YES (VAR X X0 X1 Xs Ys A B Fs Gs NF F1 F2 V Left Right G1 G2 F) (RULES intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys)) u'1'1(intersect'ii'out) -> intersect'ii'out intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys)) u'2'1(intersect'ii'out) -> intersect'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'3'1(reduce'ii'out) -> reduce'ii'out 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'4'1(reduce'ii'out) -> reduce'ii'out 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)) u'5'1(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'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'out reduce'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'out 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'14'1(reduce'ii'out) -> reduce'ii'out reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2)) u'15'1(intersect'ii'out) -> reduce'ii'out tautology'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 ) Proving termination of rewriting for cime5: -> Dependency pairs: nF_intersect'ii'in(Xs,cons(X0,Ys)) -> nF_u'1'1(intersect'ii'in(Xs,Ys)) nF_intersect'ii'in(Xs,cons(X0,Ys)) -> nF_intersect'ii'in(Xs,Ys) nF_intersect'ii'in(cons(X0,Xs),Ys) -> nF_u'2'1(intersect'ii'in(Xs,Ys)) nF_intersect'ii'in(cons(X0,Xs),Ys) -> nF_intersect'ii'in(Xs,Ys) nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF)) nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> nF_u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)) nF_reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> nF_u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF)) nF_reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(F1,Fs),Gs),NF) nF_u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> nF_u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF)) nF_u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> nF_reduce'ii'in(sequent(cons(F2,Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF)) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF)) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF) nF_u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> nF_u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF)) nF_u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> nF_reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> nF_u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF)) nF_reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> nF_reduce'ii'in(sequent(cons(G1,Fs),Gs),NF) nF_reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> nF_u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))) nF_reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))) nF_reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> nF_u'15'1(intersect'ii'in(F1,F2)) nF_reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> nF_intersect'ii'in(F1,F2) nF_tautology'i'in(F) -> nF_u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil))) nF_tautology'i'in(F) -> nF_reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)) -> Proof of termination for cime5_1_1: -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> nF_reduce'ii'in(sequent(cons(G1,Fs),Gs),NF) nF_reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))) nF_u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> nF_reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF) nF_u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> nF_reduce'ii'in(sequent(cons(F2,Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(F1,Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF) nF_reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF) UsableRules: intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys)) u'1'1(intersect'ii'out) -> intersect'ii'out intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys)) u'2'1(intersect'ii'out) -> intersect'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'3'1(reduce'ii'out) -> reduce'ii'out 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'4'1(reduce'ii'out) -> reduce'ii'out 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)) u'5'1(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'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'out reduce'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'out 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'14'1(reduce'ii'out) -> reduce'ii'out reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2)) u'15'1(intersect'ii'out) -> reduce'ii'out Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 + X2 [if](X1,X2) = X1 + X2 [u'3'1](X) = 0 [x'2b](X1,X2) = X1 + X2 [x'2d](X) = X [reduce'ii'out] = 0 [iff](X1,X2) = 2.X1 + 2.X2 + 1 [u'4'1](X) = 0 [x'2a](X1,X2) = X1 + X2 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 [nF_u'12'1](X1,X2,X3,X4,X5) = X2 + X3 + X4 [nF_u'6'1](X1,X2,X3,X4,X5) = X2 + X3 + X4 TIME: 1.1689e-2 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(F1,Fs),Gs),NF) nF_u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> nF_reduce'ii'in(sequent(cons(F2,Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF) nF_u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> nF_reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF) nF_reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))) nF_reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> nF_reduce'ii'in(sequent(cons(G1,Fs),Gs),NF) UsableRules: intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys)) u'1'1(intersect'ii'out) -> intersect'ii'out intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys)) u'2'1(intersect'ii'out) -> intersect'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'3'1(reduce'ii'out) -> reduce'ii'out 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'4'1(reduce'ii'out) -> reduce'ii'out 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)) u'5'1(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'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'out reduce'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'out 