ITP001 Axioms: ITP039^7.ax
%------------------------------------------------------------------------------
% File : ITP039^7 : 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 : defCNF.ax [Gau19]
% : HL4039^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 46 ( 6 unt; 28 typ; 0 def)
% Number of atoms : 53 ( 13 equ; 1 cnn)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 850 ( 1 ~; 1 |; 23 &; 801 @)
% ( 18 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 12 avg; 801 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 180 ( 180 >; 0 *; 0 +; 0 <<)
% Number of symbols : 28 ( 26 usr; 2 con; 0-4 aty)
% Number of variables : 118 ( 0 ^ 101 !; 4 ?; 118 :)
% ( 13 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
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(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Esum_2Esum,type,
tyop_2Esum_2Esum: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
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_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2EdefCNF_2EDEF,type,
c_2EdefCNF_2EDEF: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) ) > $o ).
thf(c_2Esum_2EINL,type,
c_2Esum_2EINL:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Esum_2EINR,type,
c_2Esum_2EINR:
!>[A_27a: $tType,A_27b: $tType] : ( A_27b > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2EdefCNF_2EOK,type,
c_2EdefCNF_2EOK: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) > $o ).
thf(c_2EdefCNF_2EOKDEF,type,
c_2EdefCNF_2EOKDEF: tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) ) > $o ).
thf(c_2Elist_2ESNOC,type,
c_2Elist_2ESNOC:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2EdefCNF_2EUNIQUE,type,
c_2EdefCNF_2EUNIQUE: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) > $o ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $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_2EdefCNF_2EDEF__def,axiom,
( ! [V0v: tyop_2Enum_2Enum > $o,V1n: tyop_2Enum_2Enum] :
( ( c_2EdefCNF_2EDEF @ V0v @ V1n @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) ) )
= c_2Ebool_2ET )
& ! [V2v: tyop_2Enum_2Enum > $o,V3n: tyop_2Enum_2Enum,V4x: tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ),V5xs: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) )] :
( ( c_2EdefCNF_2EDEF @ V2v @ V3n @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) @ V4x @ V5xs ) )
<=> ( ( c_2EdefCNF_2EUNIQUE @ V2v @ V3n @ V4x )
& ( c_2EdefCNF_2EDEF @ V2v @ ( c_2Enum_2ESUC @ V3n ) @ V5xs ) ) ) ) ).
thf(thm_2EdefCNF_2EOKDEF__def,axiom,
( ! [V0n: tyop_2Enum_2Enum] :
( ( c_2EdefCNF_2EOKDEF @ V0n @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) ) )
= c_2Ebool_2ET )
& ! [V1n: tyop_2Enum_2Enum,V2x: tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ),V3xs: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) )] :
( ( c_2EdefCNF_2EOKDEF @ V1n @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) @ V2x @ V3xs ) )
<=> ( ( c_2EdefCNF_2EOK @ V1n @ V2x )
& ( c_2EdefCNF_2EOKDEF @ ( c_2Enum_2ESUC @ V1n ) @ V3xs ) ) ) ) ).
thf(thm_2EdefCNF_2EUNIQUE__ind,axiom,
! [V0P: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) > $o] :
( ( ! [V1v: tyop_2Enum_2Enum > $o,V2n: tyop_2Enum_2Enum,V3conn: $o > $o > $o,V4i: tyop_2Enum_2Enum,V5j: tyop_2Enum_2Enum] : ( V0P @ V1v @ V2n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V3conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V4i ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V5j ) ) ) )
& ! [V6v: tyop_2Enum_2Enum > $o,V7n: tyop_2Enum_2Enum,V8conn: $o > $o > $o,V9i: tyop_2Enum_2Enum,V10b: $o] : ( V0P @ V6v @ V7n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V8conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V9i ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V10b ) ) ) )
& ! [V11v: tyop_2Enum_2Enum > $o,V12n: tyop_2Enum_2Enum,V13conn: $o > $o > $o,V14a: $o,V15j: tyop_2Enum_2Enum] : ( V0P @ V11v @ V12n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V13conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V14a ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V15j ) ) ) )
& ! [V16v: tyop_2Enum_2Enum > $o,V17n: tyop_2Enum_2Enum,V18conn: $o > $o > $o,V19a: $o,V20b: $o] : ( V0P @ V16v @ V17n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V18conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V19a ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V20b ) ) ) ) )
=> ! [V21v: tyop_2Enum_2Enum > $o,V22v1: tyop_2Enum_2Enum,V23v2: $o > $o > $o,V24v3: tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o,V25v4: tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o] : ( V0P @ V21v @ V22v1 @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V23v2 @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ V24v3 @ V25v4 ) ) ) ) ).
thf(thm_2EdefCNF_2EUNIQUE__def,axiom,
! [V0v: tyop_2Enum_2Enum > $o,V1n: tyop_2Enum_2Enum,V2j: tyop_2Enum_2Enum,V3i: tyop_2Enum_2Enum,V4conn: $o > $o > $o,V5b: $o,V6a: $o] :
( ( ( c_2EdefCNF_2EUNIQUE @ V0v @ V1n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V4conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V3i ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V2j ) ) ) )
<=> ( ( V0v @ V1n )
= ( V4conn @ ( V0v @ V3i ) @ ( V0v @ V2j ) ) ) )
& ( ( c_2EdefCNF_2EUNIQUE @ V0v @ V1n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V4conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V3i ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V5b ) ) ) )
<=> ( ( V0v @ V1n )
= ( V4conn @ ( V0v @ V3i ) @ V5b ) ) )
& ( ( c_2EdefCNF_2EUNIQUE @ V0v @ V1n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V4conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V6a ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V2j ) ) ) )
<=> ( ( V0v @ V1n )
= ( V4conn @ V6a @ ( V0v @ V2j ) ) ) )
& ( ( c_2EdefCNF_2EUNIQUE @ V0v @ V1n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V4conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V6a ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V5b ) ) ) )
<=> ( ( V0v @ V1n )
= ( V4conn @ V6a @ V5b ) ) ) ) ).
