/export/starexec/sandbox2/solver/bin/starexec_run_ttt2 /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=2 interpretation: [1 0] [s](x0) = [0 0]x0, [0] [dbl](x0) = x0 + [1], [2 0] [active](x0) = [0 1]x0, [recip](x0) = x0, [2 0] [1 0] [cons](x0, x1) = [0 2]x0 + [0 0]x1, [2 0] [0] [first](x0, x1) = [0 2]x0 + x1 + [1], [2 0] [0] [terms](x0) = [0 2]x0 + [3], [0] [0] = [0], [1 2] [0] [top](x0) = [0 0]x0 + [1], [proper](x0) = x0, [0] [nil] = [0], [0] [add](x0, x1) = x0 + x1 + [1], [2 0] [ok](x0) = [0 1]x0, [0] [mark](x0) = x0 + [1], [0] [sqr](x0) = x0 + [1] orientation: [4 0] [0] [4 0] [0] active(terms(N)) = [0 2]N + [3] >= [0 2]N + [3] = mark(cons(recip(sqr(N)),terms(s(N)))) [0] [0] active(sqr(0())) = [1] >= [1] = mark(0()) [2 0] [0] [2 0] [0] active(sqr(s(X))) = [0 0]X + [1] >= [0 0]X + [1] = mark(s(add(sqr(X),dbl(X)))) [0] [0] active(dbl(0())) = [1] >= [1] = mark(0()) [2 0] [0] [1 0] [0] active(dbl(s(X))) = [0 0]X + [1] >= [0 0]X + [1] = mark(s(s(dbl(X)))) [2 0] [0] [0] active(add(0(),X)) = [0 1]X + [1] >= X + [1] = mark(X) [2 0] [2 0] [0] [1 0] [1 0] [0] active(add(s(X),Y)) = [0 0]X + [0 1]Y + [1] >= [0 0]X + [0 0]Y + [1] = mark(s(add(X,Y))) [2 0] [0] [0] active(first(0(),X)) = [0 1]X + [1] >= [1] = mark(nil()) [4 0] [4 0] [2 0] [0] [2 0] [2 0] [1 0] [0] active(first(s(X),cons(Y,Z))) = [0 0]X + [0 2]Y + [0 0]Z + [1] >= [0 0]X + [0 2]Y + [0 0]Z + [1] = mark(cons(Y,first(X,Z))) [4 0] [0] [4 0] [0] active(terms(X)) = [0 2]X + [3] >= [0 2]X + [3] = terms(active(X)) [4 0] [2 0] [4 0] [1 0] active(cons(X1,X2)) = [0 2]X1 + [0 0]X2 >= [0 2]X1 + [0 0]X2 = cons(active(X1),X2) [2 0] [2 0] active(recip(X)) = [0 1]X >= [0 1]X = recip(active(X)) [2 0] [0] [2 0] [0] active(sqr(X)) = [0 1]X + [1] >= [0 1]X + [1] = sqr(active(X)) [2 0] [2 0] [0] [2 0] [0] active(add(X1,X2)) = [0 1]X1 + [0 1]X2 + [1] >= [0 1]X1 + X2 + [1] = add(active(X1),X2) [2 0] [2 0] [0] [2 0] [0] active(add(X1,X2)) = [0 1]X1 + [0 1]X2 + [1] >= X1 + [0 1]X2 + [1] = add(X1,active(X2)) [2 0] [0] [2 0] [0] active(dbl(X)) = [0 1]X + [1] >= [0 1]X + [1] = dbl(active(X)) [4 0] [2 0] [0] [4 0] [0] active(first(X1,X2)) = [0 2]X1 + [0 1]X2 + [1] >= [0 2]X1 + X2 + [1] = first(active(X1),X2) [4 0] [2 0] [0] [2 0] [2 0] [0] active(first(X1,X2)) = [0 2]X1 + [0 1]X2 + [1] >= [0 2]X1 + [0 1]X2 + [1] = first(X1,active(X2)) [2 0] [0] [2 0] [0] terms(mark(X)) = [0 2]X + [5] >= [0 2]X + [4] = mark(terms(X)) [2 0] [1 0] [0] [2 0] [1 0] [0] cons(mark(X1),X2) = [0 2]X1 + [0 0]X2 + [2] >= [0 2]X1 + [0 0]X2 + [1] = mark(cons(X1,X2)) [0] [0] recip(mark(X)) = X + [1] >= X + [1] = mark(recip(X)) [0] [0] sqr(mark(X)) = X + [2] >= X + [2] = mark(sqr(X)) [0] [0] add(mark(X1),X2) = X1 + X2 + [2] >= X1 + X2 + [2] = mark(add(X1,X2)) [0] [0] add(X1,mark(X2)) = X1 + X2 + [2] >= X1 + X2 + [2] = mark(add(X1,X2)) [0] [0] dbl(mark(X)) = X + [2] >= X + [2] = mark(dbl(X)) [2 0] [0] [2 0] [0] first(mark(X1),X2) = [0 2]X1 + X2 + [3] >= [0 2]X1 + X2 + [2] = mark(first(X1,X2)) [2 0] [0] [2 0] [0] first(X1,mark(X2)) = [0 2]X1 + X2 + [2] >= [0 2]X1 + X2 + [2] = mark(first(X1,X2)) [2 0] [0] [2 0] [0] proper(terms(X)) = [0 2]X + [3] >= [0 2]X + [3] = terms(proper(X)) [2 0] [1 0] [2 0] [1 0] proper(cons(X1,X2)) = [0 2]X1 + [0 0]X2 >= [0 2]X1 + [0 0]X2 = cons(proper(X1),proper(X2)) proper(recip(X)) = X >= X = recip(proper(X)) [0] [0] proper(sqr(X)) = X + [1] >= X + [1] = sqr(proper(X)) [1 