TPTP Problem File: ITP004_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP004_2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Epred__set_2EREST__SUBSET.p, bushy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Epred__set_2EREST__SUBSET.p [Gau19]
% : HL401501_2.p [TPAP]
% Status : Theorem
% Rating : 0.11 v8.2.0, 0.40 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 55 ( 15 unt; 25 typ; 0 def)
% Number of atoms : 179 ( 13 equ)
% Maximal formula atoms : 6 ( 3 avg)
% Number of connectives : 43 ( 2 ~; 0 |; 2 &)
% ( 8 <=>; 31 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 108 ( 108 fml; 0 var)
% Number of types : 4 ( 2 usr)
% Number of type conns : 24 ( 18 >; 6 *; 0 +; 0 <<)
% Number of predicates : 6 ( 5 usr; 1 prp; 0-2 aty)
% Number of functors : 21 ( 21 usr; 5 con; 0-2 aty)
% Number of variables : 54 ( 54 !; 0 ?; 54 :)
% SPC : TF0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001_2.ax').
%------------------------------------------------------------------------------
tff(stp_o,type,
tp__o: $tType ).
tff(stp_inj_o,type,
inj__o: tp__o > $i ).
tff(stp_surj_o,type,
surj__o: $i > tp__o ).
tff(stp_inj_surj_o,axiom,
! [X: tp__o] : surj__o(inj__o(X)) = X ).
tff(stp_inj_mem_o,axiom,
! [X: tp__o] : mem(inj__o(X),bool) ).
tff(stp_iso_mem_o,axiom,
! [X: $i] :
( mem(X,bool)
=> ( X = inj__o(surj__o(X)) ) ) ).
tff(tp_c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $i ).
tff(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem(c_2Emin_2E_3D_3D_3E,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Emin_2E_3D_3D_3E,type,
fo__c_2Emin_2E_3D_3D_3E: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Emin_2E_3D_3D_3E,axiom,
! [X0: tp__o,X1: tp__o] : inj__o(fo__c_2Emin_2E_3D_3D_3E(X0,X1)) = ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(X0)),inj__o(X1)) ).
tff(ax_imp_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ! [R: $i] :
( mem(R,bool)
=> ( p(ap(ap(c_2Emin_2E_3D_3D_3E,Q),R))
<=> ( p(Q)
=> p(R) ) ) ) ) ).
tff(tp_c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET: del > $i ).
tff(mem_c_2Epred__set_2ESUBSET,axiom,
! [A_27a: del] : mem(c_2Epred__set_2ESUBSET(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
tff(mem_c_2Ebool_2E_7E,axiom,
mem(c_2Ebool_2E_7E,arr(bool,bool)) ).
tff(stp_fo_c_2Ebool_2E_7E,type,
fo__c_2Ebool_2E_7E: tp__o > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_7E,axiom,
! [X0: tp__o] : inj__o(fo__c_2Ebool_2E_7E(X0)) = ap(c_2Ebool_2E_7E,inj__o(X0)) ).
tff(ax_neg_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ( p(ap(c_2Ebool_2E_7E,Q))
<=> ~ p(Q) ) ) ).
tff(tp_c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
tff(mem_c_2Ebool_2E_2F_5C,axiom,
mem(c_2Ebool_2E_2F_5C,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Ebool_2E_2F_5C,type,
fo__c_2Ebool_2E_2F_5C: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_2F_5C,axiom,
! [X0: tp__o,X1: tp__o] : inj__o(fo__c_2Ebool_2E_2F_5C(X0,X1)) = ap(ap(c_2Ebool_2E_2F_5C,inj__o(X0)),inj__o(X1)) ).
tff(ax_and_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ! [R: $i] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_2F_5C,Q),R))
<=> ( p(Q)
& p(R) ) ) ) ) ).
tff(tp_c_2Ebool_2EIN,type,
c_2Ebool_2EIN: del > $i ).
tff(mem_c_2Ebool_2EIN,axiom,
! [A_27a: del] : mem(c_2Ebool_2EIN(A_27a),arr(A_27a,arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Epred__set_2ECHOICE,type,
c_2Epred__set_2ECHOICE: del > $i ).
tff(mem_c_2Epred__set_2ECHOICE,axiom,
! [A_27a: del] : mem(c_2Epred__set_2ECHOICE(A_27a),arr(arr(A_27a,bool),A_27a)) ).
tff(tp_c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE: del > $i ).
tff(mem_c_2Epred__set_2EDELETE,axiom,
! [A_27a: del] : mem(c_2Epred__set_2EDELETE(A_27a),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Epred__set_2EREST,type,
c_2Epred__set_2EREST: del > $i ).
tff(mem_c_2Epred__set_2EREST,axiom,
! [A_27a: del] : mem(c_2Epred__set_2EREST(A_27a),arr(arr(A_27a,bool),arr(A_27a,bool))) ).
tff(tp_c_2Emin_2E_3D,type,
c_2Emin_2E_3D: del > $i ).
tff(mem_c_2Emin_2E_3D,axiom,
! [A_27a: del] : mem(c_2Emin_2E_3D(A_27a),arr(A_27a,arr(A_27a,bool))) ).
tff(ax_eq_p,axiom,
! [A: del,X: $i] :
( mem(X,A)
=> ! [Y: $i] :
( mem(Y,A)
=> ( p(ap(ap(c_2Emin_2E_3D(A),X),Y))
<=> ( X = Y ) ) ) ) ).
tff(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
tff(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : mem(c_2Ebool_2E_21(A_27a),arr(arr(A_27a,bool),bool)) ).
tff(ax_all_p,axiom,
! [A: del,Q: $i] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_21(A),Q))
<=> ! [X: $i] :
( mem(X,A)
=> p(ap(Q,X)) ) ) ) ).
tff(ax_thm_2Epred__set_2ESUBSET__DEF,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t))
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) ).
tff(conj_thm_2Epred__set_2EIN__DELETE,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V2y)))
<=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s))
& ( V1x != V2y ) ) ) ) ) ) ).
tff(ax_thm_2Epred__set_2EREST__DEF,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( ap(c_2Epred__set_2EREST(A_27a),V0s) = ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),ap(c_2Epred__set_2ECHOICE(A_27a),V0s)) ) ) ).
tff(conj_thm_2Epred__set_2EREST__SUBSET,conjecture,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Epred__set_2EREST(A_27a),V0s)),V0s)) ) ).
%------------------------------------------------------------------------------