/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [head](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1] [pairs] = [0] [0], [1 0 0] [a__tail](x0) = [0 1 0]x0 [0 1 1] , [1 0 0] [1] [a__head](x0) = [0 0 0]x0 + [0] [1 0 0] [0], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [mark](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [a__incr](x0) = [0 0 0]x0 [1 0 0] , [1] [a__odds] = [0] [1], [1] [odds] = [0] [0], [1] [a__pairs] = [0] [1], [1 0 0] [1 1 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [1 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [nats] = [0] [0], [0] [0] = [0] [0], [0] [a__nats] = [0] [0] orientation: [0] [0] a__nats() = [0] >= [0] = cons(0(),incr(nats())) [0] [0] [1] [1] a__pairs() = [0] >= [0] = cons(0(),incr(odds())) [1] [1] [1] [1] a__odds() = [0] >= [0] = a__incr(a__pairs()) [1] [1] [1 0 0] [1 1 1] [1 0 0] [1 0 0] a__incr(cons(X,XS)) = [0 0 0]X + [0 0 0]XS >= [0 0 0]X + [0 0 0]XS = cons(s(mark(X)),incr(XS)) [1 0 0] [1 1 1] [0 0 0] [1 0 0] [1 0 0] [1 1 1] [1] [1 0 0] a__head(cons(X,XS)) = [0 0 0]X + [0 0 0]XS + [0] >= [0 0 0]X = mark(X) [1 0 0] [1 1 1] [0] [1 0 0] [1 0 0] [1 1 1] [1 0 0] a__tail(cons(X,XS)) = [0 0 0]X + [0 0 0]XS >= [0 0 0]XS = mark(XS) [0 0 1] [1 0 0] [1 0 0] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 0 0] [1 0 0] mark(incr(X)) = [0 0 0]X >= [0 0 0]X = a__incr(mark(X)) [1 0 0] [1 0 0] [1] [1] mark(pairs()) = [0] >= [0] = a__pairs() [1] [1] [1] [1] mark(odds()) = [0] >= [0] = a__odds() [1] [1] [1 0 0] [1] [1 0 0] [1] mark(head(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__head(mark(X)) [1 0 0] [1] [1 0 0] [0] [1 0 0] [1 0 0] mark(tail(X)) = [0 0 0]X >= [0 0 0]X = a__tail(mark(X)) [1 0 0] [1 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [1 0 0] [1 1 1] [1 0 0] [1 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [1 0 0] [0 0 0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [1 0 0] [1 0 0] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [1 0 0] [0 0 0] [1] [1] a__pairs() = [0] >= [0] = pairs() [1] [0] [1] [1] a__odds() = [0] >= [0] = odds() [1] [0] [1 0 0] [1] [1 0 0] [1] a__head(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = head(X) [1 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] a__tail(X) = [0 1 0]X >= [0 0 0]X = tail(X) [0 1 1] [0 0 0] problem: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 4x0 + 4, [head](x0) = x0, [pairs] = 0, [a__tail](x0) = 4x0 + 4, [a__head](x0) = x0, [s](x0) = 3x0, [mark](x0) = 2x0, [a__incr](x0) = 6x0, [a__odds] = 0, [odds] = 0, [a__pairs] = 0, [cons](x0, x1) = 3x0 + x1, [incr](x0) = 6x0, [nats] = 0, [0] = 0, [a__nats] = 0 orientation: a__nats() = 0 >= 0 = cons(0(),incr(nats())) a__pairs() = 0 >= 0 = cons(0(),incr(odds())) a__odds() = 0 >= 0 = a__incr(a__pairs()) a__incr(cons(X,XS)) = 18X + 6XS >= 18X + 6XS = cons(s(mark(X)),incr(XS)) a__tail(cons(X,XS)) = 12X + 4XS + 4 >= 2XS = mark(XS) mark(nats()) = 0 >= 0 = a__nats() mark(incr(X)) = 12X >= 12X = a__incr(mark(X)) mark(pairs()) = 0 >= 0 = a__pairs() mark(odds()) = 0 >= 0 = a__odds() mark(head(X)) = 2X >= 2X = a__head(mark(X)) mark(tail(X)) = 8X + 8 >= 8X + 4 = a__tail(mark(X)) mark(cons(X1,X2)) = 6X1 + 2X2 >= 6X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 6X >= 6X = s(mark(X)) a__nats() = 0 >= 0 = nats() a__incr(X) = 6X >= 6X = incr(X) a__pairs() = 0 >= 0 = pairs() a__odds() = 0 >= 0 = odds() a__head(X) = X >= X = head(X) a__tail(X) = 4X + 4 >= 4X + 4 = tail(X) problem: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(head(X)) -> a__head(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 