0] [1 0] proper(s(X)) = [0 0]X >= [0 0]X = s(proper(X)) [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] proper(add(X1,X2)) = X1 + X2 + [1] >= X1 + X2 + [1] = add(proper(X1),proper(X2)) [0] [0] proper(dbl(X)) = X + [1] >= X + [1] = dbl(proper(X)) [2 0] [0] [2 0] [0] proper(first(X1,X2)) = [0 2]X1 + X2 + [1] >= [0 2]X1 + X2 + [1] = first(proper(X1),proper(X2)) [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [4 0] [0] [4 0] [0] terms(ok(X)) = [0 2]X + [3] >= [0 2]X + [3] = ok(terms(X)) [4 0] [2 0] [4 0] [2 0] cons(ok(X1),ok(X2)) = [0 2]X1 + [0 0]X2 >= [0 2]X1 + [0 0]X2 = ok(cons(X1,X2)) [2 0] [2 0] recip(ok(X)) = [0 1]X >= [0 1]X = ok(recip(X)) [2 0] [0] [2 0] [0] sqr(ok(X)) = [0 1]X + [1] >= [0 1]X + [1] = ok(sqr(X)) [2 0] [2 0] s(ok(X)) = [0 0]X >= [0 0]X = ok(s(X)) [2 0] [2 0] [0] [2 0] [2 0] [0] add(ok(X1),ok(X2)) = [0 1]X1 + [0 1]X2 + [1] >= [0 1]X1 + [0 1]X2 + [1] = ok(add(X1,X2)) [2 0] [0] [2 0] [0] dbl(ok(X)) = [0 1]X + [1] >= [0 1]X + [1] = ok(dbl(X)) [4 0] [2 0] [0] [4 0] [2 0] [0] first(ok(X1),ok(X2)) = [0 2]X1 + [0 1]X2 + [1] >= [0 2]X1 + [0 1]X2 + [1] = ok(first(X1,X2)) [1 2] [2] [1 2] [0] top(mark(X)) = [0 0]X + [1] >= [0 0]X + [1] = top(proper(X)) [2 2] [0] [2 2] [0] top(ok(X)) = [0 0]X + [1] >= [0 0]X + [1] = top(active(X)) problem: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0, [dbl](x0) = x0, [active](x0) = 2x0, [recip](x0) = 2x0, [cons](x0, x1) = 2x0 + x1, [first](x0, x1) = 2x0 + x1, [terms](x0) = 4x0, [0] = 4, [top](x0) = 2x0 + 2, [proper](x0) = 4x0, [nil] = 0, [add](x0, x1) = x0 + x1, [ok](x0) = 4x0, [mark](x0) = x0, [sqr](x0) = x0 orientation: active(terms(N)) = 8N >= 8N = mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) = 8 >= 4 = mark(0()) active(sqr(s(X))) = 2X >= 2X = mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) = 8 >= 4 = mark(0()) active(dbl(s(X))) = 2X >= X = mark(s(s(dbl(X)))) active(add(0(),X)) = 2X + 8 >= X = mark(X) active(add(s(X),Y)) = 2X + 2Y >= X + Y = mark(s(add(X,Y))) active(first(0(),X)) = 2X + 16 >= 0 = mark(nil()) active(first(s(X),cons(Y,Z))) = 4X + 4Y + 2Z >= 2X + 2Y + Z = mark(cons(Y,first(X,Z))) active(terms(X)) = 8X >= 8X = terms(active(X)) active(cons(X1,X2)) = 4X1 + 2X2 >= 4X1 + X2 = cons(active(X1),X2) active(recip(X)) = 4X >= 4X = recip(active(X)) active(sqr(X)) = 2X >= 2X = sqr(active(X)) active(add(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = add(active(X1),X2) active(add(X1,X2)) = 2X1 + 2X2 >= X1 + 2X2 = add(X1,active(X2)) active(dbl(X)) = 2X >= 2X = dbl(active(X)) active(first(X1,X2)) = 4X1 + 2X2 >= 4X1 + X2 = first(active(X1),X2) active(first(X1,X2)) = 4X1 + 2X2 >= 2X1 + 2X2 = first(X1,active(X2)) terms(mark(X)) = 4X >= 4X = mark(terms(X)) cons(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(cons(X1,X2)) recip(mark(X)) = 2X >= 2X = mark(recip(X)) sqr(mark(X)) = X >= X = mark(sqr(X)) add(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(add(X1,X2)) add(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(add(X1,X2)) dbl(mark(X)) = X >= X = mark(dbl(X)) first(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(first(X1,X2)) first(X1,mark(X2)) = 2X1 + X2 >= 2X1 + X2 = mark(first(X1,X2)) proper(terms(X)) = 16X >= 16X = terms(proper(X)) proper(cons(X1,X2)) = 8X1 + 4X2 >= 8X1 + 4X2 = cons(proper(X1),proper(X2)) proper(recip(X)) = 8X >= 8X = recip(proper(X)) proper(sqr(X)) = 4X >= 