ITP001 Axioms: ITP103_5.ax
%------------------------------------------------------------------------------
% File : ITP103_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 : tc_2.ax [Gau20]
% : HL4103_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 50 ( 8 unt; 8 typ; 0 def)
% Number of atoms : 815 ( 36 equ)
% Maximal formula atoms : 17 ( 16 avg)
% Number of connectives : 123 ( 2 ~; 5 |; 20 &)
% ( 6 <=>; 90 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 652 ( 652 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% Number of predicates : 30 ( 29 usr; 2 prp; 0-3 aty)
% Number of functors : 8 ( 8 usr; 0 con; 1-1 aty)
% Number of variables : 128 ( 124 !; 4 ?; 128 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Etc_2EFMAP__TO__RELN,type,
c_2Etc_2EFMAP__TO__RELN: del > $i ).
tff(mem_c_2Etc_2EFMAP__TO__RELN,axiom,
! [A_27a: del] : mem(c_2Etc_2EFMAP__TO__RELN(A_27a),arr(ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)),arr(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Etc_2ERELN__TO__FMAP,type,
c_2Etc_2ERELN__TO__FMAP: del > $i ).
tff(mem_c_2Etc_2ERELN__TO__FMAP,axiom,
! [A_27a: del] : mem(c_2Etc_2ERELN__TO__FMAP(A_27a),arr(arr(A_27a,arr(A_27a,bool)),ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))) ).
tff(tp_c_2Etc_2ETC__ITER,type,
c_2Etc_2ETC__ITER: del > $i ).
tff(mem_c_2Etc_2ETC__ITER,axiom,
! [A_27a: del] : mem(c_2Etc_2ETC__ITER(A_27a),arr(ty_2Elist_2Elist(A_27a),arr(ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)),ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool))))) ).
tff(tp_c_2Etc_2ETC__MOD,type,
c_2Etc_2ETC__MOD: del > $i ).
tff(mem_c_2Etc_2ETC__MOD,axiom,
! [A_27a: del] : mem(c_2Etc_2ETC__MOD(A_27a),arr(A_27a,arr(arr(A_27a,bool),arr(arr(A_27a,bool),arr(A_27a,bool))))) ).
tff(tp_c_2Etc_2E_5E_7C,type,
c_2Etc_2E_5E_7C: del > $i ).
tff(mem_c_2Etc_2E_5E_7C,axiom,
! [A_27a: del] : mem(c_2Etc_2E_5E_7C(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))) ).
tff(tp_c_2Etc_2E_5E_7C_5E,type,
c_2Etc_2E_5E_7C_5E: del > $i ).
tff(mem_c_2Etc_2E_5E_7C_5E,axiom,
! [A_27a: del] : mem(c_2Etc_2E_5E_7C_5E(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))) ).
tff(tp_c_2Etc_2EsubTC,type,
c_2Etc_2EsubTC: del > $i ).
tff(mem_c_2Etc_2EsubTC,axiom,
! [A_27a: del] : mem(c_2Etc_2EsubTC(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))) ).
tff(tp_c_2Etc_2E_7C_5E,type,
c_2Etc_2E_7C_5E: del > $i ).
tff(mem_c_2Etc_2E_7C_5E,axiom,
! [A_27a: del] : mem(c_2Etc_2E_7C_5E(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))) ).
tff(ax_thm_2Etc_2EDRESTR,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3b: $i] :
( mem(V3b,A_27a)
=> ( p(ap(ap(ap(ap(c_2Etc_2E_5E_7C(A_27a),V0R),V1s),V2a),V3b))
<=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s))
& p(ap(ap(V0R,V2a),V3b)) ) ) ) ) ) ) ).
tff(conj_thm_2Etc_2EDRESTR__IN,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ( ap(ap(ap(c_2Etc_2E_5E_7C(A_27a),V0R),V1s),V2a) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,bool)),ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)),ap(V0R,V2a)),c_2Epred__set_2EEMPTY(A_27a)) ) ) ) ) ).
tff(ax_thm_2Etc_2ERRESTR,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3b: $i] :
( mem(V3b,A_27a)
=> ( p(ap(ap(ap(ap(c_2Etc_2E_7C_5E(A_27a),V0R),V1s),V2a),V3b))
<=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3b),V1s))
& p(ap(ap(V0R,V2a),V3b)) ) ) ) ) ) ) ).
