ITP001 Axioms: ITP011_5.ax
%------------------------------------------------------------------------------
% File : ITP011_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : relation_2.ax [Gau20]
% : HL4011_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 366 ( 62 unt; 61 typ; 0 def)
% Number of atoms : 4495 ( 103 equ)
% Maximal formula atoms : 25 ( 12 avg)
% Number of connectives : 1259 ( 11 ~; 16 |; 162 &)
% ( 92 <=>; 978 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 2942 (2942 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 117 ( 61 >; 56 *; 0 +; 0 <<)
% Number of predicates : 19 ( 16 usr; 5 prp; 0-3 aty)
% Number of functors : 61 ( 61 usr; 0 con; 1-5 aty)
% Number of variables : 1173 (1154 !; 19 ?;1173 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Erelation_2ECR,type,
c_2Erelation_2ECR: del > $i ).
tff(mem_c_2Erelation_2ECR,axiom,
! [A_27a: del] : mem(c_2Erelation_2ECR(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2EEMPTY__REL,type,
c_2Erelation_2EEMPTY__REL: del > $i ).
tff(mem_c_2Erelation_2EEMPTY__REL,axiom,
! [A_27a: del] : mem(c_2Erelation_2EEMPTY__REL(A_27a),arr(A_27a,arr(A_27a,bool))) ).
tff(tp_c_2Erelation_2EEQC,type,
c_2Erelation_2EEQC: del > $i ).
tff(mem_c_2Erelation_2EEQC,axiom,
! [A_27a: del] : mem(c_2Erelation_2EEQC(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Erelation_2EIDEM,type,
c_2Erelation_2EIDEM: del > $i ).
tff(mem_c_2Erelation_2EIDEM,axiom,
! [A_27z: del] : mem(c_2Erelation_2EIDEM(A_27z),arr(arr(A_27z,A_27z),bool)) ).
tff(tp_c_2Erelation_2EINDUCTIVE__INVARIANT,type,
c_2Erelation_2EINDUCTIVE__INVARIANT: ( del * del ) > $i ).
tff(mem_c_2Erelation_2EINDUCTIVE__INVARIANT,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2EINDUCTIVE__INVARIANT(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(arr(A_27a,A_27b),arr(A_27a,A_27b)),bool)))) ).
tff(tp_c_2Erelation_2EINDUCTIVE__INVARIANT__ON,type,
c_2Erelation_2EINDUCTIVE__INVARIANT__ON: ( del * del ) > $i ).
tff(mem_c_2Erelation_2EINDUCTIVE__INVARIANT__ON,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2EINDUCTIVE__INVARIANT__ON(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(arr(A_27a,A_27b),arr(A_27a,A_27b)),bool))))) ).
tff(tp_c_2Erelation_2EINVOL,type,
c_2Erelation_2EINVOL: del > $i ).
tff(mem_c_2Erelation_2EINVOL,axiom,
! [A_27z: del] : mem(c_2Erelation_2EINVOL(A_27z),arr(arr(A_27z,A_27z),bool)) ).
tff(tp_c_2Erelation_2ELinearOrder,type,
c_2Erelation_2ELinearOrder: del > $i ).
tff(mem_c_2Erelation_2ELinearOrder,axiom,
! [A_27a: del] : mem(c_2Erelation_2ELinearOrder(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2EO,type,
c_2Erelation_2EO: ( del * del * del ) > $i ).
tff(mem_c_2Erelation_2EO,axiom,
! [A_27g: del,A_27h: del,A_27k: del] : mem(c_2Erelation_2EO(A_27g,A_27h,A_27k),arr(arr(A_27h,arr(A_27k,bool)),arr(arr(A_27g,arr(A_27h,bool)),arr(A_27g,arr(A_27k,bool))))) ).
tff(tp_c_2Erelation_2EOrder,type,
c_2Erelation_2EOrder: del > $i ).
tff(mem_c_2Erelation_2EOrder,axiom,
! [A_27g: del] : mem(c_2Erelation_2EOrder(A_27g),arr(arr(A_27g,arr(A_27g,bool)),bool)) ).
tff(tp_c_2Erelation_2EPreOrder,type,
c_2Erelation_2EPreOrder: del > $i ).
tff(mem_c_2Erelation_2EPreOrder,axiom,
! [A_27a: del] : mem(c_2Erelation_2EPreOrder(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2ERC,type,
c_2Erelation_2ERC: del > $i ).
tff(mem_c_2Erelation_2ERC,axiom,
! [A_27a: del] : mem(c_2Erelation_2ERC(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Erelation_2ERCOMPL,type,
c_2Erelation_2ERCOMPL: ( del * del ) > $i ).
tff(mem_c_2Erelation_2ERCOMPL,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2ERCOMPL(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(A_27a,arr(A_27b,bool)))) ).
tff(tp_c_2Erelation_2ERDOM,type,
c_2Erelation_2ERDOM: ( del * del ) > $i ).
tff(mem_c_2Erelation_2ERDOM,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2ERDOM(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(A_27a,bool))) ).
tff(tp_c_2Erelation_2ERDOM__DELETE,type,
c_2Erelation_2ERDOM__DELETE: ( del * del ) > $i ).
tff(mem_c_2Erelation_2ERDOM__DELETE,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2ERDOM__DELETE(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(A_27a,arr(A_27a,arr(A_27b,bool))))) ).
tff(tp_c_2Erelation_2ERESTRICT,type,
c_2Erelation_2ERESTRICT: ( del * del ) > $i ).
tff(mem_c_2Erelation_2ERESTRICT,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2ERESTRICT(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,bool)),arr(A_27a,arr(A_27a,A_27b))))) ).
tff(tp_c_2Erelation_2ERINTER,type,
c_2Erelation_2ERINTER: ( del * del ) > $i ).
tff(mem_c_2Erelation_2ERINTER,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2ERINTER(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(A_27a,arr(A_27b,bool)),arr(A_27a,arr(A_27b,bool))))) ).
tff(tp_c_2Erelation_2ERRANGE,type,
c_2Erelation_2ERRANGE: ( del * del ) > $i ).
tff(mem_c_2Erelation_2ERRANGE,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2ERRANGE(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(A_27b,bool))) ).
tff(tp_c_2Erelation_2ERRESTRICT,type,
c_2Erelation_2ERRESTRICT: ( del * del ) > $i ).
tff(mem_c_2Erelation_2ERRESTRICT,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2ERRESTRICT(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(A_27a,bool),arr(A_27a,arr(A_27b,bool))))) ).
tff(tp_c_2Erelation_2ERSUBSET,type,
c_2Erelation_2ERSUBSET: ( del * del ) > $i ).
tff(mem_c_2Erelation_2ERSUBSET,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2ERSUBSET(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(A_27a,arr(A_27b,bool)),bool))) ).
tff(tp_c_2Erelation_2ERTC,type,
c_2Erelation_2ERTC: del > $i ).
tff(mem_c_2Erelation_2ERTC,axiom,
! [A_27a: del] : mem(c_2Erelation_2ERTC(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Erelation_2ERUNION,type,
c_2Erelation_2ERUNION: ( del * del ) > $i ).
tff(mem_c_2Erelation_2ERUNION,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2ERUNION(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(A_27a,arr(A_27b,bool)),arr(A_27a,arr(A_27b,bool))))) ).
tff(tp_c_2Erelation_2ERUNIV,type,
c_2Erelation_2ERUNIV: ( del * del ) > $i ).
tff(mem_c_2Erelation_2ERUNIV,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2ERUNIV(A_27a,A_27b),arr(A_27a,arr(A_27b,bool))) ).
tff(tp_c_2Erelation_2ESC,type,
c_2Erelation_2ESC: del > $i ).
tff(mem_c_2Erelation_2ESC,axiom,
! [A_27a: del] : mem(c_2Erelation_2ESC(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Erelation_2ESN,type,
c_2Erelation_2ESN: del > $i ).
tff(mem_c_2Erelation_2ESN,axiom,
! [A_27a: del] : mem(c_2Erelation_2ESN(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2ESTRORD,type,
c_2Erelation_2ESTRORD: del > $i ).
tff(mem_c_2Erelation_2ESTRORD,axiom,
! [A_27a: del] : mem(c_2Erelation_2ESTRORD(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Erelation_2EStrongLinearOrder,type,
c_2Erelation_2EStrongLinearOrder: del > $i ).
tff(mem_c_2Erelation_2EStrongLinearOrder,axiom,
! [A_27a: del] : mem(c_2Erelation_2EStrongLinearOrder(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2EStrongOrder,type,
c_2Erelation_2EStrongOrder: del > $i ).
tff(mem_c_2Erelation_2EStrongOrder,axiom,
! [A_27g: del] : mem(c_2Erelation_2EStrongOrder(A_27g),arr(arr(A_27g,arr(A_27g,bool)),bool)) ).
tff(tp_c_2Erelation_2ETC,type,
c_2Erelation_2ETC: del > $i ).
tff(mem_c_2Erelation_2ETC,axiom,
! [A_27a: del] : mem(c_2Erelation_2ETC(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Erelation_2EWCR,type,
c_2Erelation_2EWCR: del > $i ).
tff(mem_c_2Erelation_2EWCR,axiom,
! [A_27a: del] : mem(c_2Erelation_2EWCR(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2EWF,type,
c_2Erelation_2EWF: del > $i ).
tff(mem_c_2Erelation_2EWF,axiom,
! [A_27a: del] : mem(c_2Erelation_2EWF(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2EWFP,type,
c_2Erelation_2EWFP: del > $i ).
tff(mem_c_2Erelation_2EWFP,axiom,
! [A_27a: del] : mem(c_2Erelation_2EWFP(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(A_27a,bool))) ).
tff(tp_c_2Erelation_2EWFREC,type,
c_2Erelation_2EWFREC: ( del * del ) > $i ).
tff(mem_c_2Erelation_2EWFREC,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2EWFREC(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(arr(A_27a,A_27b),arr(A_27a,A_27b)),arr(A_27a,A_27b)))) ).
tff(tp_c_2Erelation_2EWeakLinearOrder,type,
c_2Erelation_2EWeakLinearOrder: del > $i ).
tff(mem_c_2Erelation_2EWeakLinearOrder,axiom,
! [A_27a: del] : mem(c_2Erelation_2EWeakLinearOrder(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2EWeakOrder,type,
c_2Erelation_2EWeakOrder: del > $i ).
tff(mem_c_2Erelation_2EWeakOrder,axiom,
! [A_27g: del] : mem(c_2Erelation_2EWeakOrder(A_27g),arr(arr(A_27g,arr(A_27g,bool)),bool)) ).
tff(tp_c_2Erelation_2Eantisymmetric,type,
c_2Erelation_2Eantisymmetric: del > $i ).
tff(mem_c_2Erelation_2Eantisymmetric,axiom,
! [A_27a: del] : mem(c_2Erelation_2Eantisymmetric(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2Eapprox,type,
c_2Erelation_2Eapprox: ( del * del ) > $i ).
tff(mem_c_2Erelation_2Eapprox,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2Eapprox(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(arr(A_27a,A_27b),arr(A_27a,A_27b)),arr(A_27a,arr(arr(A_27a,A_27b),bool))))) ).
tff(tp_c_2Erelation_2Ediag,type,
c_2Erelation_2Ediag: del > $i ).
tff(mem_c_2Erelation_2Ediag,axiom,
! [A_27a: del] : mem(c_2Erelation_2Ediag(A_27a),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Erelation_2Ediamond,type,
c_2Erelation_2Ediamond: del > $i ).
