(VAR A B F F1 F2 Fs G1 G2 Gs Left NF Right V X X0 X1 Xs Ys )
(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
        
)