tff(ax_thm_2Etc_2EBRESTR,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ap(ap(c_2Etc_2E_5E_7C_5E(A_27a),V0R),V1s) = ap(ap(c_2Etc_2E_7C_5E(A_27a),ap(ap(c_2Etc_2E_5E_7C(A_27a),V0R),V1s)),V1s) ) ) ) ).
tff(conj_thm_2Etc_2EDRESTR__EMPTY,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(ap(c_2Etc_2E_5E_7C(A_27a),V0R),c_2Epred__set_2EEMPTY(A_27a)) = c_2Erelation_2EEMPTY__REL(A_27a) ) ) ).
tff(conj_thm_2Etc_2EDRESTR__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(ap(c_2Etc_2E_5E_7C(A_27a),V0R),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R)) = V0R ) ) ).
tff(conj_thm_2Etc_2EREMPTY__RRESTR,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( ap(ap(c_2Etc_2E_7C_5E(A_27a),c_2Erelation_2EEMPTY__REL(A_27a)),V0s) = c_2Erelation_2EEMPTY__REL(A_27a) ) ) ).
tff(conj_thm_2Etc_2EO__REMPTY__O,axiom,
! [A_27a: del] :
( ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(ap(c_2Erelation_2EO(A_27a,A_27a,A_27a),V0R),c_2Erelation_2EEMPTY__REL(A_27a)) = c_2Erelation_2EEMPTY__REL(A_27a) ) )
& ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(ap(c_2Erelation_2EO(A_27a,A_27a,A_27a),c_2Erelation_2EEMPTY__REL(A_27a)),V1R) = c_2Erelation_2EEMPTY__REL(A_27a) ) ) ) ).
tff(ax_thm_2Etc_2EsubTC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s),V2x),V3y))
<=> ( p(ap(ap(V0R,V2x),V3y))
| ? [V4a: $i] :
( mem(V4a,A_27a)
& ? [V5b: $i] :
( mem(V5b,A_27a)
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),ap(ap(c_2Etc_2E_5E_7C_5E(A_27a),V0R),V1s)),V4a),V5b))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4a),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5b),V1s))
& p(ap(ap(V0R,V2x),V4a))
& p(ap(ap(V0R,V5b),V3y)) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Etc_2EsubTC__thm,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s) = ap(ap(c_2Erelation_2ERUNION(A_27a,A_27a),V0R),ap(ap(c_2Erelation_2EO(A_27a,A_27a,A_27a),V0R),ap(ap(c_2Erelation_2EO(A_27a,A_27a,A_27a),ap(ap(c_2Etc_2E_5E_7C(A_27a),ap(c_2Erelation_2ERTC(A_27a),ap(ap(c_2Etc_2E_5E_7C_5E(A_27a),V0R),V1s))),V1s)),V0R))) ) ) ) ).
tff(conj_thm_2Etc_2EsubTC__EMPTY,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),c_2Epred__set_2EEMPTY(A_27a)) = V0R ) ) ).
tff(conj_thm_2Etc_2ERTC__INSERT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3w: $i] :
( mem(V3w,A_27a)
=> ! [V4z: $i] :
( mem(V4z,A_27a)
=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),ap(ap(c_2Etc_2E_5E_7C_5E(A_27a),V0R),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2a),V1s))),V3w),V4z))
<=> ( p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),ap(ap(c_2Etc_2E_5E_7C_5E(A_27a),V0R),V1s)),V3w),V4z))
| ( ( ( V2a = V3w )
| ? [V5x: $i] :
( mem(V5x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V1s))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),ap(ap(c_2Etc_2E_5E_7C_5E(A_27a),V0R),V1s)),V3w),V5x))
& p(ap(ap(V0R,V5x),V2a)) ) )
& ( ( V2a = V4z )
| ? [V6y: $i] :
( mem(V6y,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V6y),V1s))
& p(ap(ap(V0R,V2a),V6y))
& p(ap(ap(ap(c_2Erelation_2ERTC(A_27a),ap(ap(c_2Etc_2E_5E_7C_5E(A_27a),V0R),V1s)),V6y),V4z)) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Etc_2EsubTC__INSERT,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2q: $i] :
( mem(V2q,A_27a)
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( p(ap(ap(ap(ap(c_2Etc_2EsubTC(A_27a),V0R),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2q),V1s)),V3x),V4y))
<=> ( p(ap(ap(ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s),V3x),V4y))
| ( p(ap(ap(ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s),V3x),V2q))
& p(ap(ap(ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s),V2q),V4y)) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Etc_2EsubTC__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R)) = ap(c_2Erelation_2ETC(A_27a),V0R) ) ) ).
