ITP001 Axioms: ITP015^7.ax
%------------------------------------------------------------------------------
% File : ITP015^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy 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 : prim_rec.ax [Gau19]
% : HL4015^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 80 ( 20 unt; 27 typ; 0 def)
% Number of atoms : 126 ( 36 equ; 9 cnn)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 427 ( 9 ~; 4 |; 17 &; 367 @)
% ( 12 <=>; 18 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg; 367 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 119 ( 119 >; 0 *; 0 +; 0 <<)
% Number of symbols : 27 ( 25 usr; 2 con; 0-6 aty)
% Number of variables : 175 ( 20 ^ 130 !; 8 ?; 175 :)
% ( 17 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Eprim__rec_2EPRE,type,
c_2Eprim__rec_2EPRE: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2EPRIM__REC,type,
c_2Eprim__rec_2EPRIM__REC:
!>[A_27a: $tType] : ( A_27a > ( A_27a > tyop_2Enum_2Enum > A_27a ) > tyop_2Enum_2Enum > A_27a ) ).
thf(c_2Eprim__rec_2EPRIM__REC__FUN,type,
c_2Eprim__rec_2EPRIM__REC__FUN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > tyop_2Enum_2Enum > A_27a ) > tyop_2Enum_2Enum > tyop_2Enum_2Enum > A_27a ) ).
thf(c_2Erelation_2ERTC,type,
c_2Erelation_2ERTC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Eprim__rec_2ESIMP__REC,type,
c_2Eprim__rec_2ESIMP__REC:
!>[A_27a: $tType] : ( A_27a > ( A_27a > A_27a ) > tyop_2Enum_2Enum > A_27a ) ).
thf(c_2Eprim__rec_2ESIMP__REC__REL,type,
c_2Eprim__rec_2ESIMP__REC__REL:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum > A_27a ) > A_27a > ( A_27a > A_27a ) > tyop_2Enum_2Enum > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Erelation_2ETC,type,
c_2Erelation_2ETC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Erelation_2Einv__image,type,
c_2Erelation_2Einv__image:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27b > $o ) > ( A_27a > A_27b ) > A_27a > A_27a > $o ) ).
thf(c_2Eprim__rec_2Emeasure,type,
c_2Eprim__rec_2Emeasure:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > A_27a > A_27a > $o ) ).
thf(c_2Eprim__rec_2Ewellfounded,type,
c_2Eprim__rec_2Ewellfounded:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Eprim__rec_2ELESS__DEF,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
<=> ? [V2P: tyop_2Enum_2Enum > $o] :
( ! [V3n: tyop_2Enum_2Enum] :
( ( V2P @ ( c_2Enum_2ESUC @ V3n ) )
=> ( V2P @ V3n ) )
& ( V2P @ V0m )
& ( (~) @ ( V2P @ V1n ) ) ) ) ).
thf(thm_2Eprim__rec_2EPRE__DEF,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2EPRE @ V0m )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0m @ c_2Enum_2E0 ) @ c_2Enum_2E0
@ ( c_2Emin_2E_40 @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0m @ ( c_2Enum_2ESUC @ V1n ) ) ) ) ) ).
thf(thm_2Eprim__rec_2ESIMP__REC__REL,axiom,
! [A_27a: $tType,V0fun: tyop_2Enum_2Enum > A_27a,V1x: A_27a,V2f: A_27a > A_27a,V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a @ V0fun @ V1x @ V2f @ V3n )
<=> ( ( ( V0fun @ c_2Enum_2E0 )
= V1x )
& ! [V4m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V4m @ V3n )
=> ( ( V0fun @ ( c_2Enum_2ESUC @ V4m ) )
= ( V2f @ ( V0fun @ V4m ) ) ) ) ) ) ).
thf(thm_2Eprim__rec_2ESIMP__REC,axiom,
! [A_27a: $tType,V0x: A_27a,V1f_27: A_27a > A_27a,V2n: tyop_2Enum_2Enum] :
? [V3g: tyop_2Enum_2Enum > A_27a] :
( ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a @ V3g @ V0x @ V1f_27 @ ( c_2Enum_2ESUC @ V2n ) )
& ( ( c_2Eprim__rec_2ESIMP__REC @ A_27a @ V0x @ V1f_27 @ V2n )
= ( V3g @ V2n ) ) ) ).
