ITP001 Axioms: ITP091_5.ax
%------------------------------------------------------------------------------
% File : ITP091_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : EncodeVar_2.ax [Gau20]
% : HL4091_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 25 ( 2 unt; 6 typ; 0 def)
% Number of atoms : 414 ( 6 equ)
% Maximal formula atoms : 8 ( 16 avg)
% Number of connectives : 47 ( 0 ~; 0 |; 7 &)
% ( 8 <=>; 32 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 7 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 348 ( 348 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 12 ( 6 >; 6 *; 0 +; 0 <<)
% Number of predicates : 39 ( 38 usr; 15 prp; 0-4 aty)
% Number of functors : 6 ( 6 usr; 0 con; 1-3 aty)
% Number of variables : 60 ( 58 !; 2 ?; 60 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2EEncodeVar_2Efixed__width,type,
c_2EEncodeVar_2Efixed__width: del > $i ).
tff(mem_c_2EEncodeVar_2Efixed__width,axiom,
! [A_27a: del] : mem(c_2EEncodeVar_2Efixed__width(A_27a),arr(ty_2Enum_2Enum,arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))),bool))) ).
tff(tp_c_2EEncodeVar_2Eof__length,type,
c_2EEncodeVar_2Eof__length: del > $i ).
tff(mem_c_2EEncodeVar_2Eof__length,axiom,
! [A_27a: del] : mem(c_2EEncodeVar_2Eof__length(A_27a),arr(ty_2Enum_2Enum,arr(ty_2Elist_2Elist(A_27a),bool))) ).
tff(ax_thm_2EEncodeVar_2Efixed__width__def,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( mem(V1c,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
=> ( p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27a),inj__ty_2Enum_2Enum(V0n)),V1c))
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2ECoder_2Edomain(A_27a),V1c),V2x))
=> ( surj__ty_2Enum_2Enum(ap(c_2Elist_2ELENGTH(bool),ap(ap(c_2ECoder_2Eencoder(A_27a),V1c),V2x))) = V0n ) ) ) ) ) ).
tff(ax_thm_2EEncodeVar_2Eof__length__def,axiom,
! [A_27a: del,V0l: $i] :
( mem(V0l,ty_2Elist_2Elist(A_27a))
=> ! [V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Elist_2Elist(A_27a)),V0l),ap(c_2EEncodeVar_2Eof__length(A_27a),inj__ty_2Enum_2Enum(V1n))))
<=> ( surj__ty_2Enum_2Enum(ap(c_2Elist_2ELENGTH(A_27a),V0l)) = V1n ) ) ) ).
tff(lamtp_f2440,type,
f2440: ( del * $i * $i ) > $i ).
tff(lameq_f2440,axiom,
! [A_27a: del,V1c: $i] :
( mem(V1c,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
=> ! [V0phi: $i] :
( mem(V0phi,arr(A_27a,bool))
=> ! [V4w: tp__c_ty_2Elist_2Elist_o] : ( ap(f2440(A_27a,V1c,V0phi),inj__c_ty_2Elist_2Elist_o(V4w)) = ap(V0phi,ap(ap(c_2ECoder_2Edecoder(A_27a),V1c),inj__c_ty_2Elist_2Elist_o(V4w))) ) ) ) ).
tff(conj_thm_2EEncodeVar_2Efixed__width__univ,axiom,
! [A_27a: del,V0phi: $i] :
( mem(V0phi,arr(A_27a,bool))
=> ! [V1c: $i] :
( mem(V1c,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2ECoder_2Ewf__coder(A_27a),V1c))
& p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27a),inj__ty_2Enum_2Enum(V2n)),V1c)) )
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2ECoder_2Edomain(A_27a),V1c),V3x))
=> p(ap(V0phi,V3x)) ) )
<=> p(ap(ap(c_2Ebool_2ERES__FORALL(ty_2Elist_2Elist(bool)),ap(c_2EEncodeVar_2Eof__length(bool),inj__ty_2Enum_2Enum(V2n))),f2440(A_27a,V1c,V0phi))) ) ) ) ) ).
tff(lamtp_f2441,type,
f2441: ( del * $i ) > $i ).
tff(lameq_f2441,axiom,
! [A_27a: del,V0phi: $i] :
( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
=> ! [V2w: $i] : ( ap(f2441(A_27a,V0phi),V2w) = ap(V0phi,V2w) ) ) ).
tff(lamtp_f2442,type,
f2442: ( del * $i * $i ) > $i ).
tff(lameq_f2442,axiom,
! [A_27a: del,V3x: $i] :
( mem(V3x,A_27a)
=> ! [V0phi: $i] :
( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
=> ! [V4w: $i] : ( ap(f2442(A_27a,V3x,V0phi),V4w) = ap(V0phi,ap(ap(c_2Elist_2ECONS(A_27a),V3x),V4w)) ) ) ) ).
tff(conj_thm_2EEncodeVar_2Eof__length__univ__suc,axiom,
! [A_27a: del,V0phi: $i] :
( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
=> ! [V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2ERES__FORALL(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V1n)))),f2441(A_27a,V0phi)))
<=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> p(ap(ap(c_2Ebool_2ERES__FORALL(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),inj__ty_2Enum_2Enum(V1n))),f2442(A_27a,V3x,V0phi))) ) ) ) ).
