TPTP Problem File: ITP013_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP013_2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Ewords_2En2w__sub.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_2Ewords_2En2w__sub.p [Gau19]
% : HL406001_2.p [TPAP]
% Status : Theorem
% Rating : 0.44 v8.2.0, 0.60 v8.1.0, 0.73 v7.5.0
% Syntax : Number of formulae : 93 ( 30 unt; 40 typ; 0 def)
% Number of atoms : 288 ( 30 equ)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 98 ( 6 ~; 0 |; 14 &)
% ( 26 <=>; 52 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 143 ( 143 fml; 0 var)
% Number of types : 5 ( 3 usr)
% Number of type conns : 34 ( 24 >; 10 *; 0 +; 0 <<)
% Number of predicates : 8 ( 5 usr; 3 prp; 0-2 aty)
% Number of functors : 35 ( 35 usr; 13 con; 0-2 aty)
% Number of variables : 89 ( 89 !; 0 ?; 89 :)
% 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_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_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_2Ebool_2EF,type,
c_2Ebool_2EF: $i ).
tff(mem_c_2Ebool_2EF,axiom,
mem(c_2Ebool_2EF,bool) ).
tff(stp_fo_c_2Ebool_2EF,type,
fo__c_2Ebool_2EF: tp__o ).
tff(stp_eq_fo_c_2Ebool_2EF,axiom,
inj__o(fo__c_2Ebool_2EF) = c_2Ebool_2EF ).
tff(ax_false_p,axiom,
~ p(c_2Ebool_2EF) ).
tff(tp_c_2Ebool_2ET,type,
c_2Ebool_2ET: $i ).
tff(mem_c_2Ebool_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
tff(stp_fo_c_2Ebool_2ET,type,
fo__c_2Ebool_2ET: tp__o ).
tff(stp_eq_fo_c_2Ebool_2ET,axiom,
inj__o(fo__c_2Ebool_2ET) = c_2Ebool_2ET ).
tff(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
tff(tp_ty_2Efcp_2Ecart,type,
ty_2Efcp_2Ecart: ( del * del ) > del ).
tff(tp_c_2Ewords_2Eword__sub,type,
c_2Ewords_2Eword__sub: del > $i ).
tff(mem_c_2Ewords_2Eword__sub,axiom,
! [A_27a: del] : mem(c_2Ewords_2Eword__sub(A_27a),arr(ty_2Efcp_2Ecart(bool,A_27a),arr(ty_2Efcp_2Ecart(bool,A_27a),ty_2Efcp_2Ecart(bool,A_27a)))) ).
tff(tp_ty_2Enum_2Enum,type,
ty_2Enum_2Enum: del ).
tff(stp_ty_2Enum_2Enum,type,
tp__ty_2Enum_2Enum: $tType ).
tff(stp_inj_ty_2Enum_2Enum,type,
inj__ty_2Enum_2Enum: tp__ty_2Enum_2Enum > $i ).
tff(stp_surj_ty_2Enum_2Enum,type,
surj__ty_2Enum_2Enum: $i > tp__ty_2Enum_2Enum ).
tff(stp_inj_surj_ty_2Enum_2Enum,axiom,
! [X: tp__ty_2Enum_2Enum] : surj__ty_2Enum_2Enum(inj__ty_2Enum_2Enum(X)) = X ).
tff(stp_inj_mem_ty_2Enum_2Enum,axiom,
! [X: tp__ty_2Enum_2Enum] : mem(inj__ty_2Enum_2Enum(X),ty_2Enum_2Enum) ).
tff(stp_iso_mem_ty_2Enum_2Enum,axiom,
! [X: $i] :
( mem(X,ty_2Enum_2Enum)
=> ( X = inj__ty_2Enum_2Enum(surj__ty_2Enum_2Enum(X)) ) ) ).
tff(tp_c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: $i ).