thf(thm_2Eprim__rec_2EPRIM__REC__FUN,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > tyop_2Enum_2Enum > A_27a] :
( ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a @ V0x @ V1f )
= ( c_2Eprim__rec_2ESIMP__REC @ ( tyop_2Enum_2Enum > A_27a )
@ ^ [V2n: tyop_2Enum_2Enum] : V0x
@ ^ [V3fun: tyop_2Enum_2Enum > A_27a,V4n: tyop_2Enum_2Enum] : ( V1f @ ( V3fun @ ( c_2Eprim__rec_2EPRE @ V4n ) ) @ V4n ) ) ) ).
thf(thm_2Eprim__rec_2EPRIM__REC,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > tyop_2Enum_2Enum > A_27a,V2m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2EPRIM__REC @ A_27a @ V0x @ V1f @ V2m )
= ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a @ V0x @ V1f @ V2m @ ( c_2Eprim__rec_2EPRE @ V2m ) ) ) ).
thf(thm_2Eprim__rec_2Ewellfounded__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Eprim__rec_2Ewellfounded @ A_27a @ V0R )
<=> ( (~)
@ ? [V1f: tyop_2Enum_2Enum > A_27a] :
! [V2n: tyop_2Enum_2Enum] : ( V0R @ ( V1f @ ( c_2Enum_2ESUC @ V2n ) ) @ ( V1f @ V2n ) ) ) ) ).
thf(thm_2Eprim__rec_2Emeasure__def,axiom,
! [A_27a: $tType] :
( ( c_2Eprim__rec_2Emeasure @ A_27a )
= ( c_2Erelation_2Einv__image @ A_27a @ tyop_2Enum_2Enum @ c_2Eprim__rec_2E_3C ) ) ).
thf(thm_2Eprim__rec_2EINV__SUC__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Enum_2ESUC @ V0m )
= ( c_2Enum_2ESUC @ V1n ) )
<=> ( V0m = V1n ) ) ).
thf(thm_2Eprim__rec_2EPRE,axiom,
( ( ( c_2Eprim__rec_2EPRE @ c_2Enum_2E0 )
= c_2Enum_2E0 )
& ! [V0m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2EPRE @ ( c_2Enum_2ESUC @ V0m ) )
= V0m ) ) ).
thf(thm_2Eprim__rec_2ELESS__REFL,axiom,
! [V0n: tyop_2Enum_2Enum] : ( (~) @ ( c_2Eprim__rec_2E_3C @ V0n @ V0n ) ) ).
thf(thm_2Eprim__rec_2ESUC__LESS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Enum_2ESUC @ V0m ) @ V1n )
=> ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Eprim__rec_2ENOT__LESS__0,axiom,
! [V0n: tyop_2Enum_2Enum] : ( (~) @ ( c_2Eprim__rec_2E_3C @ V0n @ c_2Enum_2E0 ) ) ).
thf(thm_2Eprim__rec_2ELESS__0,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ V0n ) ) ).
thf(thm_2Eprim__rec_2ELESS__0__0,axiom,
c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ).
thf(thm_2Eprim__rec_2ELESS__MONO,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Enum_2ESUC @ V0m ) @ ( c_2Enum_2ESUC @ V1n ) ) ) ).
thf(thm_2Eprim__rec_2ELESS__MONO__REV,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Enum_2ESUC @ V0m ) @ ( c_2Enum_2ESUC @ V1n ) )
=> ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Eprim__rec_2ELESS__MONO__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Enum_2ESUC @ V0m ) @ ( c_2Enum_2ESUC @ V1n ) )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Eprim__rec_2ETC__IM__RTC__SUC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Erelation_2ETC @ tyop_2Enum_2Enum
@ ^ [V2x: tyop_2Enum_2Enum,V3y: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3y @ ( c_2Enum_2ESUC @ V2x ) )
@ V0m
@ ( c_2Enum_2ESUC @ V1n ) )
= ( c_2Erelation_2ERTC @ tyop_2Enum_2Enum
@ ^ [V4x: tyop_2Enum_2Enum,V5y: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V5y @ ( c_2Enum_2ESUC @ V4x ) )
@ V0m
@ V1n ) ) ).