tff(mem_c_2Erelation_2Ediamond,axiom,
! [A_27a: del] : mem(c_2Erelation_2Ediamond(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2Eequivalence,type,
c_2Erelation_2Eequivalence: del > $i ).
tff(mem_c_2Erelation_2Eequivalence,axiom,
! [A_27a: del] : mem(c_2Erelation_2Eequivalence(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2Einv,type,
c_2Erelation_2Einv: ( del * del ) > $i ).
tff(mem_c_2Erelation_2Einv,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2Einv(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(A_27b,arr(A_27a,bool)))) ).
tff(tp_c_2Erelation_2Einv__image,type,
c_2Erelation_2Einv__image: ( del * del ) > $i ).
tff(mem_c_2Erelation_2Einv__image,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2Einv__image(A_27a,A_27b),arr(arr(A_27b,arr(A_27b,bool)),arr(arr(A_27a,A_27b),arr(A_27a,arr(A_27a,bool))))) ).
tff(tp_c_2Erelation_2Eirreflexive,type,
c_2Erelation_2Eirreflexive: del > $i ).
tff(mem_c_2Erelation_2Eirreflexive,axiom,
! [A_27a: del] : mem(c_2Erelation_2Eirreflexive(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2Enf,type,
c_2Erelation_2Enf: ( del * del ) > $i ).
tff(mem_c_2Erelation_2Enf,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2Enf(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(A_27a,bool))) ).
tff(tp_c_2Erelation_2Ercdiamond,type,
c_2Erelation_2Ercdiamond: del > $i ).
tff(mem_c_2Erelation_2Ercdiamond,axiom,
! [A_27a: del] : mem(c_2Erelation_2Ercdiamond(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2Ereflexive,type,
c_2Erelation_2Ereflexive: del > $i ).
tff(mem_c_2Erelation_2Ereflexive,axiom,
! [A_27a: del] : mem(c_2Erelation_2Ereflexive(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2Esymmetric,type,
c_2Erelation_2Esymmetric: del > $i ).
tff(mem_c_2Erelation_2Esymmetric,axiom,
! [A_27a: del] : mem(c_2Erelation_2Esymmetric(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2Ethe__fun,type,
c_2Erelation_2Ethe__fun: ( del * del ) > $i ).
tff(mem_c_2Erelation_2Ethe__fun,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Erelation_2Ethe__fun(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(arr(A_27a,A_27b),arr(A_27a,A_27b)),arr(A_27a,arr(A_27a,A_27b))))) ).
tff(tp_c_2Erelation_2Etotal,type,
c_2Erelation_2Etotal: del > $i ).
tff(mem_c_2Erelation_2Etotal,axiom,
! [A_27a: del] : mem(c_2Erelation_2Etotal(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2Etransitive,type,
c_2Erelation_2Etransitive: del > $i ).
tff(mem_c_2Erelation_2Etransitive,axiom,
! [A_27a: del] : mem(c_2Erelation_2Etransitive(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Erelation_2Etrichotomous,type,
c_2Erelation_2Etrichotomous: del > $i ).
tff(mem_c_2Erelation_2Etrichotomous,axiom,
! [A_27a: del] : mem(c_2Erelation_2Etrichotomous(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(ax_thm_2Erelation_2Etransitive__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Etransitive(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( ( p(ap(ap(V0R,V1x),V2y))
& p(ap(ap(V0R,V2y),V3z)) )
=> p(ap(ap(V0R,V1x),V3z)) ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2Ereflexive__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Ereflexive(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(ap(V0R,V1x),V1x)) ) ) ) ).
tff(ax_thm_2Erelation_2Eirreflexive__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Eirreflexive(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ~ p(ap(ap(V0R,V1x),V1x)) ) ) ) ).
tff(ax_thm_2Erelation_2Esymmetric__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Esymmetric(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V1x),V2y))
<=> p(ap(ap(V0R,V2y),V1x)) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2Eantisymmetric__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Eantisymmetric(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( ( p(ap(ap(V0R,V1x),V2y))
& p(ap(ap(V0R,V2y),V1x)) )
=> ( V1x = V2y ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2Eequivalence__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Eequivalence(A_27a),V0R))
<=> ( p(ap(c_2Erelation_2Ereflexive(A_27a),V0R))
& p(ap(c_2Erelation_2Esymmetric(A_27a),V0R))
& p(ap(c_2Erelation_2Etransitive(A_27a),V0R)) ) ) ) ).
tff(ax_thm_2Erelation_2Etotal__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Etotal(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V1x),V2y))
| p(ap(ap(V0R,V2y),V1x)) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2Etrichotomous,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Etrichotomous(A_27a),V0R))
<=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ( p(ap(ap(V0R,V1a),V2b))
| p(ap(ap(V0R,V2b),V1a))
| ( V1a = V2b ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2ETC__DEF,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1a),V2b))
<=> ! [V3P: $i] :
( mem(V3P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(V0R,V4x),V5y))
=> p(ap(ap(V3P,V4x),V5y)) ) ) )
& ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7y: $i] :
( mem(V7y,A_27a)
=> ! [V8z: $i] :
( mem(V8z,A_27a)
=> ( ( p(ap(ap(V3P,V6x),V7y))
& p(ap(ap(V3P,V7y),V8z)) )
=> p(ap(ap(V3P,V6x),V8z)) ) ) ) ) )
=> p(ap(ap(V3P,V1a),V2b)) ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2ERTC__DEF,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1a),V2b))
<=> ! [V3P: $i] :
( mem(V3P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(ap(ap(V3P,V4x),V4x)) )
& ! [V5x: $i] :
( mem(V5x,A_27a)
=> ! [V6y: $i] :
( mem(V6y,A_27a)
=> ! [V7z: $i] :
( mem(V7z,A_27a)
=> ( ( p(ap(ap(V0R,V5x),V6y))
& p(ap(ap(V3P,V6y),V7z)) )
=> p(ap(ap(V3P,V5x),V7z)) ) ) ) ) )
=> p(ap(ap(V3P,V1a),V2b)) ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2ERC__DEF,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V0R),V1x),V2y))
<=> ( ( V1x = V2y )
| p(ap(ap(V0R,V1x),V2y)) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2ESC__DEF,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ESC(A_27a),V0R),V1x),V2y))
<=> ( p(ap(ap(V0R,V1x),V2y))
| p(ap(ap(V0R,V2y),V1x)) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2EEQC__DEF,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2EEQC(A_27a),V0R) = ap(c_2Erelation_2ERC(A_27a),ap(c_2Erelation_2ETC(A_27a),ap(c_2Erelation_2ESC(A_27a),V0R))) ) ) ).
tff(conj_thm_2Erelation_2ESC__SYMMETRIC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Esymmetric(A_27a),ap(c_2Erelation_2ESC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2ETC__TRANSITIVE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Etransitive(A_27a),ap(c_2Erelation_2ETC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2ERTC__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> p(ap(ap(V1P,V2x),V2x)) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5z: $i] :
( mem(V5z,A_27a)
=> ( ( p(ap(ap(V0R,V3x),V4y))
& p(ap(ap(V1P,V4y),V5z)) )
=> p(ap(ap(V1P,V3x),V5z)) ) ) ) ) )
=> ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7y: $i] :
( mem(V7y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V6x),V7y))
=> p(ap(ap(V1P,V6x),V7y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__RULES,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V1x),V2y))
=> p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2y)) ) ) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5z: $i] :
( mem(V5z,A_27a)
=> ( ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V3x),V4y))
& p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V4y),V5z)) )
=> p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V3x),V5z)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__RULES,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V1x)) )
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4z: $i] :
( mem(V4z,A_27a)
=> ( ( p(ap(ap(V0R,V2x),V3y))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V3y),V4z)) )
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V2x),V4z)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__REFL,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V1x)) ) ) ).
tff(conj_thm_2Erelation_2ERTC__SINGLE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V1x),V2y))
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__STRONG__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> p(ap(ap(V1P,V2x),V2x)) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5z: $i] :
( mem(V5z,A_27a)
=> ( ( p(ap(ap(V0R,V3x),V4y))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V4y),V5z))
& p(ap(ap(V1P,V4y),V5z)) )
=> p(ap(ap(V1P,V3x),V5z)) ) ) ) ) )
=> ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7y: $i] :
( mem(V7y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V6x),V7y))
=> p(ap(ap(V1P,V6x),V7y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__RTC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y))
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V2y),V3z))
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V3z)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__TRANSITIVE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Etransitive(A_27a),ap(c_2Erelation_2ERTC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2Etransitive__RTC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Etransitive(A_27a),ap(c_2Erelation_2ERTC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2ERTC__REFLEXIVE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Ereflexive(A_27a),ap(c_2Erelation_2ERTC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2Ereflexive__RTC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Ereflexive(A_27a),ap(c_2Erelation_2ERTC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2ERC__REFLEXIVE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Ereflexive(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2Ereflexive__RC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Ereflexive(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2ERC__lifts__monotonicities,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V0R,ap(V1f,V2x)),ap(V1f,V3y))) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V0R),V4x),V5y))
=> p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V0R),ap(V1f,V4x)),ap(V1f,V5y))) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERC__MONOTONE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(V0R,V4x),V5y))
=> p(ap(ap(V1Q,V4x),V5y)) ) ) )
=> ( p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V0R),V2x),V3y))
=> p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V1Q),V2x),V3y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERC__lifts__invariants,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( ( p(ap(V0P,V2x))
& p(ap(ap(V1R,V2x),V3y)) )
=> p(ap(V0P,V3y)) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( ( p(ap(V0P,V4x))
& p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V1R),V4x),V5y)) )
=> p(ap(V0P,V5y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERC__lifts__equalities,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> ( ap(V1f,V2x) = ap(V1f,V3y) ) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V0R),V4x),V5y))
=> ( ap(V1f,V4x) = ap(V1f,V5y) ) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ESC__lifts__monotonicities,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V0R,ap(V1f,V2x)),ap(V1f,V3y))) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ESC(A_27a),V0R),V4x),V5y))
=> p(ap(ap(ap(c_2Erelation_2ESC(A_27a),V0R),ap(V1f,V4x)),ap(V1f,V5y))) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ESC__lifts__equalities,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> ( ap(V1f,V2x) = ap(V1f,V3y) ) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ESC(A_27a),V0R),V4x),V5y))
=> ( ap(V1f,V4x) = ap(V1f,V5y) ) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ESC__MONOTONE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(V0R,V4x),V5y))
=> p(ap(ap(V1Q,V4x),V5y)) ) ) )
=> ( p(ap(ap(ap(c_2Erelation_2ESC(A_27a),V0R),V2x),V3y))
=> p(ap(ap(ap(c_2Erelation_2ESC(A_27a),V1Q),V2x),V3y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2Esymmetric__RC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Esymmetric(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)))
<=> p(ap(c_2Erelation_2Esymmetric(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Eantisymmetric__RC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Eantisymmetric(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)))
<=> p(ap(c_2Erelation_2Eantisymmetric(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Etransitive__RC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Etransitive(A_27a),V0R))
=> p(ap(c_2Erelation_2Etransitive(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R))) ) ) ).
