YES

The TRS could be proven terminating. The proof took 26101 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (735ms).
 | – Problem 2 was processed with processor SubtermCriterion (3ms).
 |    | – Problem 4 was processed with processor SubtermCriterion (0ms).
 | – Problem 3 was processed with processor PolynomialLinearRange4iUR (6239ms).
 |    | – Problem 5 was processed with processor PolynomialLinearRange4iUR (5690ms).
 |    |    | – Problem 6 was processed with processor PolynomialLinearRange4iUR (4181ms).
 |    |    |    | – Problem 7 was processed with processor PolynomialLinearRange4iUR (5846ms).
 |    |    |    |    | – Problem 8 was processed with processor DependencyGraph (8ms).
 |    |    |    |    |    | – Problem 9 was processed with processor PolynomialLinearRange4iUR (1043ms).
 |    |    |    |    |    |    | – Problem 10 was processed with processor PolynomialLinearRange4iUR (1019ms).
 |    |    |    |    |    |    |    | – Problem 11 was processed with processor PolynomialLinearRange4iUR (981ms).
 |    |    |    |    |    |    |    |    | – Problem 12 was processed with processor DependencyGraph (0ms).

Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

intersect'ii'in#(cons(X0, Xs), Ys)u'2'1#(intersect'ii'in(Xs, Ys))reduce'ii'in#(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1#(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
reduce'ii'in#(sequent(Fs, cons(iff(A, B), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)
reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2d(G1), Gs)), NF)reduce'ii'in#(sequent(cons(G1, Fs), Gs), NF)
reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1#(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))
intersect'ii'in#(Xs, cons(X0, Ys))u'1'1#(intersect'ii'in(Xs, Ys))u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)
reduce'ii'in#(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1#(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))reduce'ii'in#(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1#(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
intersect'ii'in#(cons(X0, Xs), Ys)intersect'ii'in#(Xs, Ys)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
reduce'ii'in#(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))reduce'ii'in#(sequent(nil, Gs), sequent(Left, cons(p(V), Right)))reduce'ii'in#(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1#(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))
reduce'ii'in#(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))reduce'ii'in#(sequent(Fs, Gs), sequent(cons(p(V), Left), Right))reduce'ii'in#(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1#(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))
reduce'ii'in#(sequent(Fs, cons(if(A, B), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF)reduce'ii'in#(sequent(nil, nil), sequent(F1, F2))intersect'ii'in#(F1, F2)
u'12'1#(reduce'ii'out, Fs, G2, Gs, NF)u'12'2#(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))reduce'ii'in#(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1#(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
reduce'ii'in#(sequent(cons(x'2d(F1), Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(F1, Gs)), NF)tautology'i'in#(F)reduce'ii'in#(sequent(nil, cons(F, nil)), sequent(nil, nil))
reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1#(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))reduce'ii'in#(sequent(cons(iff(A, B), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF)
reduce'ii'in#(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, cons(F2, Fs)), Gs), NF)reduce'ii'in#(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1#(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
reduce'ii'in#(sequent(nil, nil), sequent(F1, F2))u'15'1#(intersect'ii'in(F1, F2))tautology'i'in#(F)u'16'1#(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1#(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)u'6'2#(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))
reduce'ii'in#(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1#(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))reduce'ii'in#(sequent(cons(if(A, B), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF)
intersect'ii'in#(Xs, cons(X0, Ys))intersect'ii'in#(Xs, Ys)u'12'1#(reduce'ii'out, Fs, G2, Gs, NF)reduce'ii'in#(sequent(Fs, cons(G2, Gs)), NF)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, u'11'1, x'2a, sequent, intersect'ii'out, u'5'1, u'9'1, u'14'1, u'12'1, u'4'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, u'8'1, nil

Strategy


The following SCCs where found

intersect'ii'in#(cons(X0, Xs), Ys) → intersect'ii'in#(Xs, Ys)intersect'ii'in#(Xs, cons(X0, Ys)) → intersect'ii'in#(Xs, Ys)