thf(thm_2Eprim__rec_2ERTC__IM__TC,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a,V1m: A_27a,V2n: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a
@ ^ [V3x: A_27a,V4y: A_27a] : ( c_2Emin_2E_3D @ A_27a @ V4y @ ( V0f @ V3x ) )
@ ( V0f @ V1m )
@ V2n )
= ( c_2Erelation_2ETC @ A_27a
@ ^ [V5x: A_27a,V6y: A_27a] : ( c_2Emin_2E_3D @ A_27a @ V6y @ ( V0f @ V5x ) )
@ V1m
@ V2n ) ) ).
thf(thm_2Eprim__rec_2ELESS__ALT,axiom,
( c_2Eprim__rec_2E_3C
= ( c_2Erelation_2ETC @ tyop_2Enum_2Enum
@ ^ [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1y @ ( c_2Enum_2ESUC @ V0x ) ) ) ) ).
thf(thm_2Eprim__rec_2ELESS__SUC__REFL,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ V0n @ ( c_2Enum_2ESUC @ V0n ) ) ).
thf(thm_2Eprim__rec_2ELESS__SUC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
=> ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Enum_2ESUC @ V1n ) ) ) ).
thf(thm_2Eprim__rec_2ELESS__LEMMA1,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Enum_2ESUC @ V1n ) )
=> ( ( V0m = V1n )
| ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ) ).
thf(thm_2Eprim__rec_2ELESS__LEMMA2,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( V0m = V1n )
| ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) )
=> ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Enum_2ESUC @ V1n ) ) ) ).
thf(thm_2Eprim__rec_2ELESS__THM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Enum_2ESUC @ V1n ) )
<=> ( ( V0m = V1n )
| ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ) ).
thf(thm_2Eprim__rec_2ELESS__SUC__IMP,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Enum_2ESUC @ V1n ) )
=> ( ( (~) @ ( V0m = V1n ) )
=> ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ) ).
thf(thm_2Eprim__rec_2EEQ__LESS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Enum_2ESUC @ V0m )
= V1n )
=> ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Eprim__rec_2ESUC__ID,axiom,
! [V0n: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Enum_2ESUC @ V0n )
= V0n ) ) ).
thf(thm_2Eprim__rec_2ENOT__LESS__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( V0m = V1n )
=> ( (~) @ ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ) ).
thf(thm_2Eprim__rec_2ELESS__NOT__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
=> ( (~) @ ( V0m = V1n ) ) ) ).
thf(thm_2Eprim__rec_2ESIMP__REC__EXISTS,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > A_27a,V2n: tyop_2Enum_2Enum] :
? [V3fun: tyop_2Enum_2Enum > A_27a] : ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a @ V3fun @ V0x @ V1f @ V2n ) ).
thf(thm_2Eprim__rec_2ESIMP__REC__REL__UNIQUE,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > A_27a,V2g1: tyop_2Enum_2Enum > A_27a,V3g2: tyop_2Enum_2Enum > A_27a,V4m1: tyop_2Enum_2Enum,V5m2: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a @ V2g1 @ V0x @ V1f @ V4m1 )
& ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a @ V3g2 @ V0x @ V1f @ V5m2 ) )
=> ! [V6n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V6n @ V4m1 )
& ( c_2Eprim__rec_2E_3C @ V6n @ V5m2 ) )
=> ( ( V2g1 @ V6n )
= ( V3g2 @ V6n ) ) ) ) ).
thf(thm_2Eprim__rec_2ESIMP__REC__REL__UNIQUE__RESULT,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > A_27a,V2n: tyop_2Enum_2Enum] :
( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V3y: A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Enum_2Enum > A_27a )
@ ^ [V4g: tyop_2Enum_2Enum > A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a @ V4g @ V0x @ V1f @ ( c_2Enum_2ESUC @ V2n ) ) @ ( c_2Emin_2E_3D @ A_27a @ V3y @ ( V4g @ V2n ) ) ) ) ) ).
thf(thm_2Eprim__rec_2ELESS__SUC__SUC,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Enum_2ESUC @ V0m ) )
& ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ V0m ) ) ) ) ).
thf(thm_2Eprim__rec_2ESIMP__REC__THM,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > A_27a] :
( ( ( c_2Eprim__rec_2ESIMP__REC @ A_27a @ V0x @ V1f @ c_2Enum_2E0 )
= V0x )
& ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2ESIMP__REC @ A_27a @ V0x @ V1f @ ( c_2Enum_2ESUC @ V2m ) )
= ( V1f @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a @ V0x @ V1f @ V2m ) ) ) ) ).
