ITP001 Axioms: ITP103+5.ax


%------------------------------------------------------------------------------
% File     : ITP103+5 : TPTP v9.0.0. 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    :   42 (   0 unt;   0 def)
%            Number of atoms       :  206 (  36 equ)
%            Maximal formula atoms :   18 (   4 avg)
%            Number of connectives :  166 (   2   ~;   5   |;  20   &)
%                                         (   6 <=>; 133  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   8 avg)
%            Maximal term depth    :   11 (   2 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :   34 (  34 usr;   1 con; 0-3 aty)
%            Number of variables   :  128 ( 124   !;   4   ?)
% SPC      : FOF_SAT_RFO_SEQ

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Etc_2EFMAP__TO__RELN,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => 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)))) ) ).

fof(mem_c_2Etc_2ERELN__TO__FMAP,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => 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)))) ) ).

fof(mem_c_2Etc_2ETC__ITER,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => 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))))) ) ).

fof(mem_c_2Etc_2ETC__MOD,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Etc_2ETC__MOD(A_27a),arr(A_27a,arr(arr(A_27a,bool),arr(arr(A_27a,bool),arr(A_27a,bool))))) ) ).

fof(mem_c_2Etc_2E_5E_7C,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => 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))))) ) ).

fof(mem_c_2Etc_2E_5E_7C_5E,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => 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))))) ) ).

fof(mem_c_2Etc_2EsubTC,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => 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))))) ) ).

fof(mem_c_2Etc_2E_7C_5E,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => 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))))) ) ).

fof(ax_thm_2Etc_2EDRESTR,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( mem(V1s,arr(A_27a,bool))
             => ! [V2a] :
                  ( mem(V2a,A_27a)
                 => ! [V3b] :
                      ( 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)) ) ) ) ) ) ) ) ).

fof(conj_thm_2Etc_2EDRESTR__IN,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( mem(V1s,arr(A_27a,bool))
             => ! [V2a] :
                  ( 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)) ) ) ) ) ).

fof(ax_thm_2Etc_2ERRESTR,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( mem(V1s,arr(A_27a,bool))
             => ! [V2a] :
                  ( mem(V2a,A_27a)
                 => ! [V3b] :
                      ( 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)) ) ) ) ) ) ) ) ).

fof(ax_thm_2Etc_2EBRESTR,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( 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) ) ) ) ).

fof(conj_thm_2Etc_2EDRESTR__EMPTY,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( 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) ) ) ).

fof(conj_thm_2Etc_2EDRESTR__RDOM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( 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 ) ) ).

fof(conj_thm_2Etc_2EREMPTY__RRESTR,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0s] :
          ( 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) ) ) ).

fof(conj_thm_2Etc_2EO__REMPTY__O,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ( ! [V0R] :
            ( 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] :
            ( 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) ) ) ) ).

fof(ax_thm_2Etc_2EsubTC,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( mem(V1s,arr(A_27a,bool))
             => ! [V2x] :
                  ( mem(V2x,A_27a)
                 => ! [V3y] :
                      ( 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] :
                              ( mem(V4a,A_27a)
                              & ? [V5b] :
                                  ( 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)) ) ) ) ) ) ) ) ) ) ).

fof(conj_thm_2Etc_2EsubTC__thm,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( 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))) ) ) ) ).

fof(conj_thm_2Etc_2EsubTC__EMPTY,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ap(ap(c_2Etc_2EsubTC(A_27a),V0R),c_2Epred__set_2EEMPTY(A_27a)) = V0R ) ) ).

fof(conj_thm_2Etc_2ERTC__INSERT,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( mem(V1s,arr(A_27a,bool))
             => ! [V2a] :
                  ( mem(V2a,A_27a)
                 => ! [V3w] :
                      ( mem(V3w,A_27a)
                     => ! [V4z] :
                          ( 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] :
                                      ( 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] :
                                      ( 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)) ) ) ) ) ) ) ) ) ) ) ) ).

fof(conj_thm_2Etc_2EsubTC__INSERT,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( mem(V1s,arr(A_27a,bool))
             => ! [V2q] :
                  ( mem(V2q,A_27a)
                 => ! [V3x] :
                      ( mem(V3x,A_27a)
                     => ! [V4y] :
                          ( 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)) ) ) ) ) ) ) ) ) ) ).