reduce'ii'in#(sequent(cons(p(V), Fs), Gs), sequent(Left, Right)) → reduce'ii'in#(sequent(Fs, Gs), sequent(cons(p(V), Left), Right))reduce'ii'in#(sequent(Fs, cons(if(A, B), Gs)), NF) → reduce'ii'in#(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF)
reduce'ii'in#(sequent(Fs, cons(iff(A, B), Gs)), NF) → reduce'ii'in#(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF) → reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)
reduce'ii'in#(sequent(cons(x'2d(F1), Fs), Gs), NF) → reduce'ii'in#(sequent(Fs, cons(F1, Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2d(G1), Gs)), NF) → reduce'ii'in#(sequent(cons(G1, Fs), Gs), NF)
reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF) → reduce'ii'in#(sequent(Fs, cons(G1, Gs)), NF)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF) → reduce'ii'in#(sequent(cons(F1, Fs), Gs), NF)
reduce'ii'in#(sequent(cons(iff(A, B), Fs), Gs), NF) → reduce'ii'in#(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2a(F1, F2), Fs), Gs), NF) → reduce'ii'in#(sequent(cons(F1, cons(F2, Fs)), Gs), NF)
u'6'1#(reduce'ii'out, F2, Fs, Gs, NF) → reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF) → u'12'1#(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)
reduce'ii'in#(sequent(cons(if(A, B), Fs), Gs), NF) → reduce'ii'in#(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF) → u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
reduce'ii'in#(sequent(nil, cons(p(V), Gs)), sequent(Left, Right)) → reduce'ii'in#(sequent(nil, Gs), sequent(Left, cons(p(V), Right)))u'12'1#(reduce'ii'out, Fs, G2, Gs, NF) → reduce'ii'in#(sequent(Fs, cons(G2, Gs)), NF)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

intersect'ii'in#(cons(X0, Xs), Ys)intersect'ii'in#(Xs, Ys)intersect'ii'in#(Xs, cons(X0, Ys))intersect'ii'in#(Xs, Ys)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, u'11'1, x'2a, sequent, intersect'ii'out, u'5'1, u'9'1, u'14'1, u'12'1, u'4'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, u'8'1, nil

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

intersect'ii'in#(cons(X0, Xs), Ys)intersect'ii'in#(Xs, Ys)

Problem 4: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

intersect'ii'in#(Xs, cons(X0, Ys))intersect'ii'in#(Xs, Ys)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, x'2a, u'11'1, sequent, intersect'ii'out, u'5'1, u'14'1, u'9'1, u'4'1, u'12'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, nil, u'8'1

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

intersect'ii'in#(Xs, cons(X0, Ys))intersect'ii'in#(Xs, Ys)

Problem 3: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

reduce'ii'in#(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))reduce'ii'in#(sequent(Fs, Gs), sequent(cons(p(V), Left), Right))reduce'ii'in#(sequent(Fs, cons(if(A, B), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF)
reduce'ii'in#(sequent(Fs, cons(iff(A, B), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF)reduce'ii'in#(sequent(cons(x'2d(F1), Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(F1, Gs)), NF)
reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, Gs)), NF)
reduce'ii'in#(sequent(Fs, cons(x'2d(G1), Gs)), NF)reduce'ii'in#(sequent(cons(G1, Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, Fs), Gs), NF)
reduce'ii'in#(sequent(cons(iff(A, B), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, cons(F2, Fs)), Gs), NF)
u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1#(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)
reduce'ii'in#(sequent(cons(if(A, B), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
reduce'ii'in#(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))reduce'ii'in#(sequent(nil, Gs), sequent(Left, cons(p(V), Right)))u'12'1#(reduce'ii'out, Fs, G2, Gs, NF)reduce'ii'in#(sequent(Fs, cons(G2, Gs)), NF)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, u'11'1, x'2a, sequent, intersect'ii'out, u'5'1, u'9'1, u'14'1, u'12'1, u'4'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, u'8'1, nil

Strategy


Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

reduce'ii'in#(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))reduce'ii'in#(sequent(Fs, Gs), sequent(cons(p(V), Left), Right))reduce'ii'in#(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))reduce'ii'in#(sequent(nil, Gs), sequent(Left, cons(p(V), Right)))

Problem 5: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

reduce'ii'in#(sequent(Fs, cons(if(A, B), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(iff(A, B), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF)
reduce'ii'in#(sequent(cons(x'2d(F1), Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(F1, Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)
reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2d(G1), Gs)), NF)reduce'ii'in#(sequent(cons(G1, Fs), Gs), NF)
reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, Fs), Gs), NF)reduce'ii'in#(sequent(cons(iff(A, B), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF)
reduce'ii'in#(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, cons(F2, Fs)), Gs), NF)u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)
reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1#(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)reduce'ii'in#(sequent(cons(if(A, B), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF)
reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)u'12'1#(reduce'ii'out, Fs, G2, Gs, NF)reduce'ii'in#(sequent(Fs, cons(G2, Gs)), NF)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, x'2a, u'11'1, sequent, intersect'ii'out, u'5'1, u'14'1, u'9'1, u'4'1, u'12'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, nil, u'8'1