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'14'1(reduce'ii'out) -> reduce'ii'out reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2)) u'15'1(intersect'ii'out) -> reduce'ii'out Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 + X2 [if](X1,X2) = X1 + X2 + 1 [u'3'1](X) = 0 [x'2b](X1,X2) = X1 + X2 [x'2d](X) = X + 1 [reduce'ii'out] = 0 [iff](X1,X2) = 2.X1 + 2.X2 + 2 [u'4'1](X) = 0 [x'2a](X1,X2) = X1 + X2 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 [nF_u'6'1](X1,X2,X3,X4,X5) = X2 + X3 + X4 [nF_u'12'1](X1,X2,X3,X4,X5) = X2 + X3 + X4 TIME: 1.1595e-2 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> nF_reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF) nF_reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF) nF_u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> nF_reduce'ii'in(sequent(cons(F2,Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(F1,Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF) UsableRules: intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys)) u'1'1(intersect'ii'out) -> intersect'ii'out intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys)) u'2'1(intersect'ii'out) -> intersect'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'3'1(reduce'ii'out) -> reduce'ii'out 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'4'1(reduce'ii'out) -> reduce'ii'out 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)) u'5'1(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'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'out reduce'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'out 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'14'1(reduce'ii'out) -> reduce'ii'out reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2)) u'15'1(intersect'ii'out) -> reduce'ii'out Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 [if](X1,X2) = X1 [u'3'1](X) = 0 [x'2b](X1,X2) = X1 + X2 [x'2d](X) = 0 [reduce'ii'out] = 0 [iff](X1,X2) = 0 [u'4'1](X) = 0 [x'2a](X1,X2) = X1 + X2 + 1 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 [nF_u'12'1](X1,X2,X3,X4,X5) = X2 [nF_u'6'1](X1,X2,X3,X4,X5) = X2 + X3 TIME: 1.1105e-2 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(F1,Fs),Gs),NF) nF_u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> nF_reduce'ii'in(sequent(cons(F2,Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF) nF_reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))) nF_u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> nF_reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF) UsableRules: intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys)) u'1'1(intersect'ii'out) -> intersect'ii'out intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys)) u'2'1(intersect'ii'out) -> intersect'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'3'1(reduce'ii'out) -> reduce'ii'out 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'4'1(reduce'ii'out) -> reduce'ii'out 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)) u'5'1(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'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'out reduce'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'out 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'14'1(reduce'ii'out) -> reduce'ii'out reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2)) u'15'1(intersect'ii'out) -> reduce'ii'out Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 + X2 [if](X1,X2) = X1 + X2 [u'3'1](X) = 0 [x'2b](X1,X2) = X1 + X2 [x'2d](X) = X [reduce'ii'out] = 0 [iff](X1,X2) = 2.X1 + 2.X2 + 1 [u'4'1](X) = 0 [x'2a](X1,X2) = X1 + X2 + 1 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 [nF_u'6'1](X1,X2,X3,X4,X5) = X2 + X3 + X4 [nF_u'12'1](X1,X2,X3,X4,X5) = X2 + X3 + X4 TIME: 1.1941e-2 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF) nF_reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF) nF_u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> nF_reduce'ii'in(sequent(cons(F2,Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(F1,Fs),Gs),NF) UsableRules: intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys)) u'1'1(intersect'ii'out) -> intersect'ii'out intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys)) u'2'1(intersect'ii'out) -> intersect'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'3'1(reduce'ii'out) -> reduce'ii'out 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'4'1(reduce'ii'out) -> reduce'ii'out 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)) u'5'1(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'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'out reduce'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'out 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'14'1(reduce'ii'out) -> reduce'ii'out reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2)) u'15'1(intersect'ii'out) -> reduce'ii'out Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 [if](X1,X2) = X1 + 1 [u'3'1](X) = 0 [x'2b](X1,X2) = X1 + X2 + 1 [x'2d](X) = 0 [reduce'ii'out] = 0 [iff](X1,X2) = 0 [u'4'1](X) = 0 [x'2a](X1,X2) = 0 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 [nF_u'6'1](X1,X2,X3,X4,X5) = X2 + X3 TIME: 8.868e-3 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> nF_reduce'ii'in(sequent(cons(F2,Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) nF_reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))) nF_reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF) UsableRules: intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys)) u'1'1(intersect'ii'out) -> intersect'ii'out intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys)) u'2'1(intersect'ii'out) -> intersect'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'3'1(reduce'ii'out) -> reduce'ii'out 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'4'1(reduce'ii'out) -> reduce'ii'out 