ITP001 Axioms: ITP015^5.ax
%------------------------------------------------------------------------------
% File : ITP015^5 : 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 : prim_rec^2.ax [Gau20]
% : HL4015^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 64 ( 5 unt; 9 typ; 0 def)
% Number of atoms : 824 ( 31 equ; 0 cnn)
% Maximal formula atoms : 38 ( 12 avg)
% Number of connectives : 1236 ( 8 ~; 3 |; 23 &;1135 @)
% ( 10 <=>; 57 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 10 avg;1135 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 7 ( 7 >; 0 *; 0 +; 0 <<)
% Number of symbols : 34 ( 33 usr; 26 con; 0-2 aty)
% Number of variables : 149 ( 20 ^ 122 !; 7 ?; 149 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: $i ).
thf(mem_c_2Eprim__rec_2E_3C,axiom,
mem @ c_2Eprim__rec_2E_3C @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ).
thf(tp_c_2Eprim__rec_2EPRE,type,
c_2Eprim__rec_2EPRE: $i ).
thf(mem_c_2Eprim__rec_2EPRE,axiom,
mem @ c_2Eprim__rec_2EPRE @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Eprim__rec_2EPRE,type,
fo__c_2Eprim__rec_2EPRE: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Eprim__rec_2EPRE,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Eprim__rec_2EPRE @ X0 ) )
= ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Eprim__rec_2EPRIM__REC,type,
c_2Eprim__rec_2EPRIM__REC: del > $i ).
thf(mem_c_2Eprim__rec_2EPRIM__REC,axiom,
! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(tp_c_2Eprim__rec_2EPRIM__REC__FUN,type,
c_2Eprim__rec_2EPRIM__REC__FUN: del > $i ).
thf(mem_c_2Eprim__rec_2EPRIM__REC__FUN,axiom,
! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ) ) ).
thf(tp_c_2Eprim__rec_2ESIMP__REC,type,
c_2Eprim__rec_2ESIMP__REC: del > $i ).
thf(mem_c_2Eprim__rec_2ESIMP__REC,axiom,
! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(tp_c_2Eprim__rec_2ESIMP__REC__REL,type,
c_2Eprim__rec_2ESIMP__REC__REL: del > $i ).
thf(mem_c_2Eprim__rec_2ESIMP__REC__REL,axiom,
! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) ).
thf(tp_c_2Eprim__rec_2Emeasure,type,
c_2Eprim__rec_2Emeasure: del > $i ).
thf(mem_c_2Eprim__rec_2Emeasure,axiom,
! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2Emeasure @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eprim__rec_2Ewellfounded,type,
c_2Eprim__rec_2Ewellfounded: del > $i ).
thf(mem_c_2Eprim__rec_2Ewellfounded,axiom,
! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2Ewellfounded @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(ax_thm_2Eprim__rec_2ELESS__DEF,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ? [V2P: $i] :
( ( mem @ V2P @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ V2P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( p @ ( ap @ V2P @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ( p @ ( ap @ V2P @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
& ~ ( p @ ( ap @ V2P @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2EINV__SUC__EQ,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( V0m = V1n ) ) ).
thf(ax_thm_2Eprim__rec_2EPRE__DEF,axiom,
! [V0m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ V1n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2EPRE,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= fo__c_2Enum_2E0 )
& ! [V0m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) )
= V0m ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__REFL,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
~ ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ).
thf(conj_thm_2Eprim__rec_2ESUC__LESS,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ENOT__LESS__0,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
~ ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__0,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__0__0,axiom,
p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__MONO,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__MONO__REV,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__MONO__EQ,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ETC__IM__RTC__SUC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Erelation_2ETC @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2x: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V3y ) @ ( ap @ c_2Enum_2ESUC @ V2x ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V0m ) )
@ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Erelation_2ERTC @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4x: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V5y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V5y ) @ ( ap @ c_2Enum_2ESUC @ V4x ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V0m ) )
@ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ERTC__IM__TC,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ A_27a )
=> ! [V2n: $i] :
( ( mem @ V2n @ A_27a )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Erelation_2ERTC @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V4y ) @ ( ap @ V0f @ V3x ) ) ) ) )
@ ( ap @ V0f @ V1m ) )
@ V2n ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Erelation_2ETC @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] :
( lam @ A_27a
@ ^ [V6y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V6y ) @ ( ap @ V0f @ V5x ) ) ) ) )
@ V1m )
@ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__ALT,axiom,
( c_2Eprim__rec_2E_3C
= ( ap @ ( c_2Erelation_2ETC @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0x: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V1y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V1y ) @ ( ap @ c_2Enum_2ESUC @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__SUC__REFL,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__SUC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__LEMMA1,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( ( V0m = V1n )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__LEMMA2,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( V0m = V1n )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__THM,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( ( V0m = V1n )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__SUC__IMP,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( ( V0m != V1n )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2EEQ__LESS,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
= V1n )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ESUC__ID,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
!= V0n ) ).
