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=3 interpretation: [1 0 0] [1] [tail](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 1] [head](x0) = [0 0 0]x0 [0 0 0] , [1] [pairs] = [0] [0], [1 0 0] [1] [a__tail](x0) = [1 0 1]x0 + [0] [0 0 0] [0], [1 0 1] [a__head](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [mark](x0) = [1 0 0]x0 [0 0 0] , [1 0 1] [a__incr](x0) = [0 1 0]x0 [0 0 1] , [1] [a__odds] = [1] [0], [1] [odds] = [0] [0], [1] [a__pairs] = [1] [0], [1 0 0] [1 1 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [incr](x0) = [0 0 0]x0 [0 0 0] , [1] [nats] = [0] [0], [0] [0] = [0] [0], [1] [a__nats] = [1] [0] orientation: [1] [1] a__nats() = [1] >= [0] = cons(0(),incr(nats())) [0] [0] [1] [1] a__pairs() = [1] >= [0] = cons(0(),incr(odds())) [0] [0] [1] [1] a__odds() = [1] >= [1] = a__incr(a__pairs()) [0] [0] [1 0 0] [1 1 1] [1 0 0] [1 0 1] 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)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1] [1 0 0] a__tail(cons(X,XS)) = [1 0 0]X + [1 1 1]XS + [0] >= [1 0 0]XS = mark(XS) [0 0 0] [0 0 0] [0] [0 0 0] [1] [1] mark(nats()) = [1] >= [1] = a__nats() [0] [0] [1 0 1] [1 0 0] mark(incr(X)) = [1 0 1]X >= [1 0 0]X = a__incr(mark(X)) [0 0 0] [0 0 0] [1] [1] mark(pairs()) = [1] >= [1] = a__pairs() [0] [0] [1] [1] mark(odds()) = [1] >= [1] = a__odds() [0] [0] [1 0 1] [1 0 0] mark(head(X)) = [1 0 1]X >= [0 0 0]X = a__head(mark(X)) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] mark(tail(X)) = [1 0 0]X + [1] >= [1 0 0]X + [0] = a__tail(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] mark(cons(X1,X2)) = [1 0 0]X1 + [1 1 1]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [1 0 0]X >= [0 0 0]X = s(mark(X)) [0 0 0] [0 0 0] [1] [1] a__nats() = [1] >= [0] = nats() [0] [0] [1 0 1] [1 0 1] a__incr(X) = [0 1 0]X >= [0 0 0]X = incr(X) [0 0 1] [0 0 0] [1] [1] a__pairs() = [1] >= [0] = pairs() [0] [0] [1] [1] a__odds() = [1] >= [0] = odds() [0] [0] [1 0 1] [1 0 1] a__head(X) = [0 0 0]X >= [0 0 0]X = head(X) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] a__tail(X) = [1 0 1]X + [0] >= [0 0 0]X + [0] = tail(X) [0 0 0] [0] [0 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)) 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) = x0 + 1, [head](x0) = x0 + 4, [pairs] = 0, [a__tail](x0) = x0 + 2, [a__head](x0) = x0 + 4, [s](x0) = 2x0, [mark](x0) = 2x0, [a__incr](x0) = 4x0, [a__odds] = 0, [odds] = 0, [a__pairs] = 0, [cons](x0, x1) = x0 + 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)) = 4X + 16XS >= 4X + 16XS = cons(s(mark(X)),incr(XS)) mark(nats()) = 0 >= 0 = a__nats() mark(incr(X)) = 8X >= 8X = a__incr(mark(X)) mark(pairs()) = 0 >= 0 = a__pairs() mark(odds()) = 0 >= 0 = a__odds() mark(head(X)) = 2X + 8 >= 2X + 4 = a__head(mark(X)) mark(tail(X)) = 2X + 2 >= 2X + 2 = a__tail(mark(X)) mark(cons(X1,X2)) = 2X1 + 8X2 >= 2X1 + 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 + 4 >= X + 4 = head(X) a__tail(X) = X + 2 >= X + 1 = 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(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) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [tail](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [head](x0) = [0 0 0]x0 [0 0 0] , [0] [pairs] = [0] [0], [1 0 0] [a__tail](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [a__head](x0) = [0 