0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
↳5 QDP
↳6 QDPOrderProof (⇔)
↳7 QDP
↳8 QDPOrderProof (⇔)
↳9 QDP
↳10 PisEmptyProof (⇔)
↳11 TRUE
↳12 QDP
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
INTERSECT'II'IN(Xs, cons(X0, Ys)) → U'1'1(intersect'ii'in(Xs, Ys))
INTERSECT'II'IN(Xs, cons(X0, Ys)) → INTERSECT'II'IN(Xs, Ys)
INTERSECT'II'IN(cons(X0, Xs), Ys) → U'2'1(intersect'ii'in(Xs, Ys))
INTERSECT'II'IN(cons(X0, Xs), Ys) → INTERSECT'II'IN(Xs, Ys)
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(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(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(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) → U'5'1(reduce'ii'in(sequent(cons(F1, cons(F2, 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(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'6'1(reduce'ii'out, F2, Fs, Gs, NF) → U'6'2(reduce'ii'in(sequent(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) → U'7'1(reduce'ii'in(sequent(Fs, cons(F1, 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(if(A, B), Gs)), NF) → U'8'1(reduce'ii'in(sequent(Fs, cons(x'2b(x'2d(B), A), 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(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(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(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(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(x'2b(G1, G2), Gs)), NF) → U'11'1(reduce'ii'in(sequent(Fs, cons(G1, cons(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(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)
U'12'1(reduce'ii'out, Fs, G2, Gs, NF) → U'12'2(reduce'ii'in(sequent(Fs, cons(G2, Gs)), NF))
U'12'1(reduce'ii'out, Fs, G2, Gs, NF) → 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(Fs, cons(x'2d(G1), Gs)), NF) → REDUCE'II'IN(sequent(cons(G1, 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, cons(p(V), Gs)), sequent(Left, Right)) → 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))
REDUCE'II'IN(sequent(nil, nil), sequent(F1, F2)) → INTERSECT'II'IN(F1, F2)
TAUTOLOGY'I'IN(F) → U'16'1(reduce'ii'in(sequent(nil, cons(F, nil)), sequent(nil, nil)))
TAUTOLOGY'I'IN(F) → REDUCE'II'IN(sequent(nil, cons(F, nil)), sequent(nil, nil))
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
INTERSECT'II'IN(cons(X0, Xs), Ys) → INTERSECT'II'IN(Xs, Ys)
INTERSECT'II'IN(Xs, cons(X0, Ys)) → INTERSECT'II'IN(Xs, Ys)
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
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
INTERSECT'II'IN(Xs, cons(X0, Ys)) → INTERSECT'II'IN(Xs, Ys)
if1 > [reduce'ii'in, u'12'1] > cons1 > u'5'1 > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
if1 > [reduce'ii'in, u'12'1] > cons1 > u'6'1 > sequent1 > [intersect'ii'in, nil] > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
if1 > [reduce'ii'in, u'12'1] > cons1 > u'6'1 > sequent1 > u'4'1 > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
if1 > [reduce'ii'in, u'12'1] > cons1 > u'6'1 > sequent1 > x'2a1 > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
if1 > [reduce'ii'in, u'12'1] > cons1 > u'7'1 > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
if1 > [reduce'ii'in, u'12'1] > cons1 > u'8'1 > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
if1 > [reduce'ii'in, u'12'1] > cons1 > p > sequent1 > [intersect'ii'in, nil] > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
if1 > [reduce'ii'in, u'12'1] > cons1 > p > sequent1 > u'4'1 > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
if1 > [reduce'ii'in, u'12'1] > cons1 > p > sequent1 > x'2a1 > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
iff1 > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
[tautology'i'in, u'16'1, tautology'i'out] > [intersect'ii'out, u'3'1, x'2b, x'2d, reduce'ii'out, u'6'2, u'10'1, u'11'1, u'13'1, u'14'1]
cons1: [1]
intersect'ii'in: []
intersect'ii'out: []
reduce'ii'in: []
sequent1: [1]
if1: [1]
u'3'1: []
x'2b: []
x'2d: []
reduce'ii'out: []
iff1: [1]
u'4'1: []
x'2a1: [1]
u'5'1: []
u'6'1: []
u'6'2: []
u'7'1: []
u'8'1: []
p: []
u'10'1: []
u'11'1: []
u'12'1: []
u'13'1: []
nil: []
u'14'1: []
tautology'i'in: []
u'16'1: []
tautology'i'out: []
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
INTERSECT'II'IN(cons(X0, Xs), Ys) → INTERSECT'II'IN(Xs, Ys)
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
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
INTERSECT'II'IN(cons(X0, Xs), Ys) → INTERSECT'II'IN(Xs, Ys)
intersect'ii'in2 > [cons2, intersect'ii'out, if, x'2b, iff, u'10'1, u'11'1, nil, u'14'1] > [sequent2, u'3'1, reduce'ii'out, u'6'2, u'12'2, u'16'1, tautology'i'out]
[x'2d, u'7'1, u'13'1] > [reduce'ii'in2, u'4'1, u'5'1, u'8'1, u'9'1, u'15'1] > [cons2, intersect'ii'out, if, x'2b, iff, u'10'1, u'11'1, nil, u'14'1] > [sequent2, u'3'1, reduce'ii'out, u'6'2, u'12'2, u'16'1, tautology'i'out]
cons2: [2,1]
intersect'ii'in2: [2,1]
intersect'ii'out: []
reduce'ii'in2: [1,2]
sequent2: [2,1]
if: []
u'3'1: []
x'2b: []
x'2d: []
reduce'ii'out: []
iff: []
u'4'1: []
u'5'1: []
u'6'2: []
u'7'1: []
u'8'1: []
u'9'1: []
u'10'1: []
u'11'1: []
u'12'2: []
u'13'1: []
nil: []
u'14'1: []
u'15'1: []
u'16'1: []
tautology'i'out: []
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
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
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) → REDUCE'II'IN(sequent(cons(F2, Fs), 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) → 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)
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(if(A, B), Gs)), NF) → REDUCE'II'IN(sequent(Fs, cons(x'2b(x'2d(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(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'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) → REDUCE'II'IN(sequent(Fs, cons(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(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(nil, cons(p(V), Gs)), sequent(Left, Right)) → REDUCE'II'IN(sequent(nil, Gs), sequent(Left, cons(p(V), Right)))
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