ITP001 Axioms: ITP084^7.ax
%------------------------------------------------------------------------------
% File : ITP084^7 : 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 : hrat.ax [Gau19]
% : HL4084^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 95 ( 37 unt; 36 typ; 0 def)
% Number of atoms : 108 ( 39 equ; 3 cnn)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 600 ( 3 ~; 5 |; 10 &; 563 @)
% ( 11 <=>; 8 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg; 563 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% Number of symbols : 35 ( 33 usr; 4 con; 0-5 aty)
% Number of variables : 140 ( 4 ^ 115 !; 11 ?; 140 :)
% ( 10 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Ehrat_2Ehrat,type,
tyop_2Ehrat_2Ehrat: $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(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
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_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Eprim__rec_2EPRE,type,
c_2Eprim__rec_2EPRE: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > A_27b ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ehrat_2Ehrat__1,type,
c_2Ehrat_2Ehrat__1: tyop_2Ehrat_2Ehrat ).
thf(c_2Ehrat_2Ehrat__ABS,type,
c_2Ehrat_2Ehrat__ABS: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Ehrat_2Ehrat ).
thf(c_2Ehrat_2Ehrat__ABS__CLASS,type,
c_2Ehrat_2Ehrat__ABS__CLASS: ( ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o ) > tyop_2Ehrat_2Ehrat ).
thf(c_2Ehrat_2Ehrat__REP,type,
c_2Ehrat_2Ehrat__REP: tyop_2Ehrat_2Ehrat > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ).
thf(c_2Ehrat_2Ehrat__REP__CLASS,type,
c_2Ehrat_2Ehrat__REP__CLASS: tyop_2Ehrat_2Ehrat > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o ).
thf(c_2Ehrat_2Ehrat__add,type,
c_2Ehrat_2Ehrat__add: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).
thf(c_2Ehrat_2Ehrat__inv,type,
c_2Ehrat_2Ehrat__inv: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).
thf(c_2Ehrat_2Ehrat__mul,type,
c_2Ehrat_2Ehrat__mul: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).
thf(c_2Ehrat_2Ehrat__sucint,type,
c_2Ehrat_2Ehrat__sucint: tyop_2Enum_2Enum > tyop_2Ehrat_2Ehrat ).
thf(c_2Ehrat_2Etrat__1,type,
c_2Ehrat_2Etrat__1: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ).
thf(c_2Ehrat_2Etrat__add,type,
c_2Ehrat_2Etrat__add: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ).
thf(c_2Ehrat_2Etrat__eq,type,
c_2Ehrat_2Etrat__eq: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o ).
thf(c_2Ehrat_2Etrat__inv,type,
c_2Ehrat_2Etrat__inv: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ).
thf(c_2Ehrat_2Etrat__mul,type,
c_2Ehrat_2Etrat__mul: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ).
thf(c_2Ehrat_2Etrat__sucint,type,
c_2Ehrat_2Etrat__sucint: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ).
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_2Ehrat_2Etrat__1,axiom,
( c_2Ehrat_2Etrat__1
= ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ c_2Enum_2E0 ) ) ).
thf(thm_2Ehrat_2Etrat__inv,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__inv @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0x @ V1y ) )
= ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1y @ V0x ) ) ).
thf(thm_2Ehrat_2Etrat__add,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2x_27: tyop_2Enum_2Enum,V3y_27: tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__add @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0x @ V1y ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2x_27 @ V3y_27 ) )
= ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2EPRE @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V0x ) @ ( c_2Enum_2ESUC @ V3y_27 ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V2x_27 ) @ ( c_2Enum_2ESUC @ V1y ) ) ) ) @ ( c_2Eprim__rec_2EPRE @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V1y ) @ ( c_2Enum_2ESUC @ V3y_27 ) ) ) ) ) ).
thf(thm_2Ehrat_2Etrat__mul,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2x_27: tyop_2Enum_2Enum,V3y_27: tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__mul @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0x @ V1y ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2x_27 @ V3y_27 ) )
= ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2EPRE @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V0x ) @ ( c_2Enum_2ESUC @ V2x_27 ) ) ) @ ( c_2Eprim__rec_2EPRE @ ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V1y ) @ ( c_2Enum_2ESUC @ V3y_27 ) ) ) ) ) ).