thf(conj_thm_2Eprim__rec_2ENOT__LESS__EQ,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( V0m = V1n )
=> ~ ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__NOT__EQ,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( V0m != V1n ) ) ).
thf(ax_thm_2Eprim__rec_2ESIMP__REC__REL,axiom,
! [A_27a: del,V0fun: $i] :
( ( mem @ V0fun @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V0fun ) @ V1x ) @ V2f ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
<=> ( ( ( ap @ V0fun @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V1x )
& ! [V4m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( ( ap @ V0fun @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4m ) ) )
= ( ap @ V2f @ ( ap @ V0fun @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ESIMP__REC__EXISTS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
? [V3fun: $i] :
( ( mem @ V3fun @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V3fun ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ESIMP__REC__REL__UNIQUE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2g1: $i] :
( ( mem @ V2g1 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V3g2: $i] :
( ( mem @ V3g2 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V4m1: tp__ty_2Enum_2Enum,V5m2: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V2g1 ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V4m1 ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V3g2 ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V5m2 ) ) ) )
=> ! [V6n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( inj__ty_2Enum_2Enum @ V4m1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( inj__ty_2Enum_2Enum @ V5m2 ) ) ) )
=> ( ( ap @ V2g1 @ ( inj__ty_2Enum_2Enum @ V6n ) )
= ( ap @ V3g2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ESIMP__REC__REL__UNIQUE__RESULT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3y: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ A_27a )
@ ^ [V4g: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V4g ) @ V0x ) @ V1f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3y ) @ ( ap @ V4g @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eprim__rec_2ESIMP__REC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ A_27a @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
? [V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V3g ) @ V0x ) @ V1f_27 ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a ) @ V0x ) @ V1f_27 ) @ ( inj__ty_2Enum_2Enum @ V2n ) )
= ( ap @ V3g @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__SUC__SUC,axiom,
! [V0m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ESIMP__REC__THM,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V0x )
& ! [V2m: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a ) @ V0x ) @ V1f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
= ( ap @ V1f @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ) ) ).
thf(ax_thm_2Eprim__rec_2EPRIM__REC__FUN,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ V0x ) @ V1f )
= ( ap
@ ( ap @ ( c_2Eprim__rec_2ESIMP__REC @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : V0x ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ A_27a )
@ ^ [V3fun: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ V1f @ ( ap @ V3fun @ ( ap @ c_2Eprim__rec_2EPRE @ V4n ) ) ) @ V4n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2EPRIM__REC__EQN,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) )
= V0x )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ V0x ) @ V1f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) )
= ( ap @ ( ap @ V1f @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ).
thf(ax_thm_2Eprim__rec_2EPRIM__REC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
=> ! [V2m: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2m ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2EPRIM__REC__THM,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V0x )
& ! [V2m: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a ) @ V0x ) @ V1f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
= ( ap @ ( ap @ V1f @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2EDC,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V2a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ? [V4y: $i] :
( ( mem @ V4y @ A_27a )
& ( p @ ( ap @ V0P @ V4y ) )
& ( p @ ( ap @ ( ap @ V1R @ V3x ) @ V4y ) ) ) ) ) )
=> ? [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V2a )
& ! [V6n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ V0P @ ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
& ( p @ ( ap @ ( ap @ V1R @ ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ ( ap @ V5f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2Enum__Axiom__old,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ A_27a )
@ ^ [V2fn1: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V2fn1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0e ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V2fn1 @ ( ap @ c_2Enum_2ESUC @ V3n ) ) ) @ ( ap @ ( ap @ V1f @ ( ap @ V2fn1 @ V3n ) ) @ V3n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2Enum__Axiom,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27a ) ) )
=> ? [V2fn: $i] :
( ( mem @ V2fn @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( ( ap @ V2fn @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V0e )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( ap @ V2fn @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
= ( ap @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ V2fn @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ).
thf(ax_thm_2Eprim__rec_2Ewellfounded__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Eprim__rec_2Ewellfounded @ A_27a ) @ V0R ) )
<=> ~ ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0R @ ( ap @ V1f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2EWF__IFF__WELLFOUNDED,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
<=> ( p @ ( ap @ ( c_2Eprim__rec_2Ewellfounded @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2EWF__PRED,axiom,
( p
@ ( ap @ ( c_2Erelation_2EWF @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0x: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V1y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V1y ) @ ( ap @ c_2Enum_2ESUC @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2EWF__LESS,axiom,
p @ ( ap @ ( c_2Erelation_2EWF @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) ).
thf(ax_thm_2Eprim__rec_2Emeasure__def,axiom,
! [A_27a: del] :
( ( c_2Eprim__rec_2Emeasure @ A_27a )
= ( ap @ ( c_2Erelation_2Einv__image @ A_27a @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) ) ).
thf(conj_thm_2Eprim__rec_2EWF__measure,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Eprim__rec_2Emeasure @ A_27a ) @ V0m ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2Emeasure__thm,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2Emeasure @ A_27a ) @ V0f ) @ V1x ) @ V2y ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V0f @ V1x ) ) @ ( ap @ V0f @ V2y ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------