4x0, [head](x0) = x0, [pairs] = 4, [a__tail](x0) = 4x0 + 1, [a__head](x0) = x0, [s](x0) = x0, [mark](x0) = x0, [a__incr](x0) = x0, [a__odds] = 4, [odds] = 4, [a__pairs] = 4, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [nats] = 0, [0] = 0, [a__nats] = 0 orientation: a__nats() = 0 >= 0 = cons(0(),incr(nats())) a__pairs() = 4 >= 4 = cons(0(),incr(odds())) a__odds() = 4 >= 4 = a__incr(a__pairs()) a__incr(cons(X,XS)) = X + XS >= X + XS = cons(s(mark(X)),incr(XS)) mark(nats()) = 0 >= 0 = a__nats() mark(incr(X)) = X >= X = a__incr(mark(X)) mark(pairs()) = 4 >= 4 = a__pairs() mark(odds()) = 4 >= 4 = a__odds() mark(head(X)) = X >= X = a__head(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(mark(X)) a__nats() = 0 >= 0 = nats() a__incr(X) = X >= X = incr(X) a__pairs() = 4 >= 4 = pairs() a__odds() = 4 >= 4 = odds() a__head(X) = X >= X = head(X) a__tail(X) = 4X + 1 >= 4X = tail(X) problem: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(head(X)) -> a__head(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) Matrix Interpretation Processor: dim=1 interpretation: [head](x0) = x0 + 2, [pairs] = 0, [a__head](x0) = x0 + 2, [s](x0) = x0, [mark](x0) = 4x0, [a__incr](x0) = 4x0, [a__odds] = 0, [odds] = 0, [a__pairs] = 0, [cons](x0, x1) = 4x0 + 4x1, [incr](x0) = 4x0, [nats] = 0, [0] = 0, [a__nats] = 0 orientation: a__nats() = 0 >= 0 = cons(0(),incr(nats())) a__pairs() = 0 >= 0 = cons(0(),incr(odds())) a__odds() = 0 >= 0 = a__incr(a__pairs()) a__incr(cons(X,XS)) = 16X + 16XS >= 16X + 16XS = cons(s(mark(X)),incr(XS)) mark(nats()) = 0 >= 0 = a__nats() mark(incr(X)) = 16X >= 16X = a__incr(mark(X)) mark(pairs()) = 0 >= 0 = a__pairs() mark(odds()) = 0 >= 0 = a__odds() mark(head(X)) = 4X + 8 >= 4X + 2 = a__head(mark(X)) mark(cons(X1,X2)) = 16X1 + 16X2 >= 16X1 + 4X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 4X >= 4X = s(mark(X)) a__nats() = 0 >= 0 = nats() a__incr(X) = 4X >= 4X = incr(X) a__pairs() = 0 >= 0 = pairs() a__odds() = 0 >= 0 = odds() a__head(X) = X + 2 >= X + 2 = head(X) problem: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__nats() -> nats() a__incr(X) -> incr(X) a__pairs() -> pairs() a__odds() -> odds() a__head(X) -> head(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [head](x0) = [0 0 0]x0 [0 0 0] , [1] [pairs] = [0] [0], [1 0 0] [0] [a__head](x0) = [0 1 1]x0 + [1] [1 0 1] [0], [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 1] [a__incr](x0) = [0 1 0]x0 [0 0 1] , [1] [a__odds] = [1] [0], [0] [odds] = [1] [0], [1] [a__pairs] = [0] [0], [1 0 0] [1 0 0] [cons](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [0 1 0] [0 0 1] , [1 0 1] [incr](x0) = [0 1 0]x0 [0 0 1] , [0] [nats] = [1] [0], [0] [0] = [0] [0], [1] [a__nats] = [1] [0] orientation: [1] [0] a__nats() = [1] >= [0] = cons(0(),incr(nats())) [0] [0] [1] [0] a__pairs() = [0] >= [0] = cons(0(),incr(odds())) [0] [0] [1] [1] a__odds() = [1] >= [0] = a__incr(a__pairs()) [0] [0] [1 1 0] [1 0 1] [1 1 0] [1 0 1] a__incr(cons(X,XS)) = [0 1 1]X + [0 0 0]XS >= [0 1 0]X + [0 0 0]XS = cons(s(mark(X)),incr(XS)) [0 1 0] [0 0 1] [0 1 0] [0 0 1] [1] [1] mark(nats()) = [1] >= [1] = a__nats() [0] [0] [1 1 1] [1 1 1] mark(incr(X)) = [0 1 0]X >= [0 1 0]X = a__incr(mark(X)) [0 0 1] [0 0 1] [1] [1] mark(pairs()) = [0] >= [0] = a__pairs() [0] [0] [1] [1] mark(odds()) = [1] >= [1] = a__odds() [0] [0] [1 1 1] [1 0 0] [1 1 0] [1 0 0] mark(cons(X1,X2)) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = cons(mark(X1),X2) [0 1 0] [0 0 1] [0 1 0] [0 0 1] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 1 0] [1 1 0] mark(s(X)) = [0 1 0]X >= [0 1 0]X = s(mark(X)) [0 0 0] [0 0 0] [1] [0] a__nats() = [1] >= [1] = nats() [0] [0] [1 0 1] [1 0 1] a__incr(X) = [0 1 0]X >= [0 1 0]X = incr(X) [0 0 1] [0 0 1] [1] [1] a__pairs() = [0] >= [0] = pairs() [0] [0] [1] [0] a__odds() = [1] >= [1] = odds() [0] [0] [1 0 0] [0] [1 0 0] a__head(X) = [0 1 1]X + [1] >= [0 0 0]X = head(X) [1 0 1] [0] [0 0 0] problem: a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs() -> pairs() a__head(X) -> head(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [head](x0) = [0 0 0]x0 [0 0 0] , [0] [pairs] = [1] [0], [1 0 0] [a__head](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1] [s](x0) = [0 0 0]x0 + [0] [0 1 0] [0], [mark](x0) = x0 , [1 0 1] [a__incr](x0) = [0 0 1]x0 [0 1 0] , [1] [a__odds] = [0] [1], [1] [odds] = [0] [1], [0] [a__pairs] = [1] [0], [1 0 1] [1 0 1] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1 + [1] [0 1 0] [0 1 0] [1], [1 0 1] [incr](x0) = [0 0 1]x0 [0 1 0] , [0] [nats] = [0] [0], [1] [0] = [0] [0], [0] [a__nats] = [0] [0] orientation: [1] [0] a__odds() = [0] >= [0] = a__incr(a__pairs()) [1] [1] [1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1] a__incr(cons(X,XS)) = [0 1 0]X + [0 1 0]XS + [1] >= [0 0 0]X + [0 1 0]XS + [1] = cons(s(mark(X)),incr(XS)) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 0 1] [1 0 1] mark(incr(X)) = [0 0 1]X >= [0 0 1]X = a__incr(mark(X)) [0 1 0] [0 1 0] [0] [0] mark(pairs()) = [1] >= [1] = a__pairs() [0] [0] [1] [1] mark(odds()) = [0] >= [0] = a__odds() [1] [1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 1]X2 + [1] >= [0 0 0]X1 + [0 0 1]X2 + [1] = cons(mark(X1),X2) [0 1 0] [0 1 0] [1] [0 1 0] [0 1 0] [1] [1] [1] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 1] [1] [1 0 1] [1] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(mark(X)) [0 1 0] [0] [0 1 0] [0] [1 0 1] [1 0 1] a__incr(X) = [0 0 1]X >= [0 0 1]X = incr(X) [0 1 0] [0 1 0] [0] [0] a__pairs() = [1] >= [1] = pairs() [0] [0] [1 0 0] [1 0 0] a__head(X) = [0 0 0]X >= [0 0 0]X = head(X) [0 0 0] [0 0 0] problem: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs() -> pairs() a__head(X) -> head(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [head](x0) = [0 0 0]x0 [0 0 0] , [0] [pairs] = [0] [1], [1 0 0] [a__head](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 1] [a__incr](x0) = [0 1 0]x0 [0 0 1] , [0] [a__odds] = [0] [0], [0] [odds] = [0] [0], [0] [a__pairs] = [0] [1], [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 + [1] [0 1 0] [0 0 1] [1], [1 0 1] [incr](x0) = [0 1 0]x0 [0 0 1] , [1] [nats] = [0] [0], [0] [0] = [0] [0], [0] [a__nats] = [0] [0] orientation: [1 1 0] [1 0 1] [1] [1 1 0] [1 0 1] [0] a__incr(cons(X,XS)) = [0 1 0]X + [0 0 0]XS + [1] >= [0 1 0]X + [0 0 0]XS + [1] = cons(s(mark(X)),incr(XS)) [0 1 0] [0 0 1] [1] [0 1 0] [0 0 1] [1] [1] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 1 1] [1 1 1] mark(incr(X)) = [0 1 0]X >= [0 1 0]X = a__incr(mark(X)) [0 0 1] [0 0 1] [0] [0] mark(pairs()) = [0] >= [0] = a__pairs() [1] [1] [0] [0] mark(odds()) = [0] >= [0] = a__odds() [0] [0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] mark(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = cons(mark(X1),X2) [0 1 0] [0 0 1] [1] [0 1 0] [0 0 1] [1] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 1 0] [1 1 0] mark(s(X)) = [0 1 0]X >= [0 1 0]X = s(mark(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] a__incr(X) = [0 1 0]X >= [0 1 0]X = incr(X) [0 0 1] [0 0 1] [0] [0] a__pairs() = [0] >= [0] = pairs() [1] [1] [1 