thf(thm_2Eprim__rec_2EPRIM__REC__EQN,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > tyop_2Enum_2Enum > A_27a] :
( ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a @ V0x @ V1f @ c_2Enum_2E0 @ V2n )
= V0x )
& ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a @ V0x @ V1f @ ( c_2Enum_2ESUC @ V3m ) @ V4n )
= ( V1f @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a @ V0x @ V1f @ V3m @ ( c_2Eprim__rec_2EPRE @ V4n ) ) @ V4n ) ) ) ).
thf(thm_2Eprim__rec_2EPRIM__REC__THM,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > tyop_2Enum_2Enum > A_27a] :
( ( ( c_2Eprim__rec_2EPRIM__REC @ A_27a @ V0x @ V1f @ c_2Enum_2E0 )
= V0x )
& ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2EPRIM__REC @ A_27a @ V0x @ V1f @ ( c_2Enum_2ESUC @ V2m ) )
= ( V1f @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a @ V0x @ V1f @ V2m ) @ V2m ) ) ) ).
thf(thm_2Eprim__rec_2EDC,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1R: A_27a > A_27a > $o,V2a: A_27a] :
( ( ( V0P @ V2a )
& ! [V3x: A_27a] :
( ( V0P @ V3x )
=> ? [V4y: A_27a] :
( ( V0P @ V4y )
& ( V1R @ V3x @ V4y ) ) ) )
=> ? [V5f: tyop_2Enum_2Enum > A_27a] :
( ( ( V5f @ c_2Enum_2E0 )
= V2a )
& ! [V6n: tyop_2Enum_2Enum] :
( ( V0P @ ( V5f @ V6n ) )
& ( V1R @ ( V5f @ V6n ) @ ( V5f @ ( c_2Enum_2ESUC @ V6n ) ) ) ) ) ) ).
thf(thm_2Eprim__rec_2Enum__Axiom__old,axiom,
! [A_27a: $tType,V0e: A_27a,V1f: A_27a > tyop_2Enum_2Enum > A_27a] :
( c_2Ebool_2E_3F_21 @ ( tyop_2Enum_2Enum > A_27a )
@ ^ [V2fn1: tyop_2Enum_2Enum > A_27a] :
( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ ( V2fn1 @ c_2Enum_2E0 ) @ V0e )
@ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ A_27a @ ( V2fn1 @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V1f @ ( V2fn1 @ V3n ) @ V3n ) ) ) ) ) ).
thf(thm_2Eprim__rec_2Enum__Axiom,axiom,
! [A_27a: $tType,V0e: A_27a,V1f: tyop_2Enum_2Enum > A_27a > A_27a] :
? [V2fn: tyop_2Enum_2Enum > A_27a] :
( ( ( V2fn @ c_2Enum_2E0 )
= V0e )
& ! [V3n: tyop_2Enum_2Enum] :
( ( V2fn @ ( c_2Enum_2ESUC @ V3n ) )
= ( V1f @ V3n @ ( V2fn @ V3n ) ) ) ) ).
thf(thm_2Eprim__rec_2EWF__IFF__WELLFOUNDED,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
= ( c_2Eprim__rec_2Ewellfounded @ A_27a @ V0R ) ) ).
thf(thm_2Eprim__rec_2EWF__PRED,axiom,
( c_2Erelation_2EWF @ tyop_2Enum_2Enum
@ ^ [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1y @ ( c_2Enum_2ESUC @ V0x ) ) ) ).
thf(thm_2Eprim__rec_2EWF__LESS,axiom,
c_2Erelation_2EWF @ tyop_2Enum_2Enum @ c_2Eprim__rec_2E_3C ).
thf(thm_2Eprim__rec_2EWF__measure,axiom,
! [A_27a: $tType,V0m: A_27a > tyop_2Enum_2Enum] : ( c_2Erelation_2EWF @ A_27a @ ( c_2Eprim__rec_2Emeasure @ A_27a @ V0m ) ) ).
thf(thm_2Eprim__rec_2Emeasure__thm,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1x: A_27a,V2y: A_27a] :
( ( c_2Eprim__rec_2Emeasure @ A_27a @ V0f @ V1x @ V2y )
= ( c_2Eprim__rec_2E_3C @ ( V0f @ V1x ) @ ( V0f @ V2y ) ) ) ).
%------------------------------------------------------------------------------