4X = sqr(proper(X)) proper(s(X)) = 4X >= 4X = s(proper(X)) proper(0()) = 16 >= 16 = ok(0()) proper(add(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = add(proper(X1),proper(X2)) proper(dbl(X)) = 4X >= 4X = dbl(proper(X)) proper(first(X1,X2)) = 8X1 + 4X2 >= 8X1 + 4X2 = first(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) terms(ok(X)) = 16X >= 16X = ok(terms(X)) cons(ok(X1),ok(X2)) = 8X1 + 4X2 >= 8X1 + 4X2 = ok(cons(X1,X2)) recip(ok(X)) = 8X >= 8X = ok(recip(X)) sqr(ok(X)) = 4X >= 4X = ok(sqr(X)) s(ok(X)) = 4X >= 4X = ok(s(X)) add(ok(X1),ok(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = ok(add(X1,X2)) dbl(ok(X)) = 4X >= 4X = ok(dbl(X)) first(ok(X1),ok(X2)) = 8X1 + 4X2 >= 8X1 + 4X2 = ok(first(X1,X2)) top(ok(X)) = 8X + 2 >= 4X + 2 = top(active(X)) problem: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0, [dbl](x0) = x0, [active](x0) = 4x0, [recip](x0) = 4x0, [cons](x0, x1) = x0 + 2x1, [first](x0, x1) = x0 + x1, [terms](x0) = 2x0, [0] = 0, [top](x0) = x0 + 4, [proper](x0) = 5x0, [nil] = 1, [add](x0, x1) = 2x0 + 2x1, [ok](x0) = 4x0, [mark](x0) = x0, [sqr](x0) = x0 orientation: active(terms(N)) = 8N >= 8N = mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(s(X))) = 4X >= 4X = mark(s(add(sqr(X),dbl(X)))) active(dbl(s(X))) = 4X >= X = mark(s(s(dbl(X)))) active(add(s(X),Y)) = 8X + 8Y >= 2X + 2Y = mark(s(add(X,Y))) active(first(s(X),cons(Y,Z))) = 4X + 4Y + 8Z >= 2X + Y + 2Z = mark(cons(Y,first(X,Z))) active(terms(X)) = 8X >= 8X = terms(active(X)) active(cons(X1,X2)) = 4X1 + 8X2 >= 4X1 + 2X2 = cons(active(X1),X2) active(recip(X)) = 16X >= 16X = recip(active(X)) active(sqr(X)) = 4X >= 4X = sqr(active(X)) active(add(X1,X2)) = 8X1 + 8X2 >= 8X1 + 2X2 = add(active(X1),X2) active(add(X1,X2)) = 8X1 + 8X2 >= 2X1 + 8X2 = add(X1,active(X2)) active(dbl(X)) = 4X >= 4X = dbl(active(X)) active(first(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = first(active(X1),X2) active(first(X1,X2)) = 4X1 + 4X2 >= X1 + 4X2 = first(X1,active(X2)) terms(mark(X)) = 2X >= 2X = mark(terms(X)) cons(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(cons(X1,X2)) recip(mark(X)) = 4X >= 4X = mark(recip(X)) sqr(mark(X)) = X >= X = mark(sqr(X)) add(mark(X1),X2) = 2X1 + 2X2 >= 2X1 + 2X2 = mark(add(X1,X2)) add(X1,mark(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = mark(add(X1,X2)) dbl(mark(X)) = X >= X = mark(dbl(X)) first(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(first(X1,X2)) proper(terms(X)) = 10X >= 10X = terms(proper(X)) proper(cons(X1,X2)) = 5X1 + 10X2 >= 5X1 + 10X2 = cons(proper(X1),proper(X2)) proper(recip(X)) = 20X >= 20X = recip(proper(X)) proper(sqr(X)) = 5X >= 5X = sqr(proper(X)) proper(s(X)) = 5X >= 5X = s(proper(X)) proper(0()) = 0 >= 0 = ok(0()) proper(add(X1,X2)) = 10X1 + 10X2 >= 10X1 + 10X2 = add(proper(X1),proper(X2)) proper(dbl(X)) = 5X >= 5X = dbl(proper(X)) proper(first(X1,X2)) = 5X1 + 5X2 >= 5X1 + 5X2 = first(proper(X1),proper(X2)) proper(nil()) = 5 >= 4 = ok(nil()) terms(ok(X)) = 8X >= 8X = ok(terms(X)) cons(ok(X1),ok(X2)) = 4X1 + 8X2 >= 4X1 + 8X2 = ok(cons(X1,X2)) recip(ok(X)) = 16X >= 16X = ok(recip(X)) sqr(ok(X)) = 4X >= 4X = ok(sqr(X)) s(ok(X)) = 4X >= 4X = ok(s(X)) add(ok(X1),ok(X2)) = 8X1 + 8X2 >= 8X1 + 8X2 = ok(add(X1,X2)) dbl(ok(X)) = 4X >= 4X = ok(dbl(X)) first(ok(X1),ok(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = ok(first(X1,X2)) top(ok(X)) = 