ITP001 Axioms: ITP022+5.ax
%------------------------------------------------------------------------------
% File : ITP022+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 : ind_type+2.ax [Gau20]
% : HL4022+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 80 ( 7 unt; 0 def)
% Number of atoms : 355 ( 76 equ)
% Maximal formula atoms : 19 ( 4 avg)
% Number of connectives : 277 ( 2 ~; 1 |; 31 &)
% ( 17 <=>; 226 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 66 ( 66 usr; 22 con; 0-7 aty)
% Number of variables : 269 ( 260 !; 9 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Eind__type_2Erecspace,axiom,
! [A0] :
( ne(A0)
=> ne(ty_2Eind__type_2Erecspace(A0)) ) ).
fof(mem_c_2Eind__type_2EBOTTOM,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2EBOTTOM(A_27a),ty_2Eind__type_2Erecspace(A_27a)) ) ).
fof(mem_c_2Eind__type_2ECONSTR,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2ECONSTR(A_27a),arr(ty_2Enum_2Enum,arr(A_27a,arr(arr(ty_2Enum_2Enum,ty_2Eind__type_2Erecspace(A_27a)),ty_2Eind__type_2Erecspace(A_27a))))) ) ).
fof(mem_c_2Eind__type_2EFCONS,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2EFCONS(A_27a),arr(A_27a,arr(arr(ty_2Enum_2Enum,A_27a),arr(ty_2Enum_2Enum,A_27a)))) ) ).
fof(mem_c_2Eind__type_2EFNIL,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2EFNIL(A_27a),arr(ty_2Enum_2Enum,A_27a)) ) ).
fof(mem_c_2Eind__type_2EINJA,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2EINJA(A_27a),arr(A_27a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))) ) ).
fof(mem_c_2Eind__type_2EINJF,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2EINJF(A_27a),arr(arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))),arr(ty_2Enum_2Enum,arr(A_27a,bool)))) ) ).
fof(mem_c_2Eind__type_2EINJN,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2EINJN(A_27a),arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool)))) ) ).
fof(mem_c_2Eind__type_2EINJP,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2EINJP(A_27a),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,arr(A_27a,bool))))) ) ).
fof(mem_c_2Eind__type_2EISO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eind__type_2EISO(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27b,A_27a),bool))) ) ) ).
fof(mem_c_2Eind__type_2ENUMFST,axiom,
mem(c_2Eind__type_2ENUMFST,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Eind__type_2ENUMLEFT,axiom,
mem(c_2Eind__type_2ENUMLEFT,arr(ty_2Enum_2Enum,bool)) ).
fof(mem_c_2Eind__type_2ENUMPAIR,axiom,
mem(c_2Eind__type_2ENUMPAIR,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
fof(mem_c_2Eind__type_2ENUMRIGHT,axiom,
mem(c_2Eind__type_2ENUMRIGHT,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Eind__type_2ENUMSND,axiom,
mem(c_2Eind__type_2ENUMSND,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Eind__type_2ENUMSUM,axiom,
mem(c_2Eind__type_2ENUMSUM,arr(bool,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
fof(mem_c_2Eind__type_2EZBOT,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2EZBOT(A_27a),arr(ty_2Enum_2Enum,arr(A_27a,bool))) ) ).
fof(mem_c_2Eind__type_2EZCONSTR,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2EZCONSTR(A_27a),arr(ty_2Enum_2Enum,arr(A_27a,arr(arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))),arr(ty_2Enum_2Enum,arr(A_27a,bool)))))) ) ).
fof(mem_c_2Eind__type_2EZRECSPACE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2EZRECSPACE(A_27a),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),bool)) ) ).
fof(mem_c_2Eind__type_2Edest__rec,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2Edest__rec(A_27a),arr(ty_2Eind__type_2Erecspace(A_27a),arr(ty_2Enum_2Enum,arr(A_27a,bool)))) ) ).
fof(mem_c_2Eind__type_2Emk__rec,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eind__type_2Emk__rec(A_27a),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),ty_2Eind__type_2Erecspace(A_27a))) ) ).
