ITP001 Axioms: ITP064_5.ax


%------------------------------------------------------------------------------
% File     : ITP064_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    : dft_2.ax [Gau20]
%          : HL4064_5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   13 (   2 unt;   2 typ;   0 def)
%            Number of atoms       :  477 (   9 equ)
%            Maximal formula atoms :   21 (  36 avg)
%            Number of connectives :   80 (   2   ~;   1   |;  11   &)
%                                         (   1 <=>;  65  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   24 (  14 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of FOOLs       :  388 ( 388 fml;   0 var)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    7 (   2   >;   5   *;   0   +;   0  <<)
%            Number of predicates  :   27 (  26 usr;   3 prp; 0-2 aty)
%            Number of functors    :    2 (   2 usr;   0 con; 2-5 aty)
%            Number of variables   :   76 (  76   !;   0   ?;  76   :)
% SPC      : TF0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Edft_2EDFT,type,
    c_2Edft_2EDFT: ( del * del ) > $i ).

tff(mem_c_2Edft_2EDFT,axiom,
    ! [A_27a: del,A_27b: del] : mem(c_2Edft_2EDFT(A_27a,A_27b),arr(arr(A_27b,ty_2Elist_2Elist(A_27b)),arr(arr(A_27b,arr(A_27a,A_27a)),arr(ty_2Elist_2Elist(A_27b),arr(ty_2Elist_2Elist(A_27b),arr(A_27a,A_27a)))))) ).

tff(tp_c_2Edft_2ERel,type,
    c_2Edft_2ERel: ( del * del * del * del * del ) > $i ).

tff(mem_c_2Edft_2ERel,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del] : mem(c_2Edft_2ERel(A_27a,A_27b,A_27c,A_27d,A_27e),arr(ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(A_27b)),ty_2Epair_2Eprod(A_27c,ty_2Epair_2Eprod(ty_2Elist_2Elist(A_27a),ty_2Epair_2Eprod(ty_2Elist_2Elist(A_27d),A_27e)))),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).

tff(ax_thm_2Edft_2ERel__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,V0G: $i] :
      ( mem(V0G,arr(A_27a,ty_2Elist_2Elist(A_27b)))
     => ! [V1f: $i] :
          ( mem(V1f,A_27c)
         => ! [V2seen: $i] :
              ( mem(V2seen,ty_2Elist_2Elist(A_27a))
             => ! [V3to__visit: $i] :
                  ( mem(V3to__visit,ty_2Elist_2Elist(A_27d))
                 => ! [V4acc: $i] :
                      ( mem(V4acc,A_27e)
                     => ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum(ap(c_2Edft_2ERel(A_27a,A_27b,A_27c,A_27d,A_27e),ap(ap(c_2Epair_2E_2C(arr(A_27a,ty_2Elist_2Elist(A_27b)),ty_2Epair_2Eprod(A_27c,ty_2Epair_2Eprod(ty_2Elist_2Elist(A_27a),ty_2Epair_2Eprod(ty_2Elist_2Elist(A_27d),A_27e)))),V0G),ap(ap(c_2Epair_2E_2C(A_27c,ty_2Epair_2Eprod(ty_2Elist_2Elist(A_27a),ty_2Epair_2Eprod(ty_2Elist_2Elist(A_27d),A_27e))),V1f),ap(ap(c_2Epair_2E_2C(ty_2Elist_2Elist(A_27a),ty_2Epair_2Eprod(ty_2Elist_2Elist(A_27d),A_27e)),V2seen),ap(ap(c_2Epair_2E_2C(ty_2Elist_2Elist(A_27d),A_27e),V3to__visit),V4acc)))))) = surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum(ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2EdirGraph_2EParents(A_27a,A_27b),V0G)),ap(c_2Elist_2ELIST__TO__SET(A_27a),V2seen)))),ap(c_2Elist_2ELENGTH(A_27d),V3to__visit))) ) ) ) ) ) ) ).

tff(conj_thm_2Edft_2EDFT__def,axiom,
    ! [A_27a: del,A_27b: del,V0G: $i] :
      ( mem(V0G,arr(A_27a,ty_2Elist_2Elist(A_27a)))
     => ! [V1f: $i] :
          ( mem(V1f,arr(A_27a,arr(A_27b,A_27b)))
         => ! [V2seen: $i] :
              ( mem(V2seen,ty_2Elist_2Elist(A_27a))
             => ! [V3acc: $i] :
                  ( mem(V3acc,A_27b)
                 => ! [V4visit__now: $i] :
                      ( mem(V4visit__now,A_27a)
                     => ! [V5visit__later: $i] :
                          ( mem(V5visit__later,ty_2Elist_2Elist(A_27a))
                         => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2EdirGraph_2EParents(A_27a,A_27a),V0G)))
                           => ( ( ap(ap(ap(ap(ap(c_2Edft_2EDFT(A_27b,A_27a),V0G),V1f),V2seen),c_2Elist_2ENIL(A_27a)),V3acc) = V3acc )
                              & ( ap(ap(ap(ap(ap(c_2Edft_2EDFT(A_27b,A_27a),V0G),V1f),V2seen),ap(ap(c_2Elist_2ECONS(A_27a),V4visit__now),V5visit__later)),V3acc) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Ebool_2EIN(A_27a),V4visit__now),ap(c_2Elist_2ELIST__TO__SET(A_27a),V2seen))),ap(ap(ap(ap(ap(c_2Edft_2EDFT(A_27b,A_27a),V0G),V1f),V2seen),V5visit__later),V3acc)),ap(ap(ap(ap(ap(c_2Edft_2EDFT(A_27b,A_27a),V0G),V1f),ap(ap(c_2Elist_2ECONS(A_27a),V4visit__now),V2seen)),ap(ap(c_2Elist_2EAPPEND(A_27a),ap(V0G,V4visit__now)),V5visit__later)),ap(ap(V1f,V4visit__now),V3acc))) ) ) ) ) ) ) ) ) ) ).

