ITP001 Axioms: ITP011^7.ax
%------------------------------------------------------------------------------
% File : ITP011^7 : 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 : relation.ax [Gau19]
% : HL4011^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 320 ( 95 unt; 69 typ; 0 def)
% Number of atoms : 706 ( 117 equ; 12 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 2807 ( 12 ~; 17 |; 144 &;2325 @)
% ( 76 <=>; 233 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 9 avg;2325 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 927 ( 927 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 68 usr; 3 con; 0-7 aty)
% Number of variables : 1155 ( 10 ^1042 !; 20 ?;1155 :)
% ( 83 !>; 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(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_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_2EARB,type,
c_2Ebool_2EARB:
!>[A_27a: $tType] : A_27a ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Erelation_2ECR,type,
c_2Erelation_2ECR:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EEMPTY__REL,type,
c_2Erelation_2EEMPTY__REL:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Erelation_2EEQC,type,
c_2Erelation_2EEQC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Erelation_2EIDEM,type,
c_2Erelation_2EIDEM:
!>[A_27z: $tType] : ( ( A_27z > A_27z ) > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EINDUCTIVE__INVARIANT,type,
c_2Erelation_2EINDUCTIVE__INVARIANT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > A_27b > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > $o ) ).
thf(c_2Erelation_2EINDUCTIVE__INVARIANT__ON,type,
c_2Erelation_2EINDUCTIVE__INVARIANT__ON:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > ( A_27a > A_27b > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > $o ) ).
thf(c_2Erelation_2EINVOL,type,
c_2Erelation_2EINVOL:
!>[A_27z: $tType] : ( ( A_27z > A_27z ) > $o ) ).
thf(c_2Erelation_2ELinearOrder,type,
c_2Erelation_2ELinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EO,type,
c_2Erelation_2EO:
!>[A_27g: $tType,A_27h: $tType,A_27k: $tType] : ( ( A_27h > A_27k > $o ) > ( A_27g > A_27h > $o ) > A_27g > A_27k > $o ) ).
thf(c_2Erelation_2EOrder,type,
c_2Erelation_2EOrder:
!>[A_27g: $tType] : ( ( A_27g > A_27g > $o ) > $o ) ).
thf(c_2Erelation_2EPreOrder,type,
c_2Erelation_2EPreOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2ERC,type,
c_2Erelation_2ERC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2ERCOMPL,type,
c_2Erelation_2ERCOMPL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27a > A_27b > $o ) ).
thf(c_2Erelation_2ERDOM,type,
c_2Erelation_2ERDOM:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27a > $o ) ).
thf(c_2Erelation_2ERDOM__DELETE,type,
c_2Erelation_2ERDOM__DELETE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27a > A_27a > A_27b > $o ) ).
thf(c_2Erelation_2ERESTRICT,type,
c_2Erelation_2ERESTRICT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > A_27a > $o ) > A_27a > A_27a > A_27b ) ).
thf(c_2Erelation_2ERINTER,type,
c_2Erelation_2ERINTER:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27a > A_27b > $o ) > A_27a > A_27b > $o ) ).
thf(c_2Erelation_2ERRANGE,type,
c_2Erelation_2ERRANGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27b > $o ) ).
thf(c_2Erelation_2ERRESTRICT,type,
c_2Erelation_2ERRESTRICT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27a > $o ) > A_27a > A_27b > $o ) ).
thf(c_2Erelation_2ERSUBSET,type,
c_2Erelation_2ERSUBSET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27a > A_27b > $o ) > $o ) ).
thf(c_2Erelation_2ERTC,type,
c_2Erelation_2ERTC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2ERUNION,type,
c_2Erelation_2ERUNION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27a > A_27b > $o ) > A_27a > A_27b > $o ) ).
thf(c_2Erelation_2ERUNIV,type,
c_2Erelation_2ERUNIV:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > $o ) ).
thf(c_2Erelation_2ESC,type,
c_2Erelation_2ESC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2ESN,type,
c_2Erelation_2ESN:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2ESTRORD,type,
c_2Erelation_2ESTRORD:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2EStrongLinearOrder,type,
c_2Erelation_2EStrongLinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EStrongOrder,type,
c_2Erelation_2EStrongOrder:
!>[A_27g: $tType] : ( ( A_27g > A_27g > $o ) > $o ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Erelation_2ETC,type,
c_2Erelation_2ETC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2EWCR,type,
c_2Erelation_2EWCR:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWFP,type,
c_2Erelation_2EWFP:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > $o ) ).
thf(c_2Erelation_2EWFREC,type,
c_2Erelation_2EWFREC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Erelation_2EWeakLinearOrder,type,
c_2Erelation_2EWeakLinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWeakOrder,type,
c_2Erelation_2EWeakOrder:
!>[A_27g: $tType] : ( ( A_27g > A_27g > $o ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Erelation_2Eantisymmetric,type,
c_2Erelation_2Eantisymmetric:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Eapprox,type,
c_2Erelation_2Eapprox:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > A_27a > ( A_27a > A_27b ) > $o ) ).
thf(c_2Erelation_2Ediag,type,
c_2Erelation_2Ediag:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2Ediamond,type,
c_2Erelation_2Ediamond:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Eequivalence,type,
c_2Erelation_2Eequivalence:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Einv,type,
c_2Erelation_2Einv:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27b > A_27a > $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_2Erelation_2Eirreflexive,type,
c_2Erelation_2Eirreflexive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Enf,type,
c_2Erelation_2Enf:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27a > $o ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Erelation_2Ercdiamond,type,
c_2Erelation_2Ercdiamond:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Ereflexive,type,
c_2Erelation_2Ereflexive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Esymmetric,type,
c_2Erelation_2Esymmetric:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Ethe__fun,type,
c_2Erelation_2Ethe__fun:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > A_27a > A_27a > A_27b ) ).
thf(c_2Erelation_2Etotal,type,
c_2Erelation_2Etotal:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Etransitive,type,
c_2Erelation_2Etransitive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Etrichotomous,type,
c_2Erelation_2Etrichotomous:
!>[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_2Erelation_2Etransitive__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Etransitive @ A_27a @ V0R )
<=> ! [V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( V0R @ V1x @ V2y )
& ( V0R @ V2y @ V3z ) )
=> ( V0R @ V1x @ V3z ) ) ) ).
thf(thm_2Erelation_2Ereflexive__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ereflexive @ A_27a @ V0R )
<=> ! [V1x: A_27a] : ( V0R @ V1x @ V1x ) ) ).
thf(thm_2Erelation_2Eirreflexive__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Eirreflexive @ A_27a @ V0R )
<=> ! [V1x: A_27a] : ( (~) @ ( V0R @ V1x @ V1x ) ) ) ).
thf(thm_2Erelation_2Esymmetric__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Esymmetric @ A_27a @ V0R )
<=> ! [V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
= ( V0R @ V2y @ V1x ) ) ) ).
thf(thm_2Erelation_2Eantisymmetric__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Eantisymmetric @ A_27a @ V0R )
<=> ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0R @ V1x @ V2y )
& ( V0R @ V2y @ V1x ) )
=> ( V1x = V2y ) ) ) ).
thf(thm_2Erelation_2Eequivalence__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Eequivalence @ A_27a @ V0R )
<=> ( ( c_2Erelation_2Ereflexive @ A_27a @ V0R )
& ( c_2Erelation_2Esymmetric @ A_27a @ V0R )
& ( c_2Erelation_2Etransitive @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Etotal__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Etotal @ A_27a @ V0R )
<=> ! [V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
| ( V0R @ V2y @ V1x ) ) ) ).
thf(thm_2Erelation_2Etrichotomous,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Etrichotomous @ A_27a @ V0R )
<=> ! [V1a: A_27a,V2b: A_27a] :
( ( V0R @ V1a @ V2b )
| ( V0R @ V2b @ V1a )
| ( V1a = V2b ) ) ) ).
thf(thm_2Erelation_2ETC__DEF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1a: A_27a,V2b: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V1a @ V2b )
<=> ! [V3P: A_27a > A_27a > $o] :
( ( ! [V4x: A_27a,V5y: A_27a] :
( ( V0R @ V4x @ V5y )
=> ( V3P @ V4x @ V5y ) )
& ! [V6x: A_27a,V7y: A_27a,V8z: A_27a] :
( ( ( V3P @ V6x @ V7y )
& ( V3P @ V7y @ V8z ) )
=> ( V3P @ V6x @ V8z ) ) )
=> ( V3P @ V1a @ V2b ) ) ) ).
thf(thm_2Erelation_2ERTC__DEF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1a: A_27a,V2b: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1a @ V2b )
<=> ! [V3P: A_27a > A_27a > $o] :
( ( ! [V4x: A_27a] : ( V3P @ V4x @ V4x )
& ! [V5x: A_27a,V6y: A_27a,V7z: A_27a] :
( ( ( V0R @ V5x @ V6y )
& ( V3P @ V6y @ V7z ) )
=> ( V3P @ V5x @ V7z ) ) )
=> ( V3P @ V1a @ V2b ) ) ) ).
thf(thm_2Erelation_2ERC__DEF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERC @ A_27a @ V0R @ V1x @ V2y )
<=> ( ( V1x = V2y )
| ( V0R @ V1x @ V2y ) ) ) ).
thf(thm_2Erelation_2ESC__DEF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ESC @ A_27a @ V0R @ V1x @ V2y )
<=> ( ( V0R @ V1x @ V2y )
| ( V0R @ V2y @ V1x ) ) ) ).
thf(thm_2Erelation_2EEQC__DEF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EEQC @ A_27a @ V0R )
= ( c_2Erelation_2ERC @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ ( c_2Erelation_2ESC @ A_27a @ V0R ) ) ) ) ).
thf(thm_2Erelation_2EWF__DEF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
<=> ! [V1B: A_27a > $o] :
( ? [V2w: A_27a] : ( V1B @ V2w )
=> ? [V3min: A_27a] :
( ( V1B @ V3min )
& ! [V4b: A_27a] :
( ( V0R @ V4b @ V3min )
=> ( (~) @ ( V1B @ V4b ) ) ) ) ) ) ).
thf(thm_2Erelation_2EEMPTY__REL__DEF,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
( ( c_2Erelation_2EEMPTY__REL @ A_27a @ V0x @ V1y )
= c_2Ebool_2EF ) ).