tff(conj_thm_2Erelation_2ETC__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V1x),V2y))
=> p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V1x),V2y))
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERC__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V1x),V2y))
=> p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V0R),V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERC__RTC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V0R),V1x),V2y))
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V1P,V2x),V3y)) ) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ! [V6z: $i] :
( mem(V6z,A_27a)
=> ( ( p(ap(ap(V1P,V4x),V5y))
& p(ap(ap(V1P,V5y),V6z)) )
=> p(ap(ap(V1P,V4x),V6z)) ) ) ) ) )
=> ! [V7u: $i] :
( mem(V7u,A_27a)
=> ! [V8v: $i] :
( mem(V8v,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V7u),V8v))
=> p(ap(ap(V1P,V7u),V8v)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__INDUCT__LEFT1,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V1P,V2x),V3y)) ) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ! [V6z: $i] :
( mem(V6z,A_27a)
=> ( ( p(ap(ap(V0R,V4x),V5y))
& p(ap(ap(V1P,V5y),V6z)) )
=> p(ap(ap(V1P,V4x),V6z)) ) ) ) ) )
=> ! [V7x: $i] :
( mem(V7x,A_27a)
=> ! [V8y: $i] :
( mem(V8y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V7x),V8y))
=> p(ap(ap(V1P,V7x),V8y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__INDUCT__RIGHT1,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V1P,V2x),V3y)) ) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ! [V6z: $i] :
( mem(V6z,A_27a)
=> ( ( p(ap(ap(V1P,V4x),V5y))
& p(ap(ap(V0R,V5y),V6z)) )
=> p(ap(ap(V1P,V4x),V6z)) ) ) ) ) )
=> ! [V7x: $i] :
( mem(V7x,A_27a)
=> ! [V8y: $i] :
( mem(V8y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V7x),V8y))
=> p(ap(ap(V1P,V7x),V8y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__STRONG__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V1P,V2x),V3y)) ) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ! [V6z: $i] :
( mem(V6z,A_27a)
=> ( ( p(ap(ap(V1P,V4x),V5y))
& p(ap(ap(V1P,V5y),V6z))
& p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V4x),V5y))
& p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V5y),V6z)) )
=> p(ap(ap(V1P,V4x),V6z)) ) ) ) ) )
=> ! [V7u: $i] :
( mem(V7u,A_27a)
=> ! [V8v: $i] :
( mem(V8v,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V7u),V8v))
=> p(ap(ap(V1P,V7u),V8v)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__STRONG__INDUCT__LEFT1,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V1P,V2x),V3y)) ) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ! [V6z: $i] :
( mem(V6z,A_27a)
=> ( ( p(ap(ap(V0R,V4x),V5y))
& p(ap(ap(V1P,V5y),V6z))
& p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V5y),V6z)) )
=> p(ap(ap(V1P,V4x),V6z)) ) ) ) ) )
=> ! [V7u: $i] :
( mem(V7u,A_27a)
=> ! [V8v: $i] :
( mem(V8v,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V7u),V8v))
=> p(ap(ap(V1P,V7u),V8v)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__STRONG__INDUCT__RIGHT1,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V1P,V2x),V3y)) ) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ! [V6z: $i] :
( mem(V6z,A_27a)
=> ( ( p(ap(ap(V1P,V4x),V5y))
& p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V4x),V5y))
& p(ap(ap(V0R,V5y),V6z)) )
=> p(ap(ap(V1P,V4x),V6z)) ) ) ) ) )
=> ! [V7u: $i] :
( mem(V7u,A_27a)
=> ! [V8v: $i] :
( mem(V8v,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V7u),V8v))
=> p(ap(ap(V1P,V7u),V8v)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__INDUCT__ALT__LEFT,axiom,
! [A_27a: del,V0b: $i] :
( mem(V0b,A_27a)
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ! [V2Q: $i] :
( mem(V2Q,arr(A_27a,bool))
=> ( ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(V1R,V3x),V0b))
=> p(ap(V2Q,V3x)) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( ( p(ap(ap(V1R,V4x),V5y))
& p(ap(V2Q,V5y)) )
=> p(ap(V2Q,V4x)) ) ) ) )
=> ! [V6a: $i] :
( mem(V6a,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V1R),V6a),V0b))
=> p(ap(V2Q,V6a)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__INDUCT__ALT__RIGHT,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ! [V2Q: $i] :
( mem(V2Q,arr(A_27a,bool))
=> ( ( ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V1R,V0a),V3y))
=> p(ap(V2Q,V3y)) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( ( p(ap(V2Q,V4x))
& p(ap(ap(V1R,V4x),V5y)) )
=> p(ap(V2Q,V5y)) ) ) ) )
=> ! [V6b: $i] :
( mem(V6b,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V1R),V0a),V6b))
=> p(ap(V2Q,V6b)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__lifts__monotonicities,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V0R,ap(V1f,V2x)),ap(V1f,V3y))) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V4x),V5y))
=> p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),ap(V1f,V4x)),ap(V1f,V5y))) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__lifts__invariants,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( ( p(ap(V0P,V2x))
& p(ap(ap(V1R,V2x),V3y)) )
=> p(ap(V0P,V3y)) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( ( p(ap(V0P,V4x))
& p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V1R),V4x),V5y)) )
=> p(ap(V0P,V5y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__lifts__equalities,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> ( ap(V1f,V2x) = ap(V1f,V3y) ) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V4x),V5y))
=> ( ap(V1f,V4x) = ap(V1f,V5y) ) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__lifts__transitive__relations,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27b,arr(A_27b,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( p(ap(ap(V0R,V3x),V4y))
=> p(ap(ap(V1Q,ap(V2f,V3x)),ap(V2f,V4y))) ) ) )
& p(ap(c_2Erelation_2Etransitive(A_27b),V1Q)) )
=> ! [V5x: $i] :
( mem(V5x,A_27a)
=> ! [V6y: $i] :
( mem(V6y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V5x),V6y))
=> p(ap(ap(V1Q,ap(V2f,V5x)),ap(V2f,V6y))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__implies__one__step,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2y))
& ( V1x != V2y ) )
=> ? [V3z: $i] :
( mem(V3z,A_27a)
& p(ap(ap(V0R,V1x),V3z))
& ( V1x != V3z ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__RTC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2y))
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__TC__RC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y))
=> ( p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V0R),V1x),V2y))
| p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2y)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__RC__EQNS,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ( ap(c_2Erelation_2ERC(A_27a),ap(c_2Erelation_2ETC(A_27a),V0R)) = ap(c_2Erelation_2ERTC(A_27a),V0R) )
& ( ap(c_2Erelation_2ETC(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)) = ap(c_2Erelation_2ERTC(A_27a),V0R) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__ALT__DEF,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1a),V2b))
<=> ! [V3Q: $i] :
( mem(V3Q,arr(A_27a,bool))
=> ( ( p(ap(V3Q,V2b))
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( ( p(ap(ap(V0R,V4x),V5y))
& p(ap(V3Q,V5y)) )
=> p(ap(V3Q,V4x)) ) ) ) )
=> p(ap(V3Q,V1a)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__ALT__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ( ( p(ap(V1Q,V2b))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( ( p(ap(ap(V0R,V3x),V4y))
& p(ap(V1Q,V4y)) )
=> p(ap(V1Q,V3x)) ) ) ) )
=> ! [V5x: $i] :
( mem(V5x,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V5x),V2b))
=> p(ap(V1Q,V5x)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__ALT__RIGHT__DEF,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1a),V2b))
<=> ! [V3Q: $i] :
( mem(V3Q,arr(A_27a,bool))
=> ( ( p(ap(V3Q,V1a))
& ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5z: $i] :
( mem(V5z,A_27a)
=> ( ( p(ap(V3Q,V4y))
& p(ap(ap(V0R,V4y),V5z)) )
=> p(ap(V3Q,V5z)) ) ) ) )
=> p(ap(V3Q,V2b)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__ALT__RIGHT__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ( ( p(ap(V1Q,V2a))
& ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4z: $i] :
( mem(V4z,A_27a)
=> ( ( p(ap(V1Q,V3y))
& p(ap(ap(V0R,V3y),V4z)) )
=> p(ap(V1Q,V4z)) ) ) ) )
=> ! [V5z: $i] :
( mem(V5z,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V2a),V5z))
=> p(ap(V1Q,V5z)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__INDUCT__RIGHT1,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> p(ap(ap(V1P,V2x),V2x)) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5z: $i] :
( mem(V5z,A_27a)
=> ( ( p(ap(ap(V1P,V3x),V4y))
& p(ap(ap(V0R,V4y),V5z)) )
=> p(ap(ap(V1P,V3x),V5z)) ) ) ) ) )
=> ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7y: $i] :
( mem(V7y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V6x),V7y))
=> p(ap(ap(V1P,V6x),V7y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__RULES__RIGHT1,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V1x)) )
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4z: $i] :
( mem(V4z,A_27a)
=> ( ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V2x),V3y))
& p(ap(ap(V0R,V3y),V4z)) )
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V2x),V4z)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__STRONG__INDUCT__RIGHT1,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> p(ap(ap(V1P,V2x),V2x)) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5z: $i] :
( mem(V5z,A_27a)
=> ( ( p(ap(ap(V1P,V3x),V4y))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V3x),V4y))
& p(ap(ap(V0R,V4y),V5z)) )
=> p(ap(ap(V1P,V3x),V5z)) ) ) ) ) )
=> ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7y: $i] :
( mem(V7y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V6x),V7y))
=> p(ap(ap(V1P,V6x),V7y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EEXTEND__RTC__TC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( ( p(ap(ap(V0R,V1x),V2y))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V2y),V3z)) )
=> p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V3z)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EEXTEND__RTC__TC__EQN,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2z: $i] :
( mem(V2z,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2z))
<=> ? [V3y: $i] :
( mem(V3y,A_27a)
& p(ap(ap(V0R,V1x),V3y))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V3y),V2z)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2Ereflexive__RC__identity,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Ereflexive(A_27a),V0R))
=> ( ap(c_2Erelation_2ERC(A_27a),V0R) = V0R ) ) ) ).
tff(conj_thm_2Erelation_2Esymmetric__SC__identity,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Esymmetric(A_27a),V0R))
=> ( ap(c_2Erelation_2ESC(A_27a),V0R) = V0R ) ) ) ).
tff(conj_thm_2Erelation_2Etransitive__TC__identity,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Etransitive(A_27a),V0R))
=> ( ap(c_2Erelation_2ETC(A_27a),V0R) = V0R ) ) ) ).
tff(conj_thm_2Erelation_2ERC__IDEM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2ERC(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)) = ap(c_2Erelation_2ERC(A_27a),V0R) ) ) ).
tff(conj_thm_2Erelation_2ESC__IDEM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2ESC(A_27a),ap(c_2Erelation_2ESC(A_27a),V0R)) = ap(c_2Erelation_2ESC(A_27a),V0R) ) ) ).
tff(conj_thm_2Erelation_2ETC__IDEM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2ETC(A_27a),ap(c_2Erelation_2ETC(A_27a),V0R)) = ap(c_2Erelation_2ETC(A_27a),V0R) ) ) ).
tff(conj_thm_2Erelation_2ERC__MOVES__OUT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ( ap(c_2Erelation_2ESC(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)) = ap(c_2Erelation_2ERC(A_27a),ap(c_2Erelation_2ESC(A_27a),V0R)) )
& ( ap(c_2Erelation_2ERC(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)) = ap(c_2Erelation_2ERC(A_27a),V0R) )
& ( ap(c_2Erelation_2ETC(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)) = ap(c_2Erelation_2ERC(A_27a),ap(c_2Erelation_2ETC(A_27a),V0R)) ) ) ) ).
tff(conj_thm_2Erelation_2Esymmetric__TC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Esymmetric(A_27a),V0R))
=> p(ap(c_2Erelation_2Esymmetric(A_27a),ap(c_2Erelation_2ETC(A_27a),V0R))) ) ) ).
