &| q = q || p && not q + +p &| q = q || (p || not q) + +p &| q = not p && (p || q) + +p &| q = not p && (q || p) + +p &| q = not q && (p || q) + +p &| q = not q && (q || p) + +p &| q = not p || p && q + +p &| q = not p || q && p + +p &| q = not q || p && q + +p &| q = not q || q && p + +p &| q = (p || q) && not p + +p &| q = (p || q) && not q + +p &| q = (q || p) && not p + +p &| q = (q || p) && not q + +p &| q = p && q || not p + +p &| q = p && q || not q + +p &| q = q && p || not p + +p &| q = q && p || not q + +p &| q = p && p &| not p + +p &| q = p && p &| not q + +p &| q = p && q &| not q + +p &| q = p && False &| not q + +p &| q = p && True &| not q + +p &| q = p && not p &| p + +p &| q = p && not p &| q + +p &| q = p && not p &| False + +p &| q = p && not p &| True + +p &| q = p && not q &| q + +p &| q = q && p &| not p + +p &| q = q && p &| not q + +p &| q = q && q &| not q + +p &| q = q && False &| not q + +p &| q = q && True &| not q + +p &| q = q && not p &| p + +p &| q = q && not p &| q + +p &| q = q && not p &| False + +p &| q = q && not p &| True + +p &| q = q && not q &| q + +p &| q = p || p &| not p + +p &| q = p || p &| not q + +p &| q = p || q &| not q + +p &| q = p || False &| not q + +p &| q = p || True &| not q + +p &| q = p || not p &| p + +p &| q = p || not p &| q + +p &| q = p || not p &| False + +p &| q = p || not p &| True + +p &| q = p || not q &| q + +p &| q = q || p &| not p + +p &| q = q || p &| not q + +p &| q = q || q &| not q + +p &| q = q || False &| not q + +p &| q = q || True &| not q + +p &| q = q || not p &| p + +p &| q = q || not p &| q + +p &| q = q || not p &| False + +p &| q = q || not p &| True + +p &| q = q || not q &| q + +p &| q = p && not (p &| p) + +p &| q = p && not (p &| q) + +p &| q = p && not (p &| False) + +p &| q = p && not (p &| True) + +p &| q = p && not (q &| q) + +p &| q = p && not (False &| q) + +p &| q = p && not (True &| q) + +p &| q = q && not (p &| p) + +p &| q = q && not (p &| q) + +p &| q = q && not (p &| False) + +p &| q = q && not (p &| True) + +p &| q = q && not (q &| q) + +p &| q = q && not (False &| q) + +p &| q = q && not (True &| q) + +p &| q = not p && p &| p + +p &| q = not p && p &| q + +p &| q = not p && p &| False + +p &| q = not p && p &| True + +p &| q = not p && q &| q + +p &| q = not p && False &| q + +p &| q = not p && True &| q + +p &| q = not q && p &| p + +p &| q = not q && p &| q + +p &| q = not q && p &| False + +p &| q = not q && p &| True + +p &| q = not q && q &| q + +p &| q = not q && False &| q + +p &| q = not q && True &| q + +p &| q = p || not (p &| p) + +p &| q = p || not (p &| q) + +p &| q = p || not (p &| False) + +p &| q = p || not (p &| True) + +p &| q = p || not (q &| q) + +p &| q = p || not (False &| q) + +p &| q = p || not (True &| q) + +p &| q = q || not (p &| p) + +p &| q = q || not (p &| q) + +p &| q = q || not (p &| False) + +p &| q = q || not (p &| True) + +p &| q = q || not (q &| q) + +p &| q = q || not (False &| q) + +p &| q = q || not (True &| q) + +p &| q = not p || p &| p + +p &| q = not p || p &| q + +p &| q = not p || p &| False + +p &| q = not p || p &| True + +p &| q = not p || q &| q + +p &| q = not p || False &| q + +p &| q = not p || True &| q + +p &| q = not q || p &| p + +p &| q = not q || p &| q + +p &| q = not q || p &| False + +p &| q = not q || p &| True + +p &| q = not q || q &| q + +p &| q = not q || False &| q + +p &| q = not q || True &| q + + +pattern candidates: + +p &| q = p + +p &| q = q + +p &| q = False + +p &| q = True + +p &| q = not p + +p &| q = not q + +p &| False = p +p &| True = False + +p &| False = p +p &| True = True + +p &| False = False +p &| True = p + +p &| False = False +p &| True = True + +p &| False = True +p &| True = p + +p &| False = True +p &| True = False + +False &| p = p +True &| p = False + +False &| p = p +True &| p = True + +False &| p = False +True &| p = p + +False &| p = False +True &| p = True + +False &| p = True +True &| p = p + +False &| p = True +True &| p = False + +p &| q = p && q + +p &| q = q && p + +p &| q = p || q + +p &| q = q || p + +p &| False = p +p &| True = not p + +p &| False = False +p &| True = not p + +p &| False = True +p &| True = not p + +p &| False = not p +p &| True = p + +p &| False = not p +p &| True = False + +p &| False = not p +p &| True = True + +False &| p = p +True &| p = not p + +False &| p = False +True &| p = not p + +False &| p = True +True &| p = not p + +False &| p = not p +True &| p = p + +False &| p = not p +True &| p = False + +False &| p = not p +True &| p = True + +False &| False = False +False &| True = False +True &| False = False +True &| True = True + +False &| False = False +False &| True = False +True &| False = True +True &| True = False + +False &| False = False +False &| True = True +True &| False = False +True &| True = False + +False &| False = False +False &| True = True +True &| False = True +True &| True = False + +False &| False = False +False &| True = True +True &| False = True +True &| True = True + +False &| False = True +False &| True = False +True &| False = False +True &| True = False + +False &| False = True +False &| True = False +True &| False = False +True &| True = True + +False &| False = True +False &| True = False +True &| False = True +True &| True = True + +False &| False = True +False &| True = True +True &| False = False +True &| True = True + +False &| False = True +False &| True = True +True &| False = True +True &| True = False + +p &| q = not p && q + +p &| q = not q && p + +p &| q = not p || q + +p &| q = not q || p + +p &| q = p && not q + +p &| q = q && not p + +p &| q = p || not q + +p &| q = q || not p + +p &| q = not p && not q + +p &| q = not q && not p + +p &| q = not p || not q + +p &| q = not q || not p + +p &| q = (not p || q) && p + +p &| q = (not q || p) && q + +p &| q = (p || not q) && q + +p &| q = (q || not p) && p + +p &| q = not p && q || p + +p &| q = not q && p || q + +p &| q = p && not q || q + +p &| q = q && not p || p + +p &| q = p && (not p && q) + +p &| q = p && (not p || q) + +p &| q = p && (q && not p) + +p &| q = p && (q || not p) + +p &| q = q && (not q && p) + +p &| q = q && (not q || p) + +p &| q = q && (p && not q) + +p &| q = q && (p || not q) + +p &| q = p || not p && q + +p &| q = p || (not p || q) + +p &| q = p || q && not p + +p &| q = p || (q || not p) + +p &| q = q || not q && p + +p &| q = q || (not q || p) + +p &| q = q || p && not q + +p &| q = q || (p || not q) + +p &| q = not p && (p && q) + +p &| q = not p && (q && p) + +p &| q = not p && (p || q) + +p &| q = not p && (q || p) + +p &| q = not q && (p && q) + +p &| q = not q && (q && p) + +p &| q = not q && (p || q) + +p &| q = not q && (q || p) + +p &| q = not p || p && q + +p &| q = not p || q && p + +p &| q = not p || (p || q) + +p &| q = not p || (q || p) + +p &| q = not q || p && q + +p &| q = not q || q && p + +p &| q = not q || (p || q) + +p &| q = not q || (q || p) + +p &| q = (p || q) && not p + +p &| q = (p || q) && not q + +p &| q = (q || p) && not p + +p &| q = (q || p) && not q + +p &| q = p && q || not p + +p &| q = p && q || not q + +p &| q = q && p || not p + +p &| q = q && p || not q + + +Candidates for: gcd :: Int -> Int -> Int + pruning with 0/0 rules + [3,0,9,0,54,0,405,0,3402] direct candidates, 0 duplicates + [3,7,18,60,150,469,1245,3812,10634] pattern candidates, 0 duplicates + +no rules. + +direct candidates: + +gcd x y = x + +gcd x y = y + +gcd x y = 0 + +gcd x y = x `mod` x + +gcd x y = x `mod` y + +gcd x y = x `mod` 0 + +gcd x y = y `mod` x + +gcd x y = y `mod` y + +gcd x y = y `mod` 0 + +gcd x y = 0 `mod` x + +gcd x y = 0 `mod` y + +gcd x y = 0 `mod` 0 + +gcd x y = (x `mod` x) `mod` x + +gcd x y = (x `mod` x) `mod` y + +gcd x y = (x `mod` x) `mod` 0 + +gcd x y = (x `mod` y) `mod` x + +gcd x y = (x `mod` y) `mod` y + +gcd x y = (x `mod` y) `mod` 0 + +gcd x y = (x `mod` 0) `mod` x + +gcd x y = (x `mod` 0) `mod` y + +gcd x y = (x `mod` 0) `mod` 0 + +gcd x y = (y `mod` x) `mod` x + +gcd x y = (y `mod` x) `mod` y + +gcd x y = (y `mod` x) `mod` 0 + +gcd x y = (y `mod` y) `mod` x + +gcd x y = (y `mod` y) `mod` y + +gcd x y = (y `mod` y) `mod` 0 + +gcd x y = (y `mod` 0) `mod` x + +gcd x y = (y `mod` 0) `mod` y + +gcd x y = (y `mod` 0) `mod` 0 + +gcd x y = (0 `mod` x) `mod` x + +gcd x y = (0 `mod` x) `mod` y + +gcd x y = (0 `mod` x) `mod` 0 + +gcd x y = (0 `mod` y) `mod` x + +gcd x y = (0 `mod` y) `mod` y + +gcd x y = (0 `mod` y) `mod` 0 + +gcd x y = (0 `mod` 0) `mod` x + +gcd x y = (0 `mod` 0) `mod` y + +gcd x y = (0 `mod` 0) `mod` 0 + +gcd x y = x `mod` (x `mod` x) + +gcd x y = x `mod` (x `mod` y) + +gcd x y = x `mod` (x `mod` 0) + +gcd x y = x `mod` (y `mod` x) + +gcd x y = x `mod` (y `mod` y) + +gcd x y = x `mod` (y `mod` 0) + +gcd x y = x `mod` (0 `mod` x) + +gcd x y = x `mod` (0 `mod` y) + +gcd x y = x `mod` (0 `mod` 0) + +gcd x y = y `mod` (x `mod` x) + +gcd x y = y `mod` (x `mod` y) + +gcd x y = y `mod` (x `mod` 0) + +gcd x y = y `mod` (y `mod` x) + +gcd x y = y `mod` (y `mod` y) + +gcd x y = y `mod` (y `mod` 0) + +gcd x y = y `mod` (0 `mod` x) + +gcd x y = y `mod` (0 `mod` y) + +gcd x y = y `mod` (0 `mod` 0) + +gcd x y = 0 `mod` (x `mod` x) + +gcd x y = 0 `mod` (x `mod` y) + +gcd x y = 0 `mod` (x `mod` 0) + +gcd x y = 0 `mod` (y `mod` x) + +gcd x y = 0 `mod` (y `mod` y) + +gcd x y = 0 `mod` (y `mod` 0) + +gcd x y = 0 `mod` (0 `mod` x) + +gcd x y = 0 `mod` (0 `mod` y) + +gcd x y = 0 `mod` (0 `mod` 0) + + +pattern candidates: + +gcd x y = x + +gcd x y = y + +gcd x y = 0 + +gcd x 0 = x +gcd x y = y + +gcd x 0 = x +gcd x y = 0 + +gcd x 0 = 0 +gcd x y = x + +gcd x 0 = 0 +gcd x y = y + +gcd 0 x = x +gcd x y = 0 + +gcd 0 x = 0 +gcd x y = x + +gcd 0 x = 0 +gcd x y = y + +gcd x y = x `mod` x + +gcd x y = x `mod` y + +gcd x y = x `mod` 0 + +gcd x y = y `mod` x + +gcd x y = y `mod` y + +gcd x y = y `mod` 0 + +gcd x y = 0 `mod` x + +gcd x y = 0 `mod` y + +gcd x y = 0 `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = y + +gcd x 0 = x +gcd x y = x `mod` x + +gcd x 0 = x +gcd x y = x `mod` y + +gcd x 0 = x +gcd x y = x `mod` 0 + +gcd x 0 = x +gcd x y = y `mod` x + +gcd x 0 = x +gcd x y = y `mod` y + +gcd x 0 = x +gcd x y = y `mod` 0 + +gcd x 0 = x +gcd x y = 0 `mod` x + +gcd x 0 = x +gcd x y = 0 `mod` y + +gcd x 0 = x +gcd x y = 0 `mod` 0 + +gcd x 0 = 0 +gcd x y = x `mod` x + +gcd x 0 = 0 +gcd x y = x `mod` y + +gcd x 0 = 0 +gcd x y = x `mod` 0 + +gcd x 0 = 0 +gcd x y = y `mod` x + +gcd x 0 = 0 +gcd x y = y `mod` y + +gcd x 0 = 0 +gcd x y = y `mod` 0 + +gcd x 0 = 0 +gcd x y = 0 `mod` x + +gcd x 0 = 0 +gcd x y = 0 `mod` y + +gcd x 0 = 0 +gcd x y = 0 `mod` 0 + +gcd x 0 = x `mod` x +gcd x y = x + +gcd x 0 = x `mod` x +gcd x y = y + +gcd x 0 = x `mod` x +gcd x y = 0 + +gcd x 0 = x `mod` 0 +gcd x y = x + +gcd x 0 = x `mod` 0 +gcd x y = y + +gcd x 0 = x `mod` 0 +gcd x y = 0 + +gcd x 0 = 0 `mod` x +gcd x y = x + +gcd x 0 = 0 `mod` x +gcd x y = y + +gcd x 0 = 0 `mod` x +gcd x y = 0 + +gcd x 0 = 0 `mod` 0 +gcd x y = x + +gcd x 0 = 0 `mod` 0 +gcd x y = y + +gcd x 0 = 0 `mod` 0 +gcd x y = 0 + +gcd 0 x = x +gcd x y = x `mod` x + +gcd 0 x = x +gcd x y = x `mod` y + +gcd 0 x = x +gcd x y = x `mod` 0 + +gcd 0 x = x +gcd x y = y `mod` x + +gcd 0 x = x +gcd x y = y `mod` y + +gcd 0 x = x +gcd x y = y `mod` 0 + +gcd 0 x = x +gcd x y = 0 `mod` x + +gcd 0 x = x +gcd x y = 0 `mod` y + +gcd 0 x = x +gcd x y = 0 `mod` 0 + +gcd 0 x = 0 +gcd x y = x `mod` x + +gcd 0 x = 0 +gcd x y = x `mod` y + +gcd 0 x = 0 +gcd x y = x `mod` 0 + +gcd 0 x = 0 +gcd x y = y `mod` x + +gcd 0 x = 0 +gcd x y = y `mod` y + +gcd 0 x = 0 +gcd x y = y `mod` 0 + +gcd 0 x = 0 +gcd x y = 0 `mod` x + +gcd 0 x = 0 +gcd x y = 0 `mod` y + +gcd 0 x = 0 +gcd x y = 0 `mod` 0 + +gcd 0 x = x `mod` x +gcd x y = x + +gcd 0 x = x `mod` x +gcd x y = y + +gcd 0 x = x `mod` x +gcd x y = 0 + +gcd 0 x = x `mod` 0 +gcd x y = x + +gcd 0 x = x `mod` 0 +gcd x y = y + +gcd 0 x = x `mod` 0 +gcd x y = 0 + +gcd 0 x = 0 `mod` x +gcd x y = x + +gcd 0 x = 0 `mod` x +gcd x y = y + +gcd 0 x = 0 `mod` x +gcd x y = 0 + +gcd 0 x = 0 `mod` 0 +gcd x y = x + +gcd 0 x = 0 `mod` 0 +gcd x y = y + +gcd 0 x = 0 `mod` 0 +gcd x y = 0 + +gcd x y = (x `mod` x) `mod` x + +gcd x y = (x `mod` x) `mod` y + +gcd x y = (x `mod` x) `mod` 0 + +gcd x y = (x `mod` y) `mod` x + +gcd x y = (x `mod` y) `mod` y + +gcd x y = (x `mod` y) `mod` 0 + +gcd x y = (x `mod` 0) `mod` x + +gcd x y = (x `mod` 0) `mod` y + +gcd x y = (x `mod` 0) `mod` 0 + +gcd x y = (y `mod` x) `mod` x + +gcd x y = (y `mod` x) `mod` y + +gcd x y = (y `mod` x) `mod` 0 + +gcd x y = (y `mod` y) `mod` x + +gcd x y = (y `mod` y) `mod` y + +gcd x y = (y `mod` y) `mod` 0 + +gcd x y = (y `mod` 0) `mod` x + +gcd x y = (y `mod` 0) `mod` y + +gcd x y = (y `mod` 0) `mod` 0 + +gcd x y = (0 `mod` x) `mod` x + +gcd x y = (0 `mod` x) `mod` y + +gcd x y = (0 `mod` x) `mod` 0 + +gcd x y = (0 `mod` y) `mod` x + +gcd x y = (0 `mod` y) `mod` y + +gcd x y = (0 `mod` y) `mod` 0 + +gcd x y = (0 `mod` 0) `mod` x + +gcd x y = (0 `mod` 0) `mod` y + +gcd x y = (0 `mod` 0) `mod` 0 + +gcd x y = x `mod` (x `mod` x) + +gcd x y = x `mod` (x `mod` y) + +gcd x y = x `mod` (x `mod` 0) + +gcd x y = x `mod` (y `mod` x) + +gcd x y = x `mod` (y `mod` y) + +gcd x y = x `mod` (y `mod` 0) + +gcd x y = x `mod` (0 `mod` x) + +gcd x y = x `mod` (0 `mod` y) + +gcd x y = x `mod` (0 `mod` 0) + +gcd x y = y `mod` (x `mod` x) + +gcd x y = y `mod` (x `mod` y) + +gcd x y = y `mod` (x `mod` 0) + +gcd x y = y `mod` (y `mod` x) + +gcd x y = y `mod` (y `mod` y) + +gcd x y = y `mod` (y `mod` 0) + +gcd x y = y `mod` (0 `mod` x) + +gcd x y = y `mod` (0 `mod` y) + +gcd x y = y `mod` (0 `mod` 0) + +gcd x y = 0 `mod` (x `mod` x) + +gcd x y = 0 `mod` (x `mod` y) + +gcd x y = 0 `mod` (x `mod` 0) + +gcd x y = 0 `mod` (y `mod` x) + +gcd x y = 0 `mod` (y `mod` y) + +gcd x y = 0 `mod` (y `mod` 0) + +gcd x y = 0 `mod` (0 `mod` x) + +gcd x y = 0 `mod` (0 `mod` y) + +gcd x y = 0 `mod` (0 `mod` 0) + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = x `mod` x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = x `mod` y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = x `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = y `mod` x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = y `mod` y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = y `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = 0 `mod` x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = 0 `mod` y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = 0 `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = x `mod` x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = x `mod` y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = x `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = y `mod` x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = y `mod` y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = y `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = 0 `mod` x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = 0 `mod` y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = 0 `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x `mod` x +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x `mod` x +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x `mod` x +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x `mod` 0 +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x `mod` 0 +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = x `mod` 0 +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 `mod` x +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 `mod` x +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 `mod` x +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 `mod` 0 +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 `mod` 0 +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = x +gcd x 0 = 0 `mod` 0 +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = x `mod` x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = x `mod` y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = x `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = y `mod` x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = y `mod` y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = y `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = 0 `mod` x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = 0 `mod` y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = 0 `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = x `mod` x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = x `mod` y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = x `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = y `mod` x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = y `mod` y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = y `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = 0 `mod` x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = 0 `mod` y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = 0 `mod` 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x `mod` x +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x `mod` x +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x `mod` x +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x `mod` 0 +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x `mod` 0 +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = x `mod` 0 +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 `mod` x +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 `mod` x +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 `mod` x +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 `mod` 0 +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 `mod` 0 +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = 0 +gcd x 0 = 0 `mod` 0 +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = x `mod` x +gcd x 0 = x +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = x `mod` x +gcd x 0 = x +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = x `mod` x +gcd x 0 = x +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = x `mod` x +gcd x 0 = 0 +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = x `mod` x +gcd x 0 = 0 +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = x `mod` x +gcd x 0 = 0 +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = x `mod` 0 +gcd x 0 = x +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = x `mod` 0 +gcd x 0 = x +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = x `mod` 0 +gcd x 0 = x +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = x `mod` 0 +gcd x 0 = 0 +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = x `mod` 0 +gcd x 0 = 0 +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = x `mod` 0 +gcd x 0 = 0 +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` x +gcd x 0 = x +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` x +gcd x 0 = x +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` x +gcd x 0 = x +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` x +gcd x 0 = 0 +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` x +gcd x 0 = 0 +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` x +gcd x 0 = 0 +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` 0 +gcd x 0 = x +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` 0 +gcd x 0 = x +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` 0 +gcd x 0 = x +gcd x y = 0 + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` 0 +gcd x 0 = 0 +gcd x y = x + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` 0 +gcd x 0 = 0 +gcd x y = y + +gcd 0 0 = 0 +gcd 0 x = 0 `mod` 0 +gcd x 0 = 0 +gcd x y = 0 + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = x + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = y + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = x +gcd x 0 = x +gcd x y = 0 + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = x + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = y + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = x +gcd x 0 = 0 +gcd x y = 0 + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = x + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = y + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = 0 +gcd x 0 = x +gcd x y = 0 + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = x + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = y + +gcd 0 0 = 0 `mod` 0 +gcd 0 x = 0 +gcd x 0 = 0 +gcd x y = 0 + +gcd x 0 = x +gcd x y = gcd x (x `mod` y) + +gcd x 0 = x +gcd x y = gcd x (y `mod` x) + +gcd x 0 = x +gcd x y = gcd x (y `mod` y) + +gcd x 0 = x +gcd x y = gcd x (0 `mod` y) + +gcd x 0 = x +gcd x y = gcd y (x `mod` y) + +gcd x 0 = x +gcd x y = gcd y (y `mod` x) + +gcd x 0 = x +gcd x y = gcd y (y `mod` y) + +gcd x 0 = x +gcd x y = gcd y (0 `mod` y) + +gcd x 0 = x +gcd x y = gcd 0 (x `mod` y) + +gcd x 0 = x +gcd x y = gcd 0 (y `mod` x) + +gcd x 0 = x +gcd x y = gcd 0 (y `mod` y) + +gcd x 0 = x +gcd x y = gcd 0 (0 `mod` y) + +gcd x 0 = x +gcd x y = gcd (x `mod` x) x + +gcd x 0 = x +gcd x y = gcd (x `mod` x) y + +gcd x 0 = x +gcd x y = gcd (x `mod` x) 0 + +gcd x 0 = x +gcd x y = gcd (x `mod` y) x + +gcd x 0 = x +gcd x y = gcd (x `mod` y) y + +gcd x 0 = x +gcd x y = gcd (x `mod` y) 0 + +gcd x 0 = x +gcd x y = gcd (y `mod` x) x + +gcd x 0 = x +gcd x y = gcd (y `mod` x) y + +gcd x 0 = x +gcd x y = gcd (y `mod` x) 0 + +gcd x 0 = x +gcd x y = gcd (0 `mod` x) x + +gcd x 0 = x +gcd x y = gcd (0 `mod` x) y + +gcd x 0 = x +gcd x y = gcd (0 `mod` x) 0 + +gcd x 0 = 0 +gcd x y = gcd x (x `mod` y) + +gcd x 0 = 0 +gcd x y = gcd x (y `mod` x) + +gcd x 0 = 0 +gcd x y = gcd x (y `mod` y) + +gcd x 0 = 0 +gcd x y = gcd x (0 `mod` y) + +gcd x 0 = 0 +gcd x y = gcd y (x `mod` y) + +gcd x 0 = 0 +gcd x y = gcd y (y `mod` x) + +gcd x 0 = 0 +gcd x y = gcd y (y `mod` y) + +gcd x 0 = 0 +gcd x y = gcd y (0 `mod` y) + +gcd x 0 = 0 +gcd x y = gcd 0 (x `mod` y) + +gcd x 0 = 0 +gcd x y = gcd 0 (y `mod` x) + +gcd x 0 = 0 +gcd x y = gcd 0 (y `mod` y) + +gcd x 0 = 0 +gcd x y = gcd 0 (0 `mod` y) + +gcd x 0 = 0 +gcd x y = gcd (x `mod` x) x + +gcd x 0 = 0 +gcd x y = gcd (x `mod` x) y + +gcd x 0 = 0 +gcd x y = gcd (x `mod` x) 0 + +gcd x 0 = 0 +gcd x y = gcd (x `mod` y) x + +gcd x 0 = 0 +gcd x y = gcd (x `mod` y) y + +gcd x 0 = 0 +gcd x y = gcd (x `mod` y) 0 + +gcd x 0 = 0 +gcd x y = gcd (y `mod` x) x + +gcd x 0 = 0 +gcd x y = gcd (y `mod` x) y + +gcd x 0 = 0 +gcd x y = gcd (y `mod` x) 0 + +gcd x 0 = 0 +gcd x y = gcd (0 `mod` x) x + +gcd x 0 = 0 +gcd x y = gcd (0 `mod` x) y + +gcd x 0 = 0 +gcd x y = gcd (0 `mod` x) 0 + +gcd 0 x = x +gcd x y = gcd x (x `mod` y) + +gcd 0 x = x +gcd x y = gcd x (y `mod` x) + +gcd 0 x = x +gcd x y = gcd x (y `mod` y) + +gcd 0 x = x +gcd x y = gcd x (0 `mod` y) + +gcd 0 x = x +gcd x y = gcd y (x `mod` y) + +gcd 0 x = x +gcd x y = gcd y (y `mod` x) + +gcd 0 x = x +gcd x y = gcd y (y `mod` y) + +gcd 0 x = x +gcd x y = gcd y (0 `mod` y) + +gcd 0 x = x +gcd x y = gcd 0 (x `mod` y) + +gcd 0 x = x +gcd x y = gcd 0 (y `mod` x) + +gcd 0 x = x +gcd x y = gcd 0 (y `mod` y) + +gcd 0 x = x +gcd x y = gcd 0 (0 `mod` y) + +gcd 0 x = x +gcd x y = gcd (x `mod` x) x + +gcd 0 x = x +gcd x y = gcd (x `mod` x) y + +gcd 0 x = x +gcd x y = gcd (x `mod` x) 0 + +gcd 0 x = x +gcd x y = gcd (x `mod` y) x + +gcd 0 x = x +gcd x y = gcd (x `mod` y) y + +gcd 0 x = x +gcd x y = gcd (x `mod` y) 0 + +gcd 0 x = x +gcd x y = gcd (y `mod` x) x + +gcd 0 x = x +gcd x y = gcd (y `mod` x) y + +gcd 0 x = x +gcd x y = gcd (y `mod` x) 0 + +gcd 0 x = x +gcd x y = gcd (0 `mod` x) x + +gcd 0 x = x +gcd x y = gcd (0 `mod` x) y + +gcd 0 x = x +gcd x y = gcd (0 `mod` x) 0 + +gcd 0 x = 0 +gcd x y = gcd x (x `mod` y) + +gcd 0 x = 0 +gcd x y = gcd x (y `mod` x) + +gcd 0 x = 0 +gcd x y = gcd x (y `mod` y) + +gcd 0 x = 0 +gcd x y = gcd x (0 `mod` y) + +gcd 0 x = 0 +gcd x y = gcd y (x `mod` y) + +gcd 0 x = 0 +gcd x y = gcd y (y `mod` x) + +gcd 0 x = 0 +gcd x y = gcd y (y `mod` y) + +gcd 0 x = 0 +gcd x y = gcd y (0 `mod` y) + +gcd 0 x = 0 +gcd x y = gcd 0 (x `mod` y) + +gcd 0 x = 0 +gcd x y = gcd 0 (y `mod` x) + +gcd 0 x = 0 +gcd x y = gcd 0 (y `mod` y) + +gcd 0 x = 0 +gcd x y = gcd 0 (0 `mod` y) + +gcd 0 x = 0 +gcd x y = gcd (x `mod` x) x + +gcd 0 x = 0 +gcd x y = gcd (x `mod` x) y + +gcd 0 x = 0 +gcd x y = gcd (x `mod` x) 0 + +gcd 0 x = 0 +gcd x y = gcd (x `mod` y) x + +gcd 0 x = 0 +gcd x y = gcd (x `mod` y) y + +gcd 0 x = 0 +gcd x y = gcd (x `mod` y) 0 + +gcd 0 x = 0 +gcd x y = gcd (y `mod` x) x + +gcd 0 x = 0 +gcd x y = gcd (y `mod` x) y + +gcd 0 x = 0 +gcd x y = gcd (y `mod` x) 0 + +gcd 0 x = 0 +gcd x y = gcd (0 `mod` x) x + +gcd 0 x = 0 +gcd x y = gcd (0 `mod` x) y + +gcd 0 x = 0 +gcd x y = gcd (0 `mod` x) 0 + +gcd x 0 = x +gcd x y = (x `mod` x) `mod` x + +gcd x 0 = x +gcd x y = (x `mod` x) `mod` y + +gcd x 0 = x +gcd x y = (x `mod` x) `mod` 0 + +gcd x 0 = x +gcd x y = (x `mod` y) `mod` x + +gcd x 0 = x +gcd x y = (x `mod` y) `mod` y + +gcd x 0 = x +gcd x y = (x `mod` y) `mod` 0 + +gcd x 0 = x +gcd x y = (x `mod` 0) `mod` x + +gcd x 0 = x +gcd x y = (x `mod` 0) `mod` y + +gcd x 0 = x +gcd x y = (x `mod` 0) `mod` 0 + +gcd x 0 = x +gcd x y = (y `mod` x) `mod` x + +gcd x 0 = x +gcd x y = (y `mod` x) `mod` y + +gcd x 0 = x +gcd x y = (y `mod` x) `mod` 0 + +gcd x 0 = x +gcd x y = (y `mod` y) `mod` x + +gcd x 0 = x +gcd x y = (y `mod` y) `mod` y + +gcd x 0 = x +gcd x y = (y `mod` y) `mod` 0 + +gcd x 0 = x +gcd x y = (y `mod` 0) `mod` x + +gcd x 0 = x +gcd x y = (y `mod` 0) `mod` y + +gcd x 0 = x +gcd x y = (y `mod` 0) `mod` 0 + +gcd x 0 = x +gcd x y = (0 `mod` x) `mod` x + +gcd x 0 = x +gcd x y = (0 `mod` x) `mod` y + +gcd x 0 = x +gcd x y = (0 `mod` x) `mod` 0 + +gcd x 0 = x +gcd x y = (0 `mod` y) `mod` x + +gcd x 0 = x +gcd x y = (0 `mod` y) `mod` y + +gcd x 0 = x +gcd x y = (0 `mod` y) `mod` 0 + +gcd x 0 = x +gcd x y = (0 `mod` 0) `mod` x + +gcd x 0 = x +gcd x y = (0 `mod` 0) `mod` y + +gcd x 0 = x +gcd x y = (0 `mod` 0) `mod` 0 + +gcd x 0 = x +gcd x y = x `mod` (x `mod` x) + +gcd x 0 = x +gcd x y = x `mod` (x `mod` y) + +gcd x 0 = x +gcd x y = x `mod` (x `mod` 0) + +gcd x 0 = x +gcd x y = x `mod` (y `mod` x) + +gcd x 0 = x +gcd x y = x `mod` (y `mod` y) + +gcd x 0 = x +gcd x y = x `mod` (y `mod` 0) + +gcd x 0 = x +gcd x y = x `mod` (0 `mod` x) + +gcd x 0 = x +gcd x y = x `mod` (0 `mod` y) + +gcd x 0 = x +gcd x y = x `mod` (0 `mod` 0) + +gcd x 0 = x +gcd x y = y `mod` (x `mod` x) + +gcd x 0 = x +gcd x y = y `mod` (x `mod` y) + +gcd x 0 = x +gcd x y = y `mod` (x `mod` 0) + +gcd x 0 = x +gcd x y = y `mod` (y `mod` x) + +gcd x 0 = x +gcd x y = y `mod` (y `mod` y) + +gcd x 0 = x +gcd x y = y `mod` (y `mod` 0) + +gcd x 0 = x +gcd x y = y `mod` (0 `mod` x) + +gcd x 0 = x +gcd x y = y `mod` (0 `mod` y) + +gcd x 0 = x +gcd x y = y `mod` (0 `mod` 0) + +gcd x 0 = x +gcd x y = 0 `mod` (x `mod` x) + +gcd x 0 = x +gcd x y = 0 `mod` (x `mod` y) + +gcd x 0 = x +gcd x y = 0 `mod` (x `mod` 0) + +gcd x 0 = x +gcd x y = 0 `mod` (y `mod` x) + +gcd x 0 = x +gcd x y = 0 `mod` (y `mod` y) + +gcd x 0 = x +gcd x y = 0 `mod` (y `mod` 0) + +gcd x 0 = x +gcd x y = 0 `mod` (0 `mod` x) + +gcd x 0 = x +gcd x y = 0 `mod` (0 `mod` y) + +gcd x 0 = x +gcd x y = 0 `mod` (0 `mod` 0) + +gcd x 0 = 0 +gcd x y = (x `mod` x) `mod` x + +gcd x 0 = 0 +gcd x y = (x `mod` x) `mod` y + +gcd x 0 = 0 +gcd x y = (x `mod` x) `mod` 0 + +gcd x 0 = 0 +gcd x y = (x `mod` y) `mod` x + +gcd x 0 = 0 +gcd x y = (x `mod` y) `mod` y + +gcd x 0 = 0 +gcd x y = (x `mod` y) `mod` 0 + +gcd x 0 = 0 +gcd x y = (x `mod` 0) `mod` x + +gcd x 0 = 0 +gcd x y = (x `mod` 0) `mod` y + +gcd x 0 = 0 +gcd x y = (x `mod` 0) `mod` 0 + +gcd x 0 = 0 +gcd x y = (y `mod` x) `mod` x + +gcd x 0 = 0 +gcd x y = (y `mod` x) `mod` y + +gcd x 0 = 0 +gcd x y = (y `mod` x) `mod` 0 + +gcd x 0 = 0 +gcd x y = (y `mod` y) `mod` x + +gcd x 0 = 0 +gcd x y = (y `mod` y) `mod` y + +gcd x 0 = 0 +gcd x y = (y `mod` y) `mod` 0 + +gcd x 0 = 0 +gcd x y = (y `mod` 0) `mod` x + +gcd x 0 = 0 +gcd x y = (y `mod` 0) `mod` y + +gcd x 0 = 0 +gcd x y = (y `mod` 0) `mod` 0 + +gcd x 0 = 0 +gcd x y = (0 `mod` x) `mod` x + +gcd x 0 = 0 +gcd x y = (0 `mod` x) `mod` y + +gcd x 0 = 0 +gcd x y = (0 `mod` x) `mod` 0 + +gcd x 0 = 0 +gcd x y = (0 `mod` y) `mod` x + +gcd x 0 = 0 +gcd x y = (0 `mod` y) `mod` y + +gcd x 0 = 0 +gcd x y = (0 `mod` y) `mod` 0 + +gcd x 0 = 0 +gcd x y = (0 `mod` 0) `mod` x + +gcd x 0 = 0 +gcd x y = (0 `mod` 0) `mod` y + +gcd x 0 = 0 +gcd x y = (0 `mod` 0) `mod` 0 + +gcd x 0 = 0 +gcd x y = x `mod` (x `mod` x) + +gcd x 0 = 0 +gcd x y = x `mod` (x `mod` y) + +gcd x 0 = 0 +gcd x y = x `mod` (x `mod` 0) + +gcd x 0 = 0 +gcd x y = x `mod` (y `mod` x) + +gcd x 0 = 0 +gcd x y = x `mod` (y `mod` y) + +gcd x 0 = 0 +gcd x y = x `mod` (y `mod` 0) + +gcd x 0 = 0 +gcd x y = x `mod` (0 `mod` x) + +gcd x 0 = 0 +gcd x y = x `mod` (0 `mod` y) + +gcd x 0 = 0 +gcd x y = x `mod` (0 `mod` 0) + +gcd x 0 = 0 +gcd x y = y `mod` (x `mod` x) + +gcd x 0 = 0 +gcd x y = y `mod` (x `mod` y) + +gcd x 0 = 0 +gcd x y = y `mod` (x `mod` 0) + +gcd x 0 = 0 +gcd x y = y `mod` (y `mod` x) + +gcd x 0 = 0 +gcd x y = y `mod` (y `mod` y) + +gcd x 0 = 0 +gcd x y = y `mod` (y `mod` 0) + +gcd x 0 = 0 +gcd x y = y `mod` (0 `mod` x) + +gcd x 0 = 0 +gcd x y = y `mod` (0 `mod` y) + +gcd x 0 = 0 +gcd x y = y `mod` (0 `mod` 0) + +gcd x 0 = 0 +gcd x y = 0 `mod` (x `mod` x) + +gcd x 0 = 0 +gcd x y = 0 `mod` (x `mod` y) + +gcd x 0 = 0 +gcd x y = 0 `mod` (x `mod` 0) + +gcd x 0 = 0 +gcd x y = 0 `mod` (y `mod` x) + +gcd x 0 = 0 +gcd x y = 0 `mod` (y `mod` y) + +gcd x 0 = 0 +gcd x y = 0 `mod` (y `mod` 0) + +gcd x 0 = 0 +gcd x y = 0 `mod` (0 `mod` x) + +gcd x 0 = 0 +gcd x y = 0 `mod` (0 `mod` y) + +gcd x 0 = 0 +gcd x y = 0 `mod` (0 `mod` 0) + +gcd x 0 = x `mod` x +gcd x y = x `mod` y + +gcd x 0 = x `mod` x +gcd x y = x `mod` 0 + +gcd x 0 = x `mod` x +gcd x y = y `mod` x + +gcd x 0 = x `mod` x +gcd x y = y `mod` y + +gcd x 0 = x `mod` x +gcd x y = y `mod` 0 + +gcd x 0 = x `mod` x +gcd x y = 0 `mod` x + +gcd x 0 = x `mod` x +gcd x y = 0 `mod` y + +gcd x 0 = x `mod` x +gcd x y = 0 `mod` 0 + +gcd x 0 = x `mod` 0 +gcd x y = x `mod` x + +gcd x 0 = x `mod` 0 +gcd x y = x `mod` y + +gcd x 0 = x `mod` 0 +gcd x y = y `mod` x + +gcd x 0 = x `mod` 0 +gcd x y = y `mod` y + +gcd x 0 = x `mod` 0 +gcd x y = y `mod` 0 + +gcd x 0 = x `mod` 0 +gcd x y = 0 `mod` x + +gcd x 0 = x `mod` 0 +gcd x y = 0 `mod` y + +gcd x 0 = x `mod` 0 +gcd x y = 0 `mod` 0 + +gcd x 0 = 0 `mod` x +gcd x y = x `mod` x + +gcd x 0 = 0 `mod` x +gcd x y = x `mod` y + +gcd x 0 = 0 `mod` x +gcd x y = x `mod` 0 + +gcd x 0 = 0 `mod` x +gcd x y = y `mod` x + +gcd x 0 = 0 `mod` x +gcd x y = y `mod` y + +gcd