/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 4x0, [head](x0) = 2x0 + 1, [nats] = 0, [a__tail](x0) = 4x0, [a__head](x0) = 2x0 + 1, [zeros] = 0, [0] = 0, [a__zeros] = 0, [a__nats] = 0, [adx](x0) = x0, [a__adx](x0) = x0, [incr](x0) = x0, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = x0 + 2x1, [a__incr](x0) = x0, [nil] = 0 orientation: a__incr(nil()) = 0 >= 0 = nil() a__incr(cons(X,L)) = 2L + X >= 2L + X = cons(s(mark(X)),incr(L)) a__adx(nil()) = 0 >= 0 = nil() a__adx(cons(X,L)) = 2L + X >= 2L + X = a__incr(cons(mark(X),adx(L))) a__nats() = 0 >= 0 = a__adx(a__zeros()) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__head(cons(X,L)) = 4L + 2X + 1 >= X = mark(X) a__tail(cons(X,L)) = 8L + 4X >= L = mark(L) mark(incr(X)) = X >= X = a__incr(mark(X)) mark(adx(X)) = X >= X = a__adx(mark(X)) mark(nats()) = 0 >= 0 = a__nats() mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = 2X + 1 >= 2X + 1 = a__head(mark(X)) mark(tail(X)) = 4X >= 4X = a__tail(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__incr(X) = X >= X = incr(X) a__adx(X) = X >= X = adx(X) a__nats() = 0 >= 0 = nats() a__zeros() = 0 >= 0 = zeros() a__head(X) = 2X + 1 >= 2X + 1 = head(X) a__tail(X) = 4X >= 4X = tail(X) problem: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = x0 + 4, [head](x0) = 2x0, [nats] = 0, [a__tail](x0) = x0 + 4, [a__head](x0) = 2x0, [zeros] = 0, [0] = 0, [a__zeros] = 0, [a__nats] = 0, [adx](x0) = x0, [a__adx](x0) = x0, [incr](x0) = x0, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [a__incr](x0) = x0, [nil] = 0 orientation: a__incr(nil()) = 0 >= 0 = nil() a__incr(cons(X,L)) = L + X >= L + X = cons(s(mark(X)),incr(L)) a__adx(nil()) = 0 >= 0 = nil() a__adx(cons(X,L)) = L + X >= L + X = a__incr(cons(mark(X),adx(L))) a__nats() = 0 >= 0 = a__adx(a__zeros()) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__tail(cons(X,L)) = L + X + 4 >= L = mark(L) mark(incr(X)) = X >= X = a__incr(mark(X)) mark(adx(X)) = X >= X = a__adx(mark(X)) mark(nats()) = 0 >= 0 = a__nats() mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = 2X >= 2X = a__head(mark(X)) mark(tail(X)) = X + 4 >= X + 4 = a__tail(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__incr(X) = X >= X = incr(X) a__adx(X) = X >= X = adx(X) a__nats() = 0 >= 0 = nats() a__zeros() = 0 >= 0 = zeros() a__head(X) = 2X >= 2X = head(X) a__tail(X) = X + 4 >= X + 4 = tail(X) problem: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = x0, [head](x0) = x0, [nats] = 5, [a__tail](x0) = x0, [a__head](x0) = x0, [zeros] = 0, [0] = 0, [a__zeros] = 0, [a__nats] = 5, [adx](x0) = x0 + 1, [a__adx](x0) = x0 + 1, [incr](x0) = x0, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = 2x0 + x1, [a__incr](x0) = x0, [nil] = 0 orientation: a__incr(nil()) = 0 >= 0 = nil() a__incr(cons(X,L)) = L + 2X >= L + 2X = cons(s(mark(X)),incr(L)) a__adx(nil()) = 1 >= 0 = nil() a__adx(cons(X,L)) = L + 2X + 1 >= L + 2X + 1 = a__incr(cons(mark(X),adx(L))) a__nats() = 5 >= 1 = a__adx(a__zeros()) a__zeros() = 0 >= 0 = cons(0(),zeros()) mark(incr(X)) = X >= X = a__incr(mark(X)) mark(adx(X)) = X + 1 >= X + 1 = a__adx(mark(X)) mark(nats()) = 5 >= 5 = a__nats() mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = X >= X = a__head(mark(X)) mark(tail(X)) = X >= X = a__tail(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__incr(X) = X >= X = incr(X) a__adx(X) = X + 1 >= X + 1 = adx(X) a__nats() = 5 >= 5 = nats() a__zeros() = 0 >= 0 = zeros() a__head(X) = X >= X = head(X) a__tail(X) = X >= X = tail(X) problem: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__zeros() -> cons(0(),zeros()) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [tail](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [head](x0) = [1 0 0]x0 [0 0 0] , [0] [nats] = [0] [0], [1 0 0] [a__tail](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [a__head](x0) = [1 0 0]x0 [0 0 