thf(thm_2Ehrat_2Etrat__sucint,axiom,
( ( ( c_2Ehrat_2Etrat__sucint @ c_2Enum_2E0 )
= c_2Ehrat_2Etrat__1 )
& ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__sucint @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Ehrat_2Etrat__add @ ( c_2Ehrat_2Etrat__sucint @ V0n ) @ c_2Ehrat_2Etrat__1 ) ) ) ).
thf(thm_2Ehrat_2Etrat__eq,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum,V2x_27: tyop_2Enum_2Enum,V3y_27: tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__eq @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0x @ V1y ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2x_27 @ V3y_27 ) )
<=> ( ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V0x ) @ ( c_2Enum_2ESUC @ V3y_27 ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Enum_2ESUC @ V2x_27 ) @ ( c_2Enum_2ESUC @ V1y ) ) ) ) ).
thf(thm_2Ehrat_2Ehrat__TY__DEF,axiom,
? [V0rep: tyop_2Ehrat_2Ehrat > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o] :
( c_2Ebool_2ETYPE__DEFINITION @ ( ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o ) @ tyop_2Ehrat_2Ehrat
@ ^ [V1c: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o] :
( c_2Ebool_2E_3F @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )
@ ^ [V2r: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ebool_2E_2F_5C @ ( c_2Ehrat_2Etrat__eq @ V2r @ V2r ) @ ( c_2Emin_2E_3D @ ( ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o ) @ V1c @ ( c_2Ehrat_2Etrat__eq @ V2r ) ) ) )
@ V0rep ) ).
thf(thm_2Ehrat_2Ehrat__bijections,axiom,
( ! [V0a: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__ABS__CLASS @ ( c_2Ehrat_2Ehrat__REP__CLASS @ V0a ) )
= V0a )
& ! [V1r: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o] :
( ( ^ [V2c: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o] :
( c_2Ebool_2E_3F @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )
@ ^ [V3r: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ebool_2E_2F_5C @ ( c_2Ehrat_2Etrat__eq @ V3r @ V3r ) @ ( c_2Emin_2E_3D @ ( ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o ) @ V2c @ ( c_2Ehrat_2Etrat__eq @ V3r ) ) ) )
@ V1r )
<=> ( ( c_2Ehrat_2Ehrat__REP__CLASS @ ( c_2Ehrat_2Ehrat__ABS__CLASS @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Ehrat_2Ehrat__REP__def,axiom,
! [V0a: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__REP @ V0a )
= ( c_2Emin_2E_40 @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Ehrat_2Ehrat__REP__CLASS @ V0a ) ) ) ).
thf(thm_2Ehrat_2Ehrat__ABS__def,axiom,
! [V0r: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Ehrat__ABS @ V0r )
= ( c_2Ehrat_2Ehrat__ABS__CLASS @ ( c_2Ehrat_2Etrat__eq @ V0r ) ) ) ).
thf(thm_2Ehrat_2Ehrat__1,axiom,
( c_2Ehrat_2Ehrat__1
= ( c_2Ehrat_2Ehrat__ABS @ c_2Ehrat_2Etrat__1 ) ) ).
thf(thm_2Ehrat_2Ehrat__inv,axiom,
! [V0T1: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__inv @ V0T1 )
= ( c_2Ehrat_2Ehrat__ABS @ ( c_2Ehrat_2Etrat__inv @ ( c_2Ehrat_2Ehrat__REP @ V0T1 ) ) ) ) ).
thf(thm_2Ehrat_2Ehrat__add,axiom,
! [V0T1: tyop_2Ehrat_2Ehrat,V1T2: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__add @ V0T1 @ V1T2 )
= ( c_2Ehrat_2Ehrat__ABS @ ( c_2Ehrat_2Etrat__add @ ( c_2Ehrat_2Ehrat__REP @ V0T1 ) @ ( c_2Ehrat_2Ehrat__REP @ V1T2 ) ) ) ) ).
thf(thm_2Ehrat_2Ehrat__mul,axiom,
! [V0T1: tyop_2Ehrat_2Ehrat,V1T2: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__mul @ V0T1 @ V1T2 )
= ( c_2Ehrat_2Ehrat__ABS @ ( c_2Ehrat_2Etrat__mul @ ( c_2Ehrat_2Ehrat__REP @ V0T1 ) @ ( c_2Ehrat_2Ehrat__REP @ V1T2 ) ) ) ) ).