thf(thm_2Erelation_2Einv__image__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27b > A_27b > $o,V1f: A_27a > A_27b] :
( ( c_2Erelation_2Einv__image @ A_27a @ A_27b @ V0R @ V1f )
= ( ^ [V2x: A_27a,V3y: A_27a] : ( V0R @ ( V1f @ V2x ) @ ( V1f @ V3y ) ) ) ) ).
thf(thm_2Erelation_2ERESTRICT__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1R: A_27a > A_27a > $o,V2x: A_27a] :
( ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b @ V0f @ V1R @ V2x )
= ( ^ [V3y: A_27a] : ( c_2Ebool_2ECOND @ A_27b @ ( V1R @ V3y @ V2x ) @ ( V0f @ V3y ) @ ( c_2Ebool_2EARB @ A_27b ) ) ) ) ).
thf(thm_2Erelation_2Eapprox__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1M: ( A_27a > A_27b ) > A_27a > A_27b,V2x: A_27a,V3f: A_27a > A_27b] :
( ( c_2Erelation_2Eapprox @ A_27a @ A_27b @ V0R @ V1M @ V2x @ V3f )
<=> ( V3f
= ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b
@ ^ [V4y: A_27a] : ( V1M @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b @ V3f @ V0R @ V4y ) @ V4y )
@ V0R
@ V2x ) ) ) ).
thf(thm_2Erelation_2Ethe__fun__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1M: ( A_27a > A_27b ) > A_27a > A_27b,V2x: A_27a] :
( ( c_2Erelation_2Ethe__fun @ A_27a @ A_27b @ V0R @ V1M @ V2x )
= ( c_2Emin_2E_40 @ ( A_27a > A_27b )
@ ^ [V3f: A_27a > A_27b] : ( c_2Erelation_2Eapprox @ A_27a @ A_27b @ V0R @ V1M @ V2x @ V3f ) ) ) ).
thf(thm_2Erelation_2EWFREC__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1M: ( A_27a > A_27b ) > A_27a > A_27b] :
( ( c_2Erelation_2EWFREC @ A_27a @ A_27b @ V0R @ V1M )
= ( ^ [V2x: A_27a] :
( V1M
@ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b
@ ( c_2Erelation_2Ethe__fun @ A_27a @ A_27b @ ( c_2Erelation_2ETC @ A_27a @ V0R )
@ ^ [V3f: A_27a > A_27b,V4v: A_27a] : ( V1M @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b @ V3f @ V0R @ V4v ) @ V4v )
@ V2x )
@ V0R
@ V2x )
@ V2x ) ) ) ).
thf(thm_2Erelation_2EWFP__DEF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1a: A_27a] :
( ( c_2Erelation_2EWFP @ A_27a @ V0R @ V1a )
<=> ! [V2P: A_27a > $o] :
( ! [V3x: A_27a] :
( ! [V4y: A_27a] :
( ( V0R @ V4y @ V3x )
=> ( V2P @ V4y ) )
=> ( V2P @ V3x ) )
=> ( V2P @ V1a ) ) ) ).
thf(thm_2Erelation_2EINDUCTIVE__INVARIANT__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27b > $o,V2M: ( A_27a > A_27b ) > A_27a > A_27b] :
( ( c_2Erelation_2EINDUCTIVE__INVARIANT @ A_27a @ A_27b @ V0R @ V1P @ V2M )
<=> ! [V3f: A_27a > A_27b,V4x: A_27a] :
( ! [V5y: A_27a] :
( ( V0R @ V5y @ V4x )
=> ( V1P @ V5y @ ( V3f @ V5y ) ) )
=> ( V1P @ V4x @ ( V2M @ V3f @ V4x ) ) ) ) ).
thf(thm_2Erelation_2EINDUCTIVE__INVARIANT__ON__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1D: A_27a > $o,V2P: A_27a > A_27b > $o,V3M: ( A_27a > A_27b ) > A_27a > A_27b] :
( ( c_2Erelation_2EINDUCTIVE__INVARIANT__ON @ A_27a @ A_27b @ V0R @ V1D @ V2P @ V3M )
<=> ! [V4f: A_27a > A_27b,V5x: A_27a] :
( ( ( V1D @ V5x )
& ! [V6y: A_27a] :
( ( V1D @ V6y )
=> ( ( V0R @ V6y @ V5x )
=> ( V2P @ V6y @ ( V4f @ V6y ) ) ) ) )
=> ( V2P @ V5x @ ( V3M @ V4f @ V5x ) ) ) ) ).
thf(thm_2Erelation_2Einv__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1x: A_27b,V2y: A_27a] :
( ( c_2Erelation_2Einv @ A_27a @ A_27b @ V0R @ V1x @ V2y )
= ( V0R @ V2y @ V1x ) ) ).
thf(thm_2Erelation_2EINVOL__DEF,axiom,
! [A_27z: $tType,V0f: A_27z > A_27z] :
( ( c_2Erelation_2EINVOL @ A_27z @ V0f )
<=> ( ( c_2Ecombin_2Eo @ A_27z @ A_27z @ A_27z @ V0f @ V0f )
= ( c_2Ecombin_2EI @ A_27z ) ) ) ).
thf(thm_2Erelation_2EIDEM__DEF,axiom,
! [A_27z: $tType,V0f: A_27z > A_27z] :
( ( c_2Erelation_2EIDEM @ A_27z @ V0f )
<=> ( ( c_2Ecombin_2Eo @ A_27z @ A_27z @ A_27z @ V0f @ V0f )
= V0f ) ) ).
thf(thm_2Erelation_2EO__DEF,axiom,
! [A_27g: $tType,A_27h: $tType,A_27k: $tType,V0R1: A_27h > A_27k > $o,V1R2: A_27g > A_27h > $o,V2x: A_27g,V3z: A_27k] :
( ( c_2Erelation_2EO @ A_27g @ A_27h @ A_27k @ V0R1 @ V1R2 @ V2x @ V3z )
<=> ? [V4y: A_27h] :
( ( V1R2 @ V2x @ V4y )
& ( V0R1 @ V4y @ V3z ) ) ) ).
thf(thm_2Erelation_2ERSUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27b > $o,V1R2: A_27a > A_27b > $o] :
( ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b @ V0R1 @ V1R2 )
<=> ! [V2x: A_27a,V3y: A_27b] :
( ( V0R1 @ V2x @ V3y )
=> ( V1R2 @ V2x @ V3y ) ) ) ).
thf(thm_2Erelation_2ERUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27b > $o,V1R2: A_27a > A_27b > $o,V2x: A_27a,V3y: A_27b] :
( ( c_2Erelation_2ERUNION @ A_27a @ A_27b @ V0R1 @ V1R2 @ V2x @ V3y )
<=> ( ( V0R1 @ V2x @ V3y )
| ( V1R2 @ V2x @ V3y ) ) ) ).
thf(thm_2Erelation_2ERINTER,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27b > $o,V1R2: A_27a > A_27b > $o,V2x: A_27a,V3y: A_27b] :
( ( c_2Erelation_2ERINTER @ A_27a @ A_27b @ V0R1 @ V1R2 @ V2x @ V3y )
<=> ( ( V0R1 @ V2x @ V3y )
& ( V1R2 @ V2x @ V3y ) ) ) ).
thf(thm_2Erelation_2ERCOMPL,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1x: A_27a,V2y: A_27b] :
( ( c_2Erelation_2ERCOMPL @ A_27a @ A_27b @ V0R @ V1x @ V2y )
<=> ( (~) @ ( V0R @ V1x @ V2y ) ) ) ).
thf(thm_2Erelation_2EPreOrder,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EPreOrder @ A_27a @ V0R )
<=> ( ( c_2Erelation_2Ereflexive @ A_27a @ V0R )
& ( c_2Erelation_2Etransitive @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2EOrder,axiom,
! [A_27g: $tType,V0Z: A_27g > A_27g > $o] :
( ( c_2Erelation_2EOrder @ A_27g @ V0Z )
<=> ( ( c_2Erelation_2Eantisymmetric @ A_27g @ V0Z )
& ( c_2Erelation_2Etransitive @ A_27g @ V0Z ) ) ) ).
thf(thm_2Erelation_2EWeakOrder,axiom,
! [A_27g: $tType,V0Z: A_27g > A_27g > $o] :
( ( c_2Erelation_2EWeakOrder @ A_27g @ V0Z )
<=> ( ( c_2Erelation_2Ereflexive @ A_27g @ V0Z )
& ( c_2Erelation_2Eantisymmetric @ A_27g @ V0Z )
& ( c_2Erelation_2Etransitive @ A_27g @ V0Z ) ) ) ).
thf(thm_2Erelation_2EStrongOrder,axiom,
! [A_27g: $tType,V0Z: A_27g > A_27g > $o] :
( ( c_2Erelation_2EStrongOrder @ A_27g @ V0Z )
<=> ( ( c_2Erelation_2Eirreflexive @ A_27g @ V0Z )
& ( c_2Erelation_2Etransitive @ A_27g @ V0Z ) ) ) ).
thf(thm_2Erelation_2ESTRORD,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1a: A_27a,V2b: A_27a] :
( ( c_2Erelation_2ESTRORD @ A_27a @ V0R @ V1a @ V2b )
<=> ( ( V0R @ V1a @ V2b )
& ( (~) @ ( V1a = V2b ) ) ) ) ).
thf(thm_2Erelation_2ELinearOrder,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ELinearOrder @ A_27a @ V0R )
<=> ( ( c_2Erelation_2EOrder @ A_27a @ V0R )
& ( c_2Erelation_2Etrichotomous @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2EStrongLinearOrder,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0R )
<=> ( ( c_2Erelation_2EStrongOrder @ A_27a @ V0R )
& ( c_2Erelation_2Etrichotomous @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2EWeakLinearOrder,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWeakLinearOrder @ A_27a @ V0R )
<=> ( ( c_2Erelation_2EWeakOrder @ A_27a @ V0R )
& ( c_2Erelation_2Etrichotomous @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Ediag__def,axiom,
! [A_27a: $tType,V0A: A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2Ediag @ A_27a @ V0A @ V1x @ V2y )
<=> ( ( V1x = V2y )
& ( c_2Ebool_2EIN @ A_27a @ V1x @ V0A ) ) ) ).
thf(thm_2Erelation_2ERDOM__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1x: A_27a] :
( ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ V0R @ V1x )
<=> ? [V2y: A_27b] : ( V0R @ V1x @ V2y ) ) ).