x 0 = 0 `mod` x +gcd x y = y `mod` 0 + +gcd x 0 = 0 `mod` x +gcd x y = 0 `mod` y + +gcd x 0 = 0 `mod` x +gcd x y = 0 `mod` 0 + +gcd x 0 = 0 `mod` 0 +gcd x y = x `mod` x + +gcd x 0 = 0 `mod` 0 +gcd x y = x `mod` y + +gcd x 0 = 0 `mod` 0 +gcd x y = x `mod` 0 + +gcd x 0 = 0 `mod` 0 +gcd x y = y `mod` x + +gcd x 0 = 0 `mod` 0 +gcd x y = y `mod` y + +gcd x 0 = 0 `mod` 0 +gcd x y = y `mod` 0 + +gcd x 0 = 0 `mod` 0 +gcd x y = 0 `mod` x + +gcd x 0 = 0 `mod` 0 +gcd x y = 0 `mod` y + +gcd x 0 = (x `mod` x) `mod` x +gcd x y = x + +gcd x 0 = (x `mod` x) `mod` x +gcd x y = y + +gcd x 0 = (x `mod` x) `mod` x +gcd x y = 0 + +gcd x 0 = (x `mod` x) `mod` 0 +gcd x y = x + +gcd x 0 = (x `mod` x) `mod` 0 +gcd x y = y + +gcd x 0 = (x `mod` x) `mod` 0 +gcd x y = 0 + +gcd x 0 = (x `mod` 0) `mod` x +gcd x y = x + +gcd x 0 = (x `mod` 0) `mod` x +gcd x y = y + +gcd x 0 = (x `mod` 0) `mod` x +gcd x y = 0 + +gcd x 0 = (x `mod` 0) `mod` 0 +gcd x y = x + +gcd x 0 = (x `mod` 0) `mod` 0 +gcd x y = y + +gcd x 0 = (x `mod` 0) `mod` 0 +gcd x y = 0 + +gcd x 0 = (0 `mod` x) `mod` x +gcd x y = x + +gcd x 0 = (0 `mod` x) `mod` x +gcd x y = y + +gcd x 0 = (0 `mod` x) `mod` x +gcd x y = 0 + +gcd x 0 = (0 `mod` x) `mod` 0 +gcd x y = x + +gcd x 0 = (0 `mod` x) `mod` 0 +gcd x y = y + +gcd x 0 = (0 `mod` x) `mod` 0 +gcd x y = 0 + +gcd x 0 = (0 `mod` 0) `mod` x +gcd x y = x + +gcd x 0 = (0 `mod` 0) `mod` x +gcd x y = y + +gcd x 0 = (0 `mod` 0) `mod` x +gcd x y = 0 + +gcd x 0 = (0 `mod` 0) `mod` 0 +gcd x y = x + +gcd x 0 = (0 `mod` 0) `mod` 0 +gcd x y = y + +gcd x 0 = (0 `mod` 0) `mod` 0 +gcd x y = 0 + +gcd x 0 = x `mod` (x `mod` x) +gcd x y = x + +gcd x 0 = x `mod` (x `mod` x) +gcd x y = y + +gcd x 0 = x `mod` (x `mod` x) +gcd x y = 0 + +gcd x 0 = x `mod` (x `mod` 0) +gcd x y = x + +gcd x 0 = x `mod` (x `mod` 0) +gcd x y = y + +gcd x 0 = x `mod` (x `mod` 0) +gcd x y = 0 + +gcd x 0 = x `mod` (0 `mod` x) +gcd x y = x + +gcd x 0 = x `mod` (0 `mod` x) +gcd x y = y + +gcd x 0 = x `mod` (0 `mod` x) +gcd x y = 0 + +gcd x 0 = x `mod` (0 `mod` 0) +gcd x y = x + +gcd x 0 = x `mod` (0 `mod` 0) +gcd x y = y + +gcd x 0 = x `mod` (0 `mod` 0) +gcd x y = 0 + +gcd x 0 = 0 `mod` (x `mod` x) +gcd x y = x + +gcd x 0 = 0 `mod` (x `mod` x) +gcd x y = y + +gcd x 0 = 0 `mod` (x `mod` x) +gcd x y = 0 + +gcd x 0 = 0 `mod` (x `mod` 0) +gcd x y = x + +gcd x 0 = 0 `mod` (x `mod` 0) +gcd x y = y + +gcd x 0 = 0 `mod` (x `mod` 0) +gcd x y = 0 + +gcd x 0 = 0 `mod` (0 `mod` x) +gcd x y = x + +gcd x 0 = 0 `mod` (0 `mod` x) +gcd x y = y + +gcd x 0 = 0 `mod` (0 `mod` x) +gcd x y = 0 + +gcd x 0 = 0 `mod` (0 `mod` 0) +gcd x y = x + +gcd x 0 = 0 `mod` (0 `mod` 0) +gcd x y = y + +gcd x 0 = 0 `mod` (0 `mod` 0) +gcd x y = 0 + +gcd 0 x = x +gcd x y = (x `mod` x) `mod` x + +gcd 0 x = x +gcd x y = (x `mod` x) `mod` y + +gcd 0 x = x +gcd x y = (x `mod` x) `mod` 0 + +gcd 0 x = x +gcd x y = (x `mod` y) `mod` x + +gcd 0 x = x +gcd x y = (x `mod` y) `mod` y + +gcd 0 x = x +gcd x y = (x `mod` y) `mod` 0 + +gcd 0 x = x +gcd x y = (x `mod` 0) `mod` x + +gcd 0 x = x +gcd x y = (x `mod` 0) `mod` y + +gcd 0 x = x +gcd x y = (x `mod` 0) `mod` 0 + +gcd 0 x = x +gcd x y = (y `mod` x) `mod` x + +gcd 0 x = x +gcd x y = (y `mod` x) `mod` y + +gcd 0 x = x +gcd x y = (y `mod` x) `mod` 0 + +gcd 0 x = x +gcd x y = (y `mod` y) `mod` x + +gcd 0 x = x +gcd x y = (y `mod` y) `mod` y + +gcd 0 x = x +gcd x y = (y `mod` y) `mod` 0 + +gcd 0 x = x +gcd x y = (y `mod` 0) `mod` x + +gcd 0 x = x +gcd x y = (y `mod` 0) `mod` y + +gcd 0 x = x +gcd x y = (y `mod` 0) `mod` 0 + +gcd 0 x = x +gcd x y = (0 `mod` x) `mod` x + +gcd 0 x = x +gcd x y = (0 `mod` x) `mod` y + +gcd 0 x = x +gcd x y = (0 `mod` x) `mod` 0 + +gcd 0 x = x +gcd x y = (0 `mod` y) `mod` x + +gcd 0 x = x +gcd x y = (0 `mod` y) `mod` y + +gcd 0 x = x +gcd x y = (0 `mod` y) `mod` 0 + +gcd 0 x = x +gcd x y = (0 `mod` 0) `mod` x + +gcd 0 x = x +gcd x y = (0 `mod` 0) `mod` y + +gcd 0 x = x +gcd x y = (0 `mod` 0) `mod` 0 + +gcd 0 x = x +gcd x y = x `mod` (x `mod` x) + +gcd 0 x = x +gcd x y = x `mod` (x `mod` y) + +gcd 0 x = x +gcd x y = x `mod` (x `mod` 0) + +gcd 0 x = x +gcd x y = x `mod` (y `mod` x) + +gcd 0 x = x +gcd x y = x `mod` (y `mod` y) + +gcd 0 x = x +gcd x y = x `mod` (y `mod` 0) + +gcd 0 x = x +gcd x y = x `mod` (0 `mod` x) + +gcd 0 x = x +gcd x y = x `mod` (0 `mod` y) + +gcd 0 x = x +gcd x y = x `mod` (0 `mod` 0) + +gcd 0 x = x +gcd x y = y `mod` (x `mod` x) + +gcd 0 x = x +gcd x y = y `mod` (x `mod` y) + +gcd 0 x = x +gcd x y = y `mod` (x `mod` 0) + +gcd 0 x = x +gcd x y = y `mod` (y `mod` x) + +gcd 0 x = x +gcd x y = y `mod` (y `mod` y) + +gcd 0 x = x +gcd x y = y `mod` (y `mod` 0) + +gcd 0 x = x +gcd x y = y `mod` (0 `mod` x) + +gcd 0 x = x +gcd x y = y `mod` (0 `mod` y) + +gcd 0 x = x +gcd x y = y `mod` (0 `mod` 0) + +gcd 0 x = x +gcd x y = 0 `mod` (x `mod` x) + +gcd 0 x = x +gcd x y = 0 `mod` (x `mod` y) + +gcd 0 x = x +gcd x y = 0 `mod` (x `mod` 0) + +gcd 0 x = x +gcd x y = 0 `mod` (y `mod` x) + +gcd 0 x = x +gcd x y = 0 `mod` (y `mod` y) + +gcd 0 x = x +gcd x y = 0 `mod` (y `mod` 0) + +gcd 0 x = x +gcd x y = 0 `mod` (0 `mod` x) + +gcd 0 x = x +gcd x y = 0 `mod` (0 `mod` y) + +gcd 0 x = x +gcd x y = 0 `mod` (0 `mod` 0) + +gcd 0 x = 0 +gcd x y = (x `mod` x) `mod` x + +gcd 0 x = 0 +gcd x y = (x `mod` x) `mod` y + +gcd 0 x = 0 +gcd x y = (x `mod` x) `mod` 0 + +gcd 0 x = 0 +gcd x y = (x `mod` y) `mod` x + +gcd 0 x = 0 +gcd x y = (x `mod` y) `mod` y + +gcd 0 x = 0 +gcd x y = (x `mod` y) `mod` 0 + +gcd 0 x = 0 +gcd x y = (x `mod` 0) `mod` x + +gcd 0 x = 0 +gcd x y = (x `mod` 0) `mod` y + +gcd 0 x = 0 +gcd x y = (x `mod` 0) `mod` 0 + +gcd 0 x = 0 +gcd x y = (y `mod` x) `mod` x + +gcd 0 x = 0 +gcd x y = (y `mod` x) `mod` y + +gcd 0 x = 0 +gcd x y = (y `mod` x) `mod` 0 + +gcd 0 x = 0 +gcd x y = (y `mod` y) `mod` x + +gcd 0 x = 0 +gcd x y = (y `mod` y) `mod` y + +gcd 0 x = 0 +gcd x y = (y `mod` y) `mod` 0 + +gcd 0 x = 0 +gcd x y = (y `mod` 0) `mod` x + +gcd 0 x = 0 +gcd x y = (y `mod` 0) `mod` y + +gcd 0 x = 0 +gcd x y = (y `mod` 0) `mod` 0 + +gcd 0 x = 0 +gcd x y = (0 `mod` x) `mod` x + +gcd 0 x = 0 +gcd x y = (0 `mod` x) `mod` y + +gcd 0 x = 0 +gcd x y = (0 `mod` x) `mod` 0 + +gcd 0 x = 0 +gcd x y = (0 `mod` y) `mod` x + +gcd 0 x = 0 +gcd x y = (0 `mod` y) `mod` y + +gcd 0 x = 0 +gcd x y = (0 `mod` y) `mod` 0 + +gcd 0 x = 0 +gcd x y = (0 `mod` 0) `mod` x + +gcd 0 x = 0 +gcd x y = (0 `mod` 0) `mod` y + +gcd 0 x = 0 +gcd x y = (0 `mod` 0) `mod` 0 + +gcd 0 x = 0 +gcd x y = x `mod` (x `mod` x) + +gcd 0 x = 0 +gcd x y = x `mod` (x `mod` y) + +gcd 0 x = 0 +gcd x y = x `mod` (x `mod` 0) + +gcd 0 x = 0 +gcd x y = x `mod` (y `mod` x) + +gcd 0 x = 0 +gcd x y = x `mod` (y `mod` y) + +gcd 0 x = 0 +gcd x y = x `mod` (y `mod` 0) + +gcd 0 x = 0 +gcd x y = x `mod` (0 `mod` x) + +gcd 0 x = 0 +gcd x y = x `mod` (0 `mod` y) + +gcd 0 x = 0 +gcd x y = x `mod` (0 `mod` 0) + +gcd 0 x = 0 +gcd x y = y `mod` (x `mod` x) + +gcd 0 x = 0 +gcd x y = y `mod` (x `mod` y) + +gcd 0 x = 0 +gcd x y = y `mod` (x `mod` 0) + +gcd 0 x = 0 +gcd x y = y `mod` (y `mod` x) + +gcd 0 x = 0 +gcd x y = y `mod` (y `mod` y) + +gcd 0 x = 0 +gcd x y = y `mod` (y `mod` 0) + +gcd 0 x = 0 +gcd x y = y `mod` (0 `mod` x) + +gcd 0 x = 0 +gcd x y = y `mod` (0 `mod` y) + +gcd 0 x = 0 +gcd x y = y `mod` (0 `mod` 0) + +gcd 0 x = 0 +gcd x y = 0 `mod` (x `mod` x) + +gcd 0 x = 0 +gcd x y = 0 `mod` (x `mod` y) + +gcd 0 x = 0 +gcd x y = 0 `mod` (x `mod` 0) + +gcd 0 x = 0 +gcd x y = 0 `mod` (y `mod` x) + +gcd 0 x = 0 +gcd x y = 0 `mod` (y `mod` y) + +gcd 0 x = 0 +gcd x y = 0 `mod` (y `mod` 0) + +gcd 0 x = 0 +gcd x y = 0 `mod` (0 `mod` x) + +gcd 0 x = 0 +gcd x y = 0 `mod` (0 `mod` y) + +gcd 0 x = 0 +gcd x y = 0 `mod` (0 `mod` 0) + +gcd 0 x = x `mod` x +gcd x y = x `mod` y + +gcd 0 x = x `mod` x +gcd x y = x `mod` 0 + +gcd 0 x = x `mod` x +gcd x y = y `mod` x + +gcd 0 x = x `mod` x +gcd x y = y `mod` 0 + +gcd 0 x = x `mod` x +gcd x y = 0 `mod` x + +gcd 0 x = x `mod` x +gcd x y = 0 `mod` y + +gcd 0 x = x `mod` x +gcd x y = 0 `mod` 0 + +gcd 0 x = x `mod` 0 +gcd x y = x `mod` x + +gcd 0 x = x `mod` 0 +gcd x y = x `mod` y + +gcd 0 x = x `mod` 0 +gcd x y = y `mod` x + +gcd 0 x = x `mod` 0 +gcd x y = y `mod` y + +gcd 0 x = x `mod` 0 +gcd x y = 0 `mod` x + +gcd 0 x = x `mod` 0 +gcd x y = 0 `mod` y + +gcd 0 x = x `mod` 0 +gcd x y = 0 `mod` 0 + +gcd 0 x = 0 `mod` x +gcd x y = x `mod` x + +gcd 0 x = 0 `mod` x +gcd x y = x `mod` y + +gcd 0 x = 0 `mod` x +gcd x y = x `mod` 0 + +gcd 0 x = 0 `mod` x +gcd x y = y `mod` x + +gcd 0 x = 0 `mod` x +gcd x y = y `mod` y + +gcd 0 x = 0 `mod` x +gcd x y = y `mod` 0 + +gcd 0 x = 0 `mod` x +gcd x y = 0 `mod` 0 + +gcd 0 x = 0 `mod` 0 +gcd x y = x `mod` x + +gcd 0 x = 0 `mod` 0 +gcd x y = x `mod` y + +gcd 0 x = 0 `mod` 0 +gcd x y = x `mod` 0 + +gcd 0 x = 0 `mod` 0 +gcd x y = y `mod` x + +gcd 0 x = 0 `mod` 0 +gcd x y = y `mod` y + +gcd 0 x = 0 `mod` 0 +gcd x y = y `mod` 0 + +gcd 0 x = 0 `mod` 0 +gcd x y = 0 `mod` x + +gcd 0 x = 0 `mod` 0 +gcd x y = 0 `mod` y + +gcd 0 x = (x `mod` x) `mod` x +gcd x y = x + +gcd 0 x = (x `mod` x) `mod` x +gcd x y = y + +gcd 0 x = (x `mod` x) `mod` x +gcd x y = 0 + +gcd 0 x = (x `mod` x) `mod` 0 +gcd x y = x + +gcd 0 x = (x `mod` x) `mod` 0 +gcd x y = y + +gcd 0 x = (x `mod` x) `mod` 0 +gcd x y = 0 + +gcd 0 x = (x `mod` 0) `mod` x +gcd x y = x + +gcd 0 x = (x `mod` 0) `mod` x +gcd x y = y + +gcd 0 x = (x `mod` 0) `mod` x +gcd x y = 0 + +gcd 0 x = (x `mod` 0) `mod` 0 +gcd x y = x + +gcd 0 x = (x `mod` 0) `mod` 0 +gcd x y = y + +gcd 0 x = (x `mod` 0) `mod` 0 +gcd x y = 0 + +gcd 0 x = (0 `mod` x) `mod` x +gcd x y = x + +gcd 0 x = (0 `mod` x) `mod` x +gcd x y = y + +gcd 0 x = (0 `mod` x) `mod` x +gcd x y = 0 + +gcd 0 x = (0 `mod` x) `mod` 0 +gcd x y = x + +gcd 0 x = (0 `mod` x) `mod` 0 +gcd x y = y + +gcd 0 x = (0 `mod` x) `mod` 0 +gcd x y = 0 + +gcd 0 x = (0 `mod` 0) `mod` x +gcd x y = x + +gcd 0 x = (0 `mod` 0) `mod` x +gcd x y = y + +gcd 0 x = (0 `mod` 0) `mod` x +gcd x y = 0 + +gcd 0 x = (0 `mod` 0) `mod` 0 +gcd x y = x + +gcd 0 x = (0 `mod` 0) `mod` 0 +gcd x y = y + +gcd 0 x = (0 `mod` 0) `mod` 0 +gcd x y = 0 + +gcd 0 x = x `mod` (x `mod` x) +gcd x y = x + +gcd 0 x = x `mod` (x `mod` x) +gcd x y = y + +gcd 0 x = x `mod` (x `mod` x) +gcd x y = 0 + +gcd 0 x = x `mod` (x `mod` 0) +gcd x y = x + +gcd 0 x = x `mod` (x `mod` 0) +gcd x y = y + +gcd 0 x = x `mod` (x `mod` 0) +gcd x y = 0 + +gcd 0 x = x `mod` (0 `mod` x) +gcd x y = x + +gcd 0 x = x `mod` (0 `mod` x) +gcd x y = y + +gcd 0 x = x `mod` (0 `mod` x) +gcd x y = 0 + +gcd 0 x = x `mod` (0 `mod` 0) +gcd x y = x + +gcd 0 x = x `mod` (0 `mod` 0) +gcd x y = y + +gcd 0 x = x `mod` (0 `mod` 0) +gcd x y = 0 + +gcd 0 x = 0 `mod` (x `mod` x) +gcd x y = x + +gcd 0 x = 0 `mod` (x `mod` x) +gcd x y = y + +gcd 0 x = 0 `mod` (x `mod` x) +gcd x y = 0 + +gcd 0 x = 0 `mod` (x `mod` 0) +gcd x y = x + +gcd 0 x = 0 `mod` (x `mod` 0) +gcd x y = y + +gcd 0 x = 0 `mod` (x `mod` 0) +gcd x y = 0 + +gcd 0 x = 0 `mod` (0 `mod` x) +gcd x y = x + +gcd 0 x = 0 `mod` (0 `mod` x) +gcd x y = y + +gcd 0 x = 0 `mod` (0 `mod` x) +gcd x y = 0 + +gcd 0 x = 0 `mod` (0 `mod` 0) +gcd x y = x + +gcd 0 x = 0 `mod` (0 `mod` 0) +gcd x y = y + +gcd 0 x = 0 `mod` (0 `mod` 0) +gcd x y = 0 diff --git a/bench/gps.hs b/bench/gps.hs --- a/bench/gps.hs +++ b/bench/gps.hs @@ -42,7 +42,43 @@ , prim "fromIntegral" (fromIntegral :: Int -> Float) ] +gps1c30p :: IO () +gps1c30p = conjure "gps1" gps1p + [ pr (0 :: Float) + , pr (1 :: Float) + , pr (1/2 :: Float) + , pr (-1 :: Float) + , prim "+" ((+) :: Float -> Float -> Float) + , prim "*" ((*) :: Float -> Float -> Float) + , prim "-" ((-) :: Float -> Float -> Float) + , prim "abs" (abs :: Float -> Float) + , prim "negate" (negate :: Float -> Float) + , prim "id" (id :: Float -> Float) + , prim "fromIntegral" (fromIntegral :: Int -> Float) + , pr (0 :: Int) + , pr (1 :: Int) + , pr (-1 :: Int) + , prim "+" ((+) :: Int -> Int -> Int) + , prim "*" ((*) :: Int -> Int -> Int) + , prim "-" ((-) :: Int -> Int -> Int) + , prim "div" (div :: Int -> Int -> Int) + , prim "mod" (mod :: Int -> Int -> Int) + , prim "^" ((^) :: Int -> Int -> Int) + , prim "abs" (abs :: Int -> Int) + , prim "negate" (negate :: Int -> Int) + , prim "id" (id :: Int -> Int) + , prim "round" (round :: Float -> Int) + , prim "truncate" (truncate :: Float -> Int) + + , pr False + , pr True + , prim "&&" (&&) + , prim "||" (||) + , prim "not" not + ] + + gps2p :: Int -> Maybe String gps2p 0 = Just "small" gps2p 500 = Just "small" @@ -404,7 +440,8 @@ gps13p :: [Rational] -> Rational gps13p [0,2] = 1 -gps13p [1,2,3] = 2 +gps13p [1,3] = 2 +gps13p [1,2,3] = 2 -- not hit, but nvm gps13p [0,0,2,2] = 1 gps13g :: [Rational] -> Rational @@ -736,7 +773,8 @@ -- GPS Benchmark #23 -- Word Stats -- gps23p :: String -> ([(Int,Int)], Int, Double) -gps23p "" = ([], 0, 1.0/0.0) +gps23p = undefined +-- gps23p "" = ([], 0, 1.0/0.0) gps23c :: IO () gps23c = do diff --git a/bench/gps.out b/bench/gps.out --- a/bench/gps.out +++ b/bench/gps.out @@ -1,6 +1,6 @@ gps1 :: Int -> Float -> Float -- testing 4 combinations of argument values --- pruning with 1/2 rules +-- pruning with 0/5 rules -- looking through 1 candidates of size 1 -- looking through 1 candidates of size 2 -- looking through 1 candidates of size 3 @@ -13,18 +13,18 @@ -- pruning with 9/17 rules -- looking through 1 candidates of size 1 -- looking through 2 candidates of size 2 --- looking through 4 candidates of size 3 --- looking through 6 candidates of size 4 --- looking through 8 candidates of size 5 --- looking through 12 candidates of size 6 --- looking through 38 candidates of size 7 +-- looking through 3 candidates of size 3 +-- looking through 3 candidates of size 4 +-- looking through 2 candidates of size 5 +-- looking through 0 candidates of size 6 +-- looking through 32 candidates of size 7 -- looking through 48 candidates of size 8 --- looking through 112 candidates of size 9 --- looking through 144 candidates of size 10 --- looking through 176 candidates of size 11 --- looking through 704 candidates of size 12 --- looking through 1856 candidates of size 13 --- tested 1744 candidates +-- looking through 48 candidates of size 9 +-- looking through 48 candidates of size 10 +-- looking through 16 candidates of size 11 +-- looking through 512 candidates of size 12 +-- looking through 1792 candidates of size 13 +-- tested 1204 candidates gps2 x = if 2000 <= x then Just "large" else (if x < 1000 then Just "small" else Nothing) gps3 :: Int -> Int -> Int -> [Int] @@ -37,7 +37,7 @@ -- looking through 0 candidates of size 5 -- looking through 1536 candidates of size 6 -- looking through 0 candidates of size 7 --- looking through 26991 candidates of size 8 +-- looking through 26858 candidates of size 8 -- tested 19480 candidates gps3 x y z = enumFromThenTo x (x + z) (y - 1) @@ -67,9 +67,9 @@ -- looking through 0 candidates of size 7 -- looking through 0 candidates of size 8 -- looking through 162 candidates of size 9 --- looking through 30 candidates of size 10 +-- looking through 16 candidates of size 10 -- looking through 30 candidates of size 11 --- tested 201 candidates +-- tested 187 candidates gps4 cs ds es = length cs < length ds && length ds < length es gps5 :: [Char] -> [Char] @@ -88,12 +88,12 @@ -- testing 9 combinations of argument values -- pruning with 16/18 rules -- looking through 4 candidates of size 1 --- looking through 9 candidates of size 2 --- 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 --- tested 3392 candidates +-- looking through 3 candidates of size 2 +-- looking through 27 candidates of size 3 +-- looking through 32 candidates of size 4 +-- looking through 506 candidates of size 5 +-- looking through 530 candidates of size 6 +-- tested 1102 candidates cannot conjure gps7 :: [Char] -> ([Char],Int) @@ -242,9 +242,9 @@ -- 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 --- tested 17 candidates +-- looking through 7 candidates of size 4 +-- looking through 83 candidates of size 5 +-- tested 29 candidates odd x = x `mod` 2 /= 0 gps14 :: [Int] -> Int @@ -297,14 +297,14 @@ -- 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 --- tested 2201 candidates +-- looking through 11 candidates of size 3 +-- looking through 29 candidates of size 4 +-- looking through 60 candidates of size 5 +-- looking through 145 candidates of size 6 +-- looking through 501 candidates of size 7 +-- looking through 1098 candidates of size 8 +-- looking through 4673 candidates of size 9 +-- tested 1852 candidates gps17 0 = 0 gps17 x = x * x + gps17 (x - 1) @@ -312,15 +312,15 @@ -- 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 7 candidates of size 2 +-- looking through 9 candidates of size 3 -- looking through 44 candidates of size 4 -- looking through 116 candidates of size 5 -- looking through 92 candidates of size 6 --- looking through 791 candidates of size 7 +-- looking through 787 candidates of size 7 -- looking through 220 candidates of size 8 -- looking through 4632 candidates of size 9 --- tested 3931 candidates +-- tested 3924 candidates gps18 [] [] = [] gps18 [] (x:xs) = xs gps18 (x:xs) [] = xs @@ -330,10 +330,10 @@ -- 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 1 candidates of size 2 -- looking through 0 candidates of size 3 -- looking through 12 candidates of size 4 --- tested 14 candidates +-- tested 13 candidates gps18 xs ys = zipWith (+) xs ys gps19 :: Int -> [Char] -> [Char] @@ -348,28 +348,28 @@ -- 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 --- tested 1207 candidates +-- looking through 1 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 4 candidates of size 8 +-- looking through 4 candidates of size 9 +-- looking through 5 candidates of size 10 +-- looking through 5 candidates of size 11 +-- looking through 5 candidates of size 12 +-- looking through 5 candidates of size 13 +-- looking through 4 candidates of size 14 +-- looking through 4 candidates of size 15 +-- looking through 4 candidates of size 16 +-- looking through 3 candidates of size 17 +-- looking through 2 candidates of size 18 +-- looking through 2 candidates of size 19 +-- looking through 1 candidates of size 20 +-- looking through 1 candidates of size 21 +-- looking through 1 candidates of size 22 +-- tested 65 candidates isVowel 'a' = True isVowel 'e' = True isVowel 'i' = True @@ -438,7 +438,6 @@ 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 diff --git a/bench/gps2.out b/bench/gps2.out --- a/bench/gps2.out +++ b/bench/gps2.out @@ -33,21 +33,21 @@ -- 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 --- tested 735 candidates +-- looking through 237 candidates of size 3 +-- looking through 433 candidates of size 4 +-- tested 699 candidates cannot conjure gps2 :: Double -> Double -> Int -> Double -- testing 5 combinations of argument values --- pruning with 3/3 rules +-- pruning with 2/6 rules -- looking through 2 candidates of size 1 -- looking through 2 candidates of size 2 -- looking through 18 candidates of size 3 -- looking through 70 candidates of size 4 --- looking through 314 candidates of size 5 --- looking through 1432 candidates of size 6 --- tested 1838 candidates +-- looking through 316 candidates of size 5 +-- looking through 1440 candidates of size 6 +-- tested 1848 candidates cannot conjure gps3 :: [Char] -> Int @@ -82,12 +82,12 @@ -- 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 60 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 --- tested 2699 candidates +-- looking through 5296 candidates of size 10 +-- tested 2698 candidates tell [] x = [] tell (x:xs) y = y `div` x:tell xs (y `mod` x) @@ -107,22 +107,23 @@ -- 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 --- tested 32953 candidates +-- looking through 45 candidates of size 5 +-- looking through 92 candidates of size 6 +-- looking through 438 candidates of size 7 +-- looking through 944 candidates of size 8 +-- looking through 4413 candidates of size 9 +-- looking through 9649 candidates of size 10 +-- looking through 44336 candidates of size 11 +-- tested 32852 candidates 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 --- tested 0 candidates +-- looking through 1 candidates of size 2 +-- looking through 1 candidates of size 3 +-- tested 2 candidates cannot conjure gps7 :: Integer -> Integer -> Ratio Integer @@ -158,12 +159,12 @@ -- 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 7 candidates of size 2 +-- looking through 18 candidates of size 3 -- looking through 60 candidates of size 4 -- looking through 150 candidates of size 5 --- looking through 472 candidates of size 6 --- tested 246 candidates +-- looking through 469 candidates of size 6 +-- tested 243 candidates gcd x 0 = x gcd x y = gcd y (x `mod` y) @@ -242,26 +243,26 @@ gps18_price :: [Double] -> [Double] -> Double -- testing 4 combinations of argument values --- pruning with 28/34 rules +-- pruning with 26/35 rules -- looking through 2 candidates of size 1 -- looking through 8 candidates of size 2 --- looking through 72 candidates of size 3 +-- looking through 64 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 --- tested 2042 candidates +-- looking through 1193 candidates of size 5 +-- looking through 714 candidates of size 6 +-- tested 2053 candidates cannot conjure gps19_snowday :: Int -> Double -> Double -> Double -> Double -- testing 7 combinations of argument values --- pruning with 7/12 rules +-- pruning with 6/19 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 --- tested 4361 candidates +-- looking through 0 candidates of size 2 +-- looking through 24 candidates of size 3 +-- looking through 150 candidates of size 4 +-- looking through 546 candidates of size 5 +-- looking through 3540 candidates of size 6 +-- tested 4263 candidates cannot conjure gps20 :: [Char] -> Bool @@ -303,15 +304,15 @@ -- 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 17 candidates of size 5 -- looking through 14 candidates of size 6 --- looking through 97 candidates of size 7 +-- looking through 103 candidates of size 7 -- looking through 104 candidates of size 8 --- looking through 871 candidates of size 9 +-- looking through 901 candidates of size 9 -- looking through 935 candidates of size 10 --- looking through 9018 candidates of size 11 +-- looking through 9226 candidates of size 11 -- looking through 9686 candidates of size 12 --- tested 20746 candidates +-- tested 20992 candidates cannot conjure gps22 :: Int -> [Char] @@ -323,8 +324,8 @@ gps23 :: [Char] -> [Char] -> [Char] -> [Char] -- 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 5 candidates of size 2 +-- looking through 9 candidates of size 3 -- looking through 0 candidates of size 4 -- looking through 162 candidates of size 5 -- looking through 612 candidates of size 6 @@ -334,7 +335,7 @@ -- looking through 0 candidates of size 10 -- looking through 0 candidates of size 11 -- looking through 5184 candidates of size 12 --- tested 9174 candidates +-- tested 9143 candidates cannot conjure gps24 :: [Char] -> Twitter @@ -358,13 +359,13 @@ gps25 :: [Double] -> [Double] -> Double -- testing 6 combinations of argument values --- pruning with 33/58 rules +-- pruning with 31/59 rules -- looking through 2 candidates of size 1 -- looking through 9 candidates of size 2 --- looking through 89 candidates of size 3 +-- looking through 81 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 --- tested 12074 candidates +-- looking through 2249 candidates of size 5 +-- looking through 9419 candidates of size 6 +-- tested 12072 candidates cannot conjure diff --git a/bench/ill-hit.out b/bench/ill-hit.out --- a/bench/ill-hit.out +++ b/bench/ill-hit.out @@ -2,11 +2,11 @@ -- testing 4 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 20 candidates of size 4 --- looking through 34 candidates of size 5 --- tested 34 candidates +-- looking through 3 candidates of size 2 +-- looking through 4 candidates of size 3 +-- looking through 10 candidates of size 4 +-- looking through 19 candidates of size 5 +-- tested 20 candidates 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 @@ -9,6 +9,10 @@ -} -- reasoning produced 1 incorrect properties, please re-run with more tests for faster results +{- +invalid: +xs `isSubsequenceOf` sort xs == True +-} -- looking through 0 candidates of size 1 -- looking through 0 candidates of size 2 -- looking through 2 candidates of size 3 @@ -16,12 +20,12 @@ -- looking through 0 candidates of size 5 -- looking through 0 candidates of size 6 -- looking through 12 candidates of size 7 --- looking through 22 candidates of size 8 +-- looking through 21 candidates of size 8 -- looking through 8 candidates of size 9 -- looking through 0 candidates of size 10 -- looking through 0 candidates of size 11 -- looking through 0 candidates of size 12 --- tested 48 candidates +-- tested 47 candidates cannot conjure subset :: [Int] -> [Int] -> Bool diff --git a/bench/p12.out b/bench/p12.out --- a/bench/p12.out +++ b/bench/p12.out @@ -3,12 +3,12 @@ -- testing 6 combinations of argument values -- pruning with 67/100 rules -- looking through 3 candidates of size 1 --- looking through 7 candidates of size 2 --- 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 --- tested 406 candidates +-- looking through 5 candidates of size 2 +-- looking through 14 candidates of size 3 +-- looking through 39 candidates of size 4 +-- looking through 128 candidates of size 5 +-- looking through 497 candidates of size 6 +-- tested 200 candidates 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 @@ -9.3 +11.0 diff --git a/bench/runtime/zero/bench/gps.runtime b/bench/runtime/zero/bench/gps.runtime --- a/bench/runtime/zero/bench/gps.runtime +++ b/bench/runtime/zero/bench/gps.runtime @@ -1,1 +1,1 @@ -23.8 +23.2 diff --git a/bench/runtime/zero/bench/gps2.runtime b/bench/runtime/zero/bench/gps2.runtime --- a/bench/runtime/zero/bench/gps2.runtime +++ b/bench/runtime/zero/bench/gps2.runtime @@ -1,1 +1,1 @@ -9.3 +9.0 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 @@ -0.3 +0.4 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 @@ -3.3 +3.0 diff --git a/bench/runtime/zero/bench/terpret.runtime b/bench/runtime/zero/bench/terpret.runtime --- a/bench/runtime/zero/bench/terpret.runtime +++ b/bench/runtime/zero/bench/terpret.runtime @@ -1,1 +1,1 @@ -13.3 +12.4 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 @@ -3.3 +3.2 diff --git a/bench/runtime/zero/eg/bst.runtime b/bench/runtime/zero/eg/bst.runtime new file mode 100644 --- /dev/null +++ b/bench/runtime/zero/eg/bst.runtime @@ -0,0 +1,1 @@ +6.5 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 @@ -4.5 +4.0 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 @@ -2.4 +2.5 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 @@ -4.4 +4.5 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 @@ -8.2 +4.6 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 @@ -0.4 +1.3 diff --git a/bench/runtime/zero/eg/pow.runtime b/bench/runtime/zero/eg/pow.runtime --- a/bench/runtime/zero/eg/pow.runtime +++ b/bench/runtime/zero/eg/pow.runtime @@ -1,1 +1,1 @@ -3.4 +3.9 diff --git a/bench/runtime/zero/eg/replicate.runtime b/bench/runtime/zero/eg/replicate.runtime --- a/bench/runtime/zero/eg/replicate.runtime +++ b/bench/runtime/zero/eg/replicate.runtime @@ -1,1 +1,1 @@ -0.4 +0.5 diff --git a/bench/runtime/zero/eg/setelem.runtime b/bench/runtime/zero/eg/setelem.runtime --- a/bench/runtime/zero/eg/setelem.runtime +++ b/bench/runtime/zero/eg/setelem.runtime @@ -1,1 +1,1 @@ -2.1 +2.0 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 @@ -1.4 +2.0 diff --git a/bench/runtime/zero/proto/u-conjure.runtime b/bench/runtime/zero/proto/u-conjure.runtime --- a/bench/runtime/zero/proto/u-conjure.runtime +++ b/bench/runtime/zero/proto/u-conjure.runtime @@ -1,1 +1,1 @@ -0.3 +0.4 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 8.10.5 -leancheck-0.9.10 -express-1.0.8 +GHC 9.0.2 +leancheck-1.0.0 +express-1.0.10 speculate-0.4.14 diff --git a/bench/self.out b/bench/self.out --- a/bench/self.out +++ b/bench/self.out @@ -2,35 +2,35 @@ -- testing 360 combinations of argument values -- pruning with 0/0 rules -- looking through 4 candidates of size 1 --- looking through 18 candidates of size 2 --- looking through 120 candidates of size 3 --- tested 24 candidates +-- looking through 17 candidates of size 2 +-- looking through 80 candidates of size 3 +-- tested 23 candidates x ? y = x + y (?) :: Int -> Int -> Int -- testing 360 combinations of argument values -- pruning with 0/0 rules -- looking through 4 candidates of size 1 --- looking through 18 candidates of size 2 --- looking through 120 candidates of size 3 --- tested 40 candidates +-- looking through 17 candidates of size 2 +-- looking through 80 candidates of size 3 +-- tested 39 candidates x ? y = x * y i :: Int -> Int -- testing 360 combinations of argument values -- pruning with 0/0 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 22 candidates of size 3 --- tested 10 candidates +-- looking through 2 candidates of size 2 +-- looking through 21 candidates of size 3 +-- tested 8 candidates i x = x + 1 d :: Int -> Int -- testing 360 combinations of argument values -- pruning with 0/0 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 22 candidates of size 3 --- tested 29 candidates +-- looking through 3 candidates of size 2 +-- looking through 20 candidates of size 3 +-- tested 26 candidates cannot conjure diff --git a/bench/take-drop.out b/bench/take-drop.out --- a/bench/take-drop.out +++ b/bench/take-drop.out @@ -3,14 +3,14 @@ -- pruning with 4/7 rules -- looking through 2 candidates of size 1 -- 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 4 candidates of size 3 +-- looking through 14 candidates of size 4 +-- looking through 26 candidates of size 5 -- looking through 41 candidates of size 6 --- looking through 132 candidates of size 7 --- looking through 127 candidates of size 8 +-- looking through 130 candidates of size 7 +-- looking through 125 candidates of size 8 -- looking through 458 candidates of size 9 --- tested 495 candidates +-- tested 488 candidates drop 0 [] = [] drop 0 (x:xs) = x:xs drop x [] = [] @@ -21,14 +21,14 @@ -- pruning with 14/21 rules -- looking through 2 candidates of size 1 -- 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 4 candidates of size 3 +-- looking through 14 candidates of size 4 +-- looking through 26 candidates of size 5 -- looking through 41 candidates of size 6 --- looking through 132 candidates of size 7 --- looking through 127 candidates of size 8 +-- looking through 130 candidates of size 7 +-- looking through 125 candidates of size 8 -- looking through 458 candidates of size 9 --- tested 484 candidates +-- tested 477 candidates take 0 [] = [] take 0 (x:xs) = [] take x [] = [] diff --git a/bench/terpret.out b/bench/terpret.out --- a/bench/terpret.out +++ b/bench/terpret.out @@ -86,16 +86,16 @@ -- 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 --- tested 16290 candidates -fadder p q r = if p == q then (p,r) else (r,not r) +-- looking through 19 candidates of size 4 +-- tested 28 candidates +fadder False False False = (False,False) +fadder False False True = (False,True) +fadder False True False = (False,True) +fadder False True True = (True,False) +fadder True False False = (False,True) +fadder True False True = (True,False) +fadder True True False = (True,False) +fadder True True True = (True,True) TerpreT benchmark #6: 2-bit adder @@ -128,12 +128,12 @@ -- 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 2 candidates of size 3 +-- looking through 6 candidates of size 4 +-- looking through 18 candidates of size 5 -- looking through 23 candidates of size 6 -- looking through 68 candidates of size 7 --- tested 67 candidates +-- tested 64 candidates [] `access` 0 = undefined [] `access` x = undefined (x:xs) `access` 0 = x diff --git a/bench/versions b/bench/versions --- a/bench/versions +++ b/bench/versions @@ -7,8 +7,12 @@ get-ghc-v() { ghc --version | sed -e "s/.