tff(conj_thm_2Erelation_2Ereflexive__TC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Ereflexive(A_27a),V0R))
=> p(ap(c_2Erelation_2Ereflexive(A_27a),ap(c_2Erelation_2ETC(A_27a),V0R))) ) ) ).
tff(conj_thm_2Erelation_2EEQC__EQUIVALENCE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Eequivalence(A_27a),ap(c_2Erelation_2EEQC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2EEQC__IDEM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2EEQC(A_27a),ap(c_2Erelation_2EEQC(A_27a),V0R)) = ap(c_2Erelation_2EEQC(A_27a),V0R) ) ) ).
tff(conj_thm_2Erelation_2ERTC__IDEM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2ERTC(A_27a),ap(c_2Erelation_2ERTC(A_27a),V0R)) = ap(c_2Erelation_2ERTC(A_27a),V0R) ) ) ).
tff(conj_thm_2Erelation_2ERTC__CASES1,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y))
<=> ( ( V1x = V2y )
| ? [V3u: $i] :
( mem(V3u,A_27a)
& p(ap(ap(V0R,V1x),V3u))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V3u),V2y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__CASES__TC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y))
<=> ( ( V1x = V2y )
| p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2y)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__CASES2,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y))
<=> ( ( V1x = V2y )
| ? [V3u: $i] :
( mem(V3u,A_27a)
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V3u))
& p(ap(ap(V0R,V3u),V2y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__CASES__RTC__TWICE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y))
<=> ? [V3u: $i] :
( mem(V3u,A_27a)
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V3u))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V3u),V2y)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__CASES1__E,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2z: $i] :
( mem(V2z,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2z))
=> ( p(ap(ap(V0R,V1x),V2z))
| ? [V3y: $i] :
( mem(V3y,A_27a)
& p(ap(ap(V0R,V1x),V3y))
& p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V3y),V2z)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__CASES1,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2z: $i] :
( mem(V2z,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2z))
<=> ( p(ap(ap(V0R,V1x),V2z))
| ? [V3y: $i] :
( mem(V3y,A_27a)
& p(ap(ap(V0R,V1x),V3y))
& p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V3y),V2z)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__CASES2__E,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2z: $i] :
( mem(V2z,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2z))
=> ( p(ap(ap(V0R,V1x),V2z))
| ? [V3y: $i] :
( mem(V3y,A_27a)
& p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V3y))
& p(ap(ap(V0R,V3y),V2z)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__CASES2,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2z: $i] :
( mem(V2z,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2z))
<=> ( p(ap(ap(V0R,V1x),V2z))
| ? [V3y: $i] :
( mem(V3y,A_27a)
& p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V3y))
& p(ap(ap(V0R,V3y),V2z)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETC__MONOTONE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(V0R,V4x),V5y))
=> p(ap(ap(V1Q,V4x),V5y)) ) ) )
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V2x),V3y))
=> p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V1Q),V2x),V3y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__MONOTONE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(V0R,V4x),V5y))
=> p(ap(ap(V1Q,V4x),V5y)) ) ) )
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V2x),V3y))
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V1Q),V2x),V3y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EEQC__INDUCTION,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V1P,V2x),V3y)) ) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(ap(ap(V1P,V4x),V4x)) )
& ! [V5x: $i] :
( mem(V5x,A_27a)
=> ! [V6y: $i] :
( mem(V6y,A_27a)
=> ( p(ap(ap(V1P,V5x),V6y))
=> p(ap(ap(V1P,V6y),V5x)) ) ) )
& ! [V7x: $i] :
( mem(V7x,A_27a)
=> ! [V8y: $i] :
( mem(V8y,A_27a)
=> ! [V9z: $i] :
( mem(V9z,A_27a)
=> ( ( p(ap(ap(V1P,V7x),V8y))
& p(ap(ap(V1P,V8y),V9z)) )
=> p(ap(ap(V1P,V7x),V9z)) ) ) ) ) )
=> ! [V10x: $i] :
( mem(V10x,A_27a)
=> ! [V11y: $i] :
( mem(V11y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V10x),V11y))
=> p(ap(ap(V1P,V10x),V11y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EEQC__REFL,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V1x),V1x)) ) ) ).
tff(conj_thm_2Erelation_2EEQC__R,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V1x),V2y))
=> p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2EEQC__SYM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V1x),V2y))
=> p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V2y),V1x)) ) ) ) ) ).
tff(conj_thm_2Erelation_2EEQC__TRANS,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( ( p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V1x),V2y))
& p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V2y),V3z)) )
=> p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V1x),V3z)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2Etransitive__EQC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Etransitive(A_27a),ap(c_2Erelation_2EEQC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2Esymmetric__EQC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Esymmetric(A_27a),ap(c_2Erelation_2EEQC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2Ereflexive__EQC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Erelation_2Ereflexive(A_27a),ap(c_2Erelation_2EEQC(A_27a),V0R))) ) ).
tff(conj_thm_2Erelation_2EEQC__MOVES__IN,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ( ap(c_2Erelation_2EEQC(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)) = ap(c_2Erelation_2EEQC(A_27a),V0R) )
& ( ap(c_2Erelation_2EEQC(A_27a),ap(c_2Erelation_2ESC(A_27a),V0R)) = ap(c_2Erelation_2EEQC(A_27a),V0R) )
& ( ap(c_2Erelation_2EEQC(A_27a),ap(c_2Erelation_2ETC(A_27a),V0R)) = ap(c_2Erelation_2EEQC(A_27a),V0R) ) ) ) ).
tff(conj_thm_2Erelation_2ESTRONG__EQC__INDUCTION,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V1P,V2x),V3y)) ) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(ap(ap(V1P,V4x),V4x)) )
& ! [V5x: $i] :
( mem(V5x,A_27a)
=> ! [V6y: $i] :
( mem(V6y,A_27a)
=> ( ( p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V5x),V6y))
& p(ap(ap(V1P,V5x),V6y)) )
=> p(ap(ap(V1P,V6y),V5x)) ) ) )
& ! [V7x: $i] :
( mem(V7x,A_27a)
=> ! [V8y: $i] :
( mem(V8y,A_27a)
=> ! [V9z: $i] :
( mem(V9z,A_27a)
=> ( ( p(ap(ap(V1P,V7x),V8y))
& p(ap(ap(V1P,V8y),V9z))
& p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V7x),V8y))
& p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V8y),V9z)) )
=> p(ap(ap(V1P,V7x),V9z)) ) ) ) ) )
=> ! [V10x: $i] :
( mem(V10x,A_27a)
=> ! [V11y: $i] :
( mem(V11y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V10x),V11y))
=> p(ap(ap(V1P,V10x),V11y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EALT__equivalence,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Eequivalence(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V1x),V2y))
<=> ( ap(V0R,V1x) = ap(V0R,V2y) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EEQC__MONOTONE,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R_27: $i] :
( mem(V1R_27,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(V0R,V4x),V5y))
=> p(ap(ap(V1R_27,V4x),V5y)) ) ) )
=> ( p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V2x),V3y))
=> p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V1R_27),V2x),V3y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__EQC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V1x),V2y))
=> p(ap(ap(ap(c_2Erelation_2EEQC(A_27a),V0R),V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__lifts__monotonicities,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> p(ap(ap(V0R,ap(V1f,V2x)),ap(V1f,V3y))) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V4x),V5y))
=> p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),ap(V1f,V4x)),ap(V1f,V5y))) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__lifts__reflexive__transitive__relations,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27b,arr(A_27b,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( p(ap(ap(V0R,V3x),V4y))
=> p(ap(ap(V1Q,ap(V2f,V3x)),ap(V2f,V4y))) ) ) )
& p(ap(c_2Erelation_2Ereflexive(A_27b),V1Q))
& p(ap(c_2Erelation_2Etransitive(A_27b),V1Q)) )
=> ! [V5x: $i] :
( mem(V5x,A_27a)
=> ! [V6y: $i] :
( mem(V6y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V5x),V6y))
=> p(ap(ap(V1Q,ap(V2f,V5x)),ap(V2f,V6y))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__lifts__equalities,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V2x),V3y))
=> ( ap(V1f,V2x) = ap(V1f,V3y) ) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V4x),V5y))
=> ( ap(V1f,V4x) = ap(V1f,V5y) ) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERTC__lifts__invariants,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( ( p(ap(V0P,V2x))
& p(ap(ap(V1R,V2x),V3y)) )
=> p(ap(V0P,V3y)) ) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( ( p(ap(V0P,V4x))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V1R),V4x),V5y)) )
=> p(ap(V0P,V5y)) ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2EWF__DEF,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
<=> ! [V1B: $i] :
( mem(V1B,arr(A_27a,bool))
=> ( ? [V2w: $i] :
( mem(V2w,A_27a)
& p(ap(V1B,V2w)) )
=> ? [V3min: $i] :
( mem(V3min,A_27a)
& p(ap(V1B,V3min))
& ! [V4b: $i] :
( mem(V4b,A_27a)
=> ( p(ap(ap(V0R,V4b),V3min))
=> ~ p(ap(V1B,V4b)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EWF__INDUCTION__THM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V3y),V2x))
=> p(ap(V1P,V3y)) ) )
=> p(ap(V1P,V2x)) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(ap(V1P,V4x)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EINDUCTION__WF__THM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V3y),V2x))
=> p(ap(V1P,V3y)) ) )
=> p(ap(V1P,V2x)) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(ap(V1P,V4x)) ) ) )
=> p(ap(c_2Erelation_2EWF(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2EWF__EQ__INDUCTION__THM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
<=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V3y),V2x))
=> p(ap(V1P,V3y)) ) )
=> p(ap(V1P,V2x)) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(ap(V1P,V4x)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EWF__NOT__REFL,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
=> ( p(ap(ap(V0R,V1x),V2y))
=> ( V1x != V2y ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EWF__irreflexive,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
=> p(ap(c_2Erelation_2Eirreflexive(A_27a),V0R)) ) ) ).
tff(ax_thm_2Erelation_2EEMPTY__REL__DEF,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27a)
=> ( p(ap(ap(c_2Erelation_2EEMPTY__REL(A_27a),V0x),V1y))
<=> $false ) ) ) ).
tff(conj_thm_2Erelation_2EWF__EMPTY__REL,axiom,
! [A_27a: del] : p(ap(c_2Erelation_2EWF(A_27a),c_2Erelation_2EEMPTY__REL(A_27a))) ).
tff(conj_thm_2Erelation_2EWF__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27a,bool)))
=> ( ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V1P,V2x),V3y))
=> p(ap(ap(V0R,V2x),V3y)) ) ) ) )
=> p(ap(c_2Erelation_2EWF(A_27a),V1P)) ) ) ) ).
tff(conj_thm_2Erelation_2EWF__TC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
=> p(ap(c_2Erelation_2EWF(A_27a),ap(c_2Erelation_2ETC(A_27a),V0R))) ) ) ).
tff(conj_thm_2Erelation_2EWF__TC__EQN,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),ap(c_2Erelation_2ETC(A_27a),V0R)))
<=> p(ap(c_2Erelation_2EWF(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2EWF__noloops,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
=> ( p(ap(ap(ap(c_2Erelation_2ETC(A_27a),V0R),V1x),V2y))
=> ( V1x != V2y ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EWF__antisymmetric,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
=> p(ap(c_2Erelation_2Eantisymmetric(A_27a),V0R)) ) ) ).
tff(lamtp_f97,type,
f97: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f97,axiom,
! [A_27b: del,A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27b,arr(A_27b,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V3y: $i] : ( ap(f97(A_27b,A_27a,V0R,V2x,V1f),V3y) = ap(ap(V0R,ap(V1f,V2x)),ap(V1f,V3y)) ) ) ) ) ).