Strategy


Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

reduce'ii'in#(sequent(Fs, cons(iff(A, B), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF)reduce'ii'in#(sequent(cons(iff(A, B), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF)

Problem 6: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

reduce'ii'in#(sequent(Fs, cons(if(A, B), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF)reduce'ii'in#(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, cons(F2, Fs)), Gs), NF)
u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1#(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)
reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)reduce'ii'in#(sequent(cons(x'2d(F1), Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(F1, Gs)), NF)
reduce'ii'in#(sequent(cons(if(A, B), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(x'2d(G1), Gs)), NF)reduce'ii'in#(sequent(cons(G1, Fs), Gs), NF)
reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, Gs)), NF)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, Fs), Gs), NF)u'12'1#(reduce'ii'out, Fs, G2, Gs, NF)reduce'ii'in#(sequent(Fs, cons(G2, Gs)), NF)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, u'11'1, x'2a, sequent, intersect'ii'out, u'5'1, u'9'1, u'14'1, u'12'1, u'4'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, u'8'1, nil

Strategy


Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

reduce'ii'in#(sequent(Fs, cons(if(A, B), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF)reduce'ii'in#(sequent(cons(if(A, B), Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF)

Problem 7: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

reduce'ii'in#(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, cons(F2, Fs)), Gs), NF)u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)
reduce'ii'in#(sequent(cons(x'2d(F1), Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(F1, Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)
reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1#(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, Gs)), NF)
reduce'ii'in#(sequent(Fs, cons(x'2d(G1), Gs)), NF)reduce'ii'in#(sequent(cons(G1, Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, Fs), Gs), NF)u'12'1#(reduce'ii'out, Fs, G2, Gs, NF)reduce'ii'in#(sequent(Fs, cons(G2, Gs)), NF)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, x'2a, u'11'1, sequent, intersect'ii'out, u'5'1, u'14'1, u'9'1, u'4'1, u'12'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, nil, u'8'1

Strategy


Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

reduce'ii'in#(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, cons(F2, Fs)), Gs), NF)reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1#(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)
reduce'ii'in#(sequent(cons(x'2d(F1), Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(F1, Gs)), NF)reduce'ii'in#(sequent(Fs, cons(x'2d(G1), Gs)), NF)reduce'ii'in#(sequent(cons(G1, Fs), Gs), NF)
reduce'ii'in#(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, Gs)), NF)

Problem 8: DependencyGraph



Dependency Pair Problem

Dependency Pairs

u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)
reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, Fs), Gs), NF)
u'12'1#(reduce'ii'out, Fs, G2, Gs, NF)reduce'ii'in#(sequent(Fs, cons(G2, Gs)), NF)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, u'11'1, x'2a, sequent, intersect'ii'out, u'5'1, u'9'1, u'14'1, u'12'1, u'4'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, u'8'1, nil

Strategy


The following SCCs where found

u'6'1#(reduce'ii'out, F2, Fs, Gs, NF) → reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF) → reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)
reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF) → u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF) → reduce'ii'in#(sequent(cons(F1, Fs), Gs), NF)

Problem 9: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)
reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, Fs), Gs), NF)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, u'11'1, x'2a, sequent, intersect'ii'out, u'5'1, u'9'1, u'14'1, u'12'1, u'4'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, u'8'1, nil

Strategy


Polynomial Interpretation

Improved Usable rules

reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))reduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'5'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))u'11'1(reduce'ii'out)reduce'ii'out
u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))reduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'8'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'7'1(reduce'ii'out)reduce'ii'outu'14'1(reduce'ii'out)reduce'ii'out
u'9'1(reduce'ii'out)reduce'ii'outu'12'2(reduce'ii'out)reduce'ii'out
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))
reduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))reduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'10'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))
u'13'1(reduce'ii'out)reduce'ii'outu'6'2(reduce'ii'out)reduce'ii'out
u'15'1(intersect'ii'out)reduce'ii'outu'3'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)u'4'1(reduce'ii'out)reduce'ii'out

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)reduce'ii'in#(sequent(cons(F1, Fs), Gs), NF)