* version/GHC/" } +ghc-pkg-list() { + ghc-pkg list "$@" + ghc-pkg list --package-db ~/.cabal/store/ghc-$(ghc --version | sed -e "s/.* version //")/package.db "$@" +} get-pkg-v() { - ghc-pkg list $1 2>/dev/null | + ghc-pkg-list $1 2>/dev/null | grep " $1-" | sed -e "s/^ *//" | tail -1 diff --git a/changelog.md b/changelog.md --- a/changelog.md +++ b/changelog.md @@ -2,8 +2,21 @@ ============================ -0.5.2 ------ +v0.5.4 +------ + +This has been the "dev" version after v0.5.2 +for almost a couple of years: + +* report invalid theories-from-testing +* weed-out some redundant candidates: + - add and use `isRedundantDefn` + - update how deconstructions are handled +* add (but not use) `conjureSize` to `Conjurable` + + +v0.5.2 +------ * show number of tested candidates * complete `Conjurable` derivation functions diff --git a/code-conjure.cabal b/code-conjure.cabal --- a/code-conjure.cabal +++ b/code-conjure.cabal @@ -3,7 +3,7 @@ -- Copyright (C) 2021 Rudy Matela -- Distributed under the 3-Clause BSD licence (see the file LICENSE). name: code-conjure -version: 0.5.2 +version: 0.5.4 synopsis: synthesize Haskell functions out of partial definitions description: Conjure is a tool that synthesizes Haskell functions out of partial definitions. @@ -46,15 +46,12 @@ , bench/versions , bench/time , test/sdist -tested-with: GHC==9.0 +tested-with: GHC==9.4 + , GHC==9.2 + , GHC==9.0 , GHC==8.10 - , GHC==8.8 , GHC==8.6 - , GHC==8.4 , GHC==8.2 - , GHC==8.0 - , GHC==7.10 - , GHC==7.8 source-repository head type: git @@ -63,7 +60,7 @@ source-repository this type: git location: https://github.com/rudymatela/conjure - tag: v0.5.2 + tag: v0.5.4 library exposed-modules: Conjure @@ -76,7 +73,7 @@ , Conjure.Defn other-extensions: TemplateHaskell, CPP build-depends: base >= 4 && < 5 - , leancheck >= 0.9.10 + , leancheck >= 1.0.0 , template-haskell , speculate >= 0.4.14 , express >= 1.0.8 diff --git a/eg/arith.out b/eg/arith.out --- a/eg/arith.out +++ b/eg/arith.out @@ -2,61 +2,61 @@ -- testing 4 combinations of argument values -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 9 candidates of size 3 --- tested 8 candidates +-- looking through 2 candidates of size 2 +-- looking through 8 candidates of size 3 +-- tested 6 candidates double x = x + x triple :: Int -> Int -- testing 4 combinations of argument values -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 9 candidates of size 3 --- looking through 23 candidates of size 4 --- looking through 29 candidates of size 5 --- tested 43 candidates +-- looking through 2 candidates of size 2 +-- looking through 8 candidates of size 3 +-- looking through 8 candidates of size 4 +-- looking through 21 candidates of size 5 +-- tested 25 candidates triple x = x + (x + x) add :: Int -> Int -> Int -- testing 4 combinations of argument values -- pruning with 14/25 rules -- looking through 4 candidates of size 1 --- looking through 18 candidates of size 2 --- looking through 101 candidates of size 3 --- tested 24 candidates +-- looking through 17 candidates of size 2 +-- looking through 61 candidates of size 3 +-- tested 23 candidates add x y = x + y square :: Int -> Int -- testing 3 combinations of argument values -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 9 candidates of size 3 --- tested 12 candidates +-- looking through 2 candidates of size 2 +-- looking through 7 candidates of size 3 +-- tested 10 candidates square x = x * x cube :: Int -> Int -- testing 3 combinations of argument values -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 9 candidates of size 3 --- looking through 23 candidates of size 4 --- looking through 29 candidates of size 5 --- tested 55 candidates +-- looking through 2 candidates of size 2 +-- looking through 7 candidates of size 3 +-- looking through 8 candidates of size 4 +-- looking through 21 candidates of size 5 +-- tested 36 candidates cube x = x * (x * x) tnpo :: Int -> Int -- testing 3 combinations of argument values -- pruning with 14/25 rules -- looking through 3 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 9 candidates of size 3 --- looking through 23 candidates of size 4 --- looking through 29 candidates of size 5 --- looking through 71 candidates of size 6 --- looking through 113 candidates of size 7 --- tested 163 candidates +-- looking through 2 candidates of size 2 +-- looking through 8 candidates of size 3 +-- looking through 8 candidates of size 4 +-- looking through 21 candidates of size 5 +-- looking through 21 candidates of size 6 +-- looking through 90 candidates of size 7 +-- tested 87 candidates tnpo x = x + (x + (x + 1)) diff --git a/eg/bst.hs b/eg/bst.hs new file mode 100644 --- /dev/null +++ b/eg/bst.hs @@ -0,0 +1,218 @@ +-- bst.hs: conjuring functions over binary search trees (BSTs) +-- +-- Copyright (C) 2021 Rudy Matela +-- Distributed under the 3-Clause BSD licence (see the file LICENSE). +{-# LANGUAGE CPP, TemplateHaskell #-} +#if __GLASGOW_HASKELL__ == 708 +{-# LANGUAGE DeriveDataTypeable, StandaloneDeriving #-} +import Data.Typeable (Typeable) +#endif + +import Conjure +import Test.LeanCheck +import Data.Express hiding (height,size) + +-- TODO: remove the following import +-- and fix build on GHC 7.10 and 7.8 +-- the generation of -: and ->>>: somehow fails. +import Test.LeanCheck.Utils + +data Tree = Leaf + | Node Tree Int Tree + deriving (Eq, Ord, Show, Read) + +#if __GLASGOW_HASKELL__ == 708 +deriving instance Typeable Tree +#endif + +deriveExpress ''Tree + +unit :: Int -> Tree +unit x = Node Leaf x Leaf + +nil :: Tree -> Bool +nil Leaf = True +nil _ = False + +left :: Tree -> Tree +left (Node l _ _) = l + +right :: Tree -> Tree +right (Node _ _ r) = r + +valu :: Tree -> Int +valu (Node _ x _) = x + + +leftmost :: Tree -> Int +leftmost (Node l x _) = if nil l then x else leftmost l + +rightmost :: Tree -> Int +rightmost (Node _ x r) = if nil r then x else rightmost r + +height :: Tree -> Int +height Leaf = -1 +height (Node l _ r) = 1 + max (height l) (height r) + +size :: Tree -> Int +size Leaf = 0 +size (Node l _ r) = size l + 1 + size r + +ordered :: Tree -> Bool +ordered Leaf = True +ordered (Node l x r) = (nil l || rightmost l < x) + && (nil r || x < leftmost r) + && ordered l + && ordered r + +preorder :: Tree -> [Int] +preorder Leaf = [] +preorder (Node l x r) = [x] ++ preorder l ++ preorder r + +inorder :: Tree -> [Int] +inorder Leaf = [] +inorder (Node l x r) = inorder l ++ [x] ++ inorder r + +posorder :: Tree -> [Int] +posorder Leaf = [] +posorder (Node l x r) = posorder l ++ posorder r ++ [x] + + +-- the following assume a binary search tree +mem :: Int -> Tree -> Bool +mem _ Leaf = False +mem y (Node l x r) = y == x || (if y < x then mem y l else mem y r) + +insert :: Int -> Tree -> Tree +insert x Leaf = unit x +insert x (Node l y r) = case compare x y of + LT -> Node (insert x l) y r + EQ -> Node l y r + GT -> Node l y (insert x r) + +before :: Int -> Tree -> Tree +before _ Leaf = Leaf +before y (Node l x r) = case y `compare` x of + LT -> before y l + EQ -> l + GT -> Node l x (before y r) + +beyond :: Int -> Tree -> Tree +beyond _ Leaf = Leaf +beyond y (Node l x r) = case x `compare` y of + LT -> beyond y r + EQ -> r + GT -> Node (beyond y l) x r + +union :: Tree -> Tree -> Tree +union t Leaf = t +union t (Node l x r) = Node (union (before x t) l) x (union (beyond x t) r) + + +instance Listable Tree where + tiers = cons0 Leaf + \/ cons3 Node `suchThat` ordered + +instance Name Tree where + name _ = "t1" + +instance Conjurable Tree where + conjureExpress = reifyExpress + conjureEquality = reifyEquality + conjureTiers = reifyTiers + conjureSubTypes x = conjureType (undefined :: Int) + conjureSize = size + conjureCases t = [ val (Leaf -: t) + , value "Node" (Node ->>>: t) :$ hole l :$ hole x :$ hole r + ] + where + Node l x r = Node undefined undefined undefined -: t + + +main :: IO () +main = do + conjureWithMaxSize 15 "mem" mem + [ pr False + , prim "||" (||) + , prim "==" ((==) :: Int -> Int -> Bool) + , prim "<" ((<) :: Int -> Int -> Bool) + , prif (undefined :: Bool) + ] + + conjure "mem" mem + [ pr False + , pr True + , prim "`compare`" (compare :: Int -> Int -> Ordering) + , primOrdCaseFor (undefined :: Bool) + ] + + -- simply out of reach performance-wise (reaching 16 but need size 26) + conjureWithMaxSize 12 "insert" insert + [ pr Leaf + , prim "Node" Node + , prim "unit" unit + , prim "==" ((==) :: Int -> Int -> Bool) + , prim "<" ((<) :: Int -> Int -> Bool) + , prif (undefined :: Tree) + ] + + -- out of reach performance-wise (reaching 16 but need 19) + conjureFromSpecWith args{maxSize = 12} "before" beforeSpec + [ pr Leaf + , prim "Node" Node + , prim "==" ((==) :: Int -> Int -> Bool) + , prim "<" ((<) :: Int -> Int -> Bool) + , prif (undefined :: Tree) + ] + + -- with 15, this reaches the solution, using 12 for shorter runtime + -- using maxEquationSize = 7 reduces runtime from 13s to 11s + conjureFromSpecWith args{maxSize = 12, maxEquationSize = 7} "before" beforeSpec + [ pr Leaf + , prim "Node" Node + , prim "`compare`" (compare :: Int -> Int -> Ordering) + , primOrdCaseFor (undefined :: Tree) + ] + + -- with 15, this reaches the solution, using 12 for shorter runtime + -- using maxEquationSize = 7 reduces runtime from 13s to 11s + conjureFromSpecWith args{maxSize = 12, maxEquationSize = 7} "beyond" beyondSpec + [ pr Leaf + , prim "Node" Node + , prim "`compare`" (compare :: Int -> Int -> Ordering) + , primOrdCaseFor (undefined :: Tree) + ] + + -- out of reach (reaching 11 but need 13) + conjureWith args{maxSize = 9} "union" union + [ pr Leaf + , prim "Node" Node + , prim "before" before + , prim "beyond" beyond + ] + -- maybe with invariant following test data there will be more pruning + -- properties? + +beforeSpec :: (Int -> Tree -> Tree) -> Bool +beforeSpec before = and + [ holds n $ \x t -> ordered t ==> inorder (before x t) == takeWhile (< x) (inorder t) + ] where n = 360 + +beyondSpec :: (Int -> Tree -> Tree) -> Bool +beyondSpec beyond = and + [ holds n $ \x t -> ordered t ==> inorder (beyond x t) == dropWhile (<= x) (inorder t) + ] where n = 360 + +-- unionSpec :: (Int -> Tree -> Tree) -> Bool +-- unionSpec union = and +-- [ holds n $ \t1 t2 -> ordered t ==> inorder (union t1 t2) == merge (inorder t1) (inorder t2) +-- ] where n = 360 + +-- same as insert, but using an if instead of a case: +insertIf :: Int -> Tree -> Tree +insertIf x Leaf = unit x -- 2 +insertIf x (Node l y r) = if x == y -- 6 + then Node l y r -- 10 + else if x < y -- 14 + then Node (insert x l) y r -- 20 + else Node l y (insert x r) -- 26 diff --git a/eg/bst.out b/eg/bst.out new file mode 100644 --- /dev/null +++ b/eg/bst.out @@ -0,0 +1,122 @@ +mem :: Int -> Tree -> Bool +-- testing 360 combinations of argument values +-- pruning with 20/30 rules +-- looking through 1 candidates of size 1 +-- looking through 0 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 8 candidates of size 4 +-- looking through 0 candidates of size 5 +-- looking through 0 candidates of size 6 +-- looking through 0 candidates of size 7 +-- looking through 60 candidates of size 8 +-- looking through 120 candidates of size 9 +-- looking through 0 candidates of size 10 +-- looking through 452 candidates of size 11 +-- looking through 424 candidates of size 12 +-- tested 707 candidates +mem x Leaf = False +mem x (Node t1 y t2) = mem x t1 || (x == y || mem x t2) + +mem :: Int -> Tree -> Bool +-- testing 360 combinations of argument values +-- pruning with 1/2 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 4 candidates of size 4 +-- looking through 0 candidates of size 5 +-- looking through 0 candidates of size 6 +-- looking through 6 candidates of size 7 +-- looking through 24 candidates of size 8 +-- looking through 0 candidates of size 9 +-- looking through 192 candidates of size 10 +-- looking through 0 candidates of size 11 +-- looking through 384 candidates of size 12 +-- tested 375 candidates +mem x Leaf = False +mem x (Node t1 y t2) = case x `compare` y of LT -> mem x t1; EQ -> True; GT -> mem x t2 + +insert :: Int -> Tree -> Tree +-- testing 360 combinations of argument values +-- pruning with 6/7 rules +-- looking through 2 candidates of size 1 +-- looking through 2 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 10 candidates of size 4 +-- looking through 21 candidates of size 5 +-- looking through 0 candidates of size 6 +-- looking through 118 candidates of size 7 +-- looking through 239 candidates of size 8 +-- looking through 216 candidates of size 9 +-- looking through 2204 candidates of size 10 +-- looking through 3651 candidates of size 11 +-- looking through 8280 candidates of size 12 +-- tested 14743 candidates +cannot conjure + +before :: Int -> Tree -> Tree +-- pruning with 5/6 rules +-- looking through 2 candidates of size 1 +-- looking through 2 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 10 candidates of size 4 +-- looking through 21 candidates of size 5 +-- looking through 0 candidates of size 6 +-- looking through 118 candidates of size 7 +-- looking through 239 candidates of size 8 +-- looking through 216 candidates of size 9 +-- looking through 2204 candidates of size 10 +-- looking through 3651 candidates of size 11 +-- looking through 8280 candidates of size 12 +-- tested 14743 candidates +cannot conjure + +before :: Int -> Tree -> Tree +-- pruning with 2/4 rules +-- looking through 2 candidates of size 1 +-- looking through 2 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 10 candidates of size 4 +-- looking through 21 candidates of size 5 +-- looking through 0 candidates of size 6 +-- looking through 94 candidates of size 7 +-- looking through 287 candidates of size 8 +-- looking through 72 candidates of size 9 +-- looking through 1784 candidates of size 10 +-- looking through 5103 