tff(lamtp_f98,type,
f98: ( del * del * $i * $i ) > $i ).
tff(lameq_f98,axiom,
! [A_27b: del,A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27b,arr(A_27b,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2x: $i] : ( ap(f98(A_27b,A_27a,V0R,V1f),V2x) = f97(A_27b,A_27a,V0R,V2x,V1f) ) ) ) ).
tff(ax_thm_2Erelation_2Einv__image__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27b,arr(A_27b,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ap(ap(c_2Erelation_2Einv__image(A_27a,A_27b),V0R),V1f) = f98(A_27b,A_27a,V0R,V1f) ) ) ) ).
tff(conj_thm_2Erelation_2Einv__image__thm,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27b,arr(A_27b,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(ap(ap(c_2Erelation_2Einv__image(A_27a,A_27b),V0R),V1f),V2x),V3y))
<=> p(ap(ap(V0R,ap(V1f,V2x)),ap(V1f,V3y))) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EWF__inv__image,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27b,arr(A_27b,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ( p(ap(c_2Erelation_2EWF(A_27b),V0R))
=> p(ap(c_2Erelation_2EWF(A_27a),ap(ap(c_2Erelation_2Einv__image(A_27a,A_27b),V0R),V1f))) ) ) ) ).
tff(conj_thm_2Erelation_2Etotal__inv__image,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ( p(ap(c_2Erelation_2Etotal(A_27a),V0R))
=> p(ap(c_2Erelation_2Etotal(A_27b),ap(ap(c_2Erelation_2Einv__image(A_27b,A_27a),V0R),V1f))) ) ) ) ).
tff(conj_thm_2Erelation_2Ereflexive__inv__image,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ( p(ap(c_2Erelation_2Ereflexive(A_27a),V0R))
=> p(ap(c_2Erelation_2Ereflexive(A_27b),ap(ap(c_2Erelation_2Einv__image(A_27b,A_27a),V0R),V1f))) ) ) ) ).
tff(conj_thm_2Erelation_2Esymmetric__inv__image,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ( p(ap(c_2Erelation_2Esymmetric(A_27a),V0R))
=> p(ap(c_2Erelation_2Esymmetric(A_27b),ap(ap(c_2Erelation_2Einv__image(A_27b,A_27a),V0R),V1f))) ) ) ) ).
tff(conj_thm_2Erelation_2Etransitive__inv__image,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ( p(ap(c_2Erelation_2Etransitive(A_27a),V0R))
=> p(ap(c_2Erelation_2Etransitive(A_27b),ap(ap(c_2Erelation_2Einv__image(A_27b,A_27a),V0R),V1f))) ) ) ) ).
tff(lamtp_f99,type,
f99: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f99,axiom,
! [A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ! [V3y: $i] : ( ap(f99(A_27b,A_27a,V0f,V2x,V1R),V3y) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(V1R,V3y),V2x)),ap(V0f,V3y)),c_2Ebool_2EARB(A_27b)) ) ) ) ) ).
tff(ax_thm_2Erelation_2ERESTRICT__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ap(ap(ap(c_2Erelation_2ERESTRICT(A_27a,A_27b),V0f),V1R),V2x) = f99(A_27b,A_27a,V0f,V2x,V1R) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERESTRICT__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( p(ap(ap(V1R,V2y),V3z))
=> ( ap(ap(ap(ap(c_2Erelation_2ERESTRICT(A_27a,A_27b),V0f),V1R),V3z),V2y) = ap(V0f,V2y) ) ) ) ) ) ) ).
tff(lamtp_f100,type,
f100: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f100,axiom,
! [A_27b: del,A_27a: del,V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
=> ! [V4y: $i] : ( ap(f100(A_27b,A_27a,V1M,V0R,V3f),V4y) = ap(ap(V1M,ap(ap(ap(c_2Erelation_2ERESTRICT(A_27a,A_27b),V3f),V0R),V4y)),V4y) ) ) ) ) ).
tff(ax_thm_2Erelation_2Eapprox__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
=> ( p(ap(ap(ap(ap(c_2Erelation_2Eapprox(A_27a,A_27b),V0R),V1M),V2x),V3f))
<=> ( V3f = ap(ap(ap(c_2Erelation_2ERESTRICT(A_27a,A_27b),f100(A_27b,A_27a,V1M,V0R,V3f)),V0R),V2x) ) ) ) ) ) ) ).
tff(lamtp_f101,type,
f101: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f101,axiom,
! [A_27a: del,A_27b: del,V2x: $i] :
( mem(V2x,A_27a)
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V3f: $i] : ( ap(f101(A_27a,A_27b,V2x,V0R,V1M),V3f) = ap(ap(ap(ap(c_2Erelation_2Eapprox(A_27a,A_27b),V0R),V1M),V2x),V3f) ) ) ) ) ).
tff(ax_thm_2Erelation_2Ethe__fun__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ap(ap(ap(c_2Erelation_2Ethe__fun(A_27a,A_27b),V0R),V1M),V2x) = ap(c_2Emin_2E_40(arr(A_27a,A_27b)),f101(A_27a,A_27b,V2x,V0R,V1M)) ) ) ) ) ).
tff(lamtp_f102,type,
f102: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f102,axiom,
! [A_27b: del,A_27a: del,V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
=> ! [V4v: $i] : ( ap(f102(A_27b,A_27a,V1M,V0R,V3f),V4v) = ap(ap(V1M,ap(ap(ap(c_2Erelation_2ERESTRICT(A_27a,A_27b),V3f),V0R),V4v)),V4v) ) ) ) ) ).
tff(lamtp_f103,type,
f103: ( del * del * $i * $i ) > $i ).
tff(lameq_f103,axiom,
! [A_27b: del,A_27a: del,V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V3f: $i] : ( ap(f103(A_27b,A_27a,V1M,V0R),V3f) = f102(A_27b,A_27a,V1M,V0R,V3f) ) ) ) ).
tff(lamtp_f104,type,
f104: ( del * del * $i * $i ) > $i ).
tff(lameq_f104,axiom,
! [A_27b: del,A_27a: del,V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V2x: $i] : ( ap(f104(A_27b,A_27a,V1M,V0R),V2x) = ap(ap(V1M,ap(ap(ap(c_2Erelation_2ERESTRICT(A_27a,A_27b),ap(ap(ap(c_2Erelation_2Ethe__fun(A_27a,A_27b),ap(c_2Erelation_2ETC(A_27a),V0R)),f103(A_27b,A_27a,V1M,V0R)),V2x)),V0R),V2x)),V2x) ) ) ) ).
tff(ax_thm_2Erelation_2EWFREC__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ( ap(ap(c_2Erelation_2EWFREC(A_27a,A_27b),V0R),V1M) = f104(A_27b,A_27a,V1M,V0R) ) ) ) ).
tff(conj_thm_2Erelation_2EWFREC__THM,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ap(ap(ap(c_2Erelation_2EWFREC(A_27a,A_27b),V0R),V1M),V2x) = ap(ap(V1M,ap(ap(ap(c_2Erelation_2ERESTRICT(A_27a,A_27b),ap(ap(c_2Erelation_2EWFREC(A_27a,A_27b),V0R),V1M)),V0R),V2x)),V2x) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EWFREC__COROLLARY,axiom,
! [A_27a: del,A_27b: del,V0M: $i] :
( mem(V0M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( V2f = ap(ap(c_2Erelation_2EWFREC(A_27a,A_27b),V1R),V0M) )
=> ( p(ap(c_2Erelation_2EWF(A_27a),V1R))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ap(V2f,V3x) = ap(ap(V0M,ap(ap(ap(c_2Erelation_2ERESTRICT(A_27a,A_27b),V2f),V1R),V3x)),V3x) ) ) ) ) ) ) ) ).
tff(lamtp_f105,type,
f105: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f105,axiom,
! [A_27b: del,A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3x: $i] : ( ap(f105(A_27b,A_27a,V0R,V1M,V2f),V3x) = ap(ap(c_2Emin_2E_3D(A_27b),ap(V2f,V3x)),ap(ap(V1M,ap(ap(ap(c_2Erelation_2ERESTRICT(A_27a,A_27b),V2f),V0R),V3x)),V3x)) ) ) ) ) ).
tff(lamtp_f106,type,
f106: ( del * del * $i * $i ) > $i ).
tff(lameq_f106,axiom,
! [A_27b: del,A_27a: del,V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V2f: $i] : ( ap(f106(A_27b,A_27a,V1M,V0R),V2f) = ap(c_2Ebool_2E_21(A_27a),f105(A_27b,A_27a,V0R,V1M,V2f)) ) ) ) ).
tff(conj_thm_2Erelation_2EWF__RECURSION__THM,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
=> ! [V1M: $i] :
( mem(V1M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> p(ap(c_2Ebool_2E_3F_21(arr(A_27a,A_27b)),f106(A_27b,A_27a,V1M,V0R))) ) ) ) ).
tff(ax_thm_2Erelation_2EWFP__DEF,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ( p(ap(ap(c_2Erelation_2EWFP(A_27a),V0R),V1a))
<=> ! [V2P: $i] :
( mem(V2P,arr(A_27a,bool))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( p(ap(ap(V0R,V4y),V3x))
=> p(ap(V2P,V4y)) ) )
=> p(ap(V2P,V3x)) ) )
=> p(ap(V2P,V1a)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EWFP__RULES,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V2y),V1x))
=> p(ap(ap(c_2Erelation_2EWFP(A_27a),V0R),V2y)) ) )
=> p(ap(ap(c_2Erelation_2EWFP(A_27a),V0R),V1x)) ) ) ) ).
tff(conj_thm_2Erelation_2EWFP__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R,V3y),V2x))
=> p(ap(V1P,V3y)) ) )
=> p(ap(V1P,V2x)) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Erelation_2EWFP(A_27a),V0R),V4x))
=> p(ap(V1P,V4x)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EWFP__CASES,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Erelation_2EWFP(A_27a),V0R),V1x))
<=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V2y),V1x))
=> p(ap(ap(c_2Erelation_2EWFP(A_27a),V0R),V2y)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EWFP__STRONG__INDUCT,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ( p(ap(ap(c_2Erelation_2EWFP(A_27a),V1R),V2x))
& ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V1R,V3y),V2x))
=> p(ap(V0P,V3y)) ) ) )
=> p(ap(V0P,V2x)) ) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Erelation_2EWFP(A_27a),V1R),V4x))
=> p(ap(V0P,V4x)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EWF__EQ__WFP,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Erelation_2EWFP(A_27a),V0R),V1x)) ) ) ) ).