thf(thm_2Ehrat_2Ehrat__sucint,axiom,
! [V0T1: tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Ehrat__sucint @ V0T1 )
= ( c_2Ehrat_2Ehrat__ABS @ ( c_2Ehrat_2Etrat__sucint @ V0T1 ) ) ) ).
thf(thm_2Ehrat_2ETRAT__EQ__REFL,axiom,
! [V0p: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ V0p @ V0p ) ).
thf(thm_2Ehrat_2ETRAT__EQ__SYM,axiom,
! [V0p: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1q: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__eq @ V0p @ V1q )
= ( c_2Ehrat_2Etrat__eq @ V1q @ V0p ) ) ).
thf(thm_2Ehrat_2ETRAT__EQ__TRANS,axiom,
! [V0p: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1q: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V2r: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( ( c_2Ehrat_2Etrat__eq @ V0p @ V1q )
& ( c_2Ehrat_2Etrat__eq @ V1q @ V2r ) )
=> ( c_2Ehrat_2Etrat__eq @ V0p @ V2r ) ) ).
thf(thm_2Ehrat_2ETRAT__EQ__AP,axiom,
! [V0p: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1q: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( V0p = V1q )
=> ( c_2Ehrat_2Etrat__eq @ V0p @ V1q ) ) ).
thf(thm_2Ehrat_2ETRAT__ADD__SYM__EQ,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1i: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__add @ V0h @ V1i )
= ( c_2Ehrat_2Etrat__add @ V1i @ V0h ) ) ).
thf(thm_2Ehrat_2ETRAT__MUL__SYM__EQ,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1i: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__mul @ V0h @ V1i )
= ( c_2Ehrat_2Etrat__mul @ V1i @ V0h ) ) ).
thf(thm_2Ehrat_2ETRAT__INV__WELLDEFINED,axiom,
! [V0p: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1q: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__eq @ V0p @ V1q )
=> ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__inv @ V0p ) @ ( c_2Ehrat_2Etrat__inv @ V1q ) ) ) ).
thf(thm_2Ehrat_2ETRAT__ADD__WELLDEFINED,axiom,
! [V0p: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1q: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V2r: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__eq @ V0p @ V1q )
=> ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__add @ V0p @ V2r ) @ ( c_2Ehrat_2Etrat__add @ V1q @ V2r ) ) ) ).
thf(thm_2Ehrat_2ETRAT__ADD__WELLDEFINED2,axiom,
! [V0p1: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1p2: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V2q1: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V3q2: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( ( c_2Ehrat_2Etrat__eq @ V0p1 @ V1p2 )
& ( c_2Ehrat_2Etrat__eq @ V2q1 @ V3q2 ) )
=> ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__add @ V0p1 @ V2q1 ) @ ( c_2Ehrat_2Etrat__add @ V1p2 @ V3q2 ) ) ) ).
thf(thm_2Ehrat_2ETRAT__MUL__WELLDEFINED,axiom,
! [V0p: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1q: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V2r: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__eq @ V0p @ V1q )
=> ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__mul @ V0p @ V2r ) @ ( c_2Ehrat_2Etrat__mul @ V1q @ V2r ) ) ) ).
thf(thm_2Ehrat_2ETRAT__MUL__WELLDEFINED2,axiom,
! [V0p1: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1p2: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V2q1: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V3q2: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( ( c_2Ehrat_2Etrat__eq @ V0p1 @ V1p2 )
& ( c_2Ehrat_2Etrat__eq @ V2q1 @ V3q2 ) )
=> ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__mul @ V0p1 @ V2q1 ) @ ( c_2Ehrat_2Etrat__mul @ V1p2 @ V3q2 ) ) ) ).
thf(thm_2Ehrat_2ETRAT__ADD__SYM,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1i: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__add @ V0h @ V1i ) @ ( c_2Ehrat_2Etrat__add @ V1i @ V0h ) ) ).
thf(thm_2Ehrat_2ETRAT__ADD__ASSOC,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1i: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V2j: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__add @ V0h @ ( c_2Ehrat_2Etrat__add @ V1i @ V2j ) ) @ ( c_2Ehrat_2Etrat__add @ ( c_2Ehrat_2Etrat__add @ V0h @ V1i ) @ V2j ) ) ).