candidates of size 11 +-- looking through 3312 candidates of size 12 +-- tested 10687 candidates +cannot conjure + +beyond :: Int -> Tree -> Tree +-- pruning with 2/4 rules +-- looking through 2 candidates of size 1 +-- looking through 2 candidates of size 2 +-- looking through 0 candidates of size 3 +-- looking through 10 candidates of size 4 +-- looking through 21 candidates of size 5 +-- looking through 0 candidates of size 6 +-- looking through 94 candidates of size 7 +-- looking through 287 candidates of size 8 +-- looking through 72 candidates of size 9 +-- looking through 1784 candidates of size 10 +-- looking through 5103 candidates of size 11 +-- looking through 3312 candidates of size 12 +-- tested 10687 candidates +cannot conjure + +union :: Tree -> Tree -> Tree +-- testing 360 combinations of argument values +-- pruning with 6/8 rules +-- looking through 3 candidates of size 1 +-- looking through 11 candidates of size 2 +-- looking through 40 candidates of size 3 +-- looking through 84 candidates of size 4 +-- looking through 784 candidates of size 5 +-- looking through 1032 candidates of size 6 +-- looking through 8368 candidates of size 7 +-- looking through 24890 candidates of size 8 +-- looking through 75535 candidates of size 9 +-- tested 110747 candidates +cannot conjure + diff --git a/eg/count.hs b/eg/count.hs --- a/eg/count.hs +++ b/eg/count.hs @@ -47,5 +47,5 @@ , prim "tail" (tail :: [A] -> [A]) , prim "null" (null :: [A] -> Bool) , prim "==" ((==) :: A -> A -> Bool) - , prim "if" (\p x y -> if p then x else y :: Int) + , prif (undefined :: Int) ] diff --git a/eg/dupos.out b/eg/dupos.out --- a/eg/dupos.out +++ b/eg/dupos.out @@ -51,18 +51,18 @@ -- looking through 1 candidates of size 1 -- looking through 0 candidates of size 2 -- looking through 2 candidates of size 3 --- looking through 7 candidates of size 4 --- looking through 18 candidates of size 5 --- looking through 35 candidates of size 6 --- looking through 89 candidates of size 7 --- looking through 190 candidates of size 8 --- looking through 440 candidates of size 9 --- looking through 926 candidates of size 10 --- looking through 2113 candidates of size 11 --- looking through 4520 candidates of size 12 --- looking through 10066 candidates of size 13 --- looking through 21492 candidates of size 14 --- tested 19274 candidates +-- looking through 5 candidates of size 4 +-- looking through 15 candidates of size 5 +-- looking through 25 candidates of size 6 +-- looking through 59 candidates of size 7 +-- looking through 108 candidates of size 8 +-- looking through 256 candidates of size 9 +-- looking through 458 candidates of size 10 +-- looking through 1064 candidates of size 11 +-- looking through 1948 candidates of size 12 +-- looking through 4516 candidates of size 13 +-- looking through 8304 candidates of size 14 +-- tested 8844 candidates positionsFrom x y [] = [] positionsFrom x y (z:xs) = (if y == z then (x :) else id) (positionsFrom (x + 1) y xs) diff --git a/eg/factorial.out b/eg/factorial.out --- a/eg/factorial.out +++ b/eg/factorial.out @@ -3,12 +3,12 @@ -- 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 --- tested 343 candidates +-- looking through 11 candidates of size 3 +-- looking through 29 candidates of size 4 +-- looking through 60 candidates of size 5 +-- looking through 145 candidates of size 6 +-- looking through 501 candidates of size 7 +-- tested 283 candidates factorial 0 = 1 factorial x = x * factorial (x - 1) @@ -17,10 +17,10 @@ -- pruning with 32/72 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 247 candidates of size 6 --- tested 172 candidates +-- looking through 11 candidates of size 3 +-- looking through 29 candidates of size 4 +-- looking through 60 candidates of size 5 +-- looking through 209 candidates of size 6 +-- tested 148 candidates factorial x = foldr (*) 1 [1..x] diff --git a/eg/fib01.hs b/eg/fib01.hs --- a/eg/fib01.hs +++ b/eg/fib01.hs @@ -2,6 +2,7 @@ import Conjure fib01 :: Int -> Int -> Int -> Int +-- fib01 x y 0 = x -- even with this, out of reach fib01 0 1 0 = 1 fib01 0 1 1 = 1 fib01 0 1 2 = 2 @@ -35,3 +36,11 @@ -- expected function: -- fib01 x y z = if z <= 0 then y else fib01 y (x + y) (dec z) -- 1 2 3 4 5 6 7 8 9 10 11 12 + + -- out of reach as well: + -- conjure "fib01" fib01 + -- [ pr (0::Int) + -- , pr (1::Int) + -- , prim "+" ((+) :: Int -> Int -> Int) + -- , prim "-" ((-) :: Int -> Int -> Int) + -- ] diff --git a/eg/fib01.out b/eg/fib01.out --- a/eg/fib01.out +++ b/eg/fib01.out @@ -2,11 +2,11 @@ -- testing 8 combinations of argument values -- pruning with 6/10 rules -- looking through 4 candidates of size 1 --- looking through 31 candidates of size 2 --- looking through 295 candidates of size 3 --- looking through 1968 candidates of size 4 --- looking through 10684 candidates of size 5 --- tested 12982 candidates +-- looking through 24 candidates of size 2 +-- looking through 241 candidates of size 3 +-- looking through 1958 candidates of size 4 +-- looking through 10540 candidates of size 5 +-- tested 12767 candidates cannot conjure fib01 :: Int -> Int -> Int -> Int diff --git a/eg/fibonacci.out b/eg/fibonacci.out --- a/eg/fibonacci.out +++ b/eg/fibonacci.out @@ -2,18 +2,18 @@ -- 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 --- tested 56285 candidates +-- looking through 3 candidates of size 2 +-- looking through 14 candidates of size 3 +-- looking through 17 candidates of size 4 +-- looking through 58 candidates of size 5 +-- looking through 69 candidates of size 6 +-- looking through 357 candidates of size 7 +-- looking through 390 candidates of size 8 +-- looking through 2265 candidates of size 9 +-- looking through 2502 candidates of size 10 +-- looking through 15355 candidates of size 11 +-- looking through 17083 candidates of size 12 +-- tested 21324 candidates fibonacci 0 = 1 fibonacci 1 = 1 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 @@ -23,5 +23,16 @@ , prim "`mod`" (mod :: Int -> Int -> Int) ] -- desired function: - -- gcd a b = if b == 0 then a else gcd b (a `mod` b) - -- 1 2 3 4 5 6 7 8 9 10 + -- gcd x 0 = x + -- gcd x y = gcd y (x `mod` y) + +-- Degenerate case: +-- The number of candidates in this example is big, +-- if we set showTheory = True we can see why: +-- no properties are discovered. +-- +-- Since +-- 0 `mod` 0 = undefined +-- Speculate has no way of discovering that +-- x `mod` x = 0 +-- in all other cases. diff --git a/eg/gcd.out b/eg/gcd.out --- a/eg/gcd.out +++ b/eg/gcd.out @@ -2,12 +2,12 @@ -- testing 11 combinations of argument values -- pruning with 0/0 rules -- looking through 3 candidates of size 1 --- looking through 8 candidates of size 2 --- looking through 20 candidates of size 3 +-- looking through 7 candidates of size 2 +-- looking through 18 candidates of size 3 -- looking through 60 candidates of size 4 -- looking through 150 candidates of size 5 --- looking through 472 candidates of size 6 --- tested 246 candidates +-- looking through 469 candidates of size 6 +-- tested 243 candidates gcd x 0 = x gcd x y = gcd y (x `mod` y) diff --git a/eg/list.hs b/eg/list.hs --- a/eg/list.hs +++ b/eg/list.hs @@ -36,6 +36,22 @@ [x,y] \/ [z,w] = [x,z,y,w] [x,y,z] \/ [w,v,u] = [x,w,y,v,z,u] +merge' :: [Int] -> [Int] -> [Int] +merge' [] [] = [] +merge' xs [] = xs +merge' [] ys = ys +merge' [x] [y] | x <= y = [x,y] + | otherwise = [y,x] +merge' [0,1] [0,1] = [0,0,1,1] +merge' [0,1] [2,3] = [0,1,2,3] +merge' [0,2] [1,3] = [0,1,2,3] + +ordered' :: [Int] -> Bool +ordered' [] = True +ordered' [x] = True +ordered' [x,y] = x <= y +ordered' [x,y,z] = x <= y && y <= z + main :: IO () main = do conjure "length" length' @@ -46,7 +62,7 @@ conjure "reverse" reverse' [ pr ([] :: [Int]) - , prim "unit" ((:[]) :: Int -> [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) , prim "++" ((++) :: [Int] -> [Int] -> [Int]) ] @@ -55,29 +71,6 @@ , prim ":" ((:) :: Int -> [Int] -> [Int]) ] - -- now through fold - -- length xs = foldr (const (1 +)) 0 xs - conjure "length" length' - [ pr (0 :: Int) - , pr (1 :: Int) - , prim "+" ((+) :: Int -> Int -> Int) - , prim "foldr" (foldr :: (Int -> Int -> Int) -> Int -> [Int] -> Int) - , prim "const" (const :: (Int -> Int) -> Int -> (Int -> Int)) -- cheating? - ] - - -- now through fold and some cheating - -- reverse xs = foldr (\x xs -> xs ++ [x]) [] xs - -- reverse xs = foldr (flip (++) . unit) [] xs - conjure "reverse" reverse' - [ pr ([] :: [Int]) - , prim "unit" ((:[]) :: Int -> [Int]) - , prim "++" ((++) :: [Int] -> [Int] -> [Int]) - , prim "foldr" (foldr :: (Int->[Int]->[Int]) -> [Int] -> [Int] -> [Int]) - -- these last two are cheats: - , prim "flip" (flip :: ([Int]->[Int]->[Int]) -> [Int] -> [Int] -> [Int]) - , prim "." ((.) :: ([Int]->[Int]->[Int]) -> (Int->[Int]) -> Int -> [Int] -> [Int]) - ] - conjure "++" (+++) [ pr ([] :: [Int]) , prim ":" ((:) :: Int -> [Int] -> [Int]) @@ -87,4 +80,22 @@ conjure "\\/" (\/) [ pr ([] :: [Int]) , prim ":" ((:) :: Int -> [Int] -> [Int]) + ] + + conjure "ordered" ordered' + [ pr False + , pr True + , prim "&&" (&&) + , prim "||" (||) + , prim "<=" ((<=) :: Int -> Int -> Bool) + , prim "null" (null :: [Int] -> Bool) + , prim "head" (head :: [Int] -> Int) + , prim "tail" (tail :: [Int] -> [Int]) + ] + + -- unreachable: needs about 26, but can only reach 16 + conjureWithMaxSize 12 "merge" merge' + [ prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "compare" (compare :: Int -> Int -> Ordering) + , primOrdCaseFor (undefined :: [Int]) ] diff --git a/eg/list.out b/eg/list.out --- a/eg/list.out +++ b/eg/list.out @@ -2,85 +2,97 @@ -- testing 360 combinations of argument values -- pruning with 4/8 rules -- looking through 2 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 3 candidates of size 3 --- looking through 11 candidates of size 4 --- looking through 10 candidates of size 5 --- tested 23 candidates +-- looking through 2 candidates of size 2 +-- looking through 2 candidates of size 3 +-- looking through 4 candidates of size 4 +-- looking through 5 candidates of size 5 +-- tested 13 candidates length [] = 0 length (x:xs) = length xs + 1 reverse :: [Int] -> [Int] -- testing 360 combinations of argument values --- pruning with 3/3 rules +-- 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 2 candidates of size 4 --- looking through 5 candidates of size 5 +-- looking through 2 candidates of size 3 +-- looking through 3 candidates of size 4 +-- looking through 4 candidates of size 5 -- looking through 7 candidates of size 6 --- tested 18 candidates +-- looking through 10 candidates of size 7 +-- tested 27 candidates reverse [] = [] -reverse (x:xs) = reverse xs ++ unit x +reverse (x:xs) = reverse xs ++ [x] (++) :: [Int] -> [Int] -> [Int] -- testing 360 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 11 candidates of size 3 +-- looking through 7 candidates of size 2 +-- looking through 9 candidates of size 3 -- looking through 44 candidates of size 4 -- looking through 116 candidates of size 5 -- looking through 80 candidates of size 6 --- tested 236 candidates +-- tested 233 candidates [] ++ xs = xs (x:xs) ++ ys = x:(xs ++ ys) -length :: [Int] -> Int --- testing 360 combinations of argument values --- pruning with 6/10 rules --- looking through 2 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 3 candidates of size 3 --- looking through 13 candidates of size 4 --- looking through 16 candidates of size 5 --- tested 25 candidates -length [] = 0 -length (x:xs) = length xs + 1 - -reverse :: [Int] -> [Int] --- testing 360 combinations of argument values --- pruning with 6/6 rules --- looking through 2 candidates of size 1 --- looking through 1 candidates of size 2 --- looking through 3 candidates of size 3 --- looking through 2 candidates of size 4 --- looking through 5 candidates of size 5 --- looking through 15 candidates of size 6 --- tested 18 candidates -reverse [] = [] -reverse (x:xs) = reverse xs ++ unit x - (++) :: [Int] -> [Int] -> [Int] -- testing 360 combinations of argument values -- pruning with 2/2 rules -- looking through 3 candidates of size 1 --- looking through 8 candidates of size 2 --- looking through 11 candidates of size 3 +-- looking through 7 candidates of size 2 +-- looking through 9 candidates of size 3 -- looking through 48 candidates of size 4 --- tested 57 candidates +-- tested 54 candidates xs ++ ys = foldr (:) ys xs (\/) :: [Int] -> [Int] -> [Int] -- testing 360 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 11 candidates of size 3 +-- looking through 7 candidates of size 2 +-- looking through 9 candidates of size 3 -- looking through 44 candidates of size 4 -- looking through 116 candidates of size 5 -- looking through 80 candidates of size 6 --- tested 238 candidates +-- tested 235 candidates [] \/ xs = xs (x:xs) \/ ys = x:ys \/ xs + +ordered :: [Int] -> Bool +-- testing 360 combinations of argument values +-- pruning with 29/39 rules +-- looking through 2 candidates of size 1 +-- looking through 2 candidates of size 2 +-- looking through 3 candidates of size 3 +-- looking through 3 candidates of size 4 +-- looking through 4 candidates of size 5 +-- looking through 16 candidates of size 6 +-- looking through 28 candidates of size 7 +-- looking through 48 candidates of size 8 +-- looking through 100 candidates of size 9 +-- looking through 226 candidates of size 10 +-- looking through 482 candidates of size 11 +-- tested 616 candidates +ordered [] = True +ordered (x:xs) = (null xs || x <= head xs) && ordered xs + +merge :: [Int] -> [Int] -> [Int] +-- testing 360 combinations of argument values +-- pruning with 1/2 rules +-- looking through 2 candidates of size 1 +-- looking through 1 candidates of size 2 +-- looking through 2 candidates of size 3 +-- looking through 12 candidates of size 4 +-- looking through 16 candidates of size 5 +-- looking through 20 candidates of size 6 +-- looking through 84 candidates of size 7 +-- looking through 40 candidates of size 8 +-- looking through 368 candidates of size 9 +-- looking through 216 candidates of size 10 +-- looking through 1838 candidates of size 11 +-- looking through 1016 candidates of size 12 +-- tested 3615 candidates +cannot conjure diff --git a/eg/pow.out b/eg/pow.out --- a/eg/pow.out +++ b/eg/pow.out @@ -2,14 +2,14 @@ -- testing 5 combinations of argument values -- pruning with 14/30 rules -- looking through 4 candidates of size 1 --- looking through 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 --- tested 27290 candidates +-- looking through 17 candidates of size 2 +-- looking through 92 candidates of size 3 +-- looking through 307 candidates of size 4 +-- looking through 1632 candidates of size 5 +-- looking through 3406 candidates of size 6 +-- looking through 21046 candidates of size 7 +-- looking through 34728 candidates of size 8 +-- tested 27145 candidates pow x 0 = 1 pow x y = x * pow x (y - 1) @@ -17,11 +17,11 @@ -- testing 5 combinations of argument values -- pruning with 15/19 rules -- looking through 4 candidates of size 1 --- looking through 20 candidates of size 2 --- looking through 114 candidates of size 3 --- looking through 378 candidates of size 4 --- looking through 1690 candidates of size 5 --- looking through 4572 candidates of size 6 --- tested 