4X + 4 >= 4X + 4 = top(active(X)) problem: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0, [dbl](x0) = 4x0, [active](x0) = 5x0, [recip](x0) = x0, [cons](x0, x1) = x0 + 4x1, [first](x0, x1) = x0 + x1, [terms](x0) = 2x0, [0] = 1, [top](x0) = x0, [proper](x0) = 7x0, [add](x0, x1) = x0 + x1, [ok](x0) = 6x0, [mark](x0) = x0, [sqr](x0) = x0 orientation: active(terms(N)) = 10N >= 9N = mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(s(X))) = 5X >= 5X = mark(s(add(sqr(X),dbl(X)))) active(dbl(s(X))) = 20X >= 4X = mark(s(s(dbl(X)))) active(add(s(X),Y)) = 5X + 5Y >= X + Y = mark(s(add(X,Y))) active(first(s(X),cons(Y,Z))) = 5X + 5Y + 20Z >= 4X + Y + 4Z = mark(cons(Y,first(X,Z))) active(terms(X)) = 10X >= 10X = terms(active(X)) active(cons(X1,X2)) = 5X1 + 20X2 >= 5X1 + 4X2 = cons(active(X1),X2) active(recip(X)) = 5X >= 5X = recip(active(X)) active(sqr(X)) = 5X >= 5X = sqr(active(X)) active(add(X1,X2)) = 5X1 + 5X2 >= 5X1 + X2 = add(active(X1),X2) active(add(X1,X2)) = 5X1 + 5X2 >= X1 + 5X2 = add(X1,active(X2)) active(dbl(X)) = 20X >= 20X = dbl(active(X)) active(first(X1,X2)) = 5X1 + 5X2 >= 5X1 + X2 = first(active(X1),X2) active(first(X1,X2)) = 5X1 + 5X2 >= X1 + 5X2 = first(X1,active(X2)) terms(mark(X)) = 2X >= 2X = mark(terms(X)) cons(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(cons(X1,X2)) recip(mark(X)) = X >= X = mark(recip(X)) sqr(mark(X)) = X >= X = mark(sqr(X)) add(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(add(X1,X2)) add(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(add(X1,X2)) dbl(mark(X)) = 4X >= 4X = mark(dbl(X)) first(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(first(X1,X2)) proper(terms(X)) = 14X >= 14X = terms(proper(X)) proper(cons(X1,X2)) = 7X1 + 28X2 >= 7X1 + 28X2 = cons(proper(X1),proper(X2)) proper(recip(X)) = 7X >= 7X = recip(proper(X)) proper(sqr(X)) = 7X >= 7X = sqr(proper(X)) proper(s(X)) = 7X >= 7X = s(proper(X)) proper(0()) = 7 >= 6 = ok(0()) proper(add(X1,X2)) = 7X1 + 7X2 >= 7X1 + 7X2 = add(proper(X1),proper(X2)) proper(dbl(X)) = 28X >= 28X = dbl(proper(X)) proper(first(X1,X2)) = 7X1 + 7X2 >= 7X1 + 7X2 = first(proper(X1),proper(X2)) terms(ok(X)) = 12X >= 12X = ok(terms(X)) cons(ok(X1),ok(X2)) = 6X1 + 24X2 >= 6X1 + 24X2 = ok(cons(X1,X2)) recip(ok(X)) = 6X >= 6X = ok(recip(X)) sqr(ok(X)) = 6X >= 6X = ok(sqr(X)) s(ok(X)) = 6X >= 6X = ok(s(X)) add(ok(X1),ok(X2)) = 6X1 + 6X2 >= 6X1 + 6X2 = ok(add(X1,X2)) dbl(ok(X)) = 24X >= 24X = ok(dbl(X)) first(ok(X1),ok(X2)) = 6X1 + 6X2 >= 6X1 + 6X2 = ok(first(X1,X2)) top(ok(X)) = 6X >= 5X = top(active(X)) problem: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = 2x0, [dbl](x0) = 2x0, [active](x0) = 3x0, [recip](x0) = x0, [cons](x0, x1) = x0 + x1, [first](x0, x1) = x0 + 4x1 + 4, [terms](x0) = 4x0, [top](x0) = 4x0, [proper](x0) = x0, [add](x0, x1) = x0 + 4x1, [ok](x0) = 3x0 + 2, [mark](x0) = x0, [sqr](x0) = 4x0 orientation: active(terms(N)) = 12N >= 12N = mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(s(X))) = 24X >= 24X = mark(s(add(sqr(X),dbl(X)))) active(dbl(s(X))) = 12X >= 8X = mark(s(s(dbl(X)))) active(add(s(X),Y)) = 6X + 12Y >= 2X + 8Y = mark(s(add(X,Y))) active(first(s(X),cons(Y,Z))) = 6X + 12Y + 12Z + 12 >= X + Y + 4Z + 4 = mark(cons(Y,first(X,Z))) active(terms(X)) = 12X >= 12X = terms(active(X)) active(cons(X1,X2)) = 3X1 + 3X2 >= 3X1 + X2 = cons(active(X1),X2) active(recip(X)) = 