Problem 10: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)
reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, x'2a, u'11'1, sequent, intersect'ii'out, u'5'1, u'14'1, u'9'1, u'4'1, u'12'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, nil, u'8'1

Strategy


Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

reduce'ii'in#(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)reduce'ii'in#(sequent(Fs, cons(G1, cons(G2, Gs))), NF)

Problem 11: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, u'11'1, x'2a, sequent, intersect'ii'out, u'5'1, u'9'1, u'14'1, u'12'1, u'4'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, u'8'1, nil

Strategy


Polynomial Interpretation

Improved Usable rules

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

u'6'1#(reduce'ii'out, F2, Fs, Gs, NF)reduce'ii'in#(sequent(cons(F2, Fs), Gs), NF)

Problem 12: DependencyGraph



Dependency Pair Problem

Dependency Pairs

reduce'ii'in#(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1#(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)

Rewrite Rules

intersect'ii'in(cons(X, X0), cons(X, X1))intersect'ii'outintersect'ii'in(Xs, cons(X0, Ys))u'1'1(intersect'ii'in(Xs, Ys))
u'1'1(intersect'ii'out)intersect'ii'outintersect'ii'in(cons(X0, Xs), Ys)u'2'1(intersect'ii'in(Xs, Ys))
u'2'1(intersect'ii'out)intersect'ii'outreduce'ii'in(sequent(cons(if(A, B), Fs), Gs), NF)u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B), A), Fs), Gs), NF))
u'3'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(iff(A, B), Fs), Gs), NF)u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A, B), if(B, A)), Fs), Gs), NF))
u'4'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2a(F1, F2), Fs), Gs), NF)u'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, Fs)), Gs), NF))
u'5'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(cons(x'2b(F1, F2), Fs), Gs), NF)u'6'1(reduce'ii'in(sequent(cons(F1, Fs), Gs), NF), F2, Fs, Gs, NF)
u'6'1(reduce'ii'out, F2, Fs, Gs, NF)u'6'2(reduce'ii'in(sequent(cons(F2, Fs), Gs), NF))u'6'2(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(x'2d(F1), Fs), Gs), NF)u'7'1(reduce'ii'in(sequent(Fs, cons(F1, Gs)), NF))u'7'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(if(A, B), Gs)), NF)u'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), Gs)), NF))u'8'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(iff(A, B), Gs)), NF)u'9'1(reduce'ii'in(sequent(Fs, cons(x'2a(if(A, B), if(B, A)), Gs)), NF))u'9'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(cons(p(V), Fs), Gs), sequent(Left, Right))u'10'1(reduce'ii'in(sequent(Fs, Gs), sequent(cons(p(V), Left), Right)))u'10'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2b(G1, G2), Gs)), NF)u'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(G2, Gs))), NF))u'11'1(reduce'ii'out)reduce'ii'out
reduce'ii'in(sequent(Fs, cons(x'2a(G1, G2), Gs)), NF)u'12'1(reduce'ii'in(sequent(Fs, cons(G1, Gs)), NF), Fs, G2, Gs, NF)u'12'1(reduce'ii'out, Fs, G2, Gs, NF)u'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
u'12'2(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(Fs, cons(x'2d(G1), Gs)), NF)u'13'1(reduce'ii'in(sequent(cons(G1, Fs), Gs), NF))
u'13'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, cons(p(V), Gs)), sequent(Left, Right))u'14'1(reduce'ii'in(sequent(nil, Gs), sequent(Left, cons(p(V), Right))))
u'14'1(reduce'ii'out)reduce'ii'outreduce'ii'in(sequent(nil, nil), sequent(F1, F2))u'15'1(intersect'ii'in(F1, F2))
u'15'1(intersect'ii'out)reduce'ii'outtautology'i'in(F)u'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
u'16'1(reduce'ii'out)tautology'i'out

Original Signature

Termination of terms over the following signature is verified: x'2b, x'2a, u'11'1, sequent, intersect'ii'out, u'5'1, u'14'1, u'9'1, u'4'1, u'12'1, tautology'i'out, u'12'2, u'10'1, tautology'i'in, if, u'6'2, u'6'1, intersect'ii'in, cons, u'2'1, reduce'ii'in, reduce'ii'out, u'7'1, u'16'1, u'3'1, iff, u'15'1, u'13'1, p, u'1'1, x'2d, nil, u'8'1

Strategy


There are no SCCs!