TPTP Problem File: PRV014+1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : PRV014+1.s : ProoVer 2026
% Proof : Problems/PRV014+1.p
%------------------------------------------------------------------------------
% SZS output start Proof
fof(s0,axiom,
! [X0,X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3),
file('Problems/PRV014+1.p',s0) ).
fof(s1,axiom,
! [X6,X7] : ~ p(b),
file('Problems/PRV014+1.p',s1) ).
fof(s2,axiom,
! [X8,X10] :
? [X9] :
~ ~ p(f(b)),
file('Problems/PRV014+1.p',s2) ).
fof(s3,axiom,
~ ( q(c)
& f(b) != c ),
file('Problems/PRV014+1.p',s3) ).
fof(c,conjecture,
( ~ ~ ~ ( q(c)
& f(b) != c )
| ! [X32,X33] :
? [X34] : t ),
file('Problems/PRV014+1.p',c) ).
fof(s4,plain,
! [X10] :
? [X9] :
~ ~ p(f(b)),
inference(instantiate,[status(thm)],[s2]) ).
fof(s5,plain,
! [X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3),
inference(instantiate,[status(thm)],[s0]) ).
fof(s6,plain,
! [X8,X10] :
~ ~ p(f(b)),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(X9,sK0(X8,X10))],[s2]) ).
fof(s7,plain,
! [X11] :
? [X9] :
~ ~ p(f(b)),
inference(rename_variable,[status(thm)],[s4]) ).
fof(s8,plain,
( ~ ( q(c)
& f(b) != c )
| ~ ! [X12] :
? [X13] : q(X12) ),
inference(weaken,[status(thm)],[s3]) ).
fof(s9,plain,
! [X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3),
inference(instantiate,[status(thm)],[s0]) ).
fof(s10,plain,
~ ~ ! [X8,X10] :
? [X9] :
~ ~ p(f(b)),
inference(double_negation,[status(thm)],[s2]) ).
fof(s11,plain,
~ ~ ! [X8,X10] :
? [X9] :
~ ~ p(f(b)),
inference(double_negation,[status(thm)],[s2]) ).
fof(s12,plain,
( ~ ( q(c)
& f(b) != c )
& ! [X10] :
? [X9] :
~ ~ p(f(b)) ),
inference(conjunction,[status(thm)],[s3,s4]) ).
fof(s13,plain,
~ ( q(c)
& f(b) != c ),
inference(split_conjunct,[status(thm)],[s12]) ).
fof(s14,plain,
( ~ q(c)
| ~ ( f(b) != c ) ),
inference(de_morgan,[status(thm)],[s3]) ).
fof(s15,plain,
? [X9] :
~ ~ p(f(b)),
inference(instantiate,[status(thm),new_symbols(herbrand,[m0])],[s4]) ).
fof(s16,plain,
( ~ ( f(b) != c )
| ~ q(c) ),
inference(commute,[status(thm)],[s14]) ).
fof(s17,plain,
( ! [X10] :
? [X9] :
~ ~ p(f(b))
& ~ ( q(c)
& f(b) != c ) ),
inference(commute,[status(thm)],[s12]) ).
fof(s18,plain,
( ! [X10] :
? [X9] :
~ ~ p(f(b))
& ~ ( q(c)
& f(b) != c )
& ! [X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3) ),
inference(conjunction,[status(thm)],[s17,s9]) ).
fof(s19,plain,
( ? [X14] : r(g(X14,b),X14)
| ! [X15] :
? [X16] : q(f(c))
| ~ ( ? [X14] : r(g(X14,b),X14)
| ! [X15] :
? [X16] : q(f(c)) ) ),
inference(excluded_middle,[status(thm)],[s6]) ).
fof(s20,plain,
! [X2,X3] :
? [X4] :
! [X5] : r(f(c),X3),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1]),skolemize(X1,sK1(X2))],[s5]) ).
fof(s21,plain,
! [X17,X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3),
inference(rename_variable,[status(thm)],[s0]) ).
fof(s22,plain,
! [X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3),
inference(instantiate,[status(thm)],[s21]) ).
fof(s23,plain,
~ ~ ~ ( q(c)
& f(b) != c ),
inference(double_negation,[status(thm)],[s13]) ).