fof(conj_thm_2Etc_2EsubTC__RDOM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( 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) ) ) ).

fof(conj_thm_2Etc_2EsubTC__INSERT__COR,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( mem(V1s,arr(A_27a,bool))
             => ! [V2x] :
                  ( mem(V2x,A_27a)
                 => ! [V3a] :
                      ( 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)) ) ) ) ) ) ).

fof(ax_thm_2Etc_2EFMAP__TO__RELN,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0f] :
          ( mem(V0f,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
         => ! [V1x] :
              ( 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)) ) ) ) ).

fof(ax_thm_2Etc_2ERELN__TO__FMAP,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( 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)) ) ) ).

fof(conj_thm_2Etc_2ERDOM__SUBSET__FDOM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0f] :
          ( 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))) ) ) ).

fof(conj_thm_2Etc_2EFINITE__RDOM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0f] :
          ( 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)))) ) ) ).

fof(conj_thm_2Etc_2EFDOM__RDOM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( 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) ) ) ) ).

fof(conj_thm_2Etc_2ERELN__TO__FMAP__TO__RELN__ID,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( 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 ) ) ) ).

fof(conj_thm_2Etc_2ERDOM__subTC,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( 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) ) ) ) ).

fof(conj_thm_2Etc_2ENOT__IN__RDOM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0Q] :
          ( mem(V0Q,arr(A_27a,arr(A_27a,bool)))
         => ! [V1x] :
              ( 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))) ) ) ) ) ).

fof(ax_thm_2Etc_2ETC__MOD,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0x] :
          ( mem(V0x,A_27a)
         => ! [V1rx] :
              ( mem(V1rx,arr(A_27a,bool))
             => ! [V2ra] :
                  ( 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) ) ) ) ) ).

fof(conj_thm_2Etc_2ETC__MOD__EMPTY__ID,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0x] :
          ( mem(V0x,A_27a)
         => ! [V1ra] :
              ( 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)) ) ) ) ).

fof(conj_thm_2Etc_2EI__o__f,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [V0f] :
              ( 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 ) ) ) ).

fof(conj_thm_2Etc_2EsubTC__MAX__RDOM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( mem(V1s,arr(A_27a,bool))
             => ! [V2x] :
                  ( 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) ) ) ) ) ) ).

fof(conj_thm_2Etc_2EsubTC__SUPERSET__RDOM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( 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)) ) ) ) ) ).

fof(conj_thm_2Etc_2EsubTC__FDOM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0g] :
          ( mem(V0g,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
         => ! [V1R] :
              ( 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)) ) ) ) ) ).

fof(conj_thm_2Etc_2ESUBSET__FDOM__LEM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( mem(V1s,arr(A_27a,bool))
             => ! [V2f] :
                  ( 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))) ) ) ) ) ) ).

fof(conj_thm_2Etc_2EsubTC__FDOM__RDOM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1f] :
              ( 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) ) ) ) ) ).

fof(conj_thm_2Etc_2ETC__MOD__LEM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1s] :
              ( mem(V1s,arr(A_27a,bool))
             => ! [V2x] :
                  ( mem(V2x,A_27a)
                 => ! [V3f] :
                      ( 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)) ) ) ) ) ) ) ).

fof(ax_thm_2Etc_2ETC__ITER,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ( ! [V0r] :
            ( 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] :
            ( mem(V1x,A_27a)
           => ! [V2d] :
                ( mem(V2d,ty_2Elist_2Elist(A_27a))
               => ! [V3r] :
                    ( 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)) ) ) ) ) ) ).

fof(conj_thm_2Etc_2ETC__ITER__THM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0R] :
          ( mem(V0R,arr(A_27a,arr(A_27a,bool)))
         => ! [V1d] :
              ( mem(V1d,ty_2Elist_2Elist(A_27a))
             => ! [V2f] :
                  ( mem(V2f,ty_2Efinite__map_2Efmap(A_27a,arr(A_27a,bool)))
                 => ! [V3s] :
                      ( 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)) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------