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)) u'5'1(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'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'out reduce'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'out 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'14'1(reduce'ii'out) -> reduce'ii'out reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2)) u'15'1(intersect'ii'out) -> reduce'ii'out Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 + X2 [if](X1,X2) = X1 + X2 [u'3'1](X) = 0 [x'2b](X1,X2) = X1 + X2 [x'2d](X) = X [reduce'ii'out] = 0 [iff](X1,X2) = X1 + X2 + 1 [u'4'1](X) = 0 [x'2a](X1,X2) = X1 + 1 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 [nF_u'6'1](X1,X2,X3,X4,X5) = X2 + X3 + X4 TIME: 1.1219e-2 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) nF_reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF) nF_u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> nF_reduce'ii'in(sequent(cons(F2,Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> nF_u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF) UsableRules: intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys)) u'1'1(intersect'ii'out) -> intersect'ii'out intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys)) u'2'1(intersect'ii'out) -> intersect'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'3'1(reduce'ii'out) -> reduce'ii'out 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'4'1(reduce'ii'out) -> reduce'ii'out 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)) u'5'1(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'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'out reduce'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'out 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'14'1(reduce'ii'out) -> reduce'ii'out reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2)) u'15'1(intersect'ii'out) -> reduce'ii'out Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 [if](X1,X2) = X1 + 1 [u'3'1](X) = 0 [x'2b](X1,X2) = X2 + 1 [x'2d](X) = 0 [reduce'ii'out] = 0 [iff](X1,X2) = 0 [u'4'1](X) = 0 [x'2a](X1,X2) = 0 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 [nF_u'6'1](X1,X2,X3,X4,X5) = X2 + X3 TIME: 1.0186e-2 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) nF_reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))) There are no usable rules. Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 + X2 [if](X1,X2) = X1 + X2 [u'3'1](X) = 0 [x'2b](X1,X2) = X1 + X2 [x'2d](X) = X [reduce'ii'out] = 0 [iff](X1,X2) = 0 [u'4'1](X) = 0 [x'2a](X1,X2) = 0 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 1 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 TIME: 9.35e-4 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF) There are no usable rules. Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 + X2 [if](X1,X2) = X1 + X2 + 1 [u'3'1](X) = 0 [x'2b](X1,X2) = X1 + X2 [x'2d](X) = X [reduce'ii'out] = 0 [iff](X1,X2) = 0 [u'4'1](X) = 0 [x'2a](X1,X2) = 0 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 TIME: 9.98e-4 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF) There are no usable rules. Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 + X2 [if](X1,X2) = X1 + X2 + 1 [u'3'1](X) = 0 [x'2b](X1,X2) = X1 + X2 + 1 [x'2d](X) = X [reduce'ii'out] = 0 [iff](X1,X2) = 0 [u'4'1](X) = 0 [x'2a](X1,X2) = 0 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 TIME: 6.88e-4 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF) There are no usable rules. Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 [if](X1,X2) = 0 [u'3'1](X) = 0 [x'2b](X1,X2) = 0 [x'2d](X) = 1 [reduce'ii'out] = 0 [iff](X1,X2) = 0 [u'4'1](X) = 0 [x'2a](X1,X2) = 0 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 TIME: 4.09e-4 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) nF_reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> nF_reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)) There are no usable rules. Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 + X2 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X1 [if](X1,X2) = 0 [u'3'1](X) = 0 [x'2b](X1,X2) = 0 [x'2d](X) = 0 [reduce'ii'out] = 0 [iff](X1,X2) = 0 [u'4'1](X) = 0 [x'2a](X1,X2) = 0 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 1 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 TIME: 3.4e-4 -> -> Dependency pairs in cycle: nF_reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> nF_reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF) nF_reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> nF_reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF) There are no usable rules. Polynomial Interpretation: [intersect'ii'in](X1,X2) = 0 [cons](X1,X2) = X1 [intersect'ii'out] = 0 [u'1'1](X) = 0 [u'2'1](X) = 0 [reduce'ii'in](X1,X2) = 0 [sequent](X1,X2) = X2 [if](X1,X2) = 0 [u'3'1](X) = 0 [x'2b](X1,X2) = 0 [x'2d](X) = 0 [reduce'ii'out] = 0 [iff](X1,X2) = 1 [u'4'1](X) = 0 [x'2a](X1,X2) = 0 [u'5'1](X) = 0 [u'6'1](X1,X2,X3,X4,X5) = 0 [u'6'2](X) = 0 [u'7'1](X) = 0 [u'8'1](X) = 0 [u'9'1](X) = 0 [p](X) = 0 [u'10'1](X) = 0 [u'11'1](X) = 0 [u'12'1](X1,X2,X3,X4,X5) = 0 [u'12'2](X) = 0 [u'13'1](X) = 0 [nil] = 0 [u'14'1](X) = 0 [u'15'1](X) = 0 [tautology'i'in](X) = 0 [u'16'1](X) = 0 [tautology'i'out] = 0 [nF_reduce'ii'in](X1,X2) = X1 TIME: 2.29e-4 -> Proof of termination for cime5_1_2: -> -> Dependency pairs in cycle: nF_intersect'ii'in(Xs,cons(X0,Ys)) -> nF_intersect'ii'in(Xs,Ys) nF_intersect'ii'in(cons(X0,Xs),Ys) -> nF_intersect'ii'in(Xs,Ys) Dependency pairs oriented using subterm criterion. -> -> Dependency pairs in cycle: nF_intersect'ii'in(Xs,cons(X0,Ys)) -> nF_intersect'ii'in(Xs,Ys) Termination proved: Cycles verify subterm criterion. SETTINGS: Base ordering: Polynomial ordering Proof mode: SCCs in DG + base ordering Upper bound for coeffs: 2 Rationals below 1 for all non-replacing args: No Polynomial interpretation: Linear Coeffs in polynomials: No rationals Delta: automatic Termination was proved succesfully.