thf(thm_2Erelation_2ERRANGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1y: A_27b] :
( ( c_2Erelation_2ERRANGE @ A_27a @ A_27b @ V0R @ V1y )
<=> ? [V2x: A_27a] : ( V0R @ V2x @ V1y ) ) ).
thf(thm_2Erelation_2ERUNIV,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b] :
( ( c_2Erelation_2ERUNIV @ A_27a @ A_27b @ V0x @ V1y )
= c_2Ebool_2ET ) ).
thf(thm_2Erelation_2ERRESTRICT__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1s: A_27a > $o,V2x: A_27a,V3y: A_27b] :
( ( c_2Erelation_2ERRESTRICT @ A_27a @ A_27b @ V0R @ V1s @ V2x @ V3y )
<=> ( ( V0R @ V2x @ V3y )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) ) ) ).
thf(thm_2Erelation_2ERDOM__DELETE__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1x: A_27a,V2u: A_27a,V3v: A_27b] :
( ( c_2Erelation_2ERDOM__DELETE @ A_27a @ A_27b @ V0R @ V1x @ V2u @ V3v )
<=> ( ( V0R @ V2u @ V3v )
& ( (~) @ ( V2u = V1x ) ) ) ) ).
thf(thm_2Erelation_2Ediamond__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ediamond @ A_27a @ V0R )
<=> ! [V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( V0R @ V1x @ V2y )
& ( V0R @ V1x @ V3z ) )
=> ? [V4u: A_27a] :
( ( V0R @ V2y @ V4u )
& ( V0R @ V3z @ V4u ) ) ) ) ).
thf(thm_2Erelation_2Ercdiamond__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ercdiamond @ A_27a @ V0R )
<=> ! [V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( V0R @ V1x @ V2y )
& ( V0R @ V1x @ V3z ) )
=> ? [V4u: A_27a] :
( ( c_2Erelation_2ERC @ A_27a @ V0R @ V2y @ V4u )
& ( c_2Erelation_2ERC @ A_27a @ V0R @ V3z @ V4u ) ) ) ) ).
thf(thm_2Erelation_2ECR__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ECR @ A_27a @ V0R )
= ( c_2Erelation_2Ediamond @ A_27a @ ( c_2Erelation_2ERTC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2EWCR__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWCR @ A_27a @ V0R )
<=> ! [V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( V0R @ V1x @ V2y )
& ( V0R @ V1x @ V3z ) )
=> ? [V4u: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V2y @ V4u )
& ( c_2Erelation_2ERTC @ A_27a @ V0R @ V3z @ V4u ) ) ) ) ).
thf(thm_2Erelation_2ESN__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ESN @ A_27a @ V0R )
= ( c_2Erelation_2EWF @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Enf__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1x: A_27a] :
( ( c_2Erelation_2Enf @ A_27a @ A_27b @ V0R @ V1x )
<=> ! [V2y: A_27b] : ( (~) @ ( V0R @ V1x @ V2y ) ) ) ).
thf(thm_2Erelation_2ESC__SYMMETRIC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Esymmetric @ A_27a @ ( c_2Erelation_2ESC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2ETC__TRANSITIVE,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Etransitive @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2ERTC__INDUCT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a] : ( V1P @ V2x @ V2x )
& ! [V3x: A_27a,V4y: A_27a,V5z: A_27a] :
( ( ( V0R @ V3x @ V4y )
& ( V1P @ V4y @ V5z ) )
=> ( V1P @ V3x @ V5z ) ) )
=> ! [V6x: A_27a,V7y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V6x @ V7y )
=> ( V1P @ V6x @ V7y ) ) ) ).
thf(thm_2Erelation_2ETC__RULES,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
=> ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V2y ) )
& ! [V3x: A_27a,V4y: A_27a,V5z: A_27a] :
( ( ( c_2Erelation_2ETC @ A_27a @ V0R @ V3x @ V4y )
& ( c_2Erelation_2ETC @ A_27a @ V0R @ V4y @ V5z ) )
=> ( c_2Erelation_2ETC @ A_27a @ V0R @ V3x @ V5z ) ) ) ).
thf(thm_2Erelation_2ERTC__RULES,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ! [V1x: A_27a] : ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V1x )
& ! [V2x: A_27a,V3y: A_27a,V4z: A_27a] :
( ( ( V0R @ V2x @ V3y )
& ( c_2Erelation_2ERTC @ A_27a @ V0R @ V3y @ V4z ) )
=> ( c_2Erelation_2ERTC @ A_27a @ V0R @ V2x @ V4z ) ) ) ).
thf(thm_2Erelation_2ERTC__REFL,axiom,
! [A_27a: $tType,V0x: A_27a,V1R: A_27a > A_27a > $o] : ( c_2Erelation_2ERTC @ A_27a @ V1R @ V0x @ V0x ) ).
thf(thm_2Erelation_2ERTC__SINGLE,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
=> ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y ) ) ).
thf(thm_2Erelation_2ERTC__STRONG__INDUCT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a] : ( V1P @ V2x @ V2x )
& ! [V3x: A_27a,V4y: A_27a,V5z: A_27a] :
( ( ( V0R @ V3x @ V4y )
& ( c_2Erelation_2ERTC @ A_27a @ V0R @ V4y @ V5z )
& ( V1P @ V4y @ V5z ) )
=> ( V1P @ V3x @ V5z ) ) )
=> ! [V6x: A_27a,V7y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V6x @ V7y )
=> ( V1P @ V6x @ V7y ) ) ) ).
thf(thm_2Erelation_2ERTC__RTC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y )
=> ! [V3z: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V2y @ V3z )
=> ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V3z ) ) ) ).
thf(thm_2Erelation_2ERTC__TRANSITIVE,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Etransitive @ A_27a @ ( c_2Erelation_2ERTC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Etransitive__RTC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Etransitive @ A_27a @ ( c_2Erelation_2ERTC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2ERTC__REFLEXIVE,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Ereflexive @ A_27a @ ( c_2Erelation_2ERTC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Ereflexive__RTC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Ereflexive @ A_27a @ ( c_2Erelation_2ERTC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2ERC__REFLEXIVE,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Ereflexive @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Ereflexive__RC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Ereflexive @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2ERC__lifts__monotonicities,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a,V1R: A_27a > A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( V1R @ V2x @ V3y )
=> ( V1R @ ( V0f @ V2x ) @ ( V0f @ V3y ) ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( c_2Erelation_2ERC @ A_27a @ V1R @ V4x @ V5y )
=> ( c_2Erelation_2ERC @ A_27a @ V1R @ ( V0f @ V4x ) @ ( V0f @ V5y ) ) ) ) ).
thf(thm_2Erelation_2ERC__MONOTONE,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2R: A_27a > A_27a > $o,V3Q: A_27a > A_27a > $o] :
( ! [V4x: A_27a,V5y: A_27a] :
( ( V2R @ V4x @ V5y )
=> ( V3Q @ V4x @ V5y ) )
=> ( ( c_2Erelation_2ERC @ A_27a @ V2R @ V1x @ V0y )
=> ( c_2Erelation_2ERC @ A_27a @ V3Q @ V1x @ V0y ) ) ) ).
thf(thm_2Erelation_2ERC__lifts__invariants,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( ( V1P @ V2x )
& ( V0R @ V2x @ V3y ) )
=> ( V1P @ V3y ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( ( V1P @ V4x )
& ( c_2Erelation_2ERC @ A_27a @ V0R @ V4x @ V5y ) )
=> ( V1P @ V5y ) ) ) ).
thf(thm_2Erelation_2ERC__lifts__equalities,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1R: A_27a > A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( V1R @ V2x @ V3y )
=> ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( c_2Erelation_2ERC @ A_27a @ V1R @ V4x @ V5y )
=> ( ( V0f @ V4x )
= ( V0f @ V5y ) ) ) ) ).
thf(thm_2Erelation_2ESC__lifts__monotonicities,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a,V1R: A_27a > A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( V1R @ V2x @ V3y )
=> ( V1R @ ( V0f @ V2x ) @ ( V0f @ V3y ) ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( c_2Erelation_2ESC @ A_27a @ V1R @ V4x @ V5y )
=> ( c_2Erelation_2ESC @ A_27a @ V1R @ ( V0f @ V4x ) @ ( V0f @ V5y ) ) ) ) ).
thf(thm_2Erelation_2ESC__lifts__equalities,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1R: A_27a > A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( V1R @ V2x @ V3y )
=> ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( c_2Erelation_2ESC @ A_27a @ V1R @ V4x @ V5y )
=> ( ( V0f @ V4x )
= ( V0f @ V5y ) ) ) ) ).
thf(thm_2Erelation_2ESC__MONOTONE,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2R: A_27a > A_27a > $o,V3Q: A_27a > A_27a > $o] :
( ! [V4x: A_27a,V5y: A_27a] :
( ( V2R @ V4x @ V5y )
=> ( V3Q @ V4x @ V5y ) )
=> ( ( c_2Erelation_2ESC @ A_27a @ V2R @ V1x @ V0y )
=> ( c_2Erelation_2ESC @ A_27a @ V3Q @ V1x @ V0y ) ) ) ).
thf(thm_2Erelation_2Esymmetric__RC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Esymmetric @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= ( c_2Erelation_2Esymmetric @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Eantisymmetric__RC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Eantisymmetric @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= ( c_2Erelation_2Eantisymmetric @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Etransitive__RC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Etransitive @ A_27a @ V0R )
=> ( c_2Erelation_2Etransitive @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2ETC__SUBSET,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
=> ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V2y ) ) ).
thf(thm_2Erelation_2ERTC__SUBSET,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
=> ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y ) ) ).
thf(thm_2Erelation_2ERC__SUBSET,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
=> ( c_2Erelation_2ERC @ A_27a @ V0R @ V1x @ V2y ) ) ).
thf(thm_2Erelation_2ERC__RTC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERC @ A_27a @ V0R @ V1x @ V2y )
=> ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y ) ) ).
thf(thm_2Erelation_2ETC__INDUCT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( V0R @ V2x @ V3y )
=> ( V1P @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a,V6z: A_27a] :
( ( ( V1P @ V4x @ V5y )
& ( V1P @ V5y @ V6z ) )
=> ( V1P @ V4x @ V6z ) ) )
=> ! [V7u: A_27a,V8v: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V7u @ V8v )
=> ( V1P @ V7u @ V8v ) ) ) ).