tff(mem_c_2Earithmetic_2E_2D,axiom,
mem(c_2Earithmetic_2E_2D,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
tff(stp_fo_c_2Earithmetic_2E_2D,type,
fo__c_2Earithmetic_2E_2D: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum ) > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Earithmetic_2E_2D,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] : inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2E_2D(X0,X1)) = ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(X0)),inj__ty_2Enum_2Enum(X1)) ).
tff(tp_c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: $i ).
tff(mem_c_2Earithmetic_2E_3C_3D,axiom,
mem(c_2Earithmetic_2E_3C_3D,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) ).
tff(stp_fo_c_2Earithmetic_2E_3C_3D,type,
fo__c_2Earithmetic_2E_3C_3D: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum ) > tp__o ).
tff(stp_eq_fo_c_2Earithmetic_2E_3C_3D,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] : inj__o(fo__c_2Earithmetic_2E_3C_3D(X0,X1)) = ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(X0)),inj__ty_2Enum_2Enum(X1)) ).
tff(tp_c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND: del > $i ).
tff(mem_c_2Ebool_2ECOND,axiom,
! [A_27a: del] : mem(c_2Ebool_2ECOND(A_27a),arr(bool,arr(A_27a,arr(A_27a,A_27a)))) ).
tff(tp_c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: $i ).
tff(mem_c_2Earithmetic_2E_2B,axiom,
mem(c_2Earithmetic_2E_2B,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
tff(stp_fo_c_2Earithmetic_2E_2B,type,
fo__c_2Earithmetic_2E_2B: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum ) > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Earithmetic_2E_2B,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] : inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2E_2B(X0,X1)) = ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(X0)),inj__ty_2Enum_2Enum(X1)) ).
tff(tp_c_2Ewords_2En2w,type,
c_2Ewords_2En2w: del > $i ).
tff(mem_c_2Ewords_2En2w,axiom,
! [A_27a: del] : mem(c_2Ewords_2En2w(A_27a),arr(ty_2Enum_2Enum,ty_2Efcp_2Ecart(bool,A_27a))) ).
tff(tp_c_2Ewords_2Eword__2comp,type,
c_2Ewords_2Eword__2comp: del > $i ).
tff(mem_c_2Ewords_2Eword__2comp,axiom,
! [A_27a: del] : mem(c_2Ewords_2Eword__2comp(A_27a),arr(ty_2Efcp_2Ecart(bool,A_27a),ty_2Efcp_2Ecart(bool,A_27a))) ).
tff(tp_c_2Ewords_2Eword__add,type,
c_2Ewords_2Eword__add: del > $i ).
tff(mem_c_2Ewords_2Eword__add,axiom,
! [A_27a: del] : mem(c_2Ewords_2Eword__add(A_27a),arr(ty_2Efcp_2Ecart(bool,A_27a),arr(ty_2Efcp_2Ecart(bool,A_27a),ty_2Efcp_2Ecart(bool,A_27a)))) ).
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(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(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
tff(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
=> p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
=> $true )
<=> $true )
& ( ( $false
=> p(inj__o(V0t)) )
<=> $true )
& ( ( p(inj__o(V0t))
=> p(inj__o(V0t)) )
<=> $true )
& ( ( p(inj__o(V0t))
=> $false )
<=> ~ p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( ( V0x = V0x )
<=> $true ) ) ).
tff(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27a)
=> ( ( V0x = V1y )
<=> ( V1y = V0x ) ) ) ) ).
tff(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
<=> p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
<=> $true )
<=> p(inj__o(V0t)) )
& ( ( $false
<=> p(inj__o(V0t)) )
<=> ~ p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
<=> $false )
<=> ~ p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1: tp__o,V1t2: tp__o,V2t3: tp__o] :
( ( p(inj__o(V0t1))
=> ( p(inj__o(V1t2))
=> p(inj__o(V2t3)) ) )
<=> ( ( p(inj__o(V0t1))
& p(inj__o(V1t2)) )
=> p(inj__o(V2t3)) ) ) ).