thf(thm_2EdefCNF_2EOK__ind,axiom,
! [V0P: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) > $o] :
( ( ! [V1n: tyop_2Enum_2Enum,V2conn: $o > $o > $o,V3i: tyop_2Enum_2Enum,V4j: tyop_2Enum_2Enum] : ( V0P @ V1n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V2conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V3i ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V4j ) ) ) )
& ! [V5n: tyop_2Enum_2Enum,V6conn: $o > $o > $o,V7i: tyop_2Enum_2Enum,V8b: $o] : ( V0P @ V5n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V6conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V7i ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V8b ) ) ) )
& ! [V9n: tyop_2Enum_2Enum,V10conn: $o > $o > $o,V11a: $o,V12j: tyop_2Enum_2Enum] : ( V0P @ V9n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V10conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V11a ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V12j ) ) ) )
& ! [V13n: tyop_2Enum_2Enum,V14conn: $o > $o > $o,V15a: $o,V16b: $o] : ( V0P @ V13n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V14conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V15a ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V16b ) ) ) ) )
=> ! [V17v: tyop_2Enum_2Enum,V18v1: $o > $o > $o,V19v2: tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o,V20v3: tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o] : ( V0P @ V17v @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V18v1 @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ V19v2 @ V20v3 ) ) ) ) ).
thf(thm_2EdefCNF_2EOK__def,axiom,
! [V0n: tyop_2Enum_2Enum,V1j: tyop_2Enum_2Enum,V2i: tyop_2Enum_2Enum,V3conn: $o > $o > $o,V4b: $o,V5a: $o] :
( ( ( c_2EdefCNF_2EOK @ V0n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V3conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V2i ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V1j ) ) ) )
<=> ( ( c_2Eprim__rec_2E_3C @ V2i @ V0n )
& ( c_2Eprim__rec_2E_3C @ V1j @ V0n ) ) )
& ( ( c_2EdefCNF_2EOK @ V0n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V3conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V2i ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V4b ) ) ) )
= ( c_2Eprim__rec_2E_3C @ V2i @ V0n ) )
& ( ( c_2EdefCNF_2EOK @ V0n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V3conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V5a ) @ ( c_2Esum_2EINL @ tyop_2Enum_2Enum @ $o @ V1j ) ) ) )
= ( c_2Eprim__rec_2E_3C @ V1j @ V0n ) )
& ( ( c_2EdefCNF_2EOK @ V0n @ ( c_2Epair_2E_2C @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) @ V3conn @ ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V5a ) @ ( c_2Esum_2EINR @ tyop_2Enum_2Enum @ $o @ V4b ) ) ) )
= c_2Ebool_2ET ) ) ).
thf(thm_2EdefCNF_2EDEF__SNOC,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ),V2l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ),V3v: tyop_2Enum_2Enum > $o] :
( ( c_2EdefCNF_2EDEF @ V3v @ V0n @ ( c_2Elist_2ESNOC @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) @ V1x @ V2l ) )
<=> ( ( c_2EdefCNF_2EDEF @ V3v @ V0n @ V2l )
& ( c_2EdefCNF_2EUNIQUE @ V3v @ ( c_2Earithmetic_2E_2B @ V0n @ ( c_2Elist_2ELENGTH @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) @ V2l ) ) @ V1x ) ) ) ).
thf(thm_2EdefCNF_2EOKDEF__SNOC,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ),V2l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) )] :
( ( c_2EdefCNF_2EOKDEF @ V0n @ ( c_2Elist_2ESNOC @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) @ V1x @ V2l ) )
<=> ( ( c_2EdefCNF_2EOKDEF @ V0n @ V2l )
& ( c_2EdefCNF_2EOK @ ( c_2Earithmetic_2E_2B @ V0n @ ( c_2Elist_2ELENGTH @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) @ V2l ) ) @ V1x ) ) ) ).
thf(thm_2EdefCNF_2ECONSISTENCY,axiom,
! [V0n: tyop_2Enum_2Enum,V1l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) )] :
( ( c_2EdefCNF_2EOKDEF @ V0n @ V1l )
=> ? [V2v: tyop_2Enum_2Enum > $o] : ( c_2EdefCNF_2EDEF @ V2v @ V0n @ V1l ) ) ).
thf(thm_2EdefCNF_2EBIGSTEP,axiom,
! [V0P: ( tyop_2Enum_2Enum > $o ) > $o,V1Q: $o,V2R: ( tyop_2Enum_2Enum > $o ) > $o] :
( ! [V3v: tyop_2Enum_2Enum > $o] :
( ( V0P @ V3v )
=> ( V1Q
= ( V2R @ V3v ) ) )
=> ( ( ? [V4v: tyop_2Enum_2Enum > $o] : ( V0P @ V4v )
& V1Q )
<=> ? [V5v: tyop_2Enum_2Enum > $o] :
( ( V0P @ V5v )
& ( V2R @ V5v ) ) ) ) ).
thf(thm_2EdefCNF_2EFINAL__DEF,axiom,
! [V0v: tyop_2Enum_2Enum > $o,V1n: tyop_2Enum_2Enum,V2x: $o] :
( ( ( V0v @ V1n )
= V2x )
<=> ( ( ( V0v @ V1n )
= V2x )
& ( c_2EdefCNF_2EDEF @ V0v @ ( c_2Enum_2ESUC @ V1n ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ ( $o > $o > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ $o ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------