fof(s24,plain,
( ~ r(a,f(f(c)))
=> ~ ( q(c)
& f(b) != c ) ),
inference(add_hypothesis,[status(thm)],[s3]) ).
fof(s25,plain,
? [X18] :
( ~ r(a,f(f(X18)))
=> ~ ( q(X18)
& f(b) != X18 ) ),
inference(existential_gen,[status(thm)],[s24]) ).
fof(s26,plain,
( ! [X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3)
| ( p(b)
=> t ) ),
inference(weaken,[status(thm)],[s9]) ).
fof(s27,plain,
( ! [X0,X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3)
& ! [X10] :
? [X9] :
~ ~ p(f(b))
& ~ ( q(c)
& f(b) != c )
& ! [X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3) ),
inference(conjunction,[status(thm)],[s0,s18]) ).
fof(s28,plain,
c = c,
inference(reflexivity,[status(thm)],[s19]) ).
fof(s29,plain,
! [X8,X10] :
~ ~ p(f(b)),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2]),skolemize(X9,sK2(X8,X10))],[s2]) ).
fof(s30,plain,
~ ~ ! [X8,X10] :
~ ~ p(f(b)),
inference(double_negation,[status(thm)],[s29]) ).
fof(s31,plain,
( ! [X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3)
| ( p(b)
=> t )
| ~ ! [X19] :
? [X20] : r(f(c),a) ),
inference(weaken,[status(thm)],[s26]) ).
fof(s32,plain,
c = c,
inference(reflexivity,[status(thm)],[s14]) ).
fof(s33,plain,
~ ~ ~ ( q(c)
& f(b) != c ),
inference(double_negation,[status(thm)],[s13]) ).
fof(s34,plain,
! [X10] :
? [X9] :
~ ~ p(f(b)),
inference(split_conjunct,[status(thm)],[s17]) ).
fof(s35,plain,
( ? [X21] :
! [X22] :
? [X23] : t
| ~ ? [X21] :
! [X22] :
? [X23] : t ),
inference(excluded_middle,[status(thm)],[s15]) ).
fof(s36,plain,
( ( ! [X24] :
? [X25] : t
| ? [X26] : r(X26,g(X26,X26)) )
=> ( ! [X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3)
| ( p(b)
=> t )
| ~ ! [X19] :
? [X20] : r(f(c),a) ) ),
inference(add_hypothesis,[status(thm)],[s31]) ).
fof(s37,plain,
! [X10] :
~ ~ p(f(b)),
inference(instantiate,[status(thm)],[s6]) ).
fof(s38,plain,
! [X27,X10] :
? [X9] :
~ ~ p(f(b)),
inference(rename_variable,[status(thm)],[s2]) ).
fof(s39,plain,
! [X28,X10] :
~ ~ p(f(b)),
inference(rename_variable,[status(thm)],[s6]) ).
fof(s40,plain,
? [X29] :
( ~ ( X29 != c )
| ~ q(c) ),
inference(existential_gen,[status(thm)],[s16]) ).
fof(s41,plain,
? [X30] :
( ~ q(X30)
| ~ ( f(b) != X30 ) ),
inference(existential_gen,[status(thm)],[s14]) ).
fof(s42,plain,
! [X2,X3] :
? [X4] :
! [X5] : r(f(c),X3),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(X1,sK3(X2))],[s9]) ).
fof(s43,plain,
~ ( q(c)
& f(b) != c ),
inference(remove_double_negation,[status(thm)],[s23]) ).
fof(s44,plain,
? [X31,X9] :
~ ~ p(X31),
inference(existential_gen,[status(thm)],[s15]) ).
fof(s45,plain,
~ ~ ! [X2] :
? [X1] :
! [X3] :
? [X4] :
! [X5] : r(f(c),X3),
inference(double_negation,[status(thm)],[s9]) ).
fof(s46,plain,
( ~ ~ ~ ( q(c)
& f(b) != c )
| ! [X32,X33] :
? [X34] : t ),
inference(weaken,[status(thm)],[s33]) ).
fof(negc,negated_conjecture,
~ ( ~ ~ ~ ( q(c)
& f(b) != c )
| ! [X32,X33] :
? [X34] : t ),
inference(negated_conjecture,[status(cth)],[c]) ).
fof(bot,plain,
$false,
inference(consequence,[status(thm)],[negc,s46]) ).
% SZS output end Proof