TPTP Problem File: NUM257-2.p
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%--------------------------------------------------------------------------
% File : NUM257-2 : TPTP v8.2.0. Bugfixed v2.1.0.
% Domain : Number Theory (Ordinals)
% Problem : Transfinite recursion property 13
% Version : [Qua92] axioms : Augmented.
% English :
% Refs : [Qua92] Quaife (1992), Automated Deduction in von Neumann-Bern
% Source : [Quaife]
% Names : TREC15 [Quaife]
% Status : Unknown
% Rating : 1.00 v2.1.0
% Syntax : Number of clauses : 176 ( 49 unt; 18 nHn; 136 RR)
% Number of literals : 390 ( 85 equ; 200 neg)
% Maximal clause size : 6 ( 2 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 17 ( 16 usr; 0 prp; 1-3 aty)
% Number of functors : 65 ( 65 usr; 20 con; 0-3 aty)
% Number of variables : 349 ( 40 sgn)
% SPC : CNF_UNK_RFO_SEQ_NHN
% Comments : Not in [Qua92]. Theorem TREC15 in [Quaife].
% : Quaife proves all these problems by augmenting the axioms with
% all previously proved theorems. The user may create an augmented
% version of this problem by adding all previously proved theorems.
% These include all of [Qua92]'s set theory and Boolean algebra
% theorems, available from the SET domain.
% Bugfixes : v1.0.1 - Bugfix in SET004-1.ax.
% : v2.1.0 - Bugfix in SET004-0.ax.
%--------------------------------------------------------------------------
%----Include von Neuman-Bernays-Godel set theory axioms
include('Axioms/SET004-0.ax').
%----Include Set theory (Boolean algebra) axioms based on NBG set theory
include('Axioms/SET004-1.ax').
%----Include ordinal number theory axioms.
include('Axioms/NUM004-0.ax').
%--------------------------------------------------------------------------
%----(TREC.LEMMA0).
cnf(transfinite_recursion_lemma0,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| subclass(domain_of(intersection(complement(Y),X)),ordinal_numbers) ) ).
%----(TREC.LEMMA1).
cnf(transfinite_recursion_lemma1,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| ~ member(ordered_pair(U,V),X)
| ~ member(U,least(element_relation,domain_of(intersection(complement(Y),X))))
| member(ordered_pair(U,V),Y) ) ).
%----(TREC.LEMMA2).
cnf(transfinite_recursion_lemma2,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| ~ member(ordered_pair(U,V),Y)
| ~ member(U,least(element_relation,domain_of(intersection(complement(Y),X))))
| subclass(X,Y)
| member(ordered_pair(U,V),X) ) ).
%----(TREC.LEMMA3)
cnf(transfinite_recursion_lemma3,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| subclass(X,Y)
| restrict(X,least(element_relation,domain_of(intersection(complement(Y),X))),universal_class) = restrict(Y,least(element_relation,domain_of(intersection(complement(Y),X))),universal_class) ) ).
%----(TREC.LEMMA4).
cnf(transfinite_recursion_lemma4,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| ~ member(domain_of(X),domain_of(Y))
| subclass(X,Y)
| apply(Y,least(element_relation,domain_of(intersection(complement(Y),X)))) = apply(X,least(element_relation,domain_of(intersection(complement(Y),X)))) ) ).
%----(TREC.LEMMA5).
cnf(transfinite_recursion_lemma5,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| ~ member(domain_of(X),domain_of(Y))
| subclass(X,Y)
| member(ordered_pair(least(element_relation,domain_of(intersection(complement(Y),X))),apply(Y,least(element_relation,domain_of(intersection(complement(Y),X))))),Y) ) ).
%----(TREC.LEMMA6).
cnf(transfinite_recursion_lemma6,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| ~ member(domain_of(X),domain_of(Y))
| subclass(X,Y) ) ).
%----corollary.
cnf(corollary_1_to_transfinite_recursion_lemma6,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| member(union(X,Y),recursion_equation_functions(Z)) ) ).
cnf(corollary_2_to_transfinite_recursion_lemma6,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| function(union(X,Y)) ) ).
%----(TREC.LEMMA7).
cnf(transfinite_recursion_lemma7,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| ~ member(domain_of(X),domain_of(Y))
| ~ member(U,domain_of(X))
| restrict(X,U,universal_class) = restrict(Y,U,universal_class) ) ).
%----(TREC.LEMMA8).
cnf(transfinite_recursion_lemma8,axiom,
( ~ member(X,recursion_equation_functions(Z))
| ~ member(Y,recursion_equation_functions(Z))
| ~ member(domain_of(X),domain_of(Y))
| subclass(rest_of(X),rest_of(Y)) ) ).
%----(TREC.LEMMA9).
cnf(transfinite_recursion_lemma9_1,axiom,
( ~ member(Z,universal_class)
| image(image(composition_function,singleton(Z)),image(rest_relation,recursion_equation_functions(Z))) = recursion_equation_functions(Z) ) ).
cnf(transfinite_recursion_lemma9_2,axiom,
image(comp(Z),image(rest_relation,recursion_equation_functions(Z))) = recursion_equation_functions(Z) ).
%----next not proved.
cnf(transfinite_recursion_lemma9_3,axiom,
( ~ function(X)
| ~ function(Y)
| domain_of(X) != ordinal_numbers
| domain_of(Y) != ordinal_numbers
| X = Y
| restrict(X,least(element_relation,domain_of(intersection(complement(X),Y))),universal_class) = restrict(Y,least(element_relation,domain_of(intersection(complement(X),Y))),universal_class) ) ).
%----(TREC.LEMMA10).
cnf(transfinite_recursion_lemma10,axiom,
( ~ function(X)
| compose(Z,rest_of(X)) != X
| domain_of(X) != ordinal_numbers
| subclass(sum_class(recursion_equation_functions(Z)),X)
| apply(sum_class(recursion_equation_functions(Z)),least(element_relation,domain_of(intersection(complement(X),sum_class(recursion_equation_functions(Z)))))) = apply(X,least(element_relation,domain_of(intersection(complement(X),sum_class(recursion_equation_functions(Z)))))) ) ).
%----(TREC.LEMMA11).
cnf(transfinite_recursion_lemma11,axiom,
( ~ function(X)
| compose(Z,rest_of(X)) != X
| domain_of(X) != ordinal_numbers
| ~ member(ordered_pair(least(element_relation,domain_of(intersection(complement(X),sum_class(recursion_equation_functions(Z))))),apply(sum_class(recursion_equation_functions(Z)),least(element_relation,domain_of(intersection(complement(X),sum_class(recursion_equation_functions(Z))))))),intersection(complement(X),sum_class(recursion_equation_functions(Z))))
| subclass(sum_class(recursion_equation_functions(Z)),X) ) ).
cnf(prove_transfinite_recursion_property13_1,negated_conjecture,
member(x,recursion_equation_functions(z)) ).
cnf(prove_transfinite_recursion_property13_2,negated_conjecture,
~ member(union(singleton(ordered_pair(domain_of(x),apply(z,x))),x),recursion_equation_functions(z)) ).
%--------------------------------------------------------------------------