fof(conj_thm_2Eind__type_2EINJ__INVERSE2,axiom,
! [A_27A] :
( ne(A_27A)
=> ! [A_27B] :
( ne(A_27B)
=> ! [A_27C] :
( ne(A_27C)
=> ! [V0P] :
( mem(V0P,arr(A_27A,arr(A_27B,A_27C)))
=> ( ! [V1x1] :
( mem(V1x1,A_27A)
=> ! [V2y1] :
( mem(V2y1,A_27B)
=> ! [V3x2] :
( mem(V3x2,A_27A)
=> ! [V4y2] :
( mem(V4y2,A_27B)
=> ( ap(ap(V0P,V1x1),V2y1) = ap(ap(V0P,V3x2),V4y2)
<=> ( V1x1 = V3x2
& V2y1 = V4y2 ) ) ) ) ) )
=> ? [V5X] :
( mem(V5X,arr(A_27C,A_27A))
& ? [V6Y] :
( mem(V6Y,arr(A_27C,A_27B))
& ! [V7x] :
( mem(V7x,A_27A)
=> ! [V8y] :
( mem(V8y,A_27B)
=> ( ap(V5X,ap(ap(V0P,V7x),V8y)) = V7x
& ap(V6Y,ap(ap(V0P,V7x),V8y)) = V8y ) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Eind__type_2ENUMPAIR,axiom,
! [V0x] :
( mem(V0x,ty_2Enum_2Enum)
=> ! [V1y] :
( mem(V1y,ty_2Enum_2Enum)
=> ap(ap(c_2Eind__type_2ENUMPAIR,V0x),V1y) = ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Earithmetic_2EEXP,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V0x)),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V1y)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ) ) ).
fof(conj_thm_2Eind__type_2ENUMPAIR__INJ__LEMMA,axiom,
! [V0x1] :
( mem(V0x1,ty_2Enum_2Enum)
=> ! [V1y1] :
( mem(V1y1,ty_2Enum_2Enum)
=> ! [V2x2] :
( mem(V2x2,ty_2Enum_2Enum)
=> ! [V3y2] :
( mem(V3y2,ty_2Enum_2Enum)
=> ( ap(ap(c_2Eind__type_2ENUMPAIR,V0x1),V1y1) = ap(ap(c_2Eind__type_2ENUMPAIR,V2x2),V3y2)
=> V0x1 = V2x2 ) ) ) ) ) ).
fof(conj_thm_2Eind__type_2ENUMPAIR__INJ,axiom,
! [V0x1] :
( mem(V0x1,ty_2Enum_2Enum)
=> ! [V1y1] :
( mem(V1y1,ty_2Enum_2Enum)
=> ! [V2x2] :
( mem(V2x2,ty_2Enum_2Enum)
=> ! [V3y2] :
( mem(V3y2,ty_2Enum_2Enum)
=> ( ap(ap(c_2Eind__type_2ENUMPAIR,V0x1),V1y1) = ap(ap(c_2Eind__type_2ENUMPAIR,V2x2),V3y2)
<=> ( V0x1 = V2x2
& V1y1 = V3y2 ) ) ) ) ) ) ).
fof(ax_thm_2Eind__type_2ENUMPAIR__DEST,axiom,
! [V0x] :
( mem(V0x,ty_2Enum_2Enum)
=> ! [V1y] :
( mem(V1y,ty_2Enum_2Enum)
=> ( ap(c_2Eind__type_2ENUMFST,ap(ap(c_2Eind__type_2ENUMPAIR,V0x),V1y)) = V0x
& ap(c_2Eind__type_2ENUMSND,ap(ap(c_2Eind__type_2ENUMPAIR,V0x),V1y)) = V1y ) ) ) ).
fof(ax_thm_2Eind__type_2ENUMSUM,axiom,
! [V0b] :
( mem(V0b,bool)
=> ! [V1x] :
( mem(V1x,ty_2Enum_2Enum)
=> ap(ap(c_2Eind__type_2ENUMSUM,V0b),V1x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),V0b),ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V1x))),ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V1x)) ) ) ).
fof(conj_thm_2Eind__type_2ENUMSUM__INJ,axiom,
! [V0b1] :
( mem(V0b1,bool)
=> ! [V1x1] :
( mem(V1x1,ty_2Enum_2Enum)
=> ! [V2b2] :
( mem(V2b2,bool)
=> ! [V3x2] :
( mem(V3x2,ty_2Enum_2Enum)
=> ( ap(ap(c_2Eind__type_2ENUMSUM,V0b1),V1x1) = ap(ap(c_2Eind__type_2ENUMSUM,V2b2),V3x2)
<=> ( ( p(V0b1)
<=> p(V2b2) )
& V1x1 = V3x2 ) ) ) ) ) ) ).