0] , [0] [zeros] = [0] [0], [0] [0] = [0] [0], [0] [a__zeros] = [0] [0], [0] [a__nats] = [0] [0], [1 0 0] [1] [adx](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1] [a__adx](x0) = [1 0 0]x0 + [1] [0 0 0] [0], [1 0 1] [incr](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [mark](x0) = [1 0 0]x0 [0 0 1] , [1 0 0] [1 1 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] , [1 0 1] [a__incr](x0) = [1 0 0]x0 [0 0 1] , [0] [nil] = [0] [1] orientation: [1] [0] a__incr(nil()) = [0] >= [0] = nil() [1] [1] [1 1 1] [1 0 0] [1 0 1] [1 0 0] a__incr(cons(X,L)) = [1 1 0]L + [1 0 0]X >= [0 0 0]L + [0 0 0]X = cons(s(mark(X)),incr(L)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 1 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] a__adx(cons(X,L)) = [1 1 0]L + [1 0 0]X + [1] >= [1 0 0]L + [1 0 0]X + [1] = a__incr(cons(mark(X),adx(L))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1 0 1] [1 0 1] mark(incr(X)) = [1 0 1]X >= [1 0 0]X = a__incr(mark(X)) [0 0 1] [0 0 1] [1 0 0] [1] [1 0 0] [1] mark(adx(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = a__adx(mark(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 0 0] [1 0 0] mark(head(X)) = [1 0 0]X >= [1 0 0]X = a__head(mark(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(tail(X)) = [1 0 0]X >= [1 0 0]X = a__tail(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = nil() [1] [1] [1 0 0] [1 1 0] [1 0 0] [1 1 0] mark(cons(X1,X2)) = [1 0 0]X1 + [1 1 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [0 0 0] [0 0 1] [0 0 0] [0 0 1] [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] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 1] [1 0 1] a__incr(X) = [1 0 0]X >= [0 0 0]X = incr(X) [0 0 1] [0 0 1] [1 0 0] [1] [1 0 0] [1] a__adx(X) = [1 0 0]X + [1] >= [0 0 0]X + [0] = adx(X) [0 0 0] [0] [0 0 0] [0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [1 0 0] [1 0 0] a__head(X) = [1 0 0]X >= [1 0 0]X = head(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__tail(X) = [1 0 0]X >= [1 0 0]X = tail(X) [0 0 0] [0 0 0] problem: a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__zeros() -> cons(0(),zeros()) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [tail](x0) = [0 1 0]x0 + [0] [0 1 0] [1], [1 0 0] [0] [head](x0) = [0 1 1]x0 + [0] [0 0 0] [1], [0] [nats] = [0] [0], [1 0 0] [0] [a__tail](x0) = [0 1 0]x0 + [0] [0 1 0] [1], [1 0 0] [0] [a__head](x0) = [0 1 1]x0 + [0] [0 0 0] [1], [0] [zeros] = [0] [0], [0] [0] = [0] [1], [0] [a__zeros] = [0] [0], [0] [a__nats] = [0] [0], [1 1 0] [1] [adx](x0) = [0 1 1]x0 + [1] [0 0 1] [0], [1 1 1] [1] [a__adx](x0) = [0 1 1]x0 + [1] [0 0 1] [0], [1 0 1] [incr](x0) = [0 1 0]x0 [0 0 1] , [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 0] [cons](x0, x1) = [0 1 0]x0 + x1 [0 1 0] , [1 0 1] [a__incr](x0) = [0 1 0]x0 [0 0 1] , [0] [nil] = [0] [0] orientation: [1 0 1] [1 1 0] [1 0 1] [1 1 0] a__incr(cons(X,L)) = [0 1 0]L + [0 1 0]X >= [0 1 0]L + [0 1 0]X = cons(s(mark(X)),incr(L)) [0 0 1] [0 1 0] [0 0 1] [0 1 0] [1 1 1] [1 2 0] [1] [1 1 1] [1 2 0] [1] a__adx(cons(X,L)) = [0 1 1]L + [0 2 0]X + [1] >= [0 1 1]L + [0 1 0]X + [1] = a__incr(cons(mark(X),adx(L))) [0 0 1] [0 1 0] [0] [0 0 1] [0 1 0] [0] [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [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 2 1] [2] [1 2 1] [1] mark(adx(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = a__adx(mark(X)) [0 0 1] [0] [0 0 1] [0] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 1 1] [0] [1 1 0] [0] mark(head(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = a__head(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] mark(tail(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__tail(mark(X)) [0 1 0] [1] [0 1 0] [1] [0] [0] mark(nil()) = [0] >= [0] = nil() [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] [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] [0] [0] mark(0()) = [0] >= [0] = 