6778 candidates +-- looking through 19 candidates of size 2 +-- looking through 105 candidates of size 3 +-- looking through 360 candidates of size 4 +-- looking through 1649 candidates of size 5 +-- looking through 4534 candidates of size 6 +-- tested 6671 candidates cannot conjure diff --git a/eg/replicate.out b/eg/replicate.out --- a/eg/replicate.out +++ b/eg/replicate.out @@ -4,12 +4,12 @@ -- looking through 1 candidates of size 1 -- looking through 0 candidates of size 2 -- looking through 1 candidates of size 3 --- looking through 2 candidates of size 4 +-- looking through 1 candidates of size 4 -- looking through 3 candidates of size 5 --- looking through 7 candidates of size 6 +-- looking through 5 candidates of size 6 -- looking through 5 candidates of size 7 --- looking through 16 candidates of size 8 --- tested 20 candidates +-- looking through 8 candidates of size 8 +-- tested 17 candidates replicate 0 c = "" replicate x c = c:replicate (x - 1) c diff --git a/eg/setelem.out b/eg/setelem.out --- a/eg/setelem.out +++ b/eg/setelem.out @@ -17,14 +17,14 @@ -- testing 360 combinations of argument values -- pruning with 46/57 rules -- looking through 2 candidates of size 1 --- looking through 3 candidates of size 2 --- looking through 7 candidates of size 3 --- looking through 19 candidates of size 4 --- looking through 35 candidates of size 5 --- looking through 83 candidates of size 6 --- looking through 245 candidates of size 7 --- looking through 617 candidates of size 8 --- tested 466 candidates +-- looking through 2 candidates of size 2 +-- looking through 5 candidates of size 3 +-- looking through 11 candidates of size 4 +-- looking through 20 candidates of size 5 +-- looking through 47 candidates of size 6 +-- looking through 136 candidates of size 7 +-- looking through 348 candidates of size 8 +-- tested 256 candidates set [] = True set (x:xs) = not (elem x xs) && set xs diff --git a/eg/sort.out b/eg/sort.out --- a/eg/sort.out +++ b/eg/sort.out @@ -3,10 +3,10 @@ -- pruning with 6/7 rules -- looking through 2 candidates of size 1 -- looking through 3 candidates of size 2 --- looking through 7 candidates of size 3 --- looking through 16 candidates of size 4 --- looking through 36 candidates of size 5 --- tested 31 candidates +-- looking through 5 candidates of size 3 +-- looking through 12 candidates of size 4 +-- looking through 23 candidates of size 5 +-- tested 24 candidates sort [] = [] sort (x:xs) = insert x (sort xs) @@ -45,17 +45,17 @@ -- testing 360 combinations of argument values -- pruning with 4/4 rules -- looking through 3 candidates of size 1 --- looking through 8 candidates of size 2 --- looking through 11 candidates of size 3 +-- looking through 7 candidates of size 2 +-- looking through 9 candidates of size 3 -- looking through 44 candidates of size 4 -- looking through 116 candidates of size 5 -- looking through 80 candidates of size 6 --- looking through 719 candidates of size 7 +-- looking through 715 candidates of size 7 -- looking through 164 candidates of size 8 -- looking through 3360 candidates of size 9 -- looking through 1448 candidates of size 10 --- looking through 12905 candidates of size 11 +-- looking through 12901 candidates of size 11 -- looking through 15208 candidates of size 12 --- tested 34066 candidates +-- tested 34055 candidates cannot conjure diff --git a/eg/spec.out b/eg/spec.out --- a/eg/spec.out +++ b/eg/spec.out @@ -28,12 +28,12 @@ (++) :: [Int] -> [Int] -> [Int] -- pruning with 3/3 rules -- looking through 2 candidates of size 1 --- looking through 4 candidates of size 2 +-- looking through 3 candidates of size 2 -- looking through 10 candidates of size 3 --- looking through 28 candidates of size 4 +-- looking through 27 candidates of size 4 -- looking through 78 candidates of size 5 --- looking through 172 candidates of size 6 --- tested 196 candidates +-- looking through 171 candidates of size 6 +-- tested 194 candidates [] ++ xs = xs (x:xs) ++ ys = x:(xs ++ ys) diff --git a/eg/subset.out b/eg/subset.out --- a/eg/subset.out +++ b/eg/subset.out @@ -3,13 +3,13 @@ -- pruning with 29/39 rules -- looking through 2 candidates of size 1 -- looking through 4 candidates of size 2 --- looking through 14 candidates of size 3 +-- looking through 10 candidates of size 3 -- looking through 40 candidates of size 4 -- looking through 160 candidates of size 5 -- looking through 0 candidates of size 6 --- looking through 448 candidates of size 7 +-- looking through 444 candidates of size 7 -- looking through 784 candidates of size 8 --- tested 1318 candidates +-- tested 1310 candidates subset [] xs = True subset (x:xs) ys = elem x ys && subset xs ys diff --git a/eg/tree.hs b/eg/tree.hs --- a/eg/tree.hs +++ b/eg/tree.hs @@ -45,10 +45,10 @@ leftmost :: Tree -> Int -leftmost (Node l x _) = if nil l then x else leftmost (left l) +leftmost (Node l x _) = if nil l then x else leftmost l rightmost :: Tree -> Int -rightmost (Node _ x r) = if nil r then x else rightmost (right r) +rightmost (Node _ x r) = if nil r then x else rightmost r height :: Tree -> Int height Leaf = -1 @@ -58,21 +58,32 @@ size Leaf = 0 size (Node l _ r) = size l + 1 + size r +ordered :: Tree -> Bool +ordered Leaf = True +ordered (Node l x r) = (nil l || rightmost l < x) + && (nil r || x < leftmost r) + && ordered l + && ordered r + +preorder :: Tree -> [Int] +preorder Leaf = [] +preorder (Node l x r) = [x] ++ preorder l ++ preorder r + +inorder :: Tree -> [Int] +inorder Leaf = [] +inorder (Node l x r) = inorder l ++ [x] ++ inorder r + +posorder :: Tree -> [Int] +posorder Leaf = [] +posorder (Node l x r) = posorder l ++ posorder r ++ [x] + + -- this mem searches both sides of the tree mem :: Int -> Tree -> Bool mem _ Leaf = False mem y (Node l x r) = y == x || mem y l || mem y r -insert :: Int -> Tree -> Tree -insert x Leaf = unit x -insert x (Node l y r) = case compare x y of - LT -> Node (insert x l) y r - EQ -> Node l y r - GT -> Node l y (insert x r) --- TODO: mem alternative for binary search trees - - instance Listable Tree where tiers = cons0 Leaf \/ cons3 Node @@ -85,6 +96,7 @@ conjureEquality = reifyEquality conjureTiers = reifyTiers conjureSubTypes x = conjureType (undefined :: Int) + conjureSize = size conjureCases t = [ val (Leaf -: t) , value "Node" (Node ->>>: t) :$ hole l :$ hole x :$ hole r ] @@ -96,77 +108,73 @@ main = do conjure "leftmost" leftmost [ prim "undefined" (undefined :: Int) - , prim "if" (\p x y -> if p then x else y :: Int) + , prif (undefined :: Int) , prim "nil" nil ] conjure "rightmost" rightmost [ prim "undefined" (undefined :: Int) - , prim "if" (\p x y -> if p then x else y :: Int) + , prif (undefined :: Int) , prim "nil" nil ] - conjureWithMaxSize 13 "size" size + conjure "size" size [ pr (0 :: Int) , pr (1 :: Int) , prim "+" ((+) :: Int -> Int -> Int) , prim "nil" nil - , prim "left" left - , prim "right" right ] - conjureWithMaxSize 13 "height" height + conjure "height" height [ pr (0 :: Int) , pr (1 :: Int) , pr (-1 :: Int) , prim "+" ((+) :: Int -> Int -> Int) , prim "max" (max :: Int -> Int -> Int) , prim "nil" nil - , prim "left" left - , prim "right" right ] - -- out of reach performance-wise conjure "mem" mem [ pr False , prim "||" (||) , prim "==" ((==) :: Int -> Int -> Bool) ] - -- simply out of reach performance-wise (size 34) - conjureWithMaxSize 12 "insert" insert - [ pr Leaf - , prim "Node" Node - , prim "==" ((==) :: Int -> Int -> Bool) + -- unreachable: needs size 22 but OOMs at 18 + conjureWithMaxSize 12 "ordered" ordered + [ pr True + , pr False + , prim "&&" (&&) + , prim "||" (||) , prim "<" ((<) :: Int -> Int -> Bool) - , prim ">" ((>) :: Int -> Int -> Bool) - , prim "if" (\p t1 t2 -> if p then t1 else t2 :: Tree) + , prim "rightmost" rightmost + , prim "leftmost" leftmost ] + conjure "ordered" ordered + [ prim "strictlyOrdered" (strictlyOrdered :: [Int] -> Bool) + , prim "inorder" inorder + ] -sizeIf :: Tree -> Int -sizeIf t = if nil t -- 3 - then 0 -- 4 - else sizeIf (left t) + sizeIf (right t) - -- 5 6 7 8 9 10 11 + conjureWithMaxSize 12 "preorder" preorder + [ pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "++" ((++) :: [Int] -> [Int] -> [Int]) + ] -heightIf :: Tree -> Int -heightIf t = if nil t -- 3 - then -1 -- 4 - else 1 + max (height (left t)) (height (right t)) - -- 5 6 7 8 9 10 11 12 13 + conjureWithMaxSize 12 "inorder" inorder + [ pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "++" ((++) :: [Int] -> [Int] -> [Int]) + ] -memIf :: Int -> Tree -> Bool -memIf y t = if nil t -- 3 - then False -- 4 - else y == valu t || memIf y (left t) || memIf y (right t) - -- 5 6 7 8 9 10 11 12 13 14 15 16 17 18 + conjureWithMaxSize 12 "posorder" posorder + [ pr ([] :: [Int]) + , prim ":" ((:) :: Int -> [Int] -> [Int]) + , prim "++" ((++) :: [Int] -> [Int] -> [Int]) + ] -insertIf :: Int -> Tree -> Tree -insertIf x t = if nil t -- 3 - then unit x -- 5 - else if x == valu t -- 10 - then t -- 11 - else if x < valu t -- 16 - then Node (insert x (left t)) (valu t) (right t) -- 25 - else Node (left t) (valu t) (insert x (right t)) -- 34 + +strictlyOrdered :: [Int] -> Bool +strictlyOrdered [] = True +strictlyOrdered (x:xs) = (null xs || x < head xs) && strictlyOrdered xs diff --git a/eg/tree.out b/eg/tree.out --- a/eg/tree.out +++ b/eg/tree.out @@ -30,14 +30,14 @@ -- testing 360 combinations of argument values -- pruning with 4/8 rules -- looking through 2 candidates of size 1 --- looking through 4 candidates of size 2 --- looking through 5 candidates of size 3 --- looking through 19 candidates of size 4 --- looking through 35 candidates of size 5 --- looking through 66 candidates of size 6 --- looking through 163 candidates of size 7 --- looking through 311 candidates of size 8 --- tested 428 candidates +-- looking through 2 candidates of size 2 +-- looking through 3 candidates of size 3 +-- looking through 4 candidates of size 4 +-- looking through 9 candidates of size 5 +-- looking through 12 candidates of size 6 +-- looking through 25 candidates of size 7 +-- looking through 40 candidates of size 8 +-- tested 71 candidates size Leaf = 0 size (Node t1 x t2) = size t1 + (size t2 + 1) @@ -45,14 +45,14 @@ -- testing 360 combinations of argument values -- pruning with 49/65 rules -- looking through 3 candidates of size 1 --- looking through 9 candidates of size 2 --- looking through 8 candidates of size 3 --- looking through 59 candidates of size 4 --- looking through 114 candidates of size 5 --- looking through 472 candidates of size 6 --- looking through 1440 candidates of size 7 --- looking through 5781 candidates of size 8 --- tested 4171 candidates +-- looking through 3 candidates of size 2 +-- looking through 4 candidates of size 3 +-- looking through 13 candidates of size 4 +-- looking through 30 candidates of size 5 +-- looking through 88 candidates of size 6 +-- looking through 320 candidates of size 7 +-- looking through 1093 candidates of size 8 +-- tested 475 candidates height Leaf = -1 height (Node t1 x t2) = max (height t1) (height t2) + 1 @@ -75,21 +75,77 @@ mem x Leaf = False mem x (Node t1 y t2) = mem x t1 || (x == y || mem x t2) -insert :: Int -> Tree -> Tree +ordered :: Tree -> Bool -- testing 360 combinations of argument values --- pruning with 6/7 rules +-- pruning with 29/39 rules -- looking through 2 candidates of size 1 --- looking through 2 candidates of size 2 --- looking through 0 candidates of size 3 --- looking through 10 candidates of size 4 --- looking through 21 candidates of size 5 --- looking through 0 candidates of size 6 --- looking through 118 candidates of size 7 --- looking through 239 candidates of size 8 --- looking through 216 candidates of size 9 --- looking through 2204 candidates of size 10 --- looking through 3651 candidates of size 11 --- looking through 8280 candidates of size 12 --- tested 14743 candidates +-- looking through 1 candidates of size 2 +-- looking through 2 candidates of size 3 +-- looking through 0 candidates of size 4 +-- looking through 10 candidates of size 5 +-- looking through 20 candidates of size 6 +-- looking through 0 candidates of size 7 +-- looking through 64 candidates of size 8 +-- looking through 144 candidates of size 9 +-- looking through 112 candidates of size 10 +-- looking through 964 candidates of size 11 +-- looking through 1480 candidates of size 12 +-- tested 2799 candidates cannot conjure + +ordered :: Tree -> Bool +-- 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 +-- looking through 3 candidates of size 3 +-- tested 3 candidates +ordered t1 = strictlyOrdered (inorder t1) + +preorder :: Tree -> [Int] +-- testing 360 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 2 candidates of size 3 +-- looking through 1 candidates of size 4 +-- looking through 2 candidates of size 5 +-- looking through 5 candidates of size 6 +-- looking through 4 candidates of size 7 +-- looking through 9 candidates of size 8 +-- tested 17 candidates +preorder Leaf = [] +preorder (Node t1 x t2) = x:(preorder t1 ++ preorder t2) + +inorder :: Tree -> [Int] +-- testing 360 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 2 candidates of size 3 +-- looking through 1 candidates of size 4 +-- looking through 2 candidates of size 5 +-- looking through 5 candidates of size 6 +-- looking through 4 candidates of size 7 +-- looking through 9 candidates of size 8 +-- tested 21 candidates +inorder Leaf = [] +inorder (Node t1 x t2) = inorder t1 ++ (x:inorder t2) + +posorder :: Tree -> [Int] +-- testing 360 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 2 candidates of size 3 +-- looking through 1 candidates of size 4 +-- looking through 2 candidates of size 5 +-- looking through 5 candidates of size 6 +-- looking through 4 candidates of size 7 +-- looking through 9 candidates of size 8 +-- looking through 14 candidates of size 9 +-- looking through 17 candidates of size 10 +-- tested 52 candidates +posorder Leaf = [] +posorder (Node t1 x t2) = posorder t1 ++ (posorder t2 ++ [x]) diff --git a/mk/depend.mk b/mk/depend.mk --- a/mk/depend.mk +++ b/mk/depend.mk @@ -172,6 +172,19 @@ src/Conjure/Conjurable.hs \ src/Conjure/Conjurable/Derive.hs \ eg/bools.hs +eg/bst: \ + eg/bst.hs \ + mk/toplibs +eg/bst.o: \ + src/Conjure/Utils.hs \ + src/Conjure/Prim.hs \ + src/Conjure.hs \ + src/Conjure/Expr.hs \ + src/Conjure/Engine.hs \ + src/Conjure/Defn.hs \ + src/Conjure/Conjurable.hs \ + src/Conjure/Conjurable/Derive.hs \ + eg/bst.hs eg/count: \ eg/count.hs \ mk/toplibs diff --git a/mk/haskell.mk b/mk/haskell.mk --- a/mk/haskell.mk +++ b/mk/haskell.mk @@ -1,6 +1,6 @@ # Implicit rules for compiling Haskell code. # -# Copyright (c) 2015-2021 Rudy Matela. +# Copyright (c) 2015-2023 Rudy Matela. # Distributed under the 3-Clause BSD licence. # # You can optionally configure the "Configuration variables" below in your main @@ -21,7 +21,7 @@ GHC ?= ghc GHCCMD = $(GHC) -i$(GHCIMPORTDIRS) $(GHCFLAGS) HADDOCK ?= haddock -CABAL_INSTALL = $(shell cabal --version | grep -q "version [0-2]\." && echo 'cabal install' || echo 'cabal v1-install') +CABAL_INSTALL = $(shell cabal --version | grep -q "version [0-2]\." && echo 'cabal install' || echo 'cabal install --lib') # Hugs Parameters HUGSIMPORTDIRS ?