ITP001 Axioms: ITP019^5.ax
%------------------------------------------------------------------------------
% File : ITP019^5 : 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 : while^2.ax [Gau20]
% : HL4019^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 42 ( 2 unt; 8 typ; 0 def)
% Number of atoms : 655 ( 23 equ; 0 cnn)
% Maximal formula atoms : 46 ( 15 avg)
% Number of connectives : 1074 ( 10 ~; 0 |; 19 &; 941 @)
% ( 5 <=>; 99 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 11 avg; 941 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 6 ( 6 >; 0 *; 0 +; 0 <<)
% Number of symbols : 36 ( 35 usr; 30 con; 0-2 aty)
% Number of variables : 119 ( 13 ^ 102 !; 4 ?; 119 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Ewhile_2EHOARE__SPEC,type,
c_2Ewhile_2EHOARE__SPEC: del > del > $i ).
thf(mem_c_2Ewhile_2EHOARE__SPEC,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ewhile_2EHOARE__SPEC @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Ewhile_2ELEAST,type,
c_2Ewhile_2ELEAST: $i ).
thf(mem_c_2Ewhile_2ELEAST,axiom,
mem @ c_2Ewhile_2ELEAST @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ty_2Enum_2Enum ) ).
thf(stp_c_ty_2Eoption_2Eoption_ty_2Enum_2Enum,type,
tp__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum: $tType ).
thf(stp_inj_c_ty_2Eoption_2Eoption_ty_2Enum_2Enum,type,
inj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum: tp__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum > $i ).
thf(stp_surj_c_ty_2Eoption_2Eoption_ty_2Enum_2Enum,type,
surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum: $i > tp__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum ).
thf(stp_inj_surj_c_ty_2Eoption_2Eoption_ty_2Enum_2Enum,axiom,
! [X: tp__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum] :
( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( inj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Eoption_2Eoption_ty_2Enum_2Enum,axiom,
! [X: tp__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum] : ( mem @ ( inj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ X ) @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) ).
thf(stp_iso_mem_c_ty_2Eoption_2Eoption_ty_2Enum_2Enum,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) )
=> ( X
= ( inj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ X ) ) ) ) ).
thf(tp_c_2Ewhile_2EOLEAST,type,
c_2Ewhile_2EOLEAST: $i ).
thf(mem_c_2Ewhile_2EOLEAST,axiom,
mem @ c_2Ewhile_2EOLEAST @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Ewhile_2EOWHILE,type,
c_2Ewhile_2EOWHILE: del > $i ).
thf(mem_c_2Ewhile_2EOWHILE,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewhile_2EOWHILE @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ).
thf(tp_c_2Ewhile_2EWHILE,type,
c_2Ewhile_2EWHILE: del > $i ).
thf(mem_c_2Ewhile_2EWHILE,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewhile_2EWHILE @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) ) ) ).
thf(conj_thm_2Ewhile_2EITERATION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27a ) )
=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ap @ V2f @ V3x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ V0P @ V3x ) ) @ V3x ) @ ( ap @ V2f @ ( ap @ V1g @ V3x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewhile_2EWHILE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EWHILE @ A_27a ) @ V0P ) @ V1g ) @ V2x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ V0P @ V2x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EWHILE @ A_27a ) @ V0P ) @ V1g ) @ ( ap @ V1g @ V2x ) ) ) @ V2x ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EWHILE__INDUCTION,axiom,
! [A_27a: del,V0B: $i] :
( ( mem @ V0B @ ( arr @ A_27a @ bool ) )
=> ! [V1C: $i] :
( ( mem @ V1C @ ( arr @ A_27a @ A_27a ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V2R ) )
& ! [V3s: $i] :
( ( mem @ V3s @ A_27a )
=> ( ( p @ ( ap @ V0B @ V3s ) )
=> ( p @ ( ap @ ( ap @ V2R @ ( ap @ V1C @ V3s ) ) @ V3s ) ) ) ) )
=> ! [V4P: $i] :
( ( mem @ V4P @ ( arr @ A_27a @ bool ) )
=> ( ! [V5s: $i] :
( ( mem @ V5s @ A_27a )
=> ( ( ( p @ ( ap @ V0B @ V5s ) )
=> ( p @ ( ap @ V4P @ ( ap @ V1C @ V5s ) ) ) )
=> ( p @ ( ap @ V4P @ V5s ) ) ) )
=> ! [V6v: $i] :
( ( mem @ V6v @ A_27a )