fof(ax_thm_2Eind__type_2ENUMSUM__DEST,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1y] :
( mem(V1y,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Eind__type_2ENUMLEFT,ap(ap(c_2Eind__type_2ENUMSUM,V0x),V1y)))
<=> p(V0x) )
& ap(c_2Eind__type_2ENUMRIGHT,ap(ap(c_2Eind__type_2ENUMSUM,V0x),V1y)) = V1y ) ) ) ).
fof(lameq_f208,axiom,
! [A_27a,V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] : ap(f208(A_27a,V0m),V1n) = k(A_27a,ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1n),V0m)) ) ).
fof(ax_thm_2Eind__type_2EINJN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ap(c_2Eind__type_2EINJN(A_27a),V0m) = f208(A_27a,V0m) ) ) ).
fof(conj_thm_2Eind__type_2EINJN__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n1] :
( mem(V0n1,ty_2Enum_2Enum)
=> ! [V1n2] :
( mem(V1n2,ty_2Enum_2Enum)
=> ( ap(c_2Eind__type_2EINJN(A_27a),V0n1) = ap(c_2Eind__type_2EINJN(A_27a),V1n2)
<=> V0n1 = V1n2 ) ) ) ) ).
fof(lameq_f209,axiom,
! [A_27a,V0a] :
( mem(V0a,A_27a)
=> ! [V2b] : ap(f209(A_27a,V0a),V2b) = ap(ap(c_2Emin_2E_3D(A_27a),V2b),V0a) ) ).
fof(ax_thm_2Eind__type_2EINJA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ap(c_2Eind__type_2EINJA(A_27a),V0a) = k(ty_2Enum_2Enum,f209(A_27a,V0a)) ) ) ).
fof(conj_thm_2Eind__type_2EINJA__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a1] :
( mem(V0a1,A_27a)
=> ! [V1a2] :
( mem(V1a2,A_27a)
=> ( ap(c_2Eind__type_2EINJA(A_27a),V0a1) = ap(c_2Eind__type_2EINJA(A_27a),V1a2)
<=> V0a1 = V1a2 ) ) ) ) ).
fof(lameq_f210,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ! [V1n] : ap(f210(A_27a,V0f),V1n) = ap(ap(V0f,ap(c_2Eind__type_2ENUMFST,V1n)),ap(c_2Eind__type_2ENUMSND,V1n)) ) ).
fof(ax_thm_2Eind__type_2EINJF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ap(c_2Eind__type_2EINJF(A_27a),V0f) = f210(A_27a,V0f) ) ) ).
fof(conj_thm_2Eind__type_2EINJF__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f1] :
( mem(V0f1,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ! [V1f2] :
( mem(V1f2,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ( ap(c_2Eind__type_2EINJF(A_27a),V0f1) = ap(c_2Eind__type_2EINJF(A_27a),V1f2)
<=> V0f1 = V1f2 ) ) ) ) ).
fof(lameq_f211,axiom,
! [A_27a,V0f1] :
( mem(V0f1,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V1f2] :
( mem(V1f2,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3a] : ap(f211(A_27a,V0f1,V2n,V1f2),V3a) = ap(ap(ap(c_2Ebool_2ECOND(bool),ap(c_2Eind__type_2ENUMLEFT,V2n)),ap(ap(V0f1,ap(c_2Eind__type_2ENUMRIGHT,V2n)),V3a)),ap(ap(V1f2,ap(c_2Eind__type_2ENUMRIGHT,V2n)),V3a)) ) ) ) ).
fof(lameq_f212,axiom,
! [A_27a,V0f1] :
( mem(V0f1,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1f2] :
( mem(V1f2,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2n] : ap(f212(A_27a,V0f1,V1f2),V2n) = f211(A_27a,V0f1,V2n,V1f2) ) ) ).
fof(ax_thm_2Eind__type_2EINJP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f1] :
( mem(V0f1,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1f2] :
( mem(V1f2,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ap(ap(c_2Eind__type_2EINJP(A_27a),V0f1),V1f2) = f212(A_27a,V0f1,V1f2) ) ) ) ).