thf(thm_2Erelation_2ETC__INDUCT__LEFT1,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( V0R @ V2x @ V3y )
=> ( V1P @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a,V6z: A_27a] :
( ( ( V0R @ V4x @ V5y )
& ( V1P @ V5y @ V6z ) )
=> ( V1P @ V4x @ V6z ) ) )
=> ! [V7x: A_27a,V8y: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V7x @ V8y )
=> ( V1P @ V7x @ V8y ) ) ) ).
thf(thm_2Erelation_2ETC__INDUCT__RIGHT1,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( V0R @ V2x @ V3y )
=> ( V1P @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a,V6z: A_27a] :
( ( ( V1P @ V4x @ V5y )
& ( V0R @ V5y @ V6z ) )
=> ( V1P @ V4x @ V6z ) ) )
=> ! [V7x: A_27a,V8y: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V7x @ V8y )
=> ( V1P @ V7x @ V8y ) ) ) ).
thf(thm_2Erelation_2ETC__STRONG__INDUCT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( V0R @ V2x @ V3y )
=> ( V1P @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a,V6z: A_27a] :
( ( ( V1P @ V4x @ V5y )
& ( V1P @ V5y @ V6z )
& ( c_2Erelation_2ETC @ A_27a @ V0R @ V4x @ V5y )
& ( c_2Erelation_2ETC @ A_27a @ V0R @ V5y @ V6z ) )
=> ( V1P @ V4x @ V6z ) ) )
=> ! [V7u: A_27a,V8v: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V7u @ V8v )
=> ( V1P @ V7u @ V8v ) ) ) ).
thf(thm_2Erelation_2ETC__STRONG__INDUCT__LEFT1,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( V0R @ V2x @ V3y )
=> ( V1P @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a,V6z: A_27a] :
( ( ( V0R @ V4x @ V5y )
& ( V1P @ V5y @ V6z )
& ( c_2Erelation_2ETC @ A_27a @ V0R @ V5y @ V6z ) )
=> ( V1P @ V4x @ V6z ) ) )
=> ! [V7u: A_27a,V8v: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V7u @ V8v )
=> ( V1P @ V7u @ V8v ) ) ) ).
thf(thm_2Erelation_2ETC__STRONG__INDUCT__RIGHT1,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( V0R @ V2x @ V3y )
=> ( V1P @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a,V6z: A_27a] :
( ( ( V1P @ V4x @ V5y )
& ( c_2Erelation_2ETC @ A_27a @ V0R @ V4x @ V5y )
& ( V0R @ V5y @ V6z ) )
=> ( V1P @ V4x @ V6z ) ) )
=> ! [V7u: A_27a,V8v: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V7u @ V8v )
=> ( V1P @ V7u @ V8v ) ) ) ).
thf(thm_2Erelation_2ETC__INDUCT__ALT__LEFT,axiom,
! [A_27a: $tType,V0b: A_27a,V1R: A_27a > A_27a > $o,V2Q: A_27a > $o] :
( ( ! [V3x: A_27a] :
( ( V1R @ V3x @ V0b )
=> ( V2Q @ V3x ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( V1R @ V4x @ V5y )
& ( V2Q @ V5y ) )
=> ( V2Q @ V4x ) ) )
=> ! [V6a: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V1R @ V6a @ V0b )
=> ( V2Q @ V6a ) ) ) ).
thf(thm_2Erelation_2ETC__INDUCT__ALT__RIGHT,axiom,
! [A_27a: $tType,V0a: A_27a,V1R: A_27a > A_27a > $o,V2Q: A_27a > $o] :
( ( ! [V3y: A_27a] :
( ( V1R @ V0a @ V3y )
=> ( V2Q @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( V2Q @ V4x )
& ( V1R @ V4x @ V5y ) )
=> ( V2Q @ V5y ) ) )
=> ! [V6b: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V1R @ V0a @ V6b )
=> ( V2Q @ V6b ) ) ) ).
thf(thm_2Erelation_2ETC__lifts__monotonicities,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a,V1R: A_27a > A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( V1R @ V2x @ V3y )
=> ( V1R @ ( V0f @ V2x ) @ ( V0f @ V3y ) ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V1R @ V4x @ V5y )
=> ( c_2Erelation_2ETC @ A_27a @ V1R @ ( V0f @ V4x ) @ ( V0f @ V5y ) ) ) ) ).
thf(thm_2Erelation_2ETC__lifts__invariants,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( ( V1P @ V2x )
& ( V0R @ V2x @ V3y ) )
=> ( V1P @ V3y ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( ( V1P @ V4x )
& ( c_2Erelation_2ETC @ A_27a @ V0R @ V4x @ V5y ) )
=> ( V1P @ V5y ) ) ) ).
thf(thm_2Erelation_2ETC__lifts__equalities,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1R: A_27a > A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( V1R @ V2x @ V3y )
=> ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V1R @ V4x @ V5y )
=> ( ( V0f @ V4x )
= ( V0f @ V5y ) ) ) ) ).
thf(thm_2Erelation_2ETC__lifts__transitive__relations,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1R: A_27a > A_27a > $o,V2Q: A_27b > A_27b > $o] :
( ( ! [V3x: A_27a,V4y: A_27a] :
( ( V1R @ V3x @ V4y )
=> ( V2Q @ ( V0f @ V3x ) @ ( V0f @ V4y ) ) )
& ( c_2Erelation_2Etransitive @ A_27b @ V2Q ) )
=> ! [V5x: A_27a,V6y: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V1R @ V5x @ V6y )
=> ( V2Q @ ( V0f @ V5x ) @ ( V0f @ V6y ) ) ) ) ).
thf(thm_2Erelation_2ETC__implies__one__step,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V2y )
& ( (~) @ ( V1x = V2y ) ) )
=> ? [V3z: A_27a] :
( ( V0R @ V1x @ V3z )
& ( (~) @ ( V1x = V3z ) ) ) ) ).
thf(thm_2Erelation_2ETC__RTC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V2y )
=> ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y ) ) ).
thf(thm_2Erelation_2ERTC__TC__RC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y )
=> ( ( c_2Erelation_2ERC @ A_27a @ V0R @ V1x @ V2y )
| ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V2y ) ) ) ).
thf(thm_2Erelation_2ETC__RC__EQNS,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2ERC @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) )
= ( c_2Erelation_2ERTC @ A_27a @ V0R ) )
& ( ( c_2Erelation_2ETC @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= ( c_2Erelation_2ERTC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2ERTC__ALT__DEF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1a: A_27a,V2b: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1a @ V2b )
<=> ! [V3Q: A_27a > $o] :
( ( ( V3Q @ V2b )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( V0R @ V4x @ V5y )
& ( V3Q @ V5y ) )
=> ( V3Q @ V4x ) ) )
=> ( V3Q @ V1a ) ) ) ).
thf(thm_2Erelation_2ERTC__ALT__INDUCT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1Q: A_27a > $o,V2b: A_27a] :
( ( ( V1Q @ V2b )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0R @ V3x @ V4y )
& ( V1Q @ V4y ) )
=> ( V1Q @ V3x ) ) )
=> ! [V5x: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V5x @ V2b )
=> ( V1Q @ V5x ) ) ) ).
thf(thm_2Erelation_2ERTC__ALT__RIGHT__DEF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1a: A_27a,V2b: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1a @ V2b )
<=> ! [V3Q: A_27a > $o] :
( ( ( V3Q @ V1a )
& ! [V4y: A_27a,V5z: A_27a] :
( ( ( V3Q @ V4y )
& ( V0R @ V4y @ V5z ) )
=> ( V3Q @ V5z ) ) )
=> ( V3Q @ V2b ) ) ) ).
thf(thm_2Erelation_2ERTC__ALT__RIGHT__INDUCT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1Q: A_27a > $o,V2a: A_27a] :
( ( ( V1Q @ V2a )
& ! [V3y: A_27a,V4z: A_27a] :
( ( ( V1Q @ V3y )
& ( V0R @ V3y @ V4z ) )
=> ( V1Q @ V4z ) ) )
=> ! [V5z: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V2a @ V5z )
=> ( V1Q @ V5z ) ) ) ).
thf(thm_2Erelation_2ERTC__INDUCT__RIGHT1,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a] : ( V1P @ V2x @ V2x )
& ! [V3x: A_27a,V4y: A_27a,V5z: A_27a] :
( ( ( V1P @ V3x @ V4y )
& ( V0R @ V4y @ V5z ) )
=> ( V1P @ V3x @ V5z ) ) )
=> ! [V6x: A_27a,V7y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V6x @ V7y )
=> ( V1P @ V6x @ V7y ) ) ) ).
thf(thm_2Erelation_2ERTC__RULES__RIGHT1,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ! [V1x: A_27a] : ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V1x )
& ! [V2x: A_27a,V3y: A_27a,V4z: A_27a] :
( ( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V2x @ V3y )
& ( V0R @ V3y @ V4z ) )
=> ( c_2Erelation_2ERTC @ A_27a @ V0R @ V2x @ V4z ) ) ) ).
thf(thm_2Erelation_2ERTC__STRONG__INDUCT__RIGHT1,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a] : ( V1P @ V2x @ V2x )
& ! [V3x: A_27a,V4y: A_27a,V5z: A_27a] :
( ( ( V1P @ V3x @ V4y )
& ( c_2Erelation_2ERTC @ A_27a @ V0R @ V3x @ V4y )
& ( V0R @ V4y @ V5z ) )
=> ( V1P @ V3x @ V5z ) ) )
=> ! [V6x: A_27a,V7y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V6x @ V7y )
=> ( V1P @ V6x @ V7y ) ) ) ).
thf(thm_2Erelation_2EEXTEND__RTC__TC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( V0R @ V1x @ V2y )
& ( c_2Erelation_2ERTC @ A_27a @ V0R @ V2y @ V3z ) )
=> ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V3z ) ) ).
thf(thm_2Erelation_2EEXTEND__RTC__TC__EQN,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2z: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V2z )
<=> ? [V3y: A_27a] :
( ( V0R @ V1x @ V3y )
& ( c_2Erelation_2ERTC @ A_27a @ V0R @ V3y @ V2z ) ) ) ).
thf(thm_2Erelation_2Ereflexive__RC__identity,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ereflexive @ A_27a @ V0R )
=> ( ( c_2Erelation_2ERC @ A_27a @ V0R )
= V0R ) ) ).