=> ( p @ ( ap @ V4P @ V6v ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewhile_2EHOARE__SPEC__DEF,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1C: $i] :
( ( mem @ V1C @ ( arr @ A_27a @ A_27b ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EHOARE__SPEC @ A_27a @ A_27b ) @ V0P ) @ V1C ) @ V2Q ) )
<=> ! [V3s: $i] :
( ( mem @ V3s @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3s ) )
=> ( p @ ( ap @ V2Q @ ( ap @ V1C @ V3s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EWHILE__RULE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2B: $i] :
( ( mem @ V2B @ ( arr @ A_27a @ bool ) )
=> ! [V3C: $i] :
( ( mem @ V3C @ ( arr @ A_27a @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V1R ) )
& ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ( ( p @ ( ap @ V2B @ V4s ) )
=> ( p @ ( ap @ ( ap @ V1R @ ( ap @ V3C @ V4s ) ) @ V4s ) ) ) ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ewhile_2EHOARE__SPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5s: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V5s ) ) @ ( ap @ V2B @ V5s ) ) ) )
@ V3C )
@ V0P ) )
=> ( p
@ ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EHOARE__SPEC @ A_27a @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Ewhile_2EWHILE @ A_27a ) @ V2B ) @ V3C ) )
@ ( lam @ A_27a
@ ^ [V6s: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V6s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ V2B @ V6s ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewhile_2ELEAST__DEF,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Ewhile_2ELEAST @ V0P ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EWHILE @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V0P ) ) @ c_2Enum_2ESUC ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2ELEAST__INTRO,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1x ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ c_2Ewhile_2ELEAST @ V0P ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2ELESS__LEAST,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ c_2Ewhile_2ELEAST @ V0P ) ) )
=> ~ ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EFULL__LEAST__INTRO,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1x ) ) )
=> ( ( p @ ( ap @ V0P @ ( ap @ c_2Ewhile_2ELEAST @ V0P ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Ewhile_2ELEAST @ V0P ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2ELEAST__ELIM,axiom,
! [V0Q: $i] :
( ( mem @ V0Q @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ? [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V1P @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( ! [V4m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ~ ( p @ ( ap @ V1P @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) )
& ( p @ ( ap @ V1P @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( p @ ( ap @ V0Q @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
=> ( p @ ( ap @ V0Q @ ( ap @ c_2Ewhile_2ELEAST @ V1P ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2ELEAST__EXISTS,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ? [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0p @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ( ( p @ ( ap @ V0p @ ( ap @ c_2Ewhile_2ELEAST @ V0p ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Ewhile_2ELEAST @ V0p ) ) )
=> ~ ( p @ ( ap @ V0p @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2ELEAST__EXISTS__IMP,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ? [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0p @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( ( p @ ( ap @ V0p @ ( ap @ c_2Ewhile_2ELEAST @ V0p ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Ewhile_2ELEAST @ V0p ) ) )
=> ~ ( p @ ( ap @ V0p @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2ELEAST__EQ,axiom,
! [V0x: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum
@ ( ap @ c_2Ewhile_2ELEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V1n ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) ) )
= V0x )
& ( ( surj__ty_2Enum_2Enum
@ ( ap @ c_2Ewhile_2ELEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V2n ) ) ) )
= V0x ) ) ).
thf(conj_thm_2Ewhile_2ELEAST__T,axiom,
( ( surj__ty_2Enum_2Enum
@ ( ap @ c_2Ewhile_2ELEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0x: $i] : c_2Ebool_2ET ) ) )
= fo__c_2Enum_2E0 ) ).