tff(conj_thm_2Etc_2EsubTC__INSERT__COR,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3a: $i] :
( mem(V3a,A_27a)
=> ( ap(ap(ap(c_2Etc_2EsubTC(A_27a),V0R),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V1s)),V3a) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,bool)),ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s),V3a))),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s),V3a)),ap(ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s),V2x))),ap(ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s),V3a)) ) ) ) ) ) ).
tff(ax_thm_2Etc_2EFMAP__TO__RELN,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ap(ap(c_2Etc_2EFMAP__TO__RELN(A_27a),V0f),V1x) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,bool)),ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(c_2Efinite__map_2EFDOM(A_27a,arr(A_27a,bool)),V0f))),ap(ap(c_2Efinite__map_2EFAPPLY(A_27a,arr(A_27a,bool)),V0f),V1x)),c_2Epred__set_2EEMPTY(A_27a)) ) ) ) ).
tff(ax_thm_2Etc_2ERELN__TO__FMAP,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(c_2Etc_2ERELN__TO__FMAP(A_27a),V0R) = ap(ap(c_2Efinite__map_2EFUN__FMAP(A_27a,arr(A_27a,bool)),V0R),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R)) ) ) ).
tff(conj_thm_2Etc_2ERDOM__SUBSET__FDOM,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Erelation_2ERDOM(A_27a,A_27a),ap(c_2Etc_2EFMAP__TO__RELN(A_27a),V0f))),ap(c_2Efinite__map_2EFDOM(A_27a,arr(A_27a,bool)),V0f))) ) ).
tff(conj_thm_2Etc_2EFINITE__RDOM,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
=> p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Erelation_2ERDOM(A_27a,A_27a),ap(c_2Etc_2EFMAP__TO__RELN(A_27a),V0f)))) ) ).
tff(conj_thm_2Etc_2EFDOM__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R)))
=> ( ap(c_2Efinite__map_2EFDOM(A_27a,arr(A_27a,bool)),ap(c_2Etc_2ERELN__TO__FMAP(A_27a),V0R)) = ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R) ) ) ) ).
tff(conj_thm_2Etc_2ERELN__TO__FMAP__TO__RELN__ID,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R)))
=> ( ap(c_2Etc_2EFMAP__TO__RELN(A_27a),ap(c_2Etc_2ERELN__TO__FMAP(A_27a),V0R)) = V0R ) ) ) ).
tff(conj_thm_2Etc_2ERDOM__subTC,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ap(c_2Erelation_2ERDOM(A_27a,A_27a),ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s)) = ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R) ) ) ) ).
tff(conj_thm_2Etc_2ENOT__IN__RDOM,axiom,
! [A_27a: del,V0Q: $i] :
( mem(V0Q,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ( ap(V0Q,V1x) = c_2Epred__set_2EEMPTY(A_27a) )
<=> ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0Q))) ) ) ) ).
tff(ax_thm_2Etc_2ETC__MOD,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1rx: $i] :
( mem(V1rx,arr(A_27a,bool))
=> ! [V2ra: $i] :
( mem(V2ra,arr(A_27a,bool))
=> ( ap(ap(ap(c_2Etc_2ETC__MOD(A_27a),V0x),V1rx),V2ra) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,bool)),ap(ap(c_2Ebool_2EIN(A_27a),V0x),V2ra)),ap(ap(c_2Epred__set_2EUNION(A_27a),V2ra),V1rx)),V2ra) ) ) ) ) ).
tff(conj_thm_2Etc_2ETC__MOD__EMPTY__ID,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1ra: $i] :
( mem(V1ra,arr(A_27a,bool))
=> ( ap(ap(c_2Etc_2ETC__MOD(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a)) = c_2Ecombin_2EI(arr(A_27a,bool)) ) ) ) ).
tff(conj_thm_2Etc_2EI__o__f,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( ap(ap(c_2Efinite__map_2Eo__f(A_27a,A_27b,A_27b),c_2Ecombin_2EI(A_27b)),V0f) = V0f ) ) ).
tff(conj_thm_2Etc_2EsubTC__MAX__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R)))
=> ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V1s)) = ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s) ) ) ) ) ) ).