1 0]x0 + [0] [0 1 0] [1], [1 0 0] [0] [s](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [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] [0], [1 0 0] [cons](x0, x1) = [0 1 0]x0 + x1 [0 1 0] , [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] >= [1] = cons(0(),incr(nats())) [0] [0] [0] [0] a__pairs() = [0] >= [0] = cons(0(),incr(odds())) [0] [0] [0] [0] a__odds() = [0] >= [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 0]X + [0 1 0]XS >= [0 1 0]X + [0 1 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] [0] [0] mark(pairs()) = [0] >= [0] = a__pairs() [0] [0] [0] [0] mark(odds()) = [0] >= [0] = a__odds() [0] [0] [1 1 0] [0] [1 1 0] mark(tail(X)) = [0 0 0]X + [0] >= [0 0 0]X = a__tail(mark(X)) [0 0 0] [1] [0 0 0] [1 1 0] [1 1 0] [1 1 0] mark(cons(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + X2 = cons(mark(X1),X2) [0 1 0] [0 0 1] [0 1 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 1 0] [0] [1 1 0] [0] mark(s(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = s(mark(X)) [0 0 0] [1] [0 0 0] [1] [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] [0] [0] a__pairs() = [0] >= [0] = pairs() [0] [0] [0] [0] a__odds() = [0] >= [0] = odds() [0] [0] [1 0 0] [0] [1 0 0] a__head(X) = [0 1 0]X + [0] >= [0 0 0]X = head(X) [0 1 0] [1] [0 0 0] problem: 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(tail(X)) -> a__tail(mark(X)) 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__odds() -> odds() a__head(X) -> head(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [tail](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [head](x0) = [0 0 0]x0 [0 0 0] , [1] [pairs] = [0] [0], [1 0 0] [a__tail](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [a__head](x0) = [0 0 1]x0 + [0] [1 1 0] [0], [1 0 0] [0] [s](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [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 0]x0 + [0 0 1]x1 [0 1 0] [0 0 1] , [1 0 1] [incr](x0) = [0 1 0]x0 [0 0 1] , [0] [nats] = [1] [1], [0] [0] = [0] [1], [1] [a__nats] = [0] [0] orientation: [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 0]X + [0 0 1]XS >= [0 1 0]X + [0 0 1]XS = cons(s(mark(X)),incr(XS)) [0 1 0] [0 0 1] [0 1 0] [0 0 1] [1] [1] mark(nats()) = [1] >= [0] = a__nats() [1] [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 0] [1 1 0] mark(tail(X)) = [0 1 0]X >= [0 0 0]X = a__tail(mark(X)) [0 0 0] [0 0 0] [1 1 0] [1 0 1] [1 1 0] [1 0 0] mark(cons(X1,X2)) = [0 1 0]X1 + [0 0 1]X2 >= [0 1 0]X1 + [0 0 1]X2 = cons(mark(X1),X2) [0 1 0] [0 0 1] [0 1 0] [0 0 1] [0] [0] mark(0()) = [0] >= [0] = 0() [1] [1] [1 1 0] [0] [1 1 0] [0] mark(s(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = s(mark(X)) [0 0 0] [1] [0 0 0] [1] [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] [1] [1 0 0] a__head(X) = [0 0 1]X + [0] >= [0 0 0]X = head(X) [1 1 0] [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(tail(X)) -> a__tail(mark(X)) 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() Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [tail](x0) = [0 0 0]x0 + [1] [1 0 0] [0], [0] [pairs] = [0] [1], [1 0 0] [a__tail](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [mark](x0) = [0 1 1]x0 [0 1 1] , [1 0 0] [1] [a__incr](x0) = [1 0 1]x0 + [0] [0 1 