thf(ax_thm_2Ewhile_2EOLEAST__def,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ c_2Ewhile_2EOLEAST @ V0P ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ V0P @ V1n ) ) ) )
@ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum )
@ ( ap @ c_2Ewhile_2ELEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ V0P @ V2n ) ) ) ) )
@ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EOLEAST__INTRO,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) @ bool ) )
=> ( ( ( ! [V2n: tp__ty_2Enum_2Enum] :
~ ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ V1Q @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
& ! [V4m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ~ ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) ) )
=> ( p @ ( ap @ V1Q @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) )
=> ( p @ ( ap @ V1Q @ ( ap @ c_2Ewhile_2EOLEAST @ V0P ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EOLEAST__EQNS,axiom,
! [V0x: tp__ty_2Enum_2Enum] :
( ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum
@ ( ap @ c_2Ewhile_2EOLEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V1n ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) )
& ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum
@ ( ap @ c_2Ewhile_2EOLEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V2n ) ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) )
& ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum
@ ( ap @ c_2Ewhile_2EOLEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : c_2Ebool_2EF ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) )
& ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum
@ ( ap @ c_2Ewhile_2EOLEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : c_2Ebool_2ET ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EOLEAST__EQ__NONE,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ c_2Ewhile_2EOLEAST @ V0P ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) )
<=> ! [V1n: tp__ty_2Enum_2Enum] :
~ ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EOLEAST__EQ__SOME,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ c_2Ewhile_2EOLEAST @ V0P ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ! [V2m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ~ ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewhile_2EOWHILE__def,axiom,
! [A_27a: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EOWHILE @ A_27a ) @ V0G ) @ V1f ) @ V2s )
= ( ap
@ ( ap
@ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V0G @ ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ A_27a ) @ V1f ) @ V3n ) @ V2s ) ) ) ) ) )
@ ( ap @ ( c_2Eoption_2ESOME @ A_27a )
@ ( ap
@ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ A_27a ) @ V1f )
@ ( ap @ c_2Ewhile_2ELEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V0G @ ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ A_27a ) @ V1f ) @ V4n ) @ V2s ) ) ) ) ) )
@ V2s ) ) )
@ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EOWHILE__THM,axiom,
! [A_27a: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EOWHILE @ A_27a ) @ V0G ) @ V1f ) @ V2s )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V0G @ V2s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EOWHILE @ A_27a ) @ V0G ) @ V1f ) @ ( ap @ V1f @ V2s ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EOWHILE__EQ__NONE,axiom,
! [A_27a: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EOWHILE @ A_27a ) @ V0G ) @ V1f ) @ V2s )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0G @ ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ A_27a ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EOWHILE__ENDCOND,axiom,
! [A_27a: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ A_27a )
=> ! [V3s_27: $i] :
( ( mem @ V3s_27 @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EOWHILE @ A_27a ) @ V0G ) @ V1f ) @ V2s )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3s_27 ) )
=> ~ ( p @ ( ap @ V0G @ V3s_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EOWHILE__WHILE,axiom,
! [A_27a: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ A_27a )
=> ! [V3s_27: $i] :
( ( mem @ V3s_27 @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EOWHILE @ A_27a ) @ V0G ) @ V1f ) @ V2s )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3s_27 ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EWHILE @ A_27a ) @ V0G ) @ V1f ) @ V2s )
= V3s_27 ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EOWHILE__INV__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1G: $i] :
( ( mem @ V1G @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V3s ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V4x ) )
& ( p @ ( ap @ V1G @ V4x ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ V2f @ V4x ) ) ) ) ) )
=> ! [V5s_27: $i] :
( ( mem @ V5s_27 @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EOWHILE @ A_27a ) @ V1G ) @ V2f ) @ V3s )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V5s_27 ) )
=> ( p @ ( ap @ V0P @ V5s_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewhile_2EOWHILE__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1G: $i] :
( ( mem @ V1G @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ! [V3s: $i] :
( ( mem @ V3s @ A_27a )
=> ( ~ ( p @ ( ap @ V1G @ V3s ) )
=> ( p @ ( ap @ ( ap @ V0P @ V3s ) @ V3s ) ) ) )
& ! [V4s1: $i] :
( ( mem @ V4s1 @ A_27a )
=> ! [V5s2: $i] :
( ( mem @ V5s2 @ A_27a )
=> ( ( ( p @ ( ap @ V1G @ V4s1 ) )
& ( p @ ( ap @ ( ap @ V0P @ ( ap @ V2f @ V4s1 ) ) @ V5s2 ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V4s1 ) @ V5s2 ) ) ) ) ) )
=> ! [V6s1: $i] :
( ( mem @ V6s1 @ A_27a )
=> ! [V7s2: $i] :
( ( mem @ V7s2 @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ewhile_2EOWHILE @ A_27a ) @ V1G ) @ V2f ) @ V6s1 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V7s2 ) )
=> ( p @ ( ap @ ( ap @ V0P @ V6s1 ) @ V7s2 ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------