fof(conj_thm_2Eind__type_2EINJP__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f1] :
( mem(V0f1,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1f1_27] :
( mem(V1f1_27,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2f2] :
( mem(V2f2,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3f2_27] :
( mem(V3f2_27,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ap(ap(c_2Eind__type_2EINJP(A_27a),V0f1),V2f2) = ap(ap(c_2Eind__type_2EINJP(A_27a),V1f1_27),V3f2_27)
<=> ( V0f1 = V1f1_27
& V2f2 = V3f2_27 ) ) ) ) ) ) ) ).
fof(ax_thm_2Eind__type_2EZCONSTR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0c] :
( mem(V0c,ty_2Enum_2Enum)
=> ! [V1i] :
( mem(V1i,A_27a)
=> ! [V2r] :
( mem(V2r,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ap(ap(ap(c_2Eind__type_2EZCONSTR(A_27a),V0c),V1i),V2r) = ap(ap(c_2Eind__type_2EINJP(A_27a),ap(c_2Eind__type_2EINJN(A_27a),ap(c_2Enum_2ESUC,V0c))),ap(ap(c_2Eind__type_2EINJP(A_27a),ap(c_2Eind__type_2EINJA(A_27a),V1i)),ap(c_2Eind__type_2EINJF(A_27a),V2r))) ) ) ) ) ).
fof(ax_thm_2Eind__type_2EZBOT,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Eind__type_2EZBOT(A_27a) = ap(ap(c_2Eind__type_2EINJP(A_27a),ap(c_2Eind__type_2EINJN(A_27a),c_2Enum_2E0)),ap(c_2Emin_2E_40(arr(ty_2Enum_2Enum,arr(A_27a,bool))),k(arr(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Ebool_2ET))) ) ).
fof(conj_thm_2Eind__type_2EZCONSTR__ZBOT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0c] :
( mem(V0c,ty_2Enum_2Enum)
=> ! [V1i] :
( mem(V1i,A_27a)
=> ! [V2r] :
( mem(V2r,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ap(ap(ap(c_2Eind__type_2EZCONSTR(A_27a),V0c),V1i),V2r) != c_2Eind__type_2EZBOT(A_27a) ) ) ) ) ).
fof(lameq_f213,axiom,
! [A_27a,V5r] :
( mem(V5r,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ! [V1ZRECSPACE_27] :
( mem(V1ZRECSPACE_27,arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),bool))
=> ! [V6n] : ap(f213(A_27a,V5r,V1ZRECSPACE_27),V6n) = ap(V1ZRECSPACE_27,ap(V5r,V6n)) ) ) ).
fof(lameq_f214,axiom,
! [A_27a,V3c] :
( mem(V3c,ty_2Enum_2Enum)
=> ! [V4i] :
( mem(V4i,A_27a)
=> ! [V2a0] :
( mem(V2a0,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1ZRECSPACE_27] :
( mem(V1ZRECSPACE_27,arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),bool))
=> ! [V5r] : ap(f214(A_27a,V3c,V4i,V2a0,V1ZRECSPACE_27),V5r) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2a0),ap(ap(ap(c_2Eind__type_2EZCONSTR(A_27a),V3c),V4i),V5r))),ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f213(A_27a,V5r,V1ZRECSPACE_27))) ) ) ) ) ).
fof(lameq_f215,axiom,
! [A_27a,V1ZRECSPACE_27] :
( mem(V1ZRECSPACE_27,arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),bool))
=> ! [V2a0] :
( mem(V2a0,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3c] :
( mem(V3c,ty_2Enum_2Enum)
=> ! [V4i] : ap(f215(A_27a,V1ZRECSPACE_27,V2a0,V3c),V4i) = ap(c_2Ebool_2E_3F(arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool)))),f214(A_27a,V3c,V4i,V2a0,V1ZRECSPACE_27)) ) ) ) ).
fof(lameq_f216,axiom,
! [A_27a,V2a0] :
( mem(V2a0,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1ZRECSPACE_27] :
( mem(V1ZRECSPACE_27,arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),bool))
=> ! [V3c] : ap(f216(A_27a,V2a0,V1ZRECSPACE_27),V3c) = ap(c_2Ebool_2E_3F(A_27a),f215(A_27a,V1ZRECSPACE_27,V2a0,V3c)) ) ) ).