thf(thm_2Erelation_2Esymmetric__SC__identity,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Esymmetric @ A_27a @ V0R )
=> ( ( c_2Erelation_2ESC @ A_27a @ V0R )
= V0R ) ) ).
thf(thm_2Erelation_2Etransitive__TC__identity,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Etransitive @ A_27a @ V0R )
=> ( ( c_2Erelation_2ETC @ A_27a @ V0R )
= V0R ) ) ).
thf(thm_2Erelation_2ERC__IDEM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ERC @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= ( c_2Erelation_2ERC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2ESC__IDEM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ESC @ A_27a @ ( c_2Erelation_2ESC @ A_27a @ V0R ) )
= ( c_2Erelation_2ESC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2ETC__IDEM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ETC @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) )
= ( c_2Erelation_2ETC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2ERC__MOVES__OUT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2ESC @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= ( c_2Erelation_2ERC @ A_27a @ ( c_2Erelation_2ESC @ A_27a @ V0R ) ) )
& ( ( c_2Erelation_2ERC @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= ( c_2Erelation_2ERC @ A_27a @ V0R ) )
& ( ( c_2Erelation_2ETC @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= ( c_2Erelation_2ERC @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) ) ) ) ).
thf(thm_2Erelation_2Esymmetric__TC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Esymmetric @ A_27a @ V0R )
=> ( c_2Erelation_2Esymmetric @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Ereflexive__TC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ereflexive @ A_27a @ V0R )
=> ( c_2Erelation_2Ereflexive @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2EEQC__EQUIVALENCE,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Eequivalence @ A_27a @ ( c_2Erelation_2EEQC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2EEQC__IDEM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EEQC @ A_27a @ ( c_2Erelation_2EEQC @ A_27a @ V0R ) )
= ( c_2Erelation_2EEQC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2ERTC__IDEM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ERTC @ A_27a @ ( c_2Erelation_2ERTC @ A_27a @ V0R ) )
= ( c_2Erelation_2ERTC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2ERTC__CASES1,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y )
<=> ( ( V1x = V2y )
| ? [V3u: A_27a] :
( ( V0R @ V1x @ V3u )
& ( c_2Erelation_2ERTC @ A_27a @ V0R @ V3u @ V2y ) ) ) ) ).
thf(thm_2Erelation_2ERTC__CASES__TC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y )
<=> ( ( V1x = V2y )
| ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V2y ) ) ) ).
thf(thm_2Erelation_2ERTC__CASES2,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y )
<=> ( ( V1x = V2y )
| ? [V3u: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V3u )
& ( V0R @ V3u @ V2y ) ) ) ) ).
thf(thm_2Erelation_2ERTC__CASES__RTC__TWICE,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y )
<=> ? [V3u: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V3u )
& ( c_2Erelation_2ERTC @ A_27a @ V0R @ V3u @ V2y ) ) ) ).
thf(thm_2Erelation_2ETC__CASES1__E,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2z: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V2z )
=> ( ( V0R @ V1x @ V2z )
| ? [V3y: A_27a] :
( ( V0R @ V1x @ V3y )
& ( c_2Erelation_2ETC @ A_27a @ V0R @ V3y @ V2z ) ) ) ) ).
thf(thm_2Erelation_2ETC__CASES1,axiom,
! [A_27a: $tType,V0z: A_27a,V1x: A_27a,V2R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ETC @ A_27a @ V2R @ V1x @ V0z )
<=> ( ( V2R @ V1x @ V0z )
| ? [V3y: A_27a] :
( ( V2R @ V1x @ V3y )
& ( c_2Erelation_2ETC @ A_27a @ V2R @ V3y @ V0z ) ) ) ) ).
thf(thm_2Erelation_2ETC__CASES2__E,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2z: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V2z )
=> ( ( V0R @ V1x @ V2z )
| ? [V3y: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V0R @ V1x @ V3y )
& ( V0R @ V3y @ V2z ) ) ) ) ).
thf(thm_2Erelation_2ETC__CASES2,axiom,
! [A_27a: $tType,V0z: A_27a,V1x: A_27a,V2R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ETC @ A_27a @ V2R @ V1x @ V0z )
<=> ( ( V2R @ V1x @ V0z )
| ? [V3y: A_27a] :
( ( c_2Erelation_2ETC @ A_27a @ V2R @ V1x @ V3y )
& ( V2R @ V3y @ V0z ) ) ) ) ).
thf(thm_2Erelation_2ETC__MONOTONE,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2R: A_27a > A_27a > $o,V3Q: A_27a > A_27a > $o] :
( ! [V4x: A_27a,V5y: A_27a] :
( ( V2R @ V4x @ V5y )
=> ( V3Q @ V4x @ V5y ) )
=> ( ( c_2Erelation_2ETC @ A_27a @ V2R @ V1x @ V0y )
=> ( c_2Erelation_2ETC @ A_27a @ V3Q @ V1x @ V0y ) ) ) ).
thf(thm_2Erelation_2ERTC__MONOTONE,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2R: A_27a > A_27a > $o,V3Q: A_27a > A_27a > $o] :
( ! [V4x: A_27a,V5y: A_27a] :
( ( V2R @ V4x @ V5y )
=> ( V3Q @ V4x @ V5y ) )
=> ( ( c_2Erelation_2ERTC @ A_27a @ V2R @ V1x @ V0y )
=> ( c_2Erelation_2ERTC @ A_27a @ V3Q @ V1x @ V0y ) ) ) ).
thf(thm_2Erelation_2EEQC__INDUCTION,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( V0R @ V2x @ V3y )
=> ( V1P @ V2x @ V3y ) )
& ! [V4x: A_27a] : ( V1P @ V4x @ V4x )
& ! [V5x: A_27a,V6y: A_27a] :
( ( V1P @ V5x @ V6y )
=> ( V1P @ V6y @ V5x ) )
& ! [V7x: A_27a,V8y: A_27a,V9z: A_27a] :
( ( ( V1P @ V7x @ V8y )
& ( V1P @ V8y @ V9z ) )
=> ( V1P @ V7x @ V9z ) ) )
=> ! [V10x: A_27a,V11y: A_27a] :
( ( c_2Erelation_2EEQC @ A_27a @ V0R @ V10x @ V11y )
=> ( V1P @ V10x @ V11y ) ) ) ).
thf(thm_2Erelation_2EEQC__REFL,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a] : ( c_2Erelation_2EEQC @ A_27a @ V0R @ V1x @ V1x ) ).
thf(thm_2Erelation_2EEQC__R,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
=> ( c_2Erelation_2EEQC @ A_27a @ V0R @ V1x @ V2y ) ) ).
thf(thm_2Erelation_2EEQC__SYM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2EEQC @ A_27a @ V0R @ V1x @ V2y )
=> ( c_2Erelation_2EEQC @ A_27a @ V0R @ V2y @ V1x ) ) ).
thf(thm_2Erelation_2EEQC__TRANS,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( c_2Erelation_2EEQC @ A_27a @ V0R @ V1x @ V2y )
& ( c_2Erelation_2EEQC @ A_27a @ V0R @ V2y @ V3z ) )
=> ( c_2Erelation_2EEQC @ A_27a @ V0R @ V1x @ V3z ) ) ).
thf(thm_2Erelation_2Etransitive__EQC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Etransitive @ A_27a @ ( c_2Erelation_2EEQC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Esymmetric__EQC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Esymmetric @ A_27a @ ( c_2Erelation_2EEQC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Ereflexive__EQC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] : ( c_2Erelation_2Ereflexive @ A_27a @ ( c_2Erelation_2EEQC @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2EEQC__MOVES__IN,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2EEQC @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= ( c_2Erelation_2EEQC @ A_27a @ V0R ) )
& ( ( c_2Erelation_2EEQC @ A_27a @ ( c_2Erelation_2ESC @ A_27a @ V0R ) )
= ( c_2Erelation_2EEQC @ A_27a @ V0R ) )
& ( ( c_2Erelation_2EEQC @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) )
= ( c_2Erelation_2EEQC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2ESTRONG__EQC__INDUCTION,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( V0R @ V2x @ V3y )
=> ( V1P @ V2x @ V3y ) )
& ! [V4x: A_27a] : ( V1P @ V4x @ V4x )
& ! [V5x: A_27a,V6y: A_27a] :
( ( ( c_2Erelation_2EEQC @ A_27a @ V0R @ V5x @ V6y )
& ( V1P @ V5x @ V6y ) )
=> ( V1P @ V6y @ V5x ) )
& ! [V7x: A_27a,V8y: A_27a,V9z: A_27a] :
( ( ( V1P @ V7x @ V8y )
& ( V1P @ V8y @ V9z )
& ( c_2Erelation_2EEQC @ A_27a @ V0R @ V7x @ V8y )
& ( c_2Erelation_2EEQC @ A_27a @ V0R @ V8y @ V9z ) )
=> ( V1P @ V7x @ V9z ) ) )
=> ! [V10x: A_27a,V11y: A_27a] :
( ( c_2Erelation_2EEQC @ A_27a @ V0R @ V10x @ V11y )
=> ( V1P @ V10x @ V11y ) ) ) ).
thf(thm_2Erelation_2EALT__equivalence,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Eequivalence @ A_27a @ V0R )
<=> ! [V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
<=> ( ( V0R @ V1x )
= ( V0R @ V2y ) ) ) ) ).
thf(thm_2Erelation_2EEQC__MONOTONE,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2R_27: A_27a > A_27a > $o,V3R: A_27a > A_27a > $o] :
( ! [V4x: A_27a,V5y: A_27a] :
( ( V3R @ V4x @ V5y )
=> ( V2R_27 @ V4x @ V5y ) )
=> ( ( c_2Erelation_2EEQC @ A_27a @ V3R @ V1x @ V0y )
=> ( c_2Erelation_2EEQC @ A_27a @ V2R_27 @ V1x @ V0y ) ) ) ).
thf(thm_2Erelation_2ERTC__EQC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V0R @ V1x @ V2y )
=> ( c_2Erelation_2EEQC @ A_27a @ V0R @ V1x @ V2y ) ) ).
thf(thm_2Erelation_2ERTC__lifts__monotonicities,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a,V1R: A_27a > A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( V1R @ V2x @ V3y )
=> ( V1R @ ( V0f @ V2x ) @ ( V0f @ V3y ) ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V1R @ V4x @ V5y )
=> ( c_2Erelation_2ERTC @ A_27a @ V1R @ ( V0f @ V4x ) @ ( V0f @ V5y ) ) ) ) ).