0() [1] [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 1] [1] [1 1 0] [1] a__adx(X) = [0 1 1]X + [1] >= [0 1 1]X + [1] = adx(X) [0 0 1] [0] [0 0 1] [0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [1 0 0] [0] [1 0 0] [0] a__head(X) = [0 1 1]X + [0] >= [0 1 1]X + [0] = head(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] a__tail(X) = [0 1 0]X + [0] >= [0 1 0]X + [0] = tail(X) [0 1 0] [1] [0 1 0] [1] problem: a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__zeros() -> cons(0(),zeros()) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 4x0, [head](x0) = x0, [nats] = 0, [a__tail](x0) = 4x0, [a__head](x0) = x0, [zeros] = 0, [0] = 0, [a__zeros] = 0, [a__nats] = 0, [adx](x0) = x0, [a__adx](x0) = 4x0 + 5, [incr](x0) = 4x0, [s](x0) = 4x0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + x1, [a__incr](x0) = 4x0, [nil] = 2 orientation: a__incr(cons(X,L)) = 4L + 16X >= 4L + 16X = cons(s(mark(X)),incr(L)) a__adx(cons(X,L)) = 4L + 16X + 5 >= 4L + 16X = a__incr(cons(mark(X),adx(L))) a__zeros() = 0 >= 0 = cons(0(),zeros()) mark(incr(X)) = 4X >= 4X = a__incr(mark(X)) mark(nats()) = 0 >= 0 = a__nats() mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = X >= X = a__head(mark(X)) mark(tail(X)) = 4X >= 4X = a__tail(mark(X)) mark(nil()) = 2 >= 2 = nil() mark(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(mark(X1),X2) mark(s(X)) = 4X >= 4X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__incr(X) = 4X >= 4X = incr(X) a__adx(X) = 4X + 5 >= X = adx(X) a__nats() = 0 >= 0 = nats() a__zeros() = 0 >= 0 = zeros() a__head(X) = X >= X = head(X) a__tail(X) = 4X >= 4X = tail(X) problem: a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__zeros() -> cons(0(),zeros()) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [tail](x0) = [0 1 0]x0 + [1] [0 0 0] [1], [1 0 0] [0] [head](x0) = [0 1 1]x0 + [1] [0 0 0] [0], [0] [nats] = [1] [0], [1 0 0] [0] [a__tail](x0) = [0 1 0]x0 + [1] [0 0 0] [1], [1 0 0] [1] [a__head](x0) = [0 1 1]x0 + [1] [0 0 0] [0], [0] [zeros] = [1] [0], [1] [0] = [0] [0], [1] [a__zeros] = [1] [0], [0] [a__nats] = [1] [0], [1 0 1] [0] [incr](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 0 1] [s](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 1 1] [0 0 1] , [1 0 1] [1] [a__incr](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [0] [nil] = [0] [0] orientation: [1 0 1] [1 1 1] [1] [1 0 1] [1 1 1] a__incr(cons(X,L)) = [0 0 0]L + [0 1 0]X + [1] >= [0 0 0]L + [0 1 0]X = cons(s(mark(X)),incr(L)) [0 0 1] [0 1 1] [0] [0 0 1] [0 1 0] [1] [1] a__zeros() = [1] >= [0] = cons(0(),zeros()) [0] [0] [1 1 1] [1] [1 1 1] [1] mark(incr(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__incr(mark(X)) [0 0 1] [0] [0 0 1] [0] [1] [0] mark(nats()) = [1] >= [1] = a__nats() [0] [0] [1] [1] mark(zeros()) = [1] >= [1] = a__zeros() [0] [0] [1 1 1] [1] [1 1 0] [1] mark(head(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = a__head(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1] [1 1 0] [0] mark(tail(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__tail(mark(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] mark(nil()) = [0] >= [0] = nil() [0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] mark(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [0 1 1] [0 0 1] [0 1 1] [0 0 1] [1 1 1] [1 1 1] mark(s(X)) = [0 1 0]X >= [0 1 0]X = s(mark(X)) [0 0 0] [0 0 0] [1] [1] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 1] [1] [1 0 1] [0] a__incr(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = incr(X) [0 0 1] [0] [0 0 1] [0] [0] [0] a__nats() = [1] >= [1] = nats() [0] [0] [1] [0] a__zeros() = [1] >= [1] = zeros() [0] [0] [1 0 0] [1] [1 0 0] [0] a__head(X) = [0 1 1]X + [1] >= [0 1 1]X + [1] = head(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] a__tail(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = tail(X) [0 0 0] [1] [0 0 0] [1] problem: a__zeros() -> cons(0(),zeros()) mark(incr(X)) -> a__incr(mark(X)) mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__nats() -> nats() a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 0] , [head](x0) = x0 , [0] [nats] = [0] [0], [1 0 0] [a__tail](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__head](x0) = [0 0 0]x0 [0 0 0] , [0] [zeros] = [0] [1], [0] [0] = [1] [0], [1] [a__zeros] = [1] [1], [0] [a__nats] = [0] [0], [1 0 0] [1] [incr](x0) = [0 0 0]x0 + [0] [0 1 1] [0], [1 0 0] [0] [s](x0) = [0 0 1]x0 + [0] [0 1 0] [1], [1 1 1] [mark](x0) = [1 1 1]x0 [1 1 1] , [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 0 1]x0 + [0 0 0]x1 + [1] [0 1 0] [0 1 0] [0], [1 0 0] [1] [a__incr](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [nil] = [0] [0] orientation: [1] [0] a__zeros() = [1] >= [1] = cons(0(),zeros()) [1] [1] [1 1 1] [1] [1 1 1] [1] mark(incr(X)) = [1 1 1]X + [1] >= [0 0 0]X + [0] = a__incr(mark(X)) [1 1 1] [1] [0 0 0] [0] [1] [1] mark(zeros()) = [1] >= [1] = a__zeros() [1] [1] [1 1 1] [1 1 1] mark(head(X)) = [1 1 1]X >= [0 0 0]X = a__head(mark(X)) [1 1 1] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = nil() [0] [0] [1 1 1] [1 1 0] [1] [1 1 1] [1 0 0] [0] mark(cons(X1,X2)) = [1 1 1]X1 + [1 1 0]X2 + [1] >= [1 1 1]X1 + [0 0 0]X2 + [1] = cons(mark(X1),X2) [1 1 1] [1 1 0] [1] [1 1 1] [0 1 0] [0] [1 1 1] [1] [1 1 1] [0] mark(s(X)) = [1 1 1]X + [1] >= [1 1 1]X + [0] = s(mark(X)) [1 1 1] [1] [1 1 1] [1] [1] [0] mark(0()) = [1] >= [1] = 0() [1] [0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [1 0 0] [1 0 0] a__tail(X) = [0 0 0]X >= [0 0 0]X = tail(X) [0 0 0] [0 0 0] problem: mark(incr(X)) -> a__incr(mark(X)) mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(nil()) -> nil() a__nats() -> nats() a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = x0, [head](x0) = x0, [nats] = 0, [a__tail](x0) = x0 + 1, [a__head](x0) = x0, [zeros] = 0, [a__zeros] = 0, [a__nats] = 0, [incr](x0) = x0 + 2, [mark](x0) = x0, [a__incr](x0) = x0 + 2, [nil] = 4 orientation: mark(incr(X)) = X + 2 >= X + 2 = a__incr(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = X >= X = a__head(mark(X)) mark(nil()) = 4 >= 4 = nil() a__nats() = 0 >= 0 = nats() a__tail(X) = X + 1 >= X = tail(X) problem: mark(incr(X)) -> a__incr(mark(X)) mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(nil()) -> nil() a__nats() -> nats() Matrix Interpretation Processor: dim=1 interpretation: [head](x0) = x0, [nats] = 0, [a__head](x0) = x0, [zeros] = 0, [a__zeros] = 0, [a__nats] = 1, [incr](x0) = 4x0 + 4, [mark](x0) = x0, [a__incr](x0) = 4x0 + 4, [nil] = 0 orientation: mark(incr(X)) = 4X + 4 >= 4X + 4 = a__incr(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = X >= X = a__head(mark(X)) mark(nil()) = 0 >= 0 = nil() a__nats() = 1 >= 0 = nats() problem: mark(incr(X)) -> a__incr(mark(X)) mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(nil()) -> nil() Matrix Interpretation Processor: dim=1 interpretation: [head](x0) = 4x0 + 6, [a__head](x0) = 4x0 + 3, [zeros] = 0, [a__zeros] = 1, [incr](x0) = x0 + 3, [mark](x0) = x0 + 1, [a__incr](x0) = x0 + 3, [nil] = 0 orientation: mark(incr(X)) = X + 4 >= X + 4 = a__incr(mark(X)) mark(zeros()) = 1 >= 1 = a__zeros() mark(head(X)) = 4X + 7 >= 4X + 7 = a__head(mark(X)) mark(nil()) = 1 >= 0 = nil() problem: mark(incr(X)) -> a__incr(mark(X)) mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [head](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [a__head](x0) = [0 0 0]x0 [0 0 0] , [1] [zeros] = [0] [0], [0] [a__zeros] = [0] [0], [1 0 1] [1] [incr](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__incr](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 1] [1] [1 0 1] mark(incr(X)) = [0 0 0]X + [0] >= [0 0 0]X = a__incr(mark(X)) [0 0 0] [0] [0 0 0] [1] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 1 1] [1] [1 0 1] mark(head(X)) = [0 0 0]X + [0] >= [0 0 0]X = a__head(mark(X)) [0 0 0] [0] [0 0 0] problem: Qed