tff(conj_thm_2Ebool_2EIMP__CONG,axiom,
! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
( ( ( p(inj__o(V0x))
<=> p(inj__o(V1x_27)) )
& ( p(inj__o(V1x_27))
=> ( p(inj__o(V2y))
<=> p(inj__o(V3y_27)) ) ) )
=> ( ( p(inj__o(V0x))
=> p(inj__o(V2y)) )
<=> ( p(inj__o(V1x_27))
=> p(inj__o(V3y_27)) ) ) ) ).
tff(conj_thm_2Ebool_2ECOND__CONG,axiom,
! [A_27a: del,V0P: tp__o,V1Q: tp__o,V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3x_27: $i] :
( mem(V3x_27,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5y_27: $i] :
( mem(V5y_27,A_27a)
=> ( ( ( p(inj__o(V0P))
<=> p(inj__o(V1Q)) )
& ( p(inj__o(V1Q))
=> ( V2x = V3x_27 ) )
& ( ~ p(inj__o(V1Q))
=> ( V4y = V5y_27 ) ) )
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V0P)),V2x),V4y) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V1Q)),V3x_27),V5y_27) ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2Ebool__case__thm,axiom,
! [A_27a: del] :
( ! [V0t1: $i] :
( mem(V0t1,A_27a)
=> ! [V1t2: $i] :
( mem(V1t2,A_27a)
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2ET)),V0t1),V1t2) = V0t1 ) ) )
& ! [V2t1: $i] :
( mem(V2t1,A_27a)
=> ! [V3t2: $i] :
( mem(V3t2,A_27a)
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2EF)),V2t1),V3t2) = V3t2 ) ) ) ) ).
tff(ax_thm_2Ewords_2Eword__sub__def,axiom,
! [A_27a: del,V0v: $i] :
( mem(V0v,ty_2Efcp_2Ecart(bool,A_27a))
=> ! [V1w: $i] :
( mem(V1w,ty_2Efcp_2Ecart(bool,A_27a))
=> ( ap(ap(c_2Ewords_2Eword__sub(A_27a),V0v),V1w) = ap(ap(c_2Ewords_2Eword__add(A_27a),V0v),ap(c_2Ewords_2Eword__2comp(A_27a),V1w)) ) ) ) ).
tff(conj_thm_2Ewords_2EWORD__LITERAL__ADD,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : ap(ap(c_2Ewords_2Eword__add(A_27a),ap(c_2Ewords_2Eword__2comp(A_27a),ap(c_2Ewords_2En2w(A_27a),inj__ty_2Enum_2Enum(V0m)))),ap(c_2Ewords_2Eword__2comp(A_27a),ap(c_2Ewords_2En2w(A_27a),inj__ty_2Enum_2Enum(V1n)))) = ap(c_2Ewords_2Eword__2comp(A_27a),ap(c_2Ewords_2En2w(A_27a),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))))
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ap(ap(c_2Ewords_2Eword__add(A_27b),ap(c_2Ewords_2En2w(A_27b),inj__ty_2Enum_2Enum(V2m))),ap(c_2Ewords_2Eword__2comp(A_27b),ap(c_2Ewords_2En2w(A_27b),inj__ty_2Enum_2Enum(V3n)))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Efcp_2Ecart(bool,A_27b)),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3n)),inj__ty_2Enum_2Enum(V2m))),ap(c_2Ewords_2En2w(A_27b),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n)))),ap(c_2Ewords_2Eword__2comp(A_27b),ap(c_2Ewords_2En2w(A_27b),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V3n)),inj__ty_2Enum_2Enum(V2m))))) ) ).
tff(conj_thm_2Ewords_2En2w__sub,conjecture,
! [A_27a: del,V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1b)),inj__ty_2Enum_2Enum(V0a)))
=> ( ap(c_2Ewords_2En2w(A_27a),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V0a)),inj__ty_2Enum_2Enum(V1b))) = ap(ap(c_2Ewords_2Eword__sub(A_27a),ap(c_2Ewords_2En2w(A_27a),inj__ty_2Enum_2Enum(V0a))),ap(c_2Ewords_2En2w(A_27a),inj__ty_2Enum_2Enum(V1b))) ) ) ).
%------------------------------------------------------------------------------