tff(ax_thm_2Erelation_2EINDUCTIVE__INVARIANT__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27b,bool)))
=> ! [V2M: $i] :
( mem(V2M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ( p(ap(ap(ap(c_2Erelation_2EINDUCTIVE__INVARIANT(A_27a,A_27b),V0R),V1P),V2M))
<=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( p(ap(ap(V0R,V5y),V4x))
=> p(ap(ap(V1P,V5y),ap(V3f,V5y))) ) )
=> p(ap(ap(V1P,V4x),ap(ap(V2M,V3f),V4x))) ) ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2EINDUCTIVE__INVARIANT__ON__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1D: $i] :
( mem(V1D,arr(A_27a,bool))
=> ! [V2P: $i] :
( mem(V2P,arr(A_27a,arr(A_27b,bool)))
=> ! [V3M: $i] :
( mem(V3M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ( p(ap(ap(ap(ap(c_2Erelation_2EINDUCTIVE__INVARIANT__ON(A_27a,A_27b),V0R),V1D),V2P),V3M))
<=> ! [V4f: $i] :
( mem(V4f,arr(A_27a,A_27b))
=> ! [V5x: $i] :
( mem(V5x,A_27a)
=> ( ( p(ap(V1D,V5x))
& ! [V6y: $i] :
( mem(V6y,A_27a)
=> ( p(ap(V1D,V6y))
=> ( p(ap(ap(V0R,V6y),V5x))
=> p(ap(ap(V2P,V6y),ap(V4f,V6y))) ) ) ) )
=> p(ap(ap(V2P,V5x),ap(ap(V3M,V4f),V5x))) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EINDUCTIVE__INVARIANT__WFREC,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27b,bool)))
=> ! [V2M: $i] :
( mem(V2M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ( ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
& p(ap(ap(ap(c_2Erelation_2EINDUCTIVE__INVARIANT(A_27a,A_27b),V0R),V1P),V2M)) )
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> p(ap(ap(V1P,V3x),ap(ap(ap(c_2Erelation_2EWFREC(A_27a,A_27b),V0R),V2M),V3x))) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETFL__INDUCTIVE__INVARIANT__WFREC,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ! [V2P: $i] :
( mem(V2P,arr(A_27a,arr(A_27b,bool)))
=> ! [V3M: $i] :
( mem(V3M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( ( ( V0f = ap(ap(c_2Erelation_2EWFREC(A_27a,A_27b),V1R),V3M) )
& p(ap(c_2Erelation_2EWF(A_27a),V1R))
& p(ap(ap(ap(c_2Erelation_2EINDUCTIVE__INVARIANT(A_27a,A_27b),V1R),V2P),V3M)) )
=> p(ap(ap(V2P,V4x),ap(V0f,V4x))) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EINDUCTIVE__INVARIANT__ON__WFREC,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,arr(A_27b,bool)))
=> ! [V2M: $i] :
( mem(V2M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V3D: $i] :
( mem(V3D,arr(A_27a,bool))
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
& p(ap(ap(ap(ap(c_2Erelation_2EINDUCTIVE__INVARIANT__ON(A_27a,A_27b),V0R),V3D),V1P),V2M))
& p(ap(V3D,V4x)) )
=> p(ap(ap(V1P,V4x),ap(ap(ap(c_2Erelation_2EWFREC(A_27a,A_27b),V0R),V2M),V4x))) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ETFL__INDUCTIVE__INVARIANT__ON__WFREC,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ! [V2D: $i] :
( mem(V2D,arr(A_27a,bool))
=> ! [V3P: $i] :
( mem(V3P,arr(A_27a,arr(A_27b,bool)))
=> ! [V4M: $i] :
( mem(V4M,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V5x: $i] :
( mem(V5x,A_27a)
=> ( ( ( V0f = ap(ap(c_2Erelation_2EWFREC(A_27a,A_27b),V1R),V4M) )
& p(ap(c_2Erelation_2EWF(A_27a),V1R))
& p(ap(ap(ap(ap(c_2Erelation_2EINDUCTIVE__INVARIANT__ON(A_27a,A_27b),V1R),V2D),V3P),V4M))
& p(ap(V2D,V5x)) )
=> p(ap(ap(V3P,V5x),ap(V0f,V5x))) ) ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2Einv__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27b)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2Einv(A_27a,A_27b),V0R),V1x),V2y))
<=> p(ap(ap(V0R,V2y),V1x)) ) ) ) ) ).
tff(conj_thm_2Erelation_2Einv__inv,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ( ap(c_2Erelation_2Einv(A_27b,A_27a),ap(c_2Erelation_2Einv(A_27a,A_27b),V0R)) = V0R ) ) ).
tff(conj_thm_2Erelation_2Einv__RC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2Einv(A_27a,A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)) = ap(c_2Erelation_2ERC(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Einv__SC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ( ap(c_2Erelation_2Einv(A_27a,A_27a),ap(c_2Erelation_2ESC(A_27a),V0R)) = ap(c_2Erelation_2ESC(A_27a),V0R) )
& ( ap(c_2Erelation_2ESC(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)) = ap(c_2Erelation_2ESC(A_27a),V0R) ) ) ) ).
tff(conj_thm_2Erelation_2Einv__TC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2Einv(A_27a,A_27a),ap(c_2Erelation_2ETC(A_27a),V0R)) = ap(c_2Erelation_2ETC(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Einv__EQC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ( ap(c_2Erelation_2Einv(A_27a,A_27a),ap(c_2Erelation_2EEQC(A_27a),V0R)) = ap(c_2Erelation_2EEQC(A_27a),V0R) )
& ( ap(c_2Erelation_2EEQC(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)) = ap(c_2Erelation_2EEQC(A_27a),V0R) ) ) ) ).
tff(conj_thm_2Erelation_2Einv__MOVES__OUT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ( ap(c_2Erelation_2Einv(A_27a,A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)) = V0R )
& ( ap(c_2Erelation_2ESC(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)) = ap(c_2Erelation_2ESC(A_27a),V0R) )
& ( ap(c_2Erelation_2ERC(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)) = ap(c_2Erelation_2Einv(A_27a,A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)) )
& ( ap(c_2Erelation_2ETC(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)) = ap(c_2Erelation_2Einv(A_27a,A_27a),ap(c_2Erelation_2ETC(A_27a),V0R)) )
& ( ap(c_2Erelation_2ERTC(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)) = ap(c_2Erelation_2Einv(A_27a,A_27a),ap(c_2Erelation_2ERTC(A_27a),V0R)) )
& ( ap(c_2Erelation_2EEQC(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)) = ap(c_2Erelation_2EEQC(A_27a),V0R) ) ) ) ).
tff(conj_thm_2Erelation_2Ereflexive__inv,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Ereflexive(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)))
<=> p(ap(c_2Erelation_2Ereflexive(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Eirreflexive__inv,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Eirreflexive(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)))
<=> p(ap(c_2Erelation_2Eirreflexive(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Esymmetric__inv,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Esymmetric(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)))
<=> p(ap(c_2Erelation_2Esymmetric(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Eantisymmetric__inv,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Eantisymmetric(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)))
<=> p(ap(c_2Erelation_2Eantisymmetric(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Etransitive__inv,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Etransitive(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)))
<=> p(ap(c_2Erelation_2Etransitive(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Esymmetric__inv__identity,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Esymmetric(A_27a),V0R))
=> ( ap(c_2Erelation_2Einv(A_27a,A_27a),V0R) = V0R ) ) ) ).
tff(conj_thm_2Erelation_2Eequivalence__inv__identity,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Eequivalence(A_27a),V0R))
=> ( ap(c_2Erelation_2Einv(A_27a,A_27a),V0R) = V0R ) ) ) ).
tff(ax_thm_2Erelation_2EINVOL__DEF,axiom,
! [A_27z: del,V0f: $i] :
( mem(V0f,arr(A_27z,A_27z))
=> ( p(ap(c_2Erelation_2EINVOL(A_27z),V0f))
<=> ( ap(ap(c_2Ecombin_2Eo(A_27z,A_27z,A_27z),V0f),V0f) = c_2Ecombin_2EI(A_27z) ) ) ) ).
tff(conj_thm_2Erelation_2EINVOL,axiom,
! [A_27z: del,V0f: $i] :
( mem(V0f,arr(A_27z,A_27z))
=> ( p(ap(c_2Erelation_2EINVOL(A_27z),V0f))
<=> ! [V1x: $i] :
( mem(V1x,A_27z)
=> ( ap(V0f,ap(V0f,V1x)) = V1x ) ) ) ) ).
tff(conj_thm_2Erelation_2EINVOL__ONE__ONE,axiom,
! [A_27z: del,V0f: $i] :
( mem(V0f,arr(A_27z,A_27z))
=> ( p(ap(c_2Erelation_2EINVOL(A_27z),V0f))
=> ! [V1a: $i] :
( mem(V1a,A_27z)
=> ! [V2b: $i] :
( mem(V2b,A_27z)
=> ( ( ap(V0f,V1a) = ap(V0f,V2b) )
<=> ( V1a = V2b ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EINVOL__ONE__ENO,axiom,
! [A_27z: del,V0f: $i] :
( mem(V0f,arr(A_27z,A_27z))
=> ( p(ap(c_2Erelation_2EINVOL(A_27z),V0f))
=> ! [V1a: $i] :
( mem(V1a,A_27z)
=> ! [V2b: $i] :
( mem(V2b,A_27z)
=> ( ( ap(V0f,V1a) = V2b )
<=> ( V1a = ap(V0f,V2b) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ENOT__INVOL,axiom,
p(ap(c_2Erelation_2EINVOL(bool),c_2Ebool_2E_7E)) ).
tff(ax_thm_2Erelation_2EIDEM__DEF,axiom,
! [A_27z: del,V0f: $i] :
( mem(V0f,arr(A_27z,A_27z))
=> ( p(ap(c_2Erelation_2EIDEM(A_27z),V0f))
<=> ( ap(ap(c_2Ecombin_2Eo(A_27z,A_27z,A_27z),V0f),V0f) = V0f ) ) ) ).
tff(conj_thm_2Erelation_2EIDEM,axiom,
! [A_27z: del,V0f: $i] :
( mem(V0f,arr(A_27z,A_27z))
=> ( p(ap(c_2Erelation_2EIDEM(A_27z),V0f))
<=> ! [V1x: $i] :
( mem(V1x,A_27z)
=> ( ap(V0f,ap(V0f,V1x)) = ap(V0f,V1x) ) ) ) ) ).
tff(conj_thm_2Erelation_2Einv__INVOL,axiom,
! [A_27a: del] : p(ap(c_2Erelation_2EINVOL(arr(A_27a,arr(A_27a,bool))),c_2Erelation_2Einv(A_27a,A_27a))) ).
tff(ax_thm_2Erelation_2EO__DEF,axiom,
! [A_27g: del,A_27h: del,A_27k: del,V0R1: $i] :
( mem(V0R1,arr(A_27h,arr(A_27k,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27g,arr(A_27h,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27g)
=> ! [V3z: $i] :
( mem(V3z,A_27k)
=> ( p(ap(ap(ap(ap(c_2Erelation_2EO(A_27g,A_27h,A_27k),V0R1),V1R2),V2x),V3z))
<=> ? [V4y: $i] :
( mem(V4y,A_27h)
& p(ap(ap(V1R2,V2x),V4y))
& p(ap(ap(V0R1,V4y),V3z)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2Einv__O,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1R_27: $i] :
( mem(V1R_27,arr(A_27c,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2Einv(A_27c,A_27b),ap(ap(c_2Erelation_2EO(A_27c,A_27a,A_27b),V0R),V1R_27)) = ap(ap(c_2Erelation_2EO(A_27b,A_27a,A_27c),ap(c_2Erelation_2Einv(A_27c,A_27a),V1R_27)),ap(c_2Erelation_2Einv(A_27a,A_27b),V0R)) ) ) ) ).
tff(ax_thm_2Erelation_2ERSUBSET,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27b,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27b),V0R1),V1R2))
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27b)
=> ( p(ap(ap(V0R1,V2x),V3y))
=> p(ap(ap(V1R2,V2x),V3y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2Eirreflexive__RSUBSET,axiom,
! [A_27a: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27a,bool)))
=> ( ( p(ap(c_2Erelation_2Eirreflexive(A_27a),V1R2))
& p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27a),V0R1),V1R2)) )
=> p(ap(c_2Erelation_2Eirreflexive(A_27a),V0R1)) ) ) ) ).