fof(lameq_f217,axiom,
! [A_27a,V1ZRECSPACE_27] :
( mem(V1ZRECSPACE_27,arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),bool))
=> ! [V2a0] : ap(f217(A_27a,V1ZRECSPACE_27),V2a0) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Emin_2E_3D(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2a0),c_2Eind__type_2EZBOT(A_27a))),ap(c_2Ebool_2E_3F(ty_2Enum_2Enum),f216(A_27a,V2a0,V1ZRECSPACE_27)))),ap(V1ZRECSPACE_27,V2a0)) ) ).
fof(lameq_f218,axiom,
! [A_27a,V0a0] :
( mem(V0a0,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1ZRECSPACE_27] : ap(f218(A_27a,V0a0),V1ZRECSPACE_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(arr(ty_2Enum_2Enum,arr(A_27a,bool))),f217(A_27a,V1ZRECSPACE_27))),ap(V1ZRECSPACE_27,V0a0)) ) ).
fof(lameq_f219,axiom,
! [A_27a,V0a0] : ap(f219(A_27a),V0a0) = ap(c_2Ebool_2E_21(arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),bool)),f218(A_27a,V0a0)) ).
fof(ax_thm_2Eind__type_2EZRECSPACE__def,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Eind__type_2EZRECSPACE(A_27a) = f219(A_27a) ) ).
fof(conj_thm_2Eind__type_2EZRECSPACE__rules,axiom,
! [A_27a] :
( ne(A_27a)
=> ( p(ap(c_2Eind__type_2EZRECSPACE(A_27a),c_2Eind__type_2EZBOT(A_27a)))
& ! [V0c] :
( mem(V0c,ty_2Enum_2Enum)
=> ! [V1i] :
( mem(V1i,A_27a)
=> ! [V2r] :
( mem(V2r,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ( ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(c_2Eind__type_2EZRECSPACE(A_27a),ap(V2r,V3n))) )
=> p(ap(c_2Eind__type_2EZRECSPACE(A_27a),ap(ap(ap(c_2Eind__type_2EZCONSTR(A_27a),V0c),V1i),V2r))) ) ) ) ) ) ) ).
fof(conj_thm_2Eind__type_2EZRECSPACE__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0ZRECSPACE_27] :
( mem(V0ZRECSPACE_27,arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),bool))
=> ( ( p(ap(V0ZRECSPACE_27,c_2Eind__type_2EZBOT(A_27a)))
& ! [V1c] :
( mem(V1c,ty_2Enum_2Enum)
=> ! [V2i] :
( mem(V2i,A_27a)
=> ! [V3r] :
( mem(V3r,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ( ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(V0ZRECSPACE_27,ap(V3r,V4n))) )
=> p(ap(V0ZRECSPACE_27,ap(ap(ap(c_2Eind__type_2EZCONSTR(A_27a),V1c),V2i),V3r))) ) ) ) ) )
=> ! [V5a0] :
( mem(V5a0,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(c_2Eind__type_2EZRECSPACE(A_27a),V5a0))
=> p(ap(V0ZRECSPACE_27,V5a0)) ) ) ) ) ) ).
fof(conj_thm_2Eind__type_2EZRECSPACE__strongind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0ZRECSPACE_27] :
( mem(V0ZRECSPACE_27,arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),bool))
=> ( ( p(ap(V0ZRECSPACE_27,c_2Eind__type_2EZBOT(A_27a)))
& ! [V1c] :
( mem(V1c,ty_2Enum_2Enum)
=> ! [V2i] :
( mem(V2i,A_27a)
=> ! [V3r] :
( mem(V3r,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ( ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( p(ap(c_2Eind__type_2EZRECSPACE(A_27a),ap(V3r,V4n)))
& p(ap(V0ZRECSPACE_27,ap(V3r,V4n))) ) )
=> p(ap(V0ZRECSPACE_27,ap(ap(ap(c_2Eind__type_2EZCONSTR(A_27a),V1c),V2i),V3r))) ) ) ) ) )
=> ! [V5a0] :
( mem(V5a0,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(c_2Eind__type_2EZRECSPACE(A_27a),V5a0))
=> p(ap(V0ZRECSPACE_27,V5a0)) ) ) ) ) ) ).