= "/usr/lib/hugs/packages/*" @@ -52,7 +52,6 @@ PKGNAME = $(shell cat *.cabal | grep "^name:" | sed -e "s/name: *//") HADDOCK_HAS = haddock --help | grep -q -- -HADDOCK_FILTER ?= cat # Implicit rules @@ -91,7 +90,10 @@ find $(ALL_HSS) | ./mk/ghcdeps -i$(GHCIMPORTDIRS) $(GHCFLAGS) > $(DEPMK) install-dependencies: - $(CABAL_INSTALL) $(INSTALL_DEPS) + if [ -n "$(INSTALL_DEPS)" ]; then \ + cabal update && \ + $(CABAL_INSTALL) $(INSTALL_DEPS); \ + fi # haddock rules haddock: doc/index.html @@ -112,8 +114,7 @@ $(shell $(HADDOCK_HAS) --package-name && echo "--package-name=$(PKGNAME)" ) \ $(shell $(HADDOCK_HAS) --hyperlinked-source && echo "--hyperlinked-source" ) \ $(shell $(HADDOCK_HAS) --no-print-missing-docs && echo --no-print-missing-docs ) \ - $(HADDOCKFLAGS) | \ - $(HADDOCK_FILTER) + $(HADDOCKFLAGS) clean-cabal: rm -rf dist/ dist-newstyle/ cabal.project.local cabal.project.local~ diff --git a/src/Conjure.hs b/src/Conjure.hs --- a/src/Conjure.hs +++ b/src/Conjure.hs @@ -48,6 +48,7 @@ , pr , prim , prif + , primOrdCaseFor -- * Advanced use , conjureWithMaxSize @@ -63,7 +64,7 @@ , conjureFromSpecWith -- * When using custom types - , Conjurable (conjureExpress, conjureEquality, conjureTiers, conjureCases, conjureSubTypes) + , Conjurable (conjureExpress, conjureEquality, conjureTiers, conjureCases, conjureSubTypes, conjureSize) , reifyExpress , reifyEquality , reifyTiers diff --git a/src/Conjure/Conjurable.hs b/src/Conjure/Conjurable.hs --- a/src/Conjure/Conjurable.hs +++ b/src/Conjure/Conjurable.hs @@ -29,6 +29,7 @@ , conjureIsDeconstructor , conjureIsDeconstruction , candidateDeconstructionsFrom + , candidateDeconstructionsFromHoled , conjureIsUnbreakable , conjureReification , conjureReification1 @@ -361,11 +362,18 @@ $ e `match` ed err = error "conjureIsDeconstructor: the impossible happened" + -- | Compute candidate deconstructions from an 'Expr'. -- --- This is used in the implementation of --- 'Conjure.Engine.candidateDefnsC' and 'Conjure.Engine.candidateExprs' +-- This is used in the implementation of 'Conjure.Engine.candidateDefnsC' -- followed by 'conjureIsDeconstruction'. +-- +-- > > candidateDeconstructionsFrom (xx `mod'` yy) +-- > [ _ `mod` y +-- > , x `mod` _ +-- > ] +-- +-- To be constrasted with 'candidateDeconstructionsFromHoled'. candidateDeconstructionsFrom :: Expr -> [Expr] candidateDeconstructionsFrom e = [ e' @@ -374,6 +382,31 @@ , let e' = e //- [(v, holeAsTypeOf v)] , length (holes e') == 1 ] + +-- | Compute candidate deconstructions from an 'Expr'. +-- +-- This is used in the implementation of 'Conjure.Engine.candidateExprs' +-- followed by 'conjureIsDeconstruction'. +-- +-- This is similar to 'canonicalVariations' +-- but always leaves a hole +-- of the same return type as the given expression. +-- +-- > > candidateDeconstructionsFrom (i_ `mod'` i_) +-- > [ _ `mod` x +-- > , x `mod` _ +-- > ] +-- +-- To be contrasted with 'candidateDeconstructionsFrom' +candidateDeconstructionsFromHoled :: Expr -> [Expr] +candidateDeconstructionsFromHoled e = map (//- [(v, h)]) + $ concatMap canonicalVariations + $ deholings v e + where + h = holeAsTypeOf e + v = "_#_" `varAsTypeOf` e -- a marker variable with an invalid name + -- at some point I should get rid of candidateDeconstructionsFrom in favour + -- of this one -- | Checks if an 'Expr' is of an unbreakable type. conjureIsUnbreakable :: Conjurable f => f -> Expr -> Bool diff --git a/src/Conjure/Defn.hs b/src/Conjure/Defn.hs --- a/src/Conjure/Defn.hs +++ b/src/Conjure/Defn.hs @@ -20,6 +20,7 @@ , devalFast , showDefn , defnApparentlyTerminates + , isRedundantDefn , module Conjure.Expr ) where @@ -31,7 +32,7 @@ import Data.Express.Fixtures import Data.Dynamic import Control.Applicative ((<$>)) -- for older GHCs -import Test.LeanCheck.Utils ((-:>)) -- for toDynamicWithDefn +import Test.LeanCheck.Utils ((-:>), classifyOn) -- | A function definition as a list of top-level case bindings ('Bndn'). -- @@ -134,6 +135,7 @@ -- should only be used to evaluate an expr of the form: -- ef' :$ exprExpr ex :$ exprExpr ey :$ ... red :: Int -> Memo -> Expr -> Maybe (Int, Memo, Dynamic) + red n m e | size e > n = error "toDynamicWithDefn: argument-size limit reached" red n m e = case lookup e m of Just Nothing -> error $ "toDynamicWithDefn: loop detected " ++ show e Just (Just d) -> Just (n,m,d) @@ -194,3 +196,40 @@ defnApparentlyTerminates :: Defn -> Bool defnApparentlyTerminates [(efxs, e)] = apparentlyTerminates efxs e defnApparentlyTerminates _ = True + +-- | Returns whether the given 'Defn' is redundant +-- with regards to its patterns. +-- +-- Here is an example of a redundant 'Defn': +-- +-- > 0 ? 0 = 0 +-- > 0 ? x = 0 +-- > x ? 0 = x +-- > x ? y = x +-- +-- It is redundant because it is equivalent to: +-- +-- > 0 ? _ = 0 +-- > x ? _ = x +-- +-- If the given expression is incomplete ('hasHole') +-- this function returns 'True' as nothing can be said. +isRedundantDefn :: Defn -> Bool +isRedundantDefn d = all isComplete (map snd d) + && any isRedundant1 (unfoldDefnArgs d) + where + isRedundant1 :: [(Expr,Expr)] -> Bool + isRedundant1 = all allEqual + . map (map snd) + . classifyOn fst + . map (unfoldPair . canonicalize . foldPair) + + -- Returns a list of degenerate 'Defn' + -- whose patterns have been "lensed" on each argument. + unfoldDefnArgs :: Defn -> [[(Expr,Expr)]] + unfoldDefnArgs = transpose . map unfoldBndnArgs + where + unfoldBndnArgs :: Bndn -> [(Expr,Expr)] + unfoldBndnArgs (p,r) = map (\a -> (a,r)) as + where + (_:as) = unfoldApp p diff --git a/src/Conjure/Engine.hs b/src/Conjure/Engine.hs --- a/src/Conjure/Engine.hs +++ b/src/Conjure/Engine.hs @@ -49,6 +49,7 @@ import Test.LeanCheck import Test.LeanCheck.Tiers import Test.LeanCheck.Error (errorToTrue, errorToFalse, errorToNothing) +import Test.LeanCheck.Utils (classifyOn) import Test.Speculate.Reason (Thy, rules, equations, invalid, canReduceTo, printThy, closureLimit, doubleCheck) import Test.Speculate.Engine (theoryFromAtoms, grounds, groundBinds, boolTy) @@ -163,6 +164,7 @@ , 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 + , copyBindings :: Bool -- ^ copy partial definition bindings in candidates , showTheory :: Bool -- ^ show theory discovered by Speculate used in pruning , uniqueCandidates :: Bool -- ^ unique-modulo-testing candidates } @@ -189,6 +191,7 @@ , maxDeconstructionSize = 4 , requireDescent = True , usePatterns = True + , copyBindings = True , showTheory = False , uniqueCandidates = False } @@ -217,12 +220,18 @@ putStrLn $ "{-" printThy thy putStrLn $ "-}" - when (not . null $ invalid thy) $ + when (not . null $ invalid thy) $ do putStrLn $ "-- reasoning produced " ++ show (length (invalid thy)) ++ " incorrect properties," ++ " please re-run with more tests for faster results" + when (showTheory args) $ do + putStrLn $ "{-" + putStrLn $ "invalid:" + putStr $ unlines $ map showEq $ invalid thy + putStrLn $ "-}" pr 1 0 rs where + showEq eq = showExpr (fst eq) ++ " == " ++ showExpr (snd eq) pr n t [] = do putStrLn $ "-- tested " ++ show t ++ " candidates" putStrLn $ "cannot conjure\n" pr n t ((is,cs):rs) = do @@ -395,6 +404,7 @@ $ cjHoles (prim nm f:ps) ++ [val False, val True] ++ es (===) = cjAreEqual (prim nm f:ps) maxTests + -- | Return apparently unique candidate definitions -- using pattern matching. candidateDefnsC :: Conjurable f => Args -> String -> f -> [Prim] -> ([[Defn]], Thy) @@ -413,11 +423,17 @@ appsWith :: Expr -> [Expr] -> [[Expr]] appsWith eh vs = enumerateAppsFor eh keep $ vs ++ es - ps2fss :: [Expr] -> [[Defn]] - ps2fss pats = discardT (allEqual . map snd) . products $ map p2eess pats + ps2fss pats = discardT isRedundantDefn + . discardT (allEqual . map snd) + . products + $ map p2eess pats where p2eess :: Expr -> [[Bndn]] + -- the following guarded line is an optional optimization + -- if the function is defined for the given pattern, + -- simply use its return value as the only possible result + p2eess pat | copyBindings && isGroundPat f pat = [[(pat, toValPat f pat)]] p2eess pat = mapT (pat,) . appsWith pat . tail @@ -436,7 +452,6 @@ fillingsFor :: Defn -> [[Defn]] fillingsFor = products . map fillingsFor1 - ds = filter (conjureIsDeconstructor f maxTests) es keepR ep | requireDescent = descends isDecOf ep | otherwise = const True where @@ -444,13 +459,18 @@ [ () | d <- deconstructions , m <- maybeToList (e `match` d) - , filter (uncurry (/=)) m == [(holeAsTypeOf e', e')] + -- h (_) is bound to e' + , lookup h m == Just e' + -- other than (h,e') we only accept (var,var) + , all (\(e1,e2) -> e1 == h || isVar e2) m ] + where + h = holeAsTypeOf e' deconstructions :: [Expr] deconstructions = filter (conjureIsDeconstruction f maxTests) - $ concatMap candidateDeconstructionsFrom + $ concatMap candidateDeconstructionsFromHoled $ concat . take maxDeconstructionSize - $ concatMapT (`appsWith` vars ep) [hs] + $ concatMapT (`appsWith` hs) [hs] where hs = nub $ conjureArgumentHoles f recs ep = filterT (keepR ep) @@ -547,6 +567,42 @@ 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 + + +-- | Checks if the given pattern is a ground pattern. +-- +-- A pattern is a ground pattern when its arguments are fully defined +-- and evaluating the function returns a defined value. +-- +-- This is to be used on values returned by conjurePats. +-- +-- For now, this is only used on 'candidateDefnsC'. +isGroundPat :: Conjurable f => f -> Expr -> Bool +isGroundPat f pat = errorToFalse . eval False $ gpat -==- gpat + where + gpat = toGroundPat f pat + (-==-) = conjureMkEquation f + + +-- | Given a complete "pattern", i.e. application encoded as expr, +-- converts it from using a "variable" function, +-- to an actual "value" function. +-- +-- This function is used on 'isGroundPat' and 'toValPat' +toGroundPat :: Conjurable f => f -> Expr -> Expr +toGroundPat f pat = foldApp (value "f" f : tail (unfoldApp pat)) + +-- | Evaluates a pattern to its final value. +-- +-- Only to be used when the function is defined for the given set of arguments. +-- +-- For now, this is only used on 'candidateDefnsC'. +toValPat :: Conjurable f => f -> Expr -> Expr +toValPat f = conjureExpress f . toGroundPat f +-- NOTE: the use of conjureExpress above is a hack. +-- Here, one could have used a conjureVal function, +-- that lifts 'val' over 'Expr's. +-- However this function does not exist. -- hardcoded filtering rules diff --git a/src/Conjure/Expr.hs b/src/Conjure/Expr.hs --- a/src/Conjure/Expr.hs +++ b/src/Conjure/Expr.hs @@ -19,6 +19,7 @@ , mayNotEvaluateArgument , compareSimplicity , ifFor + , caseForOrd , valuesBFS , holesBFS , fillBFS @@ -28,6 +29,7 @@ , revaluate , reval , useMatches + , deholings , enumerateAppsFor , enumerateFillings @@ -179,12 +181,19 @@ -- -- > > ifFor (undefined :: String) -- > if :: Bool -> [Char] -> [Char] -> [Char] --- --- You need to provide this as part of your building blocks on the primitives --- if you want recursive functions to be considered and produced. ifFor :: Typeable a => a -> Expr ifFor a = value "if" (\p x y -> if p then x else y `asTypeOf` a) +-- | Creates a case 'Expr' of the type of the given proxy. +-- +-- > > caseForOrd (undefined :: Int) +-- > case :: Bool -> Int -> Int -> Int -> Int +-- +-- > > caseForOrd (undefined :: String) +-- > case :: Bool -> [Char] -> [Char] -> [Char] -> [Char] +caseForOrd :: Typeable a => a -> Expr +caseForOrd a = value "case" (\o x y z -> case o of LT -> x; EQ -> y; GT -> z `asTypeOf` a) + -- | Application cross-product between lists of Exprs (>$$<) :: [Expr] -> [Expr] -> [Expr] exs >$$< eys = catMaybes [ex $$ ey | ex <- exs, ey <- eys] @@ -422,6 +431,27 @@ rehole (e1 :$ e2) = rehole e1 :$ rehole e2 rehole e | isVar e = "" `varAsTypeOf` e | otherwise = e + +-- | Takes two expressions and returns all possible ways +-- in which the first expression can appear once in +-- one of the holes of the second expression. +-- +-- > > deholings zero (i_ -+- i_ -+- i_) +-- > [ (0 + _) + _ :: Int +-- > , (_ + 0) + _ :: Int +-- > , (_ + _) + 0 :: Int +-- > ] +-- +-- > > deholings zero (i_ -+- one -+- ord' c_) +-- > [(0 + 1) + ord _ :: Int] +deholings :: Expr -> Expr -> [Expr] +deholings e' = deh + where + deh (e1 :$ e2) = map (:$ e2) (deh e1) + ++ map (e1 :$) (deh e2) + deh e = if typ e == typ e' && isHole e + then [e'] + else [] instance Express A where expr = val instance Express B where expr = val diff --git a/src/Conjure/Prim.hs b/src/Conjure/Prim.hs --- a/src/Conjure/Prim.hs +++ b/src/Conjure/Prim.hs @@ -14,6 +14,7 @@ , prim , pr , prif + , primOrdCaseFor , cjHoles , cjTiersFor , cjAreEqual @@ -51,6 +52,11 @@ -- | Provides an if condition bound to the given return type. prif :: Conjurable a => a -> Prim prif x = (ifFor x, conjureType x) + + +-- | Provides a case condition bound to the given return type. +primOrdCaseFor :: Conjurable a => a -> Prim +primOrdCaseFor x = (caseForOrd x, conjureType x) -- the following functions mirror their "conjure" counterparts from diff --git a/src/Conjure/Utils.hs b/src/Conjure/Utils.hs --- a/src/Conjure/Utils.hs +++ b/src/Conjure/Utils.hs @@ -45,6 +45,8 @@ import System.IO.Unsafe -- | Checks if all elements of a list are equal. +-- +-- Exceptionally this function returns false for an empty or unit list. allEqual :: Eq a => [a] -> Bool allEqual [] = False allEqual [x] = False diff --git a/stack.yaml b/stack.yaml --- a/stack.yaml +++ b/stack.yaml @@ -3,12 +3,12 @@ # Copyright (C) 2021 Rudy Matela # Distributed under the 3-Clause BSD licence (see the file LICENSE). -resolver: lts-18.14 # or ghc-8.10.7 +resolver: lts-19.19 # or ghc-9.0.2 packages: - . extra-deps: -- leancheck-0.9.10 +- leancheck-1.0.0 - speculate-0.4.14 -- express-1.0.8 +- express-1.0.10 diff --git a/test/conjurable.hs b/test/conjurable.hs --- a/test/conjurable.hs +++ b/test/conjurable.hs @@ -320,6 +320,19 @@ , candidateDeconstructionsFrom (div' xx xx) == [] , candidateDeconstructionsFrom ((xx -+- xx) -+- yy) == [(xx -+- xx) -+- i_] + , candidateDeconstructionsFromHoled (div' i_ i_) == [ div' i_ xx + , div' xx i_ + ] + , candidateDeconstructionsFromHoled (div' xx yy) == [] + , candidateDeconstructionsFromHoled ((i_ -+- i_) -+- i_) == + [ (i_ -+- xx) -+- yy + , (i_ -+- xx) -+- xx + , (xx -+- i_) -+- yy + , (xx -+- i_) -+- xx + , (xx -+- yy) -+- i_ + , (xx -+- xx) -+- i_ + ] + , fromDynamic (conjureDynamicEq ((+) :: Int -> Int -> Int) `dynApp` toDyn (1::Int) `dynApp` toDyn (2::Int)) == Just False , fromDynamic (conjureDynamicEq ((+) :: Int -> Int -> Int) `dynApp` toDyn (1::Int) `dynApp` toDyn (1::Int)) diff --git a/test/sdist b/test/sdist --- a/test/sdist +++ b/test/sdist @@ -2,7 +2,7 @@ # # test/sdist: tests the package generated by "cabal sdist". # -# Copyright (c) 2015-2021 Rudy Matela. +# Copyright (c) 2015-2023 Rudy Matela. # Distributed under the 3-Clause BSD licence. set -xe @@ -29,7 +29,11 @@ tar -tf $pkg | sort --ignore-case --unique > $tmp/ls-cabal-iu diff -rud $tmp/ls-cabal-i $tmp/ls-cabal-iu -if [ -d .git ] +# Check if we have a clone of the repo and git is available +# The check that we can run git ls-files is needed to avoid: +# fatal: detected dubious ownership in repository at '/__w/.../hello-haskell' +# on CI. +if [ -d .git ] && git --version && git ls-files >/dev/null then # Test if files included by cabal are the same as files tracked in git. git ls-files | sort > $tmp/ls-git