3X >= 3X = recip(active(X)) active(sqr(X)) = 12X >= 12X = sqr(active(X)) active(add(X1,X2)) = 3X1 + 12X2 >= 3X1 + 4X2 = add(active(X1),X2) active(add(X1,X2)) = 3X1 + 12X2 >= X1 + 12X2 = add(X1,active(X2)) active(dbl(X)) = 6X >= 6X = dbl(active(X)) active(first(X1,X2)) = 3X1 + 12X2 + 12 >= 3X1 + 4X2 + 4 = first(active(X1),X2) active(first(X1,X2)) = 3X1 + 12X2 + 12 >= X1 + 12X2 + 4 = first(X1,active(X2)) terms(mark(X)) = 4X >= 4X = mark(terms(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) recip(mark(X)) = X >= X = mark(recip(X)) sqr(mark(X)) = 4X >= 4X = mark(sqr(X)) add(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(add(X1,X2)) add(X1,mark(X2)) = X1 + 4X2 >= X1 + 4X2 = mark(add(X1,X2)) dbl(mark(X)) = 2X >= 2X = mark(dbl(X)) first(mark(X1),X2) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = mark(first(X1,X2)) proper(terms(X)) = 4X >= 4X = terms(proper(X)) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(recip(X)) = X >= X = recip(proper(X)) proper(sqr(X)) = 4X >= 4X = sqr(proper(X)) proper(s(X)) = 2X >= 2X = s(proper(X)) proper(add(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = add(proper(X1),proper(X2)) proper(dbl(X)) = 2X >= 2X = dbl(proper(X)) proper(first(X1,X2)) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = first(proper(X1),proper(X2)) terms(ok(X)) = 12X + 8 >= 12X + 2 = ok(terms(X)) cons(ok(X1),ok(X2)) = 3X1 + 3X2 + 4 >= 3X1 + 3X2 + 2 = ok(cons(X1,X2)) recip(ok(X)) = 3X + 2 >= 3X + 2 = ok(recip(X)) sqr(ok(X)) = 12X + 8 >= 12X + 2 = ok(sqr(X)) s(ok(X)) = 6X + 4 >= 6X + 2 = ok(s(X)) add(ok(X1),ok(X2)) = 3X1 + 12X2 + 10 >= 3X1 + 12X2 + 2 = ok(add(X1,X2)) dbl(ok(X)) = 6X + 4 >= 6X + 2 = ok(dbl(X)) first(ok(X1),ok(X2)) = 3X1 + 12X2 + 14 >= 3X1 + 12X2 + 14 = ok(first(X1,X2)) top(ok(X)) = 12X + 8 >= 12X = top(active(X)) problem: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) recip(ok(X)) -> ok(recip(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = 2x0, [dbl](x0) = x0, [active](x0) = 5x0, [recip](x0) = 3x0 + 5, [cons](x0, x1) = 2x0 + 2x1 + 5, [first](x0, x1) = 2x0 + 3x1 + 2, [terms](x0) = 6x0 + 6, [proper](x0) = x0, [add](x0, x1) = x0 + 4x1, [ok](x0) = 2x0 + 3, [mark](x0) = x0, [sqr](x0) = x0 orientation: active(terms(N)) = 30N + 30 >= 30N + 27 = mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(s(X))) = 10X >= 10X = mark(s(add(sqr(X),dbl(X)))) active(dbl(s(X))) = 10X >= 4X = mark(s(s(dbl(X)))) active(add(s(X),Y)) = 10X + 20Y >= 2X + 8Y = mark(s(add(X,Y))) active(terms(X)) = 30X + 30 >= 30X + 6 = terms(active(X)) active(cons(X1,X2)) = 10X1 + 10X2 + 25 >= 10X1 + 2X2 + 5 = cons(active(X1),X2) active(recip(X)) = 15X + 25 >= 15X + 5 = recip(active(X)) active(sqr(X)) = 5X >= 5X = sqr(active(X)) active(add(X1,X2)) = 5X1 + 20X2 >= 5X1 + 4X2 = add(active(X1),X2) active(add(X1,X2)) = 5X1 + 20X2 >= X1 + 20X2 = add(X1,active(X2)) active(dbl(X)) = 5X >= 5X = dbl(active(X)) terms(mark(X)) = 6X + 6 >= 6X + 6 = mark(terms(X)) cons(mark(X1),X2) = 2X1 + 2X2 + 5 >= 2X1 + 2X2 + 5 = mark(cons(X1,X2)) recip(mark(X)) = 3X + 5 >= 3X + 5 = mark(recip(X)) sqr(mark(X)) = X >= X = mark(sqr(X)) add(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(add(X1,X2)) add(X1,mark(X2)) = X1 + 4X2 >= X1 + 4X2 = mark(add(X1,X2)) dbl(mark(X)) = X >= X = mark(dbl(X)) first(mark(X1),X2) = 2X1 + 3X2 + 2 >= 2X1 + 3X2 + 2 = mark(first(X1,X2)) first(X1,mark(X2)) = 2X1 + 3X2 + 2 >= 2X1 + 3X2 + 2 = mark(first(X1,X2)) proper(terms(X)) = 6X + 6 >= 6X + 6 = terms(proper(X)) proper(cons(X1,X2)) = 2X1 + 2X2 + 5 >= 2X1 + 2X2 + 5 = cons(proper(X1),proper(X2)) proper(recip(X)) = 3X + 5 >= 3X + 5 = recip(proper(X)) proper(sqr(X)) = X >= X = sqr(proper(X)) proper(s(X)) = 2X >= 2X = s(proper(X)) proper(add(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = add(proper(X1),proper(X2)) proper(dbl(X)) = X >= X = dbl(proper(X)) proper(first(X1,X2)) = 2X1 + 3X2 + 2 >= 2X1 + 3X2 + 2 = first(proper(X1),proper(X2)) recip(ok(X)) = 6X + 14 >= 6X + 13 = ok(recip(X)) first(ok(X1),ok(X2)) = 4X1 + 6X2 + 17 >= 4X1 + 6X2 + 7 = ok(first(X1,X2)) problem: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0 + 4, [dbl](x0) = x0 + 2, [active](x0) = 3x0 + 4, [recip](x0) = x0, [cons](x0, x1) = x0 + x1 + 2, [first](x0, x1) = x0 + x1 + 6, [terms](x0) = x0 + 2, [proper](x0) = x0, [add](x0, x1) = x0 + x1 + 3, [mark](x0) = x0 + 5, [sqr](x0) = x0 orientation: active(sqr(s(X))) = 3X + 16 >= 2X + 14 = mark(s(add(sqr(X),dbl(X)))) active(dbl(s(X))) = 3X + 22 >= X + 15 = mark(s(s(dbl(X)))) active(add(s(X),Y)) = 3X + 3Y + 25 >= X + Y + 12 = mark(s(add(X,Y))) active(sqr(X)) = 3X + 4 >= 3X + 4 = sqr(active(X)) active(add(X1,X2)) = 3X1 + 3X2 + 13 >= 3X1 + X2 + 7 = add(active(X1),X2) active(add(X1,X2)) = 3X1 + 3X2 + 13 >= X1 + 3X2 + 7 = add(X1,active(X2)) active(dbl(X)) = 3X + 10 >= 3X + 6 = dbl(active(X)) terms(mark(X)) = X + 7 >= X + 7 = mark(terms(X)) cons(mark(X1),X2) = X1 + X2 + 7 >= X1 + X2 + 7 = mark(cons(X1,X2)) recip(mark(X)) = X + 5 >= X + 5 = mark(recip(X)) sqr(mark(X)) = X + 5 >= X + 5 = mark(sqr(X)) add(mark(X1),X2) = X1 + X2 + 8 >= X1 + X2 + 8 = mark(add(X1,X2)) add(X1,mark(X2)) = X1 + X2 + 8 >= X1 + X2 + 8 = mark(add(X1,X2)) dbl(mark(X)) = X + 7 >= X + 7 = mark(dbl(X)) first(mark(X1),X2) = X1 + X2 + 11 >= X1 + X2 + 11 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 + 11 >= X1 + X2 + 11 = mark(first(X1,X2)) proper(terms(X)) = X + 2 >= X + 2 = terms(proper(X)) proper(cons(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = cons(proper(X1),proper(X2)) proper(recip(X)) = X >= X = recip(proper(X)) proper(sqr(X)) = X >= X = sqr(proper(X)) proper(s(X)) = X + 4 >= X + 4 = s(proper(X)) proper(add(X1,X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = add(proper(X1),proper(X2)) proper(dbl(X)) = X + 2 >= X + 2 = dbl(proper(X)) proper(first(X1,X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = first(proper(X1),proper(X2)) problem: active(sqr(X)) -> sqr(active(X)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0 + 3, [dbl](x0) = 2x0 + 2, [active](x0) = x0, [recip](x0) = 4x0, [cons](x0, x1) = 4x0 + 4x1, [first](x0, x1) = 2x0 + 2x1, [terms](x0) = 2x0 + 1, [proper](x0) = 4x0, [add](x0, x1) = 4x0 + 2x1 + 2, [mark](x0) = x0, [sqr](x0) = 2x0 + 2 orientation: active(sqr(X)) = 2X + 2 >= 2X + 2 = sqr(active(X)) terms(mark(X)) = 2X + 1 >= 2X + 1 = mark(terms(X)) cons(mark(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = mark(cons(X1,X2)) recip(mark(X)) = 4X >= 4X = mark(recip(X)) sqr(mark(X)) = 2X + 2 >= 2X + 2 = mark(sqr(X)) add(mark(X1),X2) = 4X1 + 2X2 + 2 >= 4X1 + 2X2 + 2 = mark(add(X1,X2)) add(X1,mark(X2)) = 4X1 + 2X2 + 2 >= 4X1 + 2X2 + 2 = mark(add(X1,X2)) dbl(mark(X)) = 2X + 2 >= 2X + 2 = mark(dbl(X)) first(mark(X1),X2) = 2X1 + 2X2 >= 2X1 + 2X2 = mark(first(X1,X2)) first(X1,mark(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = mark(first(X1,X2)) proper(terms(X)) = 8X + 4 >= 8X + 1 = terms(proper(X)) proper(cons(X1,X2)) = 16X1 + 16X2 >= 16X1 + 16X2 = cons(proper(X1),proper(X2)) proper(recip(X)) = 16X >= 16X = recip(proper(X)) proper(sqr(X)) = 8X + 8 >= 8X + 2 = sqr(proper(X)) proper(s(X)) = 4X + 12 >= 4X + 3 = s(proper(X)) proper(add(X1,X2)) = 16X1 + 8X2 + 8 >= 16X1 + 8X2 + 2 = add(proper(X1),proper(X2)) proper(dbl(X)) = 8X + 8 >= 8X + 2 = dbl(proper(X)) proper(first(X1,X2)) = 8X1 + 8X2 >= 8X1 + 8X2 = first(proper(X1),proper(X2)) problem: active(sqr(X)) -> sqr(active(X)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) Matrix Interpretation Processor: dim=1 interpretation: [dbl](x0) = x0 + 4, [active](x0) = 5x0, [recip](x0) = 2x0 + 5, [cons](x0, x1) = 3x0 + x1 + 2, [first](x0, x1) = 4x0 + x1 + 2, [terms](x0) = 4x0 + 2, [proper](x0) = 5x0 + 2, [add](x0, x1) = x0 + x1, [mark](x0) = x0, [sqr](x0) = 4x0 + 4 orientation: active(sqr(X)) = 20X + 20 >= 20X + 4 = sqr(active(X)) terms(mark(X)) = 4X + 2 >= 4X + 2 = mark(terms(X)) cons(mark(X1),X2) = 3X1 + X2 + 2 >= 3X1 + X2 + 2 = mark(cons(X1,X2)) recip(mark(X)) = 2X + 5 >= 2X + 5 = mark(recip(X)) sqr(mark(X)) = 4X + 4 >= 4X + 4 = mark(sqr(X)) add(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(add(X1,X2)) add(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(add(X1,X2)) dbl(mark(X)) = X + 4 >= X + 4 = mark(dbl(X)) first(mark(X1),X2) = 4X1 + X2 + 2 >= 4X1 + X2 + 2 = mark(first(X1,X2)) first(X1,mark(X2)) = 4X1 + X2 + 2 >= 4X1 + X2 + 2 = mark(first(X1,X2)) proper(cons(X1,X2)) = 15X1 + 5X2 + 12 >= 15X1 + 5X2 + 10 = cons(proper(X1),proper(X2)) proper(recip(X)) = 10X + 27 >= 10X + 9 = recip(proper(X)) proper(first(X1,X2)) = 20X1 + 5X2 + 12 >= 20X1 + 5X2 + 12 = first(proper(X1),proper(X2)) problem: terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) Matrix Interpretation Processor: dim=3 interpretation: [dbl](x0) = x0 , [1 1 1] [0] [recip](x0) = [0 1 1]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 1 1]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 1 0] [1 0 0] [0] [first](x0, x1) = [0 1 1]x0 + [1 1 1]x1 + [0] [0 0 0] [0 0 0] [1], [1 1 1] [0] [terms](x0) = [0 1 1]x0 + [0] [0 0 0] [1], [proper](x0) = x0 , [1 0 0] [1 0 0] [1] [add](x0, x1) = [0 1 1]x0 + [0 1 1]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 0] [0] [mark](x0) = [0 1 1]x0 + [0] [0 0 0] [1], [0] [sqr](x0) = x0 + [1] [0] orientation: [1 1 1] [1] [1 1 1] [0] terms(mark(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = mark(terms(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1] = mark(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 1] [1] [1 1 1] [0] recip(mark(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = mark(recip(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] sqr(mark(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = mark(sqr(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] add(mark(X1),X2) = [0 1 1]X1 + [0 1 1]X2 + [1] >= [0 1 1]X1 + [0 1 1]X2 + [1] = mark(add(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] add(X1,mark(X2)) = [0 1 1]X1 + [0 1 1]X2 + [1] >= [0 1 1]X1 + [0 1 1]X2 + [1] = mark(add(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] dbl(mark(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = mark(dbl(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1 0 