0] [0], [1] [a__odds] = [1] [1], [1] [odds] = [1] [0], [0] [a__pairs] = [0] [1], [1 0 0] [1 0 0] [cons](x0, x1) = [1 1 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 0] [1] [incr](x0) = [1 0 1]x0 + [0] [0 1 0] [0], [0] [nats] = [0] [0], [0] [0] = [0] [1], [0] [a__nats] = [0] [0] orientation: [1] [1] a__odds() = [1] >= [1] = a__incr(a__pairs()) [1] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] a__incr(cons(X,XS)) = [1 0 1]X + [1 0 0]XS + [0] >= [1 0 0]X + [0 0 0]XS + [0] = cons(s(mark(X)),incr(XS)) [1 1 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 0 0] [1] [1 0 0] [1] mark(incr(X)) = [1 1 1]X + [0] >= [1 1 1]X + [0] = a__incr(mark(X)) [1 1 1] [0] [0 1 1] [0] [0] [0] mark(pairs()) = [1] >= [0] = a__pairs() [1] [1] [1] [1] mark(odds()) = [1] >= [1] = a__odds() [1] [1] [1 0 0] [1] [1 0 0] mark(tail(X)) = [1 0 0]X + [1] >= [1 0 0]X = a__tail(mark(X)) [1 0 0] [1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [1 1 1]X1 + [0 0 0]X2 >= [1 1 1]X1 + [0 0 0]X2 = cons(mark(X1),X2) [1 1 1] [0 0 0] [0 1 1] [0 0 0] [0] [0] mark(0()) = [1] >= [0] = 0() [1] [1] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] a__incr(X) = [1 0 1]X + [0] >= [1 0 1]X + [0] = incr(X) [0 1 0] [0] [0 1 0] [0] [0] [0] a__pairs() = [0] >= [0] = pairs() [1] [1] 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() Matrix Interpretation Processor: dim=3 interpretation: [0] [pairs] = [1] [0], [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [mark](x0) = x0 + [0] [1], [1 0 0] [0] [a__incr](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1] [a__odds] = [1] [1], [1] [odds] = [1] [0], [0] [a__pairs] = [1] [1], [1 0 0] [1 1 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 1 0] [0 0 0] , [1 0 0] [incr](x0) = [0 1 0]x0 [0 0 0] , [0] [nats] = [1] [0], [1] [0] = [0] [0], [0] [a__nats] = [0] [1] orientation: [1] [0] a__odds() = [1] >= [1] = a__incr(a__pairs()) [1] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] a__incr(cons(X,XS)) = [0 0 0]X + [0 0 0]XS + [0] >= [0 0 0]X + [0 0 0]XS + [0] = cons(s(mark(X)),incr(XS)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] mark(nats()) = [1] >= [0] = a__nats() [1] [1] [1 0 0] [0] [1 0 0] [0] mark(incr(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__incr(mark(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] mark(pairs()) = [1] >= [1] = a__pairs() [1] [1] [1] [1] mark(odds()) = [1] >= [1] = a__odds() [1] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [0 1 0] [0 0 0] [1] [0 1 0] [0 0 0] [1] [1] mark(0()) = [0] >= [0] = 0() [1] [0] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(mark(X)) [0 0 0] [1] [0 0 0] [0] [1 0 0] [0] [1 0 0] a__incr(X) = [0 1 0]X + [0] >= [0 1 0]X = incr(X) [0 0 0] [1] [0 0 0] [0] [0] a__pairs() = [1] >= [1] = pairs() [1] [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() Matrix Interpretation Processor: dim=3 interpretation: [1] [pairs] = [0] [0], [1 1 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [mark](x0) = [0 1 0]x0 [1 0 0] , [1 1 0] [a__incr](x0) = [0 1 0]x0 [0 0 0] , [0] [a__odds] = [0] [0], [1] [odds] = [0] [0], [1] [a__pairs] = [0] [0], [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 1 0] [incr](x0) = [0 1 0]x0 [0 0 0] , [0] [nats] = [0] [0], [1] [0] = [0] [0], [0] [a__nats] = [0] [0] orientation: [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [0] 