tff(lamtp_f2443,type,
f2443: ( del * $i ) > $i ).
tff(lameq_f2443,axiom,
! [A_27a: del,V0phi: $i] :
( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
=> ! [V1w: $i] : ( ap(f2443(A_27a,V0phi),V1w) = ap(V0phi,V1w) ) ) ).
tff(conj_thm_2EEncodeVar_2Eof__length__univ__zero,axiom,
! [A_27a: del,V0phi: $i] :
( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
=> ( p(ap(ap(c_2Ebool_2ERES__FORALL(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),f2443(A_27a,V0phi)))
<=> p(ap(V0phi,c_2Elist_2ENIL(A_27a))) ) ) ).
tff(conj_thm_2EEncodeVar_2Efixed__width__exists,axiom,
! [A_27a: del,V0phi: $i] :
( mem(V0phi,arr(A_27a,bool))
=> ! [V1c: $i] :
( mem(V1c,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2ECoder_2Ewf__coder(A_27a),V1c))
& p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27a),inj__ty_2Enum_2Enum(V2n)),V1c)) )
=> ( ? [V3x: $i] :
( mem(V3x,A_27a)
& p(ap(ap(c_2ECoder_2Edomain(A_27a),V1c),V3x))
& p(ap(V0phi,V3x)) )
<=> p(ap(ap(c_2Ebool_2ERES__EXISTS(ty_2Elist_2Elist(bool)),ap(c_2EEncodeVar_2Eof__length(bool),inj__ty_2Enum_2Enum(V2n))),f2440(A_27a,V1c,V0phi))) ) ) ) ) ).
tff(conj_thm_2EEncodeVar_2Eof__length__exists__suc,axiom,
! [A_27a: del,V0phi: $i] :
( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
=> ! [V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2ERES__EXISTS(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V1n)))),f2441(A_27a,V0phi)))
<=> ? [V3x: $i] :
( mem(V3x,A_27a)
& p(ap(ap(c_2Ebool_2ERES__EXISTS(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),inj__ty_2Enum_2Enum(V1n))),f2442(A_27a,V3x,V0phi))) ) ) ) ).
tff(conj_thm_2EEncodeVar_2Eof__length__exists__zero,axiom,
! [A_27a: del,V0phi: $i] :
( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),f2443(A_27a,V0phi)))
<=> p(ap(V0phi,c_2Elist_2ENIL(A_27a))) ) ) ).
tff(conj_thm_2EEncodeVar_2Efixed__width__unit,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eone_2Eone,bool))
=> p(ap(ap(c_2EEncodeVar_2Efixed__width(ty_2Eone_2Eone),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(c_2ECoder_2Eunit__coder,V0p))) ) ).
tff(conj_thm_2EEncodeVar_2Efixed__width__bool,axiom,
! [V0p: $i] :
( mem(V0p,arr(bool,bool))
=> p(ap(ap(c_2EEncodeVar_2Efixed__width(bool),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),ap(c_2ECoder_2Ebool__coder,V0p))) ) ).
tff(conj_thm_2EEncodeVar_2Efixed__width__prod,axiom,
! [A_27a: del,A_27b: del,V0c1: $i] :
( mem(V0c1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
=> ! [V1c2: $i] :
( mem(V1c2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(A_27b,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27b,ty_2Elist_2Elist(bool)))))))
=> ! [V2n1: tp__ty_2Enum_2Enum,V3n2: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27a),inj__ty_2Enum_2Enum(V2n1)),V0c1))
& p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27b),inj__ty_2Enum_2Enum(V3n2)),V1c2)) )
=> p(ap(ap(c_2EEncodeVar_2Efixed__width(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n1)),inj__ty_2Enum_2Enum(V3n2))),ap(ap(c_2ECoder_2Eprod__coder(A_27a,A_27b,A_27a,A_27b),V0c1),V1c2))) ) ) ) ).
tff(conj_thm_2EEncodeVar_2Efixed__width__sum,axiom,
! [A_27a: del,A_27b: del,V0c1: $i] :
( mem(V0c1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
=> ! [V1c2: $i] :
( mem(V1c2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(A_27b,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27b,ty_2Elist_2Elist(bool)))))))
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27a),inj__ty_2Enum_2Enum(V2n)),V0c1))
& p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27b),inj__ty_2Enum_2Enum(V2n)),V1c2)) )
=> p(ap(ap(c_2EEncodeVar_2Efixed__width(ty_2Esum_2Esum(A_27a,A_27b)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n))),ap(ap(c_2ECoder_2Esum__coder(A_27a,A_27b,A_27a,A_27b),V0c1),V1c2))) ) ) ) ).
tff(conj_thm_2EEncodeVar_2Efixed__width__bnum,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1p: $i] :
( mem(V1p,arr(ty_2Enum_2Enum,bool))
=> p(ap(ap(c_2EEncodeVar_2Efixed__width(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2ECoder_2Ebnum__coder,inj__ty_2Enum_2Enum(V0m)),V1p))) ) ).
%------------------------------------------------------------------------------