fof(conj_thm_2Eind__type_2EZRECSPACE__cases,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a0] :
( mem(V0a0,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(c_2Eind__type_2EZRECSPACE(A_27a),V0a0))
<=> ( V0a0 = c_2Eind__type_2EZBOT(A_27a)
| ? [V1c] :
( mem(V1c,ty_2Enum_2Enum)
& ? [V2i] :
( mem(V2i,A_27a)
& ? [V3r] :
( mem(V3r,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
& V0a0 = ap(ap(ap(c_2Eind__type_2EZCONSTR(A_27a),V1c),V2i),V3r)
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(c_2Eind__type_2EZRECSPACE(A_27a),ap(V3r,V4n))) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Eind__type_2Erecspace__TY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ? [V0rep] :
( mem(V0rep,arr(ty_2Eind__type_2Erecspace(A_27a),arr(ty_2Enum_2Enum,arr(A_27a,bool))))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(arr(ty_2Enum_2Enum,arr(A_27a,bool)),ty_2Eind__type_2Erecspace(A_27a)),c_2Eind__type_2EZRECSPACE(A_27a)),V0rep)) ) ) ).
fof(ax_thm_2Eind__type_2Erecspace__repfns,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0a] :
( mem(V0a,ty_2Eind__type_2Erecspace(A_27a))
=> ap(c_2Eind__type_2Emk__rec(A_27a),ap(c_2Eind__type_2Edest__rec(A_27a),V0a)) = V0a )
& ! [V1r] :
( mem(V1r,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(c_2Eind__type_2EZRECSPACE(A_27a),V1r))
<=> ap(c_2Eind__type_2Edest__rec(A_27a),ap(c_2Eind__type_2Emk__rec(A_27a),V1r)) = V1r ) ) ) ) ).
fof(ax_thm_2Eind__type_2EBOTTOM,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Eind__type_2EBOTTOM(A_27a) = ap(c_2Eind__type_2Emk__rec(A_27a),c_2Eind__type_2EZBOT(A_27a)) ) ).
fof(lameq_f220,axiom,
! [A_27a,V2r] :
( mem(V2r,arr(ty_2Enum_2Enum,ty_2Eind__type_2Erecspace(A_27a)))
=> ! [V3n] : ap(f220(A_27a,V2r),V3n) = ap(c_2Eind__type_2Edest__rec(A_27a),ap(V2r,V3n)) ) ).
fof(ax_thm_2Eind__type_2ECONSTR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0c] :
( mem(V0c,ty_2Enum_2Enum)
=> ! [V1i] :
( mem(V1i,A_27a)
=> ! [V2r] :
( mem(V2r,arr(ty_2Enum_2Enum,ty_2Eind__type_2Erecspace(A_27a)))
=> ap(ap(ap(c_2Eind__type_2ECONSTR(A_27a),V0c),V1i),V2r) = ap(c_2Eind__type_2Emk__rec(A_27a),ap(ap(ap(c_2Eind__type_2EZCONSTR(A_27a),V0c),V1i),f220(A_27a,V2r))) ) ) ) ) ).
fof(conj_thm_2Eind__type_2EMK__REC__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1y] :
( mem(V1y,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ap(c_2Eind__type_2Emk__rec(A_27a),V0x) = ap(c_2Eind__type_2Emk__rec(A_27a),V1y)
=> ( ( p(ap(c_2Eind__type_2EZRECSPACE(A_27a),V0x))
& p(ap(c_2Eind__type_2EZRECSPACE(A_27a),V1y)) )
=> V0x = V1y ) ) ) ) ) ).
fof(conj_thm_2Eind__type_2EDEST__REC__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,ty_2Eind__type_2Erecspace(A_27a))
=> ! [V1y] :
( mem(V1y,ty_2Eind__type_2Erecspace(A_27a))
=> ( ap(c_2Eind__type_2Edest__rec(A_27a),V0x) = ap(c_2Eind__type_2Edest__rec(A_27a),V1y)
<=> V0x = V1y ) ) ) ) ).
fof(conj_thm_2Eind__type_2ECONSTR__BOT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0c] :
( mem(V0c,ty_2Enum_2Enum)
=> ! [V1i] :
( mem(V1i,A_27a)
=> ! [V2r] :
( mem(V2r,arr(ty_2Enum_2Enum,ty_2Eind__type_2Erecspace(A_27a)))
=> ap(ap(ap(c_2Eind__type_2ECONSTR(A_27a),V0c),V1i),V2r) != c_2Eind__type_2EBOTTOM(A_27a) ) ) ) ) ).