thf(thm_2Erelation_2ERTC__lifts__reflexive__transitive__relations,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1R: A_27a > A_27a > $o,V2Q: A_27b > A_27b > $o] :
( ( ! [V3x: A_27a,V4y: A_27a] :
( ( V1R @ V3x @ V4y )
=> ( V2Q @ ( V0f @ V3x ) @ ( V0f @ V4y ) ) )
& ( c_2Erelation_2Ereflexive @ A_27b @ V2Q )
& ( c_2Erelation_2Etransitive @ A_27b @ V2Q ) )
=> ! [V5x: A_27a,V6y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V1R @ V5x @ V6y )
=> ( V2Q @ ( V0f @ V5x ) @ ( V0f @ V6y ) ) ) ) ).
thf(thm_2Erelation_2ERTC__lifts__equalities,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1R: A_27a > A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( V1R @ V2x @ V3y )
=> ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ V1R @ V4x @ V5y )
=> ( ( V0f @ V4x )
= ( V0f @ V5y ) ) ) ) ).
thf(thm_2Erelation_2ERTC__lifts__invariants,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( ( V1P @ V2x )
& ( V0R @ V2x @ V3y ) )
=> ( V1P @ V3y ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( ( V1P @ V4x )
& ( c_2Erelation_2ERTC @ A_27a @ V0R @ V4x @ V5y ) )
=> ( V1P @ V5y ) ) ) ).
thf(thm_2Erelation_2EWF__INDUCTION__THM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
=> ! [V1P: A_27a > $o] :
( ! [V2x: A_27a] :
( ! [V3y: A_27a] :
( ( V0R @ V3y @ V2x )
=> ( V1P @ V3y ) )
=> ( V1P @ V2x ) )
=> ! [V4x: A_27a] : ( V1P @ V4x ) ) ) ).
thf(thm_2Erelation_2EINDUCTION__WF__THM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ! [V1P: A_27a > $o] :
( ! [V2x: A_27a] :
( ! [V3y: A_27a] :
( ( V0R @ V3y @ V2x )
=> ( V1P @ V3y ) )
=> ( V1P @ V2x ) )
=> ! [V4x: A_27a] : ( V1P @ V4x ) )
=> ( c_2Erelation_2EWF @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2EWF__EQ__INDUCTION__THM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
<=> ! [V1P: A_27a > $o] :
( ! [V2x: A_27a] :
( ! [V3y: A_27a] :
( ( V0R @ V3y @ V2x )
=> ( V1P @ V3y ) )
=> ( V1P @ V2x ) )
=> ! [V4x: A_27a] : ( V1P @ V4x ) ) ) ).
thf(thm_2Erelation_2EWF__NOT__REFL,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
=> ( ( V0R @ V1x @ V2y )
=> ( (~) @ ( V1x = V2y ) ) ) ) ).
thf(thm_2Erelation_2EWF__irreflexive,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
=> ( c_2Erelation_2Eirreflexive @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2EWF__EMPTY__REL,axiom,
! [A_27a: $tType] : ( c_2Erelation_2EWF @ A_27a @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ).
thf(thm_2Erelation_2EWF__SUBSET,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2EWF @ A_27a @ V0R )
& ! [V2x: A_27a,V3y: A_27a] :
( ( V1P @ V2x @ V3y )
=> ( V0R @ V2x @ V3y ) ) )
=> ( c_2Erelation_2EWF @ A_27a @ V1P ) ) ).
thf(thm_2Erelation_2EWF__TC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
=> ( c_2Erelation_2EWF @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2EWF__TC__EQN,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) )
= ( c_2Erelation_2EWF @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2EWF__noloops,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V2R )
=> ( ( c_2Erelation_2ETC @ A_27a @ V2R @ V1x @ V0y )
=> ( (~) @ ( V1x = V0y ) ) ) ) ).
thf(thm_2Erelation_2EWF__antisymmetric,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
=> ( c_2Erelation_2Eantisymmetric @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Einv__image__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27b > A_27b > $o,V1f: A_27a > A_27b,V2x: A_27a,V3y: A_27a] :
( ( c_2Erelation_2Einv__image @ A_27a @ A_27b @ V0R @ V1f @ V2x @ V3y )
= ( V0R @ ( V1f @ V2x ) @ ( V1f @ V3y ) ) ) ).
thf(thm_2Erelation_2EWF__inv__image,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27b > A_27b > $o,V1f: A_27a > A_27b] :
( ( c_2Erelation_2EWF @ A_27b @ V0R )
=> ( c_2Erelation_2EWF @ A_27a @ ( c_2Erelation_2Einv__image @ A_27a @ A_27b @ V0R @ V1f ) ) ) ).
thf(thm_2Erelation_2Etotal__inv__image,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1f: A_27b > A_27a] :
( ( c_2Erelation_2Etotal @ A_27a @ V0R )
=> ( c_2Erelation_2Etotal @ A_27b @ ( c_2Erelation_2Einv__image @ A_27b @ A_27a @ V0R @ V1f ) ) ) ).
thf(thm_2Erelation_2Ereflexive__inv__image,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1f: A_27b > A_27a] :
( ( c_2Erelation_2Ereflexive @ A_27a @ V0R )
=> ( c_2Erelation_2Ereflexive @ A_27b @ ( c_2Erelation_2Einv__image @ A_27b @ A_27a @ V0R @ V1f ) ) ) ).
thf(thm_2Erelation_2Esymmetric__inv__image,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1f: A_27b > A_27a] :
( ( c_2Erelation_2Esymmetric @ A_27a @ V0R )
=> ( c_2Erelation_2Esymmetric @ A_27b @ ( c_2Erelation_2Einv__image @ A_27b @ A_27a @ V0R @ V1f ) ) ) ).
thf(thm_2Erelation_2Etransitive__inv__image,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1f: A_27b > A_27a] :
( ( c_2Erelation_2Etransitive @ A_27a @ V0R )
=> ( c_2Erelation_2Etransitive @ A_27b @ ( c_2Erelation_2Einv__image @ A_27b @ A_27a @ V0R @ V1f ) ) ) ).
thf(thm_2Erelation_2ERESTRICT__LEMMA,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1R: A_27a > A_27a > $o,V2y: A_27a,V3z: A_27a] :
( ( V1R @ V2y @ V3z )
=> ( ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b @ V0f @ V1R @ V3z @ V2y )
= ( V0f @ V2y ) ) ) ).
thf(thm_2Erelation_2EWFREC__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1M: ( A_27a > A_27b ) > A_27a > A_27b] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
=> ! [V2x: A_27a] :
( ( c_2Erelation_2EWFREC @ A_27a @ A_27b @ V0R @ V1M @ V2x )
= ( V1M @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b @ V0R @ V1M ) @ V0R @ V2x ) @ V2x ) ) ) ).
thf(thm_2Erelation_2EWFREC__COROLLARY,axiom,
! [A_27a: $tType,A_27b: $tType,V0M: ( A_27a > A_27b ) > A_27a > A_27b,V1R: A_27a > A_27a > $o,V2f: A_27a > A_27b] :
( ( V2f
= ( c_2Erelation_2EWFREC @ A_27a @ A_27b @ V1R @ V0M ) )
=> ( ( c_2Erelation_2EWF @ A_27a @ V1R )
=> ! [V3x: A_27a] :
( ( V2f @ V3x )
= ( V0M @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b @ V2f @ V1R @ V3x ) @ V3x ) ) ) ) ).
thf(thm_2Erelation_2EWF__RECURSION__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
=> ! [V1M: ( A_27a > A_27b ) > A_27a > A_27b] :
( c_2Ebool_2E_3F_21 @ ( A_27a > A_27b )
@ ^ [V2f: A_27a > A_27b] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V3x: A_27a] : ( c_2Emin_2E_3D @ A_27b @ ( V2f @ V3x ) @ ( V1M @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b @ V2f @ V0R @ V3x ) @ V3x ) ) ) ) ) ).
thf(thm_2Erelation_2EWFP__RULES,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a] :
( ! [V2y: A_27a] :
( ( V0R @ V2y @ V1x )
=> ( c_2Erelation_2EWFP @ A_27a @ V0R @ V2y ) )
=> ( c_2Erelation_2EWFP @ A_27a @ V0R @ V1x ) ) ).
thf(thm_2Erelation_2EWFP__INDUCT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > $o] :
( ! [V2x: A_27a] :
( ! [V3y: A_27a] :
( ( V0R @ V3y @ V2x )
=> ( V1P @ V3y ) )
=> ( V1P @ V2x ) )
=> ! [V4x: A_27a] :
( ( c_2Erelation_2EWFP @ A_27a @ V0R @ V4x )
=> ( V1P @ V4x ) ) ) ).
thf(thm_2Erelation_2EWFP__CASES,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a] :
( ( c_2Erelation_2EWFP @ A_27a @ V0R @ V1x )
<=> ! [V2y: A_27a] :
( ( V0R @ V2y @ V1x )
=> ( c_2Erelation_2EWFP @ A_27a @ V0R @ V2y ) ) ) ).
thf(thm_2Erelation_2EWFP__STRONG__INDUCT,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1R: A_27a > A_27a > $o] :
( ! [V2x: A_27a] :
( ( ( c_2Erelation_2EWFP @ A_27a @ V1R @ V2x )
& ! [V3y: A_27a] :
( ( V1R @ V3y @ V2x )
=> ( V0P @ V3y ) ) )
=> ( V0P @ V2x ) )
=> ! [V4x: A_27a] :
( ( c_2Erelation_2EWFP @ A_27a @ V1R @ V4x )
=> ( V0P @ V4x ) ) ) ).
thf(thm_2Erelation_2EWF__EQ__WFP,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
<=> ! [V1x: A_27a] : ( c_2Erelation_2EWFP @ A_27a @ V0R @ V1x ) ) ).
thf(thm_2Erelation_2EINDUCTIVE__INVARIANT__WFREC,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27b > $o,V2M: ( A_27a > A_27b ) > A_27a > A_27b] :
( ( ( c_2Erelation_2EWF @ A_27a @ V0R )
& ( c_2Erelation_2EINDUCTIVE__INVARIANT @ A_27a @ A_27b @ V0R @ V1P @ V2M ) )
=> ! [V3x: A_27a] : ( V1P @ V3x @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b @ V0R @ V2M @ V3x ) ) ) ).