thf(thm_2Ehrat_2ETRAT__MUL__SYM,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1i: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__mul @ V0h @ V1i ) @ ( c_2Ehrat_2Etrat__mul @ V1i @ V0h ) ) ).
thf(thm_2Ehrat_2ETRAT__MUL__ASSOC,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1i: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V2j: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__mul @ V0h @ ( c_2Ehrat_2Etrat__mul @ V1i @ V2j ) ) @ ( c_2Ehrat_2Etrat__mul @ ( c_2Ehrat_2Etrat__mul @ V0h @ V1i ) @ V2j ) ) ).
thf(thm_2Ehrat_2ETRAT__LDISTRIB,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1i: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V2j: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__mul @ V0h @ ( c_2Ehrat_2Etrat__add @ V1i @ V2j ) ) @ ( c_2Ehrat_2Etrat__add @ ( c_2Ehrat_2Etrat__mul @ V0h @ V1i ) @ ( c_2Ehrat_2Etrat__mul @ V0h @ V2j ) ) ) ).
thf(thm_2Ehrat_2ETRAT__MUL__LID,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__mul @ c_2Ehrat_2Etrat__1 @ V0h ) @ V0h ) ).
thf(thm_2Ehrat_2ETRAT__MUL__LINV,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__mul @ ( c_2Ehrat_2Etrat__inv @ V0h ) @ V0h ) @ c_2Ehrat_2Etrat__1 ) ).
thf(thm_2Ehrat_2ETRAT__NOZERO,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1i: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( (~) @ ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__add @ V0h @ V1i ) @ V0h ) ) ).
thf(thm_2Ehrat_2ETRAT__ADD__TOTAL,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1i: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__eq @ V0h @ V1i )
| ? [V2d: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ V0h @ ( c_2Ehrat_2Etrat__add @ V1i @ V2d ) )
| ? [V3d: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ V1i @ ( c_2Ehrat_2Etrat__add @ V0h @ V3d ) ) ) ).
thf(thm_2Ehrat_2ETRAT__SUCINT__0,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__sucint @ V0n ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) ) ).
thf(thm_2Ehrat_2ETRAT__ARCH,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
? [V1n: tyop_2Enum_2Enum,V2d: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__sucint @ V1n ) @ ( c_2Ehrat_2Etrat__add @ V0h @ V2d ) ) ).
thf(thm_2Ehrat_2ETRAT__SUCINT,axiom,
( ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__sucint @ c_2Enum_2E0 ) @ c_2Ehrat_2Etrat__1 )
& ! [V0n: tyop_2Enum_2Enum] : ( c_2Ehrat_2Etrat__eq @ ( c_2Ehrat_2Etrat__sucint @ ( c_2Enum_2ESUC @ V0n ) ) @ ( c_2Ehrat_2Etrat__add @ ( c_2Ehrat_2Etrat__sucint @ V0n ) @ c_2Ehrat_2Etrat__1 ) ) ) ).
thf(thm_2Ehrat_2ETRAT__EQ__EQUIV,axiom,
! [V0p: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum,V1q: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__eq @ V0p @ V1q )
<=> ( ( c_2Ehrat_2Etrat__eq @ V0p )
= ( c_2Ehrat_2Etrat__eq @ V1q ) ) ) ).
thf(thm_2Ehrat_2Ehrat__ABS__REP__CLASS,axiom,
( ! [V0a: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__ABS__CLASS @ ( c_2Ehrat_2Ehrat__REP__CLASS @ V0a ) )
= V0a )
& ! [V1c: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > $o] :
( ? [V2r: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Etrat__eq @ V2r @ V2r )
& ( V1c
= ( c_2Ehrat_2Etrat__eq @ V2r ) ) )
<=> ( ( c_2Ehrat_2Ehrat__REP__CLASS @ ( c_2Ehrat_2Ehrat__ABS__CLASS @ V1c ) )
= V1c ) ) ) ).
thf(thm_2Ehrat_2Ehrat__QUOTIENT,axiom,
c_2Equotient_2EQUOTIENT @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ tyop_2Ehrat_2Ehrat @ c_2Ehrat_2Etrat__eq @ c_2Ehrat_2Ehrat__ABS @ c_2Ehrat_2Ehrat__REP ).