fof(conj_thm_2Eind__type_2ECONSTR__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0c1] :
( mem(V0c1,ty_2Enum_2Enum)
=> ! [V1i1] :
( mem(V1i1,A_27a)
=> ! [V2r1] :
( mem(V2r1,arr(ty_2Enum_2Enum,ty_2Eind__type_2Erecspace(A_27a)))
=> ! [V3c2] :
( mem(V3c2,ty_2Enum_2Enum)
=> ! [V4i2] :
( mem(V4i2,A_27a)
=> ! [V5r2] :
( mem(V5r2,arr(ty_2Enum_2Enum,ty_2Eind__type_2Erecspace(A_27a)))
=> ( ap(ap(ap(c_2Eind__type_2ECONSTR(A_27a),V0c1),V1i1),V2r1) = ap(ap(ap(c_2Eind__type_2ECONSTR(A_27a),V3c2),V4i2),V5r2)
<=> ( V0c1 = V3c2
& V1i1 = V4i2
& V2r1 = V5r2 ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eind__type_2ECONSTR__IND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Eind__type_2Erecspace(A_27a),bool))
=> ( ( p(ap(V0P,c_2Eind__type_2EBOTTOM(A_27a)))
& ! [V1c] :
( mem(V1c,ty_2Enum_2Enum)
=> ! [V2i] :
( mem(V2i,A_27a)
=> ! [V3r] :
( mem(V3r,arr(ty_2Enum_2Enum,ty_2Eind__type_2Erecspace(A_27a)))
=> ( ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(V0P,ap(V3r,V4n))) )
=> p(ap(V0P,ap(ap(ap(c_2Eind__type_2ECONSTR(A_27a),V1c),V2i),V3r))) ) ) ) ) )
=> ! [V5x] :
( mem(V5x,ty_2Eind__type_2Erecspace(A_27a))
=> p(ap(V0P,V5x)) ) ) ) ) ).
fof(lameq_f221,axiom,
! [A_27a,A_27b,V4r] :
( mem(V4r,arr(ty_2Enum_2Enum,ty_2Eind__type_2Erecspace(A_27a)))
=> ! [V1f] :
( mem(V1f,arr(ty_2Eind__type_2Erecspace(A_27a),A_27b))
=> ! [V5n] : ap(f221(A_27a,A_27b,V4r,V1f),V5n) = ap(V1f,ap(V4r,V5n)) ) ) ).
fof(conj_thm_2Eind__type_2ECONSTR__REC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0Fn] :
( mem(V0Fn,arr(ty_2Enum_2Enum,arr(A_27a,arr(arr(ty_2Enum_2Enum,ty_2Eind__type_2Erecspace(A_27a)),arr(arr(ty_2Enum_2Enum,A_27b),A_27b)))))
=> ? [V1f] :
( mem(V1f,arr(ty_2Eind__type_2Erecspace(A_27a),A_27b))
& ! [V2c] :
( mem(V2c,ty_2Enum_2Enum)
=> ! [V3i] :
( mem(V3i,A_27a)
=> ! [V4r] :
( mem(V4r,arr(ty_2Enum_2Enum,ty_2Eind__type_2Erecspace(A_27a)))
=> ap(V1f,ap(ap(ap(c_2Eind__type_2ECONSTR(A_27a),V2c),V3i),V4r)) = ap(ap(ap(ap(V0Fn,V2c),V3i),V4r),f221(A_27a,A_27b,V4r,V1f)) ) ) ) ) ) ) ) ).
fof(ax_thm_2Eind__type_2EFCONS,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
=> ap(ap(ap(c_2Eind__type_2EFCONS(A_27a),V0a),V1f),c_2Enum_2E0) = V0a ) )
& ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3f] :
( mem(V3f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ap(ap(ap(c_2Eind__type_2EFCONS(A_27a),V2a),V3f),ap(c_2Enum_2ESUC,V4n)) = ap(V3f,V4n) ) ) ) ) ) ).
fof(ax_thm_2Eind__type_2EFNIL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Eind__type_2EFNIL(A_27a),V0n) = c_2Ebool_2EARB(A_27a) ) ) ).
fof(conj_thm_2Eind__type_2EFCONS__DEST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(ap(c_2Eind__type_2EFCONS(A_27a),V0a),V1f),V2n) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V2n),c_2Enum_2E0)),V0a),ap(V1f,ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ) ) ).