thf(thm_2Erelation_2ETFL__INDUCTIVE__INVARIANT__WFREC,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1R: A_27a > A_27a > $o,V2P: A_27a > A_27b > $o,V3M: ( A_27a > A_27b ) > A_27a > A_27b,V4x: A_27a] :
( ( ( V0f
= ( c_2Erelation_2EWFREC @ A_27a @ A_27b @ V1R @ V3M ) )
& ( c_2Erelation_2EWF @ A_27a @ V1R )
& ( c_2Erelation_2EINDUCTIVE__INVARIANT @ A_27a @ A_27b @ V1R @ V2P @ V3M ) )
=> ( V2P @ V4x @ ( V0f @ V4x ) ) ) ).
thf(thm_2Erelation_2EINDUCTIVE__INVARIANT__ON__WFREC,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > A_27b > $o,V2M: ( A_27a > A_27b ) > A_27a > A_27b,V3D: A_27a > $o,V4x: A_27a] :
( ( ( c_2Erelation_2EWF @ A_27a @ V0R )
& ( c_2Erelation_2EINDUCTIVE__INVARIANT__ON @ A_27a @ A_27b @ V0R @ V3D @ V1P @ V2M )
& ( V3D @ V4x ) )
=> ( V1P @ V4x @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b @ V0R @ V2M @ V4x ) ) ) ).
thf(thm_2Erelation_2ETFL__INDUCTIVE__INVARIANT__ON__WFREC,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1R: A_27a > A_27a > $o,V2D: A_27a > $o,V3P: A_27a > A_27b > $o,V4M: ( A_27a > A_27b ) > A_27a > A_27b,V5x: A_27a] :
( ( ( V0f
= ( c_2Erelation_2EWFREC @ A_27a @ A_27b @ V1R @ V4M ) )
& ( c_2Erelation_2EWF @ A_27a @ V1R )
& ( c_2Erelation_2EINDUCTIVE__INVARIANT__ON @ A_27a @ A_27b @ V1R @ V2D @ V3P @ V4M )
& ( V2D @ V5x ) )
=> ( V3P @ V5x @ ( V0f @ V5x ) ) ) ).
thf(thm_2Erelation_2Einv__inv,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Erelation_2Einv @ A_27b @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27b @ V0R ) )
= V0R ) ).
thf(thm_2Erelation_2Einv__RC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= ( c_2Erelation_2ERC @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Einv__SC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Erelation_2ESC @ A_27a @ V0R ) )
= ( c_2Erelation_2ESC @ A_27a @ V0R ) )
& ( ( c_2Erelation_2ESC @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2ESC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Einv__TC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) )
= ( c_2Erelation_2ETC @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Einv__EQC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Erelation_2EEQC @ A_27a @ V0R ) )
= ( c_2Erelation_2EEQC @ A_27a @ V0R ) )
& ( ( c_2Erelation_2EEQC @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2EEQC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Einv__MOVES__OUT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= V0R )
& ( ( c_2Erelation_2ESC @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2ESC @ A_27a @ V0R ) )
& ( ( c_2Erelation_2ERC @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) ) )
& ( ( c_2Erelation_2ETC @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) ) )
& ( ( c_2Erelation_2ERTC @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Erelation_2ERTC @ A_27a @ V0R ) ) )
& ( ( c_2Erelation_2EEQC @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2EEQC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Ereflexive__inv,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ereflexive @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2Ereflexive @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Eirreflexive__inv,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Eirreflexive @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2Eirreflexive @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Esymmetric__inv,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Esymmetric @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2Esymmetric @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Eantisymmetric__inv,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Eantisymmetric @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2Eantisymmetric @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Etransitive__inv,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Etransitive @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2Etransitive @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Esymmetric__inv__identity,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Esymmetric @ A_27a @ V0R )
=> ( ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R )
= V0R ) ) ).
thf(thm_2Erelation_2Eequivalence__inv__identity,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Eequivalence @ A_27a @ V0R )
=> ( ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R )
= V0R ) ) ).
thf(thm_2Erelation_2EINVOL,axiom,
! [A_27z: $tType,V0f: A_27z > A_27z] :
( ( c_2Erelation_2EINVOL @ A_27z @ V0f )
<=> ! [V1x: A_27z] :
( ( V0f @ ( V0f @ V1x ) )
= V1x ) ) ).
thf(thm_2Erelation_2EINVOL__ONE__ONE,axiom,
! [A_27z: $tType,V0f: A_27z > A_27z] :
( ( c_2Erelation_2EINVOL @ A_27z @ V0f )
=> ! [V1a: A_27z,V2b: A_27z] :
( ( ( V0f @ V1a )
= ( V0f @ V2b ) )
<=> ( V1a = V2b ) ) ) ).
thf(thm_2Erelation_2EINVOL__ONE__ENO,axiom,
! [A_27z: $tType,V0f: A_27z > A_27z] :
( ( c_2Erelation_2EINVOL @ A_27z @ V0f )
=> ! [V1a: A_27z,V2b: A_27z] :
( ( ( V0f @ V1a )
= V2b )
<=> ( V1a
= ( V0f @ V2b ) ) ) ) ).
thf(thm_2Erelation_2ENOT__INVOL,axiom,
c_2Erelation_2EINVOL @ $o @ c_2Ebool_2E_7E ).
thf(thm_2Erelation_2EIDEM,axiom,
! [A_27z: $tType,V0f: A_27z > A_27z] :
( ( c_2Erelation_2EIDEM @ A_27z @ V0f )
<=> ! [V1x: A_27z] :
( ( V0f @ ( V0f @ V1x ) )
= ( V0f @ V1x ) ) ) ).
thf(thm_2Erelation_2Einv__INVOL,axiom,
! [A_27a: $tType] : ( c_2Erelation_2EINVOL @ ( A_27a > A_27a > $o ) @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) ) ).
thf(thm_2Erelation_2Einv__O,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0R: A_27a > A_27b > $o,V1R_27: A_27c > A_27a > $o] :
( ( c_2Erelation_2Einv @ A_27c @ A_27b @ ( c_2Erelation_2EO @ A_27c @ A_27a @ A_27b @ V0R @ V1R_27 ) )
= ( c_2Erelation_2EO @ A_27b @ A_27a @ A_27c @ ( c_2Erelation_2Einv @ A_27c @ A_27a @ V1R_27 ) @ ( c_2Erelation_2Einv @ A_27a @ A_27b @ V0R ) ) ) ).
thf(thm_2Erelation_2Eirreflexive__RSUBSET,axiom,
! [A_27a: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Eirreflexive @ A_27a @ V1R2 )
& ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a @ V0R1 @ V1R2 ) )
=> ( c_2Erelation_2Eirreflexive @ A_27a @ V0R1 ) ) ).
thf(thm_2Erelation_2ERUNION__COMM,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2: A_27a > A_27b > $o,V1R1: A_27a > A_27b > $o] :
( ( c_2Erelation_2ERUNION @ A_27a @ A_27b @ V1R1 @ V0R2 )
= ( c_2Erelation_2ERUNION @ A_27a @ A_27b @ V0R2 @ V1R1 ) ) ).
thf(thm_2Erelation_2ERUNION__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,V0R3: A_27a > A_27b > $o,V1R2: A_27a > A_27b > $o,V2R1: A_27a > A_27b > $o] :
( ( c_2Erelation_2ERUNION @ A_27a @ A_27b @ V2R1 @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b @ V1R2 @ V0R3 ) )
= ( c_2Erelation_2ERUNION @ A_27a @ A_27b @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b @ V2R1 @ V1R2 ) @ V0R3 ) ) ).
thf(thm_2Erelation_2ERINTER__COMM,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2: A_27a > A_27b > $o,V1R1: A_27a > A_27b > $o] :
( ( c_2Erelation_2ERINTER @ A_27a @ A_27b @ V1R1 @ V0R2 )
= ( c_2Erelation_2ERINTER @ A_27a @ A_27b @ V0R2 @ V1R1 ) ) ).
thf(thm_2Erelation_2ERINTER__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,V0R3: A_27a > A_27b > $o,V1R2: A_27a > A_27b > $o,V2R1: A_27a > A_27b > $o] :
( ( c_2Erelation_2ERINTER @ A_27a @ A_27b @ V2R1 @ ( c_2Erelation_2ERINTER @ A_27a @ A_27b @ V1R2 @ V0R3 ) )
= ( c_2Erelation_2ERINTER @ A_27a @ A_27b @ ( c_2Erelation_2ERINTER @ A_27a @ A_27b @ V2R1 @ V1R2 ) @ V0R3 ) ) ).
thf(thm_2Erelation_2Eantisymmetric__RINTER,axiom,
! [A_27a: $tType,V0R2: A_27a > A_27a > $o,V1R1: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Eantisymmetric @ A_27a @ V1R1 )
=> ( c_2Erelation_2Eantisymmetric @ A_27a @ ( c_2Erelation_2ERINTER @ A_27a @ A_27a @ V1R1 @ V0R2 ) ) )
& ( ( c_2Erelation_2Eantisymmetric @ A_27a @ V0R2 )
=> ( c_2Erelation_2Eantisymmetric @ A_27a @ ( c_2Erelation_2ERINTER @ A_27a @ A_27a @ V1R1 @ V0R2 ) ) ) ) ).
thf(thm_2Erelation_2Etransitive__RINTER,axiom,
! [A_27a: $tType,V0R2: A_27a > A_27a > $o,V1R1: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Etransitive @ A_27a @ V1R1 )
& ( c_2Erelation_2Etransitive @ A_27a @ V0R2 ) )
=> ( c_2Erelation_2Etransitive @ A_27a @ ( c_2Erelation_2ERINTER @ A_27a @ A_27a @ V1R1 @ V0R2 ) ) ) ).
thf(thm_2Erelation_2Ereflexive__Id__RSUBSET,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ereflexive @ A_27a @ V0R )
= ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a @ ( c_2Emin_2E_3D @ A_27a ) @ V0R ) ) ).
thf(thm_2Erelation_2Esymmetric__inv__RSUBSET,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Esymmetric @ A_27a @ V0R )
= ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) @ V0R ) ) ).
thf(thm_2Erelation_2Etransitive__O__RSUBSET,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Etransitive @ A_27a @ V0R )
= ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a @ V0R @ V0R ) @ V0R ) ) ).