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 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 1 0] [1 1 0] mark(incr(X)) = [0 1 0]X >= [0 1 0]X = a__incr(mark(X)) [1 1 0] [0 0 0] [1] [1] mark(pairs()) = [0] >= [0] = a__pairs() [1] [0] [1] [0] mark(odds()) = [0] >= [0] = a__odds() [1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] mark(cons(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = cons(mark(X1),X2) [1 0 0] [1 0 0] [0] [0 0 0] [0 0 0] [0] [1] [1] mark(0()) = [0] >= [0] = 0() [1] [0] [1 1 0] [1 1 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [1 1 0] [0 0 0] [1 1 0] [1 1 0] a__incr(X) = [0 1 0]X >= [0 1 0]X = incr(X) [0 0 0] [0 0 0] [1] [1] a__pairs() = [0] >= [0] = pairs() [0] [0] problem: mark(nats()) -> a__nats() mark(incr(X)) -> a__incr(mark(X)) mark(pairs()) -> a__pairs() 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() Matrix Interpretation Processor: dim=1 interpretation: [pairs] = 2, [s](x0) = 4x0, [mark](x0) = 5x0, [a__incr](x0) = x0 + 6, [a__pairs] = 2, [cons](x0, x1) = 3x0 + 6x1, [incr](x0) = x0 + 5, [nats] = 0, [0] = 2, [a__nats] = 0 orientation: mark(nats()) = 0 >= 0 = a__nats() mark(incr(X)) = 5X + 25 >= 5X + 6 = a__incr(mark(X)) mark(pairs()) = 10 >= 2 = a__pairs() mark(cons(X1,X2)) = 15X1 + 30X2 >= 15X1 + 6X2 = cons(mark(X1),X2) mark(0()) = 10 >= 2 = 0() mark(s(X)) = 20X >= 20X = s(mark(X)) a__incr(X) = X + 6 >= X + 5 = incr(X) a__pairs() = 2 >= 2 = pairs() problem: mark(nats()) -> a__nats() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__pairs() -> pairs() Matrix Interpretation Processor: dim=3 interpretation: [0] [pairs] = [0] [0], [1 0 1] [1] [s](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [mark](x0) = x0 , [1] [a__pairs] = [0] [0], [1 0 0] [1 0 1] [cons](x0, x1) = [1 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [nats] = [0] [0], [0] [a__nats] = [0] [0] orientation: [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] mark(cons(X1,X2)) = [1 0 0]X1 + [0 0 0]X2 >= [1 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1] [1 0 1] [1] mark(s(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = s(mark(X)) [0 0 0] [0] [0 0 0] [0] [1] [0] a__pairs() = [0] >= [0] = pairs() [0] [0] problem: mark(nats()) -> a__nats() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [1 0 0] [1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 1 1] [1 1 0] [0], [1] [nats] = [0] [0], [0] [a__nats] = [0] [0] orientation: [1] [0] mark(nats()) = [0] >= [0] = a__nats() [1] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(mark(X1),X2) [0 1 1] [1 1 0] [1] [0 0 1] [1 1 0] [1] [1 0 0] [0] [1 0 0] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(mark(X)) [0 0 0] [1] [0 0 0] problem: mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) Matrix Interpretation Processor: dim=3 interpretation: [0] [s](x0) = x0 + [1] [1], [1 0 1] [mark](x0) = [1 0 1]x0 [0 0 1] , [1 0 0] [1 0 0] [0] [cons](x0, x1) = [1 0 0]x0 + [1 0 0]x1 + [1] [0 0 1] [0 0 0] [1] orientation: [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] mark(cons(X1,X2)) = [1 0 1]X1 + [1 0 0]X2 + [1] >= [1 0 1]X1 + [1 0 0]X2 + [1] = cons(mark(X1),X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [0] mark(s(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = s(mark(X)) [0 0 1] [1] [0 0 1] [1] problem: Qed