fof(ax_thm_2Eind__type_2EISO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27a))
=> ( p(ap(ap(c_2Eind__type_2EISO(A_27a,A_27b),V0f),V1g))
<=> ( ! [V2x] :
( mem(V2x,A_27b)
=> ap(V0f,ap(V1g,V2x)) = V2x )
& ! [V3y] :
( mem(V3y,A_27a)
=> ap(V1g,ap(V0f,V3y)) = V3y ) ) ) ) ) ) ) ).
fof(conj_thm_2Eind__type_2EISO__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> p(ap(ap(c_2Eind__type_2EISO(A_27a,A_27a),i(A_27a)),i(A_27a))) ) ).
fof(lameq_f222,axiom,
! [A_27d,A_27b,A_27a,A_27c,V4h] :
( mem(V4h,arr(A_27a,A_27b))
=> ! [V1f_27] :
( mem(V1f_27,arr(A_27c,A_27a))
=> ! [V2g] :
( mem(V2g,arr(A_27b,A_27d))
=> ! [V5a_27] : ap(f222(A_27d,A_27b,A_27a,A_27c,V4h,V1f_27,V2g),V5a_27) = ap(V2g,ap(V4h,ap(V1f_27,V5a_27))) ) ) ) ).
fof(lameq_f223,axiom,
! [A_27c,A_27d,A_27a,A_27b,V1f_27] :
( mem(V1f_27,arr(A_27c,A_27a))
=> ! [V2g] :
( mem(V2g,arr(A_27b,A_27d))
=> ! [V4h] : ap(f223(A_27c,A_27d,A_27a,A_27b,V1f_27,V2g),V4h) = f222(A_27d,A_27b,A_27a,A_27c,V4h,V1f_27,V2g) ) ) ).
fof(lameq_f224,axiom,
! [A_27b,A_27d,A_27c,A_27a,V6h] :
( mem(V6h,arr(A_27c,A_27d))
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V3g_27] :
( mem(V3g_27,arr(A_27d,A_27b))
=> ! [V7a] : ap(f224(A_27b,A_27d,A_27c,A_27a,V6h,V0f,V3g_27),V7a) = ap(V3g_27,ap(V6h,ap(V0f,V7a))) ) ) ) ).
fof(lameq_f225,axiom,
! [A_27a,A_27b,A_27c,A_27d,V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V3g_27] :
( mem(V3g_27,arr(A_27d,A_27b))
=> ! [V6h] : ap(f225(A_27a,A_27b,A_27c,A_27d,V0f,V3g_27),V6h) = f224(A_27b,A_27d,A_27c,A_27a,V6h,V0f,V3g_27) ) ) ).
fof(conj_thm_2Eind__type_2EISO__FUN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V1f_27] :
( mem(V1f_27,arr(A_27c,A_27a))
=> ! [V2g] :
( mem(V2g,arr(A_27b,A_27d))
=> ! [V3g_27] :
( mem(V3g_27,arr(A_27d,A_27b))
=> ( ( p(ap(ap(c_2Eind__type_2EISO(A_27a,A_27c),V0f),V1f_27))
& p(ap(ap(c_2Eind__type_2EISO(A_27b,A_27d),V2g),V3g_27)) )
=> p(ap(ap(c_2Eind__type_2EISO(arr(A_27a,A_27b),arr(A_27c,A_27d)),f223(A_27c,A_27d,A_27a,A_27b,V1f_27,V2g)),f225(A_27a,A_27b,A_27c,A_27d,V0f,V3g_27))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eind__type_2EISO__USAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27a))
=> ( p(ap(ap(c_2Eind__type_2EISO(A_27a,A_27b),V0f),V1g))
=> ( ! [V2P] :
( mem(V2P,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(V2P,V3x)) )
<=> ! [V4x] :
( mem(V4x,A_27b)
=> p(ap(V2P,ap(V1g,V4x))) ) ) )
& ! [V5P] :
( mem(V5P,arr(A_27a,bool))
=> ( ? [V6x] :
( mem(V6x,A_27a)
& p(ap(V5P,V6x)) )
<=> ? [V7x] :
( mem(V7x,A_27b)
& p(ap(V5P,ap(V1g,V7x))) ) ) )
& ! [V8a] :
( mem(V8a,A_27a)
=> ! [V9b] :
( mem(V9b,A_27b)
=> ( V8a = ap(V1g,V9b)
<=> ap(V0f,V8a) = V9b ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------