0] [0] [1 1 0] [1 0 0] [0] first(mark(X1),X2) = [0 1 1]X1 + [1 1 1]X2 + [1] >= [0 1 1]X1 + [1 1 1]X2 + [1] = mark(first(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1 0 0] [0] [1 1 0] [1 0 0] [0] first(X1,mark(X2)) = [0 1 1]X1 + [1 1 1]X2 + [1] >= [0 1 1]X1 + [1 1 1]X2 + [1] = mark(first(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1 0 0] [0] [1 1 0] [1 0 0] [0] proper(first(X1,X2)) = [0 1 1]X1 + [1 1 1]X2 + [0] >= [0 1 1]X1 + [1 1 1]X2 + [0] = first(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] problem: cons(mark(X1),X2) -> mark(cons(X1,X2)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) Matrix Interpretation Processor: dim=1 interpretation: [dbl](x0) = x0, [cons](x0, x1) = x0 + x1, [first](x0, x1) = x0 + x1 + 4, [proper](x0) = 2x0 + 2, [add](x0, x1) = 4x0 + x1, [mark](x0) = x0, [sqr](x0) = x0 orientation: cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) sqr(mark(X)) = X >= X = mark(sqr(X)) add(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(add(X1,X2)) add(X1,mark(X2)) = 4X1 + X2 >= 4X1 + X2 = mark(add(X1,X2)) dbl(mark(X)) = X >= X = mark(dbl(X)) first(mark(X1),X2) = X1 + X2 + 4 >= X1 + X2 + 4 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = mark(first(X1,X2)) proper(first(X1,X2)) = 2X1 + 2X2 + 10 >= 2X1 + 2X2 + 8 = first(proper(X1),proper(X2)) problem: cons(mark(X1),X2) -> mark(cons(X1,X2)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [dbl](x0) = [0 1 0]x0 [0 1 0] , [1 0 1] [1 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 1 0] [0 0 0] , [1 0 0] [1 1 0] [first](x0, x1) = [0 0 1]x0 + [0 0 1]x1 [0 0 1] [0 0 1] , [1 1 0] [1 0 1] [add](x0, x1) = [0 0 1]x0 + [0 1 0]x1 [0 0 1] [1 1 0] , [0] [mark](x0) = x0 + [1] [1], [1 0 1] [sqr](x0) = [0 0 1]x0 [0 0 1] orientation: [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] cons(mark(X1),X2) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = mark(cons(X1,X2)) [0 1 0] [0 0 0] [1] [0 1 0] [0 0 0] [1] [1 0 1] [1] [1 0 1] [0] sqr(mark(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = mark(sqr(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1 0 1] [1] [1 1 0] [1 0 1] [0] add(mark(X1),X2) = [0 0 1]X1 + [0 1 0]X2 + [1] >= [0 0 1]X1 + [0 1 0]X2 + [1] = mark(add(X1,X2)) [0 0 1] [1 1 0] [1] [0 0 1] [1 1 0] [1] [1 1 0] [1 0 1] [1] [1 1 0] [1 0 1] [0] add(X1,mark(X2)) = [0 0 1]X1 + [0 1 0]X2 + [1] >= [0 0 1]X1 + [0 1 0]X2 + [1] = mark(add(X1,X2)) [0 0 1] [1 1 0] [1] [0 0 1] [1 1 0] [1] [1 0 1] [1] [1 0 1] [0] dbl(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(dbl(X)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] first(mark(X1),X2) = [0 0 1]X1 + [0 0 1]X2 + [1] >= [0 0 1]X1 + [0 0 1]X2 + [1] = mark(first(X1,X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [0] first(X1,mark(X2)) = [0 0 1]X1 + [0 0 1]X2 + [1] >= [0 0 1]X1 + [0 0 1]X2 + [1] = mark(first(X1,X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] problem: first(mark(X1),X2) -> mark(first(X1,X2)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [1 0 0] [first](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 1 1] [1 0 0] , [1 0 0] [0] [mark](x0) = [0 1 0]x0 + [1] [1 0 0] [0] orientation: [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] first(mark(X1),X2) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = mark(first(X1,X2)) [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] problem: Qed