ITP001 Axioms: ITP021^5.ax
%------------------------------------------------------------------------------
% File : ITP021^5 : 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 : divides^2.ax [Gau20]
% : HL4021^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 50 ( 1 unt; 4 typ; 0 def)
% Number of atoms : 614 ( 19 equ; 0 cnn)
% Maximal formula atoms : 66 ( 12 avg)
% Number of connectives : 759 ( 5 ~; 3 |; 17 &; 706 @)
% ( 7 <=>; 21 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg; 706 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 1 ( 1 >; 0 *; 0 +; 0 <<)
% Number of symbols : 34 ( 32 usr; 32 con; 0-2 aty)
% Number of variables : 71 ( 1 ^ 63 !; 7 ?; 71 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Edivides_2EPRIMES,type,
c_2Edivides_2EPRIMES: $i ).
thf(mem_c_2Edivides_2EPRIMES,axiom,
mem @ c_2Edivides_2EPRIMES @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Edivides_2EPRIMES,type,
fo__c_2Edivides_2EPRIMES: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Edivides_2EPRIMES,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Edivides_2EPRIMES @ X0 ) )
= ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Edivides_2Edivides,type,
c_2Edivides_2Edivides: $i ).
thf(mem_c_2Edivides_2Edivides,axiom,
mem @ c_2Edivides_2Edivides @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ).
thf(tp_c_2Edivides_2Eprime,type,
c_2Edivides_2Eprime: $i ).
thf(mem_c_2Edivides_2Eprime,axiom,
mem @ c_2Edivides_2Eprime @ ( arr @ ty_2Enum_2Enum @ bool ) ).
thf(ax_thm_2Edivides_2Edivides__def,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
<=> ? [V2q: tp__ty_2Enum_2Enum] :
( V1b
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2q ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EALL__DIVIDES__0,axiom,
! [V0a: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Edivides_2EZERO__DIVIDES,axiom,
! [V0m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
<=> ( V0m = fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__REFL,axiom,
! [V0a: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__TRANS,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ).
thf(conj_thm_2Edivides_2EONE__DIVIDES__ALL,axiom,
! [V0a: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__ONE,axiom,
! [V0x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
<=> ( V0x
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__ADD__1,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__ADD__2,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__SUB,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__LE,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__LEQ__OR__ZERO,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
| ( V1n = fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Edivides_2ENOT__LT__DIVIDES,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) )
=> ~ ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__ANTISYM,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) )
=> ( V0a = V1b ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__MULT,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
=> ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__MULT__LEFT,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
<=> ( ( V1m = fo__c_2Enum_2E0 )
| ( V0n
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EDIVIDES__FACT,axiom,
! [V0b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
=> ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ c_2Earithmetic_2EFACT @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) ) ) ).
thf(conj_thm_2Edivides_2ELEQ__DIVIDES__FACT,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Earithmetic_2EFACT @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(ax_thm_2Edivides_2Eprime__def,axiom,
! [V0a: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
<=> ( ( V0a
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V1b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
=> ( ( V1b = V0a )
| ( V1b
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Edivides_2ENOT__PRIME__0,axiom,
~ ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Edivides_2ENOT__PRIME__1,axiom,
~ ( p @ ( ap @ c_2Edivides_2Eprime @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EPRIME__2,axiom,
p @ ( ap @ c_2Edivides_2Eprime @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ).
thf(conj_thm_2Edivides_2EPRIME__3,axiom,
p @ ( ap @ c_2Edivides_2Eprime @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EPRIME__POS,axiom,
! [V0p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V0p ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) ) ) ).
thf(conj_thm_2Edivides_2EONE__LT__PRIME,axiom,
! [V0p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V0p ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) ) ) ).
thf(conj_thm_2Edivides_2Eprime__divides__only__self,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
& ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( V0m = V1n ) ) ).
thf(conj_thm_2Edivides_2EPRIME__FACTOR,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( V0n
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
=> ? [V1p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V1p ) ) )
& ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EEUCLID,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
? [V1p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1p ) ) )
& ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V1p ) ) ) ) ).
thf(ax_thm_2Edivides_2EPRIMES__def,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Edivides_2EPRIMES @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ c_2Ewhile_2ELEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1p: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Edivides_2Eprime @ V1p ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1p ) ) ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EprimePRIMES,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ c_2Edivides_2Eprime @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Edivides_2EINFINITE__PRIMES,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Edivides_2EPRIMES @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Edivides_2ELT__PRIMES,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_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EPRIMES__11,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( V0m = V1n ) ) ).
thf(conj_thm_2Edivides_2EINDEX__LESS__PRIMES,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Edivides_2EEUCLID__PRIMES,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
? [V1i: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V1i ) ) ) ) ).
thf(conj_thm_2Edivides_2ENEXT__LARGER__PRIME,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
? [V1i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V1i ) ) ) )
& ! [V2j: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V2j ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EPRIMES__NO__GAP,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1p: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ V1p ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( ap @ c_2Edivides_2EPRIMES @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V1p ) ) ) )
=> $false ) ).
thf(conj_thm_2Edivides_2EPRIMES__ONTO,axiom,
! [V0p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V0p ) ) )
=> ? [V1i: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V1i ) ) )
= V0p ) ) ).
thf(conj_thm_2Edivides_2EPRIME__INDEX,axiom,
! [V0p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Edivides_2Eprime @ ( inj__ty_2Enum_2Enum @ V0p ) ) )
<=> ? [V1i: tp__ty_2Enum_2Enum] :
( V0p
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V1i ) ) ) ) ) ).
thf(conj_thm_2Edivides_2EONE__LT__PRIMES,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Edivides_2EZERO__LT__PRIMES,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Edivides_2EPRIMES @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Edivides_2Ecompute__divides,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ c_2Ebool_2ET ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ c_2Ebool_2ET ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------