0 0] [1 0 0] a__head(X) = [0 0 0]X >= [0 0 0]X = head(X) [0 0 0] [0 0 0] problem: mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs() -> pairs() a__head(X) -> head(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [head](x0) = [0 0 0]x0 [0 0 0] , [0] [pairs] = [0] [0], [1 1 0] [1] [a__head](x0) = [1 0 0]x0 + [0] [1 1 0] [1], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 1] [1] [a__incr](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [a__odds] = [0] [0], [0] [odds] = [0] [0], [0] [a__pairs] = [1] [0], [1 0 0] [1] [incr](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [0] = [1] [0] orientation: [1 0 0] [1] [1 0 0] [1] mark(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = a__incr(mark(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] mark(pairs()) = [1] >= [1] = a__pairs() [0] [0] [0] [0] mark(odds()) = [1] >= [0] = a__odds() [0] [0] [0] [0] mark(0()) = [1] >= [1] = 0() [0] [0] [1 0 0] [0] [1 0 0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X = s(mark(X)) [0 0 0] [0] [0 0 0] [1 0 1] [1] [1 0 0] [1] a__incr(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = incr(X) [0 0 0] [0] [0 0 0] [0] [0] [0] a__pairs() = [1] >= [0] = pairs() [0] [0] [1 1 0] [1] [1 0 0] a__head(X) = [1 0 0]X + [0] >= [0 0 0]X = head(X) [1 1 0] [1] [0 0 0] problem: mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs() -> pairs() Matrix Interpretation Processor: dim=3 interpretation: [0] [pairs] = [1] [0], [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [mark](x0) = [0 1 0]x0 [0 1 0] , [0] [a__incr](x0) = x0 + [1] [1], [0] [a__odds] = [0] [0], [1] [odds] = [0] [0], [0] [a__pairs] = [1] [1], [0] [incr](x0) = x0 + [1] [0], [0] [0] = [0] [0] orientation: [1 1 0] [1] [1 1 0] [0] mark(incr(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__incr(mark(X)) [0 1 0] [1] [0 1 0] [1] [1] [0] mark(pairs()) = [1] >= [1] = a__pairs() [1] [1] [1] [0] mark(odds()) = [0] >= [0] = a__odds() [0] [0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 1 0] [1 1 0] mark(s(X)) = [0 1 0]X >= [0 1 0]X = s(mark(X)) [0 1 0] [0 0 0] [0] [0] a__incr(X) = X + [1] >= X + [1] = incr(X) [1] [0] [0] [0] a__pairs() = [1] >= [1] = pairs() [1] [0] problem: mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) a__pairs() -> pairs() Matrix Interpretation Processor: dim=3 interpretation: [0] [pairs] = [0] [0], [1 0 0] [0] [s](x0) = [0 0 1]x0 + [0] [0 0 1] [1], [1 0 1] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [a__incr](x0) = [0 0 0]x0 [0 0 0] , [0] [a__pairs] = [0] [0], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [1] orientation: [1] [0] mark(0()) = [0] >= [0] = 0() [1] [1] [1 0 1] [1] [1 0 1] [0] mark(s(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = s(mark(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1 0 0] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [0] [0] a__pairs() = [0] >= [0] = pairs() [0] [0] problem: a__incr(X) -> incr(X) a__pairs() -> pairs() Matrix Interpretation Processor: dim=3 interpretation: [0] [pairs] = [0] [0], [1 0 0] [a__incr](x0) = [0 0 0]x0 [0 0 0] , [1] [a__pairs] = [0] [0], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1 0 0] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1] [0] a__pairs() = [0] >= [0] = pairs() [0] [0] problem: a__incr(X) -> incr(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [a__incr](x0) = [0 0 0]x0 + [0] [0 0 1] [0], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1] [1 0 0] a__incr(X) = [0 0 0]X + [0] >= [0 0 0]X = incr(X) [0 0 1] [0] [0 0 0] problem: Qed