tff(ax_thm_2Erelation_2ERUNION,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27b,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27b)
=> ( p(ap(ap(ap(ap(c_2Erelation_2ERUNION(A_27a,A_27b),V0R1),V1R2),V2x),V3y))
<=> ( p(ap(ap(V0R1,V2x),V3y))
| p(ap(ap(V1R2,V2x),V3y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERUNION__COMM,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27b,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27b,bool)))
=> ( ap(ap(c_2Erelation_2ERUNION(A_27a,A_27b),V0R1),V1R2) = ap(ap(c_2Erelation_2ERUNION(A_27a,A_27b),V1R2),V0R1) ) ) ) ).
tff(conj_thm_2Erelation_2ERUNION__ASSOC,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27b,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27b,bool)))
=> ! [V2R3: $i] :
( mem(V2R3,arr(A_27a,arr(A_27b,bool)))
=> ( ap(ap(c_2Erelation_2ERUNION(A_27a,A_27b),V0R1),ap(ap(c_2Erelation_2ERUNION(A_27a,A_27b),V1R2),V2R3)) = ap(ap(c_2Erelation_2ERUNION(A_27a,A_27b),ap(ap(c_2Erelation_2ERUNION(A_27a,A_27b),V0R1),V1R2)),V2R3) ) ) ) ) ).
tff(ax_thm_2Erelation_2ERINTER,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27b,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27b)
=> ( p(ap(ap(ap(ap(c_2Erelation_2ERINTER(A_27a,A_27b),V0R1),V1R2),V2x),V3y))
<=> ( p(ap(ap(V0R1,V2x),V3y))
& p(ap(ap(V1R2,V2x),V3y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERINTER__COMM,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27b,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27b,bool)))
=> ( ap(ap(c_2Erelation_2ERINTER(A_27a,A_27b),V0R1),V1R2) = ap(ap(c_2Erelation_2ERINTER(A_27a,A_27b),V1R2),V0R1) ) ) ) ).
tff(conj_thm_2Erelation_2ERINTER__ASSOC,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27b,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27b,bool)))
=> ! [V2R3: $i] :
( mem(V2R3,arr(A_27a,arr(A_27b,bool)))
=> ( ap(ap(c_2Erelation_2ERINTER(A_27a,A_27b),V0R1),ap(ap(c_2Erelation_2ERINTER(A_27a,A_27b),V1R2),V2R3)) = ap(ap(c_2Erelation_2ERINTER(A_27a,A_27b),ap(ap(c_2Erelation_2ERINTER(A_27a,A_27b),V0R1),V1R2)),V2R3) ) ) ) ) ).
tff(conj_thm_2Erelation_2Eantisymmetric__RINTER,axiom,
! [A_27a: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27a,bool)))
=> ( ( p(ap(c_2Erelation_2Eantisymmetric(A_27a),V0R1))
=> p(ap(c_2Erelation_2Eantisymmetric(A_27a),ap(ap(c_2Erelation_2ERINTER(A_27a,A_27a),V0R1),V1R2))) )
& ( p(ap(c_2Erelation_2Eantisymmetric(A_27a),V1R2))
=> p(ap(c_2Erelation_2Eantisymmetric(A_27a),ap(ap(c_2Erelation_2ERINTER(A_27a,A_27a),V0R1),V1R2))) ) ) ) ) ).
tff(conj_thm_2Erelation_2Etransitive__RINTER,axiom,
! [A_27a: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27a,bool)))
=> ( ( p(ap(c_2Erelation_2Etransitive(A_27a),V0R1))
& p(ap(c_2Erelation_2Etransitive(A_27a),V1R2)) )
=> p(ap(c_2Erelation_2Etransitive(A_27a),ap(ap(c_2Erelation_2ERINTER(A_27a,A_27a),V0R1),V1R2))) ) ) ) ).
tff(ax_thm_2Erelation_2ERCOMPL,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27b)
=> ( p(ap(ap(ap(c_2Erelation_2ERCOMPL(A_27a,A_27b),V0R),V1x),V2y))
<=> ~ p(ap(ap(V0R,V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2Ereflexive__Id__RSUBSET,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Ereflexive(A_27a),V0R))
<=> p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27a),c_2Emin_2E_3D(A_27a)),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Esymmetric__inv__RSUBSET,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Esymmetric(A_27a),V0R))
<=> p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R)),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Etransitive__O__RSUBSET,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Etransitive(A_27a),V0R))
<=> p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27a),ap(ap(c_2Erelation_2EO(A_27a,A_27a,A_27a),V0R),V0R)),V0R)) ) ) ).
tff(ax_thm_2Erelation_2EPreOrder,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EPreOrder(A_27a),V0R))
<=> ( p(ap(c_2Erelation_2Ereflexive(A_27a),V0R))
& p(ap(c_2Erelation_2Etransitive(A_27a),V0R)) ) ) ) ).
tff(ax_thm_2Erelation_2EOrder,axiom,
! [A_27g: del,V0Z: $i] :
( mem(V0Z,arr(A_27g,arr(A_27g,bool)))
=> ( p(ap(c_2Erelation_2EOrder(A_27g),V0Z))
<=> ( p(ap(c_2Erelation_2Eantisymmetric(A_27g),V0Z))
& p(ap(c_2Erelation_2Etransitive(A_27g),V0Z)) ) ) ) ).
tff(ax_thm_2Erelation_2EWeakOrder,axiom,
! [A_27g: del,V0Z: $i] :
( mem(V0Z,arr(A_27g,arr(A_27g,bool)))
=> ( p(ap(c_2Erelation_2EWeakOrder(A_27g),V0Z))
<=> ( p(ap(c_2Erelation_2Ereflexive(A_27g),V0Z))
& p(ap(c_2Erelation_2Eantisymmetric(A_27g),V0Z))
& p(ap(c_2Erelation_2Etransitive(A_27g),V0Z)) ) ) ) ).
tff(ax_thm_2Erelation_2EStrongOrder,axiom,
! [A_27g: del,V0Z: $i] :
( mem(V0Z,arr(A_27g,arr(A_27g,bool)))
=> ( p(ap(c_2Erelation_2EStrongOrder(A_27g),V0Z))
<=> ( p(ap(c_2Erelation_2Eirreflexive(A_27g),V0Z))
& p(ap(c_2Erelation_2Etransitive(A_27g),V0Z)) ) ) ) ).
tff(conj_thm_2Erelation_2Eirrefl__trans__implies__antisym,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ( p(ap(c_2Erelation_2Eirreflexive(A_27a),V0R))
& p(ap(c_2Erelation_2Etransitive(A_27a),V0R)) )
=> p(ap(c_2Erelation_2Eantisymmetric(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2EStrongOrd__Ord,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EStrongOrder(A_27a),V0R))
=> p(ap(c_2Erelation_2EOrder(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2EWeakOrd__Ord,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWeakOrder(A_27a),V0R))
=> p(ap(c_2Erelation_2EOrder(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2EWeakOrder__EQ,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWeakOrder(A_27a),V0R))
=> ! [V1y: $i] :
( mem(V1y,A_27a)
=> ! [V2z: $i] :
( mem(V2z,A_27a)
=> ( ( V1y = V2z )
<=> ( p(ap(ap(V0R,V1y),V2z))
& p(ap(ap(V0R,V2z),V1y)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ERSUBSET__ANTISYM,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27b,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27b,bool)))
=> ( ( p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27b),V0R1),V1R2))
& p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27b),V1R2),V0R1)) )
=> ( V0R1 = V1R2 ) ) ) ) ).
tff(conj_thm_2Erelation_2ERSUBSET__antisymmetric,axiom,
! [A_27a: del,A_27b: del] : p(ap(c_2Erelation_2Eantisymmetric(arr(A_27a,arr(A_27b,bool))),c_2Erelation_2ERSUBSET(A_27a,A_27b))) ).
tff(conj_thm_2Erelation_2ERSUBSET__WeakOrder,axiom,
! [A_27a: del,A_27b: del] : p(ap(c_2Erelation_2EWeakOrder(arr(A_27a,arr(A_27b,bool))),c_2Erelation_2ERSUBSET(A_27a,A_27b))) ).
tff(conj_thm_2Erelation_2EEqIsBothRSUBSET,axiom,
! [A_27a: del,A_27b: del,V0y: $i] :
( mem(V0y,arr(A_27a,arr(A_27b,bool)))
=> ! [V1z: $i] :
( mem(V1z,arr(A_27a,arr(A_27b,bool)))
=> ( ( V0y = V1z )
<=> ( p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27b),V0y),V1z))
& p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27b),V1z),V0y)) ) ) ) ) ).
tff(ax_thm_2Erelation_2ESTRORD,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ESTRORD(A_27a),V0R),V1a),V2b))
<=> ( p(ap(ap(V0R,V1a),V2b))
& ( V1a != V2b ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2ESTRORD__AND__NOT__Id,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2ESTRORD(A_27a),V0R) = ap(ap(c_2Erelation_2ERINTER(A_27a,A_27a),V0R),ap(c_2Erelation_2ERCOMPL(A_27a,A_27a),c_2Emin_2E_3D(A_27a))) ) ) ).
tff(conj_thm_2Erelation_2ERC__OR__Id,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Erelation_2ERC(A_27a),V0R) = ap(ap(c_2Erelation_2ERUNION(A_27a,A_27a),V0R),c_2Emin_2E_3D(A_27a)) ) ) ).
tff(conj_thm_2Erelation_2ERC__Weak,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EOrder(A_27a),V0R))
<=> p(ap(c_2Erelation_2EWeakOrder(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R))) ) ) ).
tff(conj_thm_2Erelation_2ESTRORD__Strong,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EOrder(A_27a),V0R))
<=> p(ap(c_2Erelation_2EStrongOrder(A_27a),ap(c_2Erelation_2ESTRORD(A_27a),V0R))) ) ) ).
tff(conj_thm_2Erelation_2ESTRORD__RC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EStrongOrder(A_27a),V0R))
=> ( ap(c_2Erelation_2ESTRORD(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)) = V0R ) ) ) ).
tff(conj_thm_2Erelation_2ERC__STRORD,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWeakOrder(A_27a),V0R))
=> ( ap(c_2Erelation_2ERC(A_27a),ap(c_2Erelation_2ESTRORD(A_27a),V0R)) = V0R ) ) ) ).
tff(conj_thm_2Erelation_2EIDEM__STRORD,axiom,
! [A_27a: del] : p(ap(c_2Erelation_2EIDEM(arr(A_27a,arr(A_27a,bool))),c_2Erelation_2ESTRORD(A_27a))) ).
tff(conj_thm_2Erelation_2EIDEM__RC,axiom,
! [A_27a: del] : p(ap(c_2Erelation_2EIDEM(arr(A_27a,arr(A_27a,bool))),c_2Erelation_2ERC(A_27a))) ).
tff(conj_thm_2Erelation_2EIDEM__SC,axiom,
! [A_27a: del] : p(ap(c_2Erelation_2EIDEM(arr(A_27a,arr(A_27a,bool))),c_2Erelation_2ESC(A_27a))) ).
tff(conj_thm_2Erelation_2EIDEM__TC,axiom,
! [A_27a: del] : p(ap(c_2Erelation_2EIDEM(arr(A_27a,arr(A_27a,bool))),c_2Erelation_2ETC(A_27a))) ).
tff(conj_thm_2Erelation_2EIDEM__RTC,axiom,
! [A_27a: del] : p(ap(c_2Erelation_2EIDEM(arr(A_27a,arr(A_27a,bool))),c_2Erelation_2ERTC(A_27a))) ).