tff(conj_thm_2Etc_2EsubTC__SUPERSET__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R)),V1s)) = ap(ap(c_2Etc_2EsubTC(A_27a),V0R),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R)) ) ) ) ) ).
tff(conj_thm_2Etc_2EsubTC__FDOM,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27a,bool)))
=> ( ( ap(ap(c_2Etc_2EsubTC(A_27a),V1R),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V1R)) = ap(c_2Etc_2EFMAP__TO__RELN(A_27a),V0g) )
=> ( ap(ap(c_2Etc_2EsubTC(A_27a),V1R),ap(c_2Efinite__map_2EFDOM(A_27a,arr(A_27a,bool)),V0g)) = ap(ap(c_2Etc_2EsubTC(A_27a),V1R),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V1R)) ) ) ) ) ).
tff(conj_thm_2Etc_2ESUBSET__FDOM__LEM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
=> ( ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s) = ap(c_2Etc_2EFMAP__TO__RELN(A_27a),V2f) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R)),ap(c_2Efinite__map_2EFDOM(A_27a,arr(A_27a,bool)),V2f))) ) ) ) ) ).
tff(conj_thm_2Etc_2EsubTC__FDOM__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
=> ( ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),ap(c_2Efinite__map_2EFDOM(A_27a,arr(A_27a,bool)),V1f)) = ap(c_2Etc_2EFMAP__TO__RELN(A_27a),V1f) )
=> ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),ap(c_2Erelation_2ERDOM(A_27a,A_27a),V0R)) = ap(c_2Etc_2EFMAP__TO__RELN(A_27a),V1f) ) ) ) ) ).
tff(conj_thm_2Etc_2ETC__MOD__LEM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3f: $i] :
( mem(V3f,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Efinite__map_2EFDOM(A_27a,arr(A_27a,bool)),V3f)))
& ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V1s) = ap(c_2Etc_2EFMAP__TO__RELN(A_27a),V3f) ) )
=> ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),V1s)) = ap(c_2Etc_2EFMAP__TO__RELN(A_27a),ap(ap(c_2Efinite__map_2Eo__f(A_27a,arr(A_27a,bool),arr(A_27a,bool)),ap(ap(c_2Etc_2ETC__MOD(A_27a),V2x),ap(ap(c_2Efinite__map_2EFAPPLY(A_27a,arr(A_27a,bool)),V3f),V2x))),V3f)) ) ) ) ) ) ) ).
tff(ax_thm_2Etc_2ETC__ITER,axiom,
! [A_27a: del] :
( ! [V0r: $i] :
( mem(V0r,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
=> ( ap(ap(c_2Etc_2ETC__ITER(A_27a),c_2Elist_2ENIL(A_27a)),V0r) = V0r ) )
& ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2d: $i] :
( mem(V2d,ty_2Elist_2Elist(A_27a))
=> ! [V3r: $i] :
( mem(V3r,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
=> ( ap(ap(c_2Etc_2ETC__ITER(A_27a),ap(ap(c_2Elist_2ECONS(A_27a),V1x),V2d)),V3r) = ap(ap(c_2Etc_2ETC__ITER(A_27a),V2d),ap(ap(c_2Efinite__map_2Eo__f(A_27a,arr(A_27a,bool),arr(A_27a,bool)),ap(ap(c_2Etc_2ETC__MOD(A_27a),V1x),ap(ap(c_2Efinite__map_2EFAPPLY(A_27a,arr(A_27a,bool)),V3r),V1x))),V3r)) ) ) ) ) ) ).
tff(conj_thm_2Etc_2ETC__ITER__THM,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1d: $i] :
( mem(V1d,ty_2Elist_2Elist(A_27a))
=> ! [V2f: $i] :
( mem(V2f,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( ( ( ap(ap(c_2Epred__set_2EUNION(A_27a),V3s),ap(c_2Elist_2ELIST__TO__SET(A_27a),V1d)) = ap(c_2Efinite__map_2EFDOM(A_27a,arr(A_27a,bool)),V2f) )
& ( ap(ap(c_2Etc_2EsubTC(A_27a),V0R),V3s) = ap(c_2Etc_2EFMAP__TO__RELN(A_27a),V2f) ) )
=> ( ap(c_2Erelation_2ETC(A_27a),V0R) = ap(c_2Etc_2EFMAP__TO__RELN(A_27a),ap(ap(c_2Etc_2ETC__ITER(A_27a),V1d),V2f)) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------