thf(thm_2Erelation_2Eirrefl__trans__implies__antisym,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Eirreflexive @ A_27a @ V0R )
& ( c_2Erelation_2Etransitive @ A_27a @ V0R ) )
=> ( c_2Erelation_2Eantisymmetric @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2EStrongOrd__Ord,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EStrongOrder @ A_27a @ V0R )
=> ( c_2Erelation_2EOrder @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2EWeakOrd__Ord,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWeakOrder @ A_27a @ V0R )
=> ( c_2Erelation_2EOrder @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2EWeakOrder__EQ,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWeakOrder @ A_27a @ V0R )
=> ! [V1y: A_27a,V2z: A_27a] :
( ( V1y = V2z )
<=> ( ( V0R @ V1y @ V2z )
& ( V0R @ V2z @ V1y ) ) ) ) ).
thf(thm_2Erelation_2ERSUBSET__ANTISYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27b > $o,V1R2: A_27a > A_27b > $o] :
( ( ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b @ V0R1 @ V1R2 )
& ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b @ V1R2 @ V0R1 ) )
=> ( V0R1 = V1R2 ) ) ).
thf(thm_2Erelation_2ERSUBSET__antisymmetric,axiom,
! [A_27a: $tType,A_27b: $tType] : ( c_2Erelation_2Eantisymmetric @ ( A_27a > A_27b > $o ) @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) ) ).
thf(thm_2Erelation_2ERSUBSET__WeakOrder,axiom,
! [A_27a: $tType,A_27b: $tType] : ( c_2Erelation_2EWeakOrder @ ( A_27a > A_27b > $o ) @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) ) ).
thf(thm_2Erelation_2EEqIsBothRSUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27a > A_27b > $o,V1z: A_27a > A_27b > $o] :
( ( V0y = V1z )
<=> ( ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b @ V0y @ V1z )
& ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b @ V1z @ V0y ) ) ) ).
thf(thm_2Erelation_2ESTRORD__AND__NOT__Id,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ESTRORD @ A_27a @ V0R )
= ( c_2Erelation_2ERINTER @ A_27a @ A_27a @ V0R @ ( c_2Erelation_2ERCOMPL @ A_27a @ A_27a @ ( c_2Emin_2E_3D @ A_27a ) ) ) ) ).
thf(thm_2Erelation_2ERC__OR__Id,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ERC @ A_27a @ V0R )
= ( c_2Erelation_2ERUNION @ A_27a @ A_27a @ V0R @ ( c_2Emin_2E_3D @ A_27a ) ) ) ).
thf(thm_2Erelation_2ERC__Weak,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EOrder @ A_27a @ V0R )
= ( c_2Erelation_2EWeakOrder @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2ESTRORD__Strong,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EOrder @ A_27a @ V0R )
= ( c_2Erelation_2EStrongOrder @ A_27a @ ( c_2Erelation_2ESTRORD @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2ESTRORD__RC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EStrongOrder @ A_27a @ V0R )
=> ( ( c_2Erelation_2ESTRORD @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= V0R ) ) ).
thf(thm_2Erelation_2ERC__STRORD,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWeakOrder @ A_27a @ V0R )
=> ( ( c_2Erelation_2ERC @ A_27a @ ( c_2Erelation_2ESTRORD @ A_27a @ V0R ) )
= V0R ) ) ).
thf(thm_2Erelation_2EIDEM__STRORD,axiom,
! [A_27a: $tType] : ( c_2Erelation_2EIDEM @ ( A_27a > A_27a > $o ) @ ( c_2Erelation_2ESTRORD @ A_27a ) ) ).
thf(thm_2Erelation_2EIDEM__RC,axiom,
! [A_27a: $tType] : ( c_2Erelation_2EIDEM @ ( A_27a > A_27a > $o ) @ ( c_2Erelation_2ERC @ A_27a ) ) ).
thf(thm_2Erelation_2EIDEM__SC,axiom,
! [A_27a: $tType] : ( c_2Erelation_2EIDEM @ ( A_27a > A_27a > $o ) @ ( c_2Erelation_2ESC @ A_27a ) ) ).
thf(thm_2Erelation_2EIDEM__TC,axiom,
! [A_27a: $tType] : ( c_2Erelation_2EIDEM @ ( A_27a > A_27a > $o ) @ ( c_2Erelation_2ETC @ A_27a ) ) ).
thf(thm_2Erelation_2EIDEM__RTC,axiom,
! [A_27a: $tType] : ( c_2Erelation_2EIDEM @ ( A_27a > A_27a > $o ) @ ( c_2Erelation_2ERTC @ A_27a ) ) ).
thf(thm_2Erelation_2Etrichotomous__STRORD,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Etrichotomous @ A_27a @ ( c_2Erelation_2ESTRORD @ A_27a @ V0R ) )
= ( c_2Erelation_2Etrichotomous @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2Etrichotomous__RC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Etrichotomous @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) )
= ( c_2Erelation_2Etrichotomous @ A_27a @ V0R ) ) ).
thf(thm_2Erelation_2EWeakLinearOrder__dichotomy,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWeakLinearOrder @ A_27a @ V0R )
<=> ( ( c_2Erelation_2EWeakOrder @ A_27a @ V0R )
& ! [V1a: A_27a,V2b: A_27a] :
( ( V0R @ V1a @ V2b )
| ( V0R @ V2b @ V1a ) ) ) ) ).
thf(thm_2Erelation_2EO__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R3: A_27a > A_27d > $o,V1R2: A_27d > A_27c > $o,V2R1: A_27c > A_27b > $o] :
( ( c_2Erelation_2EO @ A_27a @ A_27c @ A_27b @ V2R1 @ ( c_2Erelation_2EO @ A_27a @ A_27d @ A_27c @ V1R2 @ V0R3 ) )
= ( c_2Erelation_2EO @ A_27a @ A_27d @ A_27b @ ( c_2Erelation_2EO @ A_27d @ A_27c @ A_27b @ V2R1 @ V1R2 ) @ V0R3 ) ) ).
thf(thm_2Erelation_2EId__O,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Erelation_2EO @ A_27a @ A_27b @ A_27b @ ( c_2Emin_2E_3D @ A_27b ) @ V0R )
= V0R ) ).
thf(thm_2Erelation_2EO__Id,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27b @ V0R @ ( c_2Emin_2E_3D @ A_27a ) )
= V0R ) ).
thf(thm_2Erelation_2EO__MONO,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0S2: A_27c > A_27a > $o,V1S1: A_27c > A_27a > $o,V2R2: A_27a > A_27b > $o,V3R1: A_27a > A_27b > $o] :
( ( ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b @ V3R1 @ V2R2 )
& ( c_2Erelation_2ERSUBSET @ A_27c @ A_27a @ V1S1 @ V0S2 ) )
=> ( c_2Erelation_2ERSUBSET @ A_27c @ A_27b @ ( c_2Erelation_2EO @ A_27c @ A_27a @ A_27b @ V3R1 @ V1S1 ) @ ( c_2Erelation_2EO @ A_27c @ A_27a @ A_27b @ V2R2 @ V0S2 ) ) ) ).
thf(thm_2Erelation_2Einv__Id,axiom,
! [A_27a: $tType] :
( ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Emin_2E_3D @ A_27a ) )
= ( c_2Emin_2E_3D @ A_27a ) ) ).
thf(thm_2Erelation_2Einv__diag,axiom,
! [A_27a: $tType,V0A: A_27a > $o] :
( ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Erelation_2Ediag @ A_27a @ V0A ) )
= ( c_2Erelation_2Ediag @ A_27a @ V0A ) ) ).
thf(thm_2Erelation_2EIN__RDOM,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1R: A_27a > A_27b > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ V1R ) )
<=> ? [V2y: A_27b] : ( V1R @ V0x @ V2y ) ) ).
thf(thm_2Erelation_2EIN__RRANGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27a,V1R: A_27b > A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0y @ ( c_2Erelation_2ERRANGE @ A_27b @ A_27a @ V1R ) )
<=> ? [V2x: A_27b] : ( V1R @ V2x @ V0y ) ) ).
thf(thm_2Erelation_2EIN__RDOM__RUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1R2: A_27a > A_27b > $o,V2R1: A_27a > A_27b > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b @ V2R1 @ V1R2 ) ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ V2R1 ) )
| ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ V1R2 ) ) ) ) ).
thf(thm_2Erelation_2ERUNIV__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b @ ( c_2Erelation_2ERUNIV @ A_27a @ A_27b ) @ V0R )
<=> ( V0R
= ( c_2Erelation_2ERUNIV @ A_27a @ A_27b ) ) )
& ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b @ V0R @ ( c_2Erelation_2ERUNIV @ A_27a @ A_27b ) ) ) ).
thf(thm_2Erelation_2EREMPTY__SUBSET,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) @ V0R )
& ( ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a @ V0R @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) )
<=> ( V0R
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ) ) ).
thf(thm_2Erelation_2EIN__RDOM__RRESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1s: A_27a > $o,V2R: A_27a > A_27b > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ ( c_2Erelation_2ERRESTRICT @ A_27a @ A_27b @ V2R @ V1s ) ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ V2R ) )
& ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s ) ) ) ).
thf(thm_2Erelation_2EIN__RDOM__DELETE,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1k: A_27a,V2R: A_27a > A_27b > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ ( c_2Erelation_2ERDOM__DELETE @ A_27a @ A_27b @ V2R @ V1k ) ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ V2R ) )
& ( (~) @ ( V0x = V1k ) ) ) ) ).
thf(thm_2Erelation_2Ercdiamond__diamond,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ercdiamond @ A_27a @ V0R )
= ( c_2Erelation_2Ediamond @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Ediamond__RC__diamond,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ediamond @ A_27a @ V0R )
=> ( c_2Erelation_2Ediamond @ A_27a @ ( c_2Erelation_2ERC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Ediamond__TC__diamond,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ediamond @ A_27a @ V0R )
=> ( c_2Erelation_2Ediamond @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2Eestablish__CR,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Ercdiamond @ A_27a @ V0R )
=> ( c_2Erelation_2ECR @ A_27a @ V0R ) )
& ( ( c_2Erelation_2Ediamond @ A_27a @ V0R )
=> ( c_2Erelation_2ECR @ A_27a @ V0R ) ) ) ).
thf(thm_2Erelation_2ENewmans__lemma,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2EWCR @ A_27a @ V0R )
& ( c_2Erelation_2ESN @ A_27a @ V0R ) )
=> ( c_2Erelation_2ECR @ A_27a @ V0R ) ) ).
%------------------------------------------------------------------------------