tff(conj_thm_2Erelation_2Etrichotomous__STRORD,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Etrichotomous(A_27a),ap(c_2Erelation_2ESTRORD(A_27a),V0R)))
<=> p(ap(c_2Erelation_2Etrichotomous(A_27a),V0R)) ) ) ).
tff(conj_thm_2Erelation_2Etrichotomous__RC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Etrichotomous(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R)))
<=> p(ap(c_2Erelation_2Etrichotomous(A_27a),V0R)) ) ) ).
tff(ax_thm_2Erelation_2ELinearOrder,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2ELinearOrder(A_27a),V0R))
<=> ( p(ap(c_2Erelation_2EOrder(A_27a),V0R))
& p(ap(c_2Erelation_2Etrichotomous(A_27a),V0R)) ) ) ) ).
tff(ax_thm_2Erelation_2EStrongLinearOrder,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EStrongLinearOrder(A_27a),V0R))
<=> ( p(ap(c_2Erelation_2EStrongOrder(A_27a),V0R))
& p(ap(c_2Erelation_2Etrichotomous(A_27a),V0R)) ) ) ) ).
tff(ax_thm_2Erelation_2EWeakLinearOrder,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWeakLinearOrder(A_27a),V0R))
<=> ( p(ap(c_2Erelation_2EWeakOrder(A_27a),V0R))
& p(ap(c_2Erelation_2Etrichotomous(A_27a),V0R)) ) ) ) ).
tff(conj_thm_2Erelation_2EWeakLinearOrder__dichotomy,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWeakLinearOrder(A_27a),V0R))
<=> ( p(ap(c_2Erelation_2EWeakOrder(A_27a),V0R))
& ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ( p(ap(ap(V0R,V1a),V2b))
| p(ap(ap(V0R,V2b),V1a)) ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2Ediag__def,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2Ediag(A_27a),V0A),V1x),V2y))
<=> ( ( V1x = V2y )
& p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0A)) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EO__ASSOC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27c,arr(A_27b,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27d,arr(A_27c,bool)))
=> ! [V2R3: $i] :
( mem(V2R3,arr(A_27a,arr(A_27d,bool)))
=> ( ap(ap(c_2Erelation_2EO(A_27a,A_27c,A_27b),V0R1),ap(ap(c_2Erelation_2EO(A_27a,A_27d,A_27c),V1R2),V2R3)) = ap(ap(c_2Erelation_2EO(A_27a,A_27d,A_27b),ap(ap(c_2Erelation_2EO(A_27d,A_27c,A_27b),V0R1),V1R2)),V2R3) ) ) ) ) ).
tff(conj_thm_2Erelation_2EId__O,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ( ap(ap(c_2Erelation_2EO(A_27a,A_27b,A_27b),c_2Emin_2E_3D(A_27b)),V0R) = V0R ) ) ).
tff(conj_thm_2Erelation_2EO__Id,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ( ap(ap(c_2Erelation_2EO(A_27a,A_27a,A_27b),V0R),c_2Emin_2E_3D(A_27a)) = V0R ) ) ).
tff(conj_thm_2Erelation_2EO__MONO,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27b,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27a,arr(A_27b,bool)))
=> ! [V2S1: $i] :
( mem(V2S1,arr(A_27c,arr(A_27a,bool)))
=> ! [V3S2: $i] :
( mem(V3S2,arr(A_27c,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27b),V0R1),V1R2))
& p(ap(ap(c_2Erelation_2ERSUBSET(A_27c,A_27a),V2S1),V3S2)) )
=> p(ap(ap(c_2Erelation_2ERSUBSET(A_27c,A_27b),ap(ap(c_2Erelation_2EO(A_27c,A_27a,A_27b),V0R1),V2S1)),ap(ap(c_2Erelation_2EO(A_27c,A_27a,A_27b),V1R2),V3S2))) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2Einv__Id,axiom,
! [A_27a: del] : ( ap(c_2Erelation_2Einv(A_27a,A_27a),c_2Emin_2E_3D(A_27a)) = c_2Emin_2E_3D(A_27a) ) ).
tff(conj_thm_2Erelation_2Einv__diag,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(A_27a,bool))
=> ( ap(c_2Erelation_2Einv(A_27a,A_27a),ap(c_2Erelation_2Ediag(A_27a),V0A)) = ap(c_2Erelation_2Ediag(A_27a),V0A) ) ) ).
tff(ax_thm_2Erelation_2ERDOM__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Erelation_2ERDOM(A_27a,A_27b),V0R),V1x))
<=> ? [V2y: $i] :
( mem(V2y,A_27b)
& p(ap(ap(V0R,V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2EIN__RDOM,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Erelation_2ERDOM(A_27a,A_27b),V1R)))
<=> ? [V2y: $i] :
( mem(V2y,A_27b)
& p(ap(ap(V1R,V0x),V2y)) ) ) ) ) ).
tff(ax_thm_2Erelation_2ERRANGE,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1y: $i] :
( mem(V1y,A_27b)
=> ( p(ap(ap(c_2Erelation_2ERRANGE(A_27a,A_27b),V0R),V1y))
<=> ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(ap(V0R,V2x),V1y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2EIN__RRANGE,axiom,
! [A_27a: del,A_27b: del,V0y: $i] :
( mem(V0y,A_27a)
=> ! [V1R: $i] :
( mem(V1R,arr(A_27b,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0y),ap(c_2Erelation_2ERRANGE(A_27b,A_27a),V1R)))
<=> ? [V2x: $i] :
( mem(V2x,A_27b)
& p(ap(ap(V1R,V2x),V0y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2EIN__RDOM__RUNION,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1R1: $i] :
( mem(V1R1,arr(A_27a,arr(A_27b,bool)))
=> ! [V2R2: $i] :
( mem(V2R2,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Erelation_2ERDOM(A_27a,A_27b),ap(ap(c_2Erelation_2ERUNION(A_27a,A_27b),V1R1),V2R2))))
<=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Erelation_2ERDOM(A_27a,A_27b),V1R1)))
| p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Erelation_2ERDOM(A_27a,A_27b),V2R2))) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2ERUNIV,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27b)
=> ( p(ap(ap(c_2Erelation_2ERUNIV(A_27a,A_27b),V0x),V1y))
<=> $true ) ) ) ).
tff(conj_thm_2Erelation_2ERUNIV__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ( ( p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27b),c_2Erelation_2ERUNIV(A_27a,A_27b)),V0R))
<=> ( V0R = c_2Erelation_2ERUNIV(A_27a,A_27b) ) )
& p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27b),V0R),c_2Erelation_2ERUNIV(A_27a,A_27b))) ) ) ).
tff(conj_thm_2Erelation_2EREMPTY__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27a),c_2Erelation_2EEMPTY__REL(A_27a)),V0R))
& ( p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27a),V0R),c_2Erelation_2EEMPTY__REL(A_27a)))
<=> ( V0R = c_2Erelation_2EEMPTY__REL(A_27a) ) ) ) ) ).
tff(ax_thm_2Erelation_2ERRESTRICT__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27b)
=> ( p(ap(ap(ap(ap(c_2Erelation_2ERRESTRICT(A_27a,A_27b),V0R),V1s),V2x),V3y))
<=> ( p(ap(ap(V0R,V2x),V3y))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EIN__RDOM__RRESTRICT,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27b,bool)))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Erelation_2ERDOM(A_27a,A_27b),ap(ap(c_2Erelation_2ERRESTRICT(A_27a,A_27b),V1R),V2s))))
<=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Erelation_2ERDOM(A_27a,A_27b),V1R)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2s)) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2ERDOM__DELETE__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2u: $i] :
( mem(V2u,A_27a)
=> ! [V3v: $i] :
( mem(V3v,A_27b)
=> ( p(ap(ap(ap(ap(c_2Erelation_2ERDOM__DELETE(A_27a,A_27b),V0R),V1x),V2u),V3v))
<=> ( p(ap(ap(V0R,V2u),V3v))
& ( V2u != V1x ) ) ) ) ) ) ) ).
tff(conj_thm_2Erelation_2EIN__RDOM__DELETE,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27b,bool)))
=> ! [V2k: $i] :
( mem(V2k,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Erelation_2ERDOM(A_27a,A_27b),ap(ap(c_2Erelation_2ERDOM__DELETE(A_27a,A_27b),V1R),V2k))))
<=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Erelation_2ERDOM(A_27a,A_27b),V1R)))
& ( V0x != V2k ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2Ediamond__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Ediamond(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( ( p(ap(ap(V0R,V1x),V2y))
& p(ap(ap(V0R,V1x),V3z)) )
=> ? [V4u: $i] :
( mem(V4u,A_27a)
& p(ap(ap(V0R,V2y),V4u))
& p(ap(ap(V0R,V3z),V4u)) ) ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2Ercdiamond__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Ercdiamond(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( ( p(ap(ap(V0R,V1x),V2y))
& p(ap(ap(V0R,V1x),V3z)) )
=> ? [V4u: $i] :
( mem(V4u,A_27a)
& p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V0R),V2y),V4u))
& p(ap(ap(ap(c_2Erelation_2ERC(A_27a),V0R),V3z),V4u)) ) ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2ECR__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2ECR(A_27a),V0R))
<=> p(ap(c_2Erelation_2Ediamond(A_27a),ap(c_2Erelation_2ERTC(A_27a),V0R))) ) ) ).
tff(ax_thm_2Erelation_2EWCR__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWCR(A_27a),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( ( p(ap(ap(V0R,V1x),V2y))
& p(ap(ap(V0R,V1x),V3z)) )
=> ? [V4u: $i] :
( mem(V4u,A_27a)
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V2y),V4u))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V3z),V4u)) ) ) ) ) ) ) ) ).
tff(ax_thm_2Erelation_2ESN__def,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2ESN(A_27a),V0R))
<=> p(ap(c_2Erelation_2EWF(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R))) ) ) ).
tff(ax_thm_2Erelation_2Enf__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Erelation_2Enf(A_27a,A_27b),V0R),V1x))
<=> ! [V2y: $i] :
( mem(V2y,A_27b)
=> ~ p(ap(ap(V0R,V1x),V2y)) ) ) ) ) ).
tff(conj_thm_2Erelation_2Ercdiamond__diamond,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Ercdiamond(A_27a),V0R))
<=> p(ap(c_2Erelation_2Ediamond(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R))) ) ) ).
tff(conj_thm_2Erelation_2Ediamond__RC__diamond,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Ediamond(A_27a),V0R))
=> p(ap(c_2Erelation_2Ediamond(A_27a),ap(c_2Erelation_2ERC(A_27a),V0R))) ) ) ).
tff(conj_thm_2Erelation_2Ediamond__TC__diamond,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2Ediamond(A_27a),V0R))
=> p(ap(c_2Erelation_2Ediamond(A_27a),ap(c_2Erelation_2ETC(A_27a),V0R))) ) ) ).
tff(conj_thm_2Erelation_2Eestablish__CR,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ( p(ap(c_2Erelation_2Ercdiamond(A_27a),V0R))
=> p(ap(c_2Erelation_2ECR(A_27a),V0R)) )
& ( p(ap(c_2Erelation_2Ediamond(A_27a),V0R))
=> p(ap(c_2Erelation_2ECR(A_27a),V0R)) ) ) ) ).
tff(conj_thm_2Erelation_2ENewmans__lemma,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ( p(ap(c_2Erelation_2EWCR(A_27a),V0R))
& p(ap(c_2Erelation_2ESN(A_27a),V0R)) )
=> p(ap(c_2Erelation_2ECR(A_27a),V0R)) ) ) ).
%------------------------------------------------------------------------------