thf(thm_2Ehrat_2EHRAT__ADD__SYM,axiom,
! [V0h: tyop_2Ehrat_2Ehrat,V1i: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__add @ V0h @ V1i )
= ( c_2Ehrat_2Ehrat__add @ V1i @ V0h ) ) ).
thf(thm_2Ehrat_2EHRAT__ADD__ASSOC,axiom,
! [V0h: tyop_2Ehrat_2Ehrat,V1i: tyop_2Ehrat_2Ehrat,V2j: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__add @ V0h @ ( c_2Ehrat_2Ehrat__add @ V1i @ V2j ) )
= ( c_2Ehrat_2Ehrat__add @ ( c_2Ehrat_2Ehrat__add @ V0h @ V1i ) @ V2j ) ) ).
thf(thm_2Ehrat_2EHRAT__MUL__SYM,axiom,
! [V0h: tyop_2Ehrat_2Ehrat,V1i: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__mul @ V0h @ V1i )
= ( c_2Ehrat_2Ehrat__mul @ V1i @ V0h ) ) ).
thf(thm_2Ehrat_2EHRAT__MUL__ASSOC,axiom,
! [V0h: tyop_2Ehrat_2Ehrat,V1i: tyop_2Ehrat_2Ehrat,V2j: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__mul @ V0h @ ( c_2Ehrat_2Ehrat__mul @ V1i @ V2j ) )
= ( c_2Ehrat_2Ehrat__mul @ ( c_2Ehrat_2Ehrat__mul @ V0h @ V1i ) @ V2j ) ) ).
thf(thm_2Ehrat_2EHRAT__LDISTRIB,axiom,
! [V0h: tyop_2Ehrat_2Ehrat,V1i: tyop_2Ehrat_2Ehrat,V2j: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__mul @ V0h @ ( c_2Ehrat_2Ehrat__add @ V1i @ V2j ) )
= ( c_2Ehrat_2Ehrat__add @ ( c_2Ehrat_2Ehrat__mul @ V0h @ V1i ) @ ( c_2Ehrat_2Ehrat__mul @ V0h @ V2j ) ) ) ).
thf(thm_2Ehrat_2EHRAT__MUL__LID,axiom,
! [V0h: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__mul @ c_2Ehrat_2Ehrat__1 @ V0h )
= V0h ) ).
thf(thm_2Ehrat_2EHRAT__MUL__LINV,axiom,
! [V0h: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__mul @ ( c_2Ehrat_2Ehrat__inv @ V0h ) @ V0h )
= c_2Ehrat_2Ehrat__1 ) ).
thf(thm_2Ehrat_2EHRAT__NOZERO,axiom,
! [V0h: tyop_2Ehrat_2Ehrat,V1i: tyop_2Ehrat_2Ehrat] :
( (~)
@ ( ( c_2Ehrat_2Ehrat__add @ V0h @ V1i )
= V0h ) ) ).
thf(thm_2Ehrat_2EHRAT__ADD__TOTAL,axiom,
! [V0h: tyop_2Ehrat_2Ehrat,V1i: tyop_2Ehrat_2Ehrat] :
( ( V0h = V1i )
| ? [V2d: tyop_2Ehrat_2Ehrat] :
( V0h
= ( c_2Ehrat_2Ehrat__add @ V1i @ V2d ) )
| ? [V3d: tyop_2Ehrat_2Ehrat] :
( V1i
= ( c_2Ehrat_2Ehrat__add @ V0h @ V3d ) ) ) ).
thf(thm_2Ehrat_2EHRAT__ARCH,axiom,
! [V0h: tyop_2Ehrat_2Ehrat] :
? [V1n: tyop_2Enum_2Enum,V2d: tyop_2Ehrat_2Ehrat] :
( ( c_2Ehrat_2Ehrat__sucint @ V1n )
= ( c_2Ehrat_2Ehrat__add @ V0h @ V2d ) ) ).
thf(thm_2Ehrat_2EHRAT__SUCINT,axiom,
( ( ( c_2Ehrat_2Ehrat__sucint @ c_2Enum_2E0 )
= c_2Ehrat_2Ehrat__1 )
& ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ehrat_2Ehrat__sucint @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Ehrat_2Ehrat__add @ ( c_2Ehrat_2Ehrat__sucint @ V0n ) @ c_2Ehrat_2Ehrat__1 ) ) ) ).
%------------------------------------------------------------------------------