tff(conj_thm_2Edft_2EDFT__ind,axiom,
    ! [A_27a: del,A_27b: del,V0P: $i] :
      ( mem(V0P,arr(arr(A_27a,ty_2Elist_2Elist(A_27a)),arr(arr(A_27a,arr(A_27b,A_27b)),arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27a),arr(A_27b,bool))))))
     => ( ! [V1G: $i] :
            ( mem(V1G,arr(A_27a,ty_2Elist_2Elist(A_27a)))
           => ! [V2f: $i] :
                ( mem(V2f,arr(A_27a,arr(A_27b,A_27b)))
               => ! [V3seen: $i] :
                    ( mem(V3seen,ty_2Elist_2Elist(A_27a))
                   => ! [V4visit__now: $i] :
                        ( mem(V4visit__now,A_27a)
                       => ! [V5visit__later: $i] :
                            ( mem(V5visit__later,ty_2Elist_2Elist(A_27a))
                           => ! [V6acc: $i] :
                                ( mem(V6acc,A_27b)
                               => ( p(ap(ap(ap(ap(ap(V0P,V1G),V2f),V3seen),c_2Elist_2ENIL(A_27a)),V6acc))
                                  & ( ( ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2EdirGraph_2EParents(A_27a,A_27a),V1G)))
                                          & ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V4visit__now),ap(c_2Elist_2ELIST__TO__SET(A_27a),V3seen))) )
                                       => p(ap(ap(ap(ap(ap(V0P,V1G),V2f),ap(ap(c_2Elist_2ECONS(A_27a),V4visit__now),V3seen)),ap(ap(c_2Elist_2EAPPEND(A_27a),ap(V1G,V4visit__now)),V5visit__later)),ap(ap(V2f,V4visit__now),V6acc))) )
                                      & ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2EdirGraph_2EParents(A_27a,A_27a),V1G)))
                                          & p(ap(ap(c_2Ebool_2EIN(A_27a),V4visit__now),ap(c_2Elist_2ELIST__TO__SET(A_27a),V3seen))) )
                                       => p(ap(ap(ap(ap(ap(V0P,V1G),V2f),V3seen),V5visit__later),V6acc)) ) )
                                   => p(ap(ap(ap(ap(ap(V0P,V1G),V2f),V3seen),ap(ap(c_2Elist_2ECONS(A_27a),V4visit__now),V5visit__later)),V6acc)) ) ) ) ) ) ) ) )
       => ! [V7v: $i] :
            ( mem(V7v,arr(A_27a,ty_2Elist_2Elist(A_27a)))
           => ! [V8v1: $i] :
                ( mem(V8v1,arr(A_27a,arr(A_27b,A_27b)))
               => ! [V9v2: $i] :
                    ( mem(V9v2,ty_2Elist_2Elist(A_27a))
                   => ! [V10v3: $i] :
                        ( mem(V10v3,ty_2Elist_2Elist(A_27a))
                       => ! [V11v4: $i] :
                            ( mem(V11v4,A_27b)
                           => p(ap(ap(ap(ap(ap(V0P,V7v),V8v1),V9v2),V10v3),V11v4)) ) ) ) ) ) ) ) ).

tff(conj_thm_2Edft_2EDFT__CONS,axiom,
    ! [A_27a: del,V0G: $i] :
      ( mem(V0G,arr(A_27a,ty_2Elist_2Elist(A_27a)))
     => ! [V1f: $i] :
          ( mem(V1f,arr(A_27a,arr(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27a))))
         => ! [V2seen: $i] :
              ( mem(V2seen,ty_2Elist_2Elist(A_27a))
             => ! [V3to__visit: $i] :
                  ( mem(V3to__visit,ty_2Elist_2Elist(A_27a))
                 => ! [V4acc: $i] :
                      ( mem(V4acc,ty_2Elist_2Elist(A_27a))
                     => ! [V5a: $i] :
                          ( mem(V5a,ty_2Elist_2Elist(A_27a))
                         => ! [V6b: $i] :
                              ( mem(V6b,ty_2Elist_2Elist(A_27a))
                             => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2EdirGraph_2EParents(A_27a,A_27a),V0G)))
                                  & ( V1f = c_2Elist_2ECONS(A_27a) )
                                  & ( V4acc = ap(ap(c_2Elist_2EAPPEND(A_27a),V5a),V6b) ) )
                               => ( ap(ap(ap(ap(ap(c_2Edft_2EDFT(ty_2Elist_2Elist(A_27a),A_27a),V0G),V1f),V2seen),V3to__visit),V4acc) = ap(ap(c_2Elist_2EAPPEND(A_27a),ap(ap(ap(ap(ap(c_2Edft_2EDFT(ty_2Elist_2Elist(A_27a),A_27a),V0G),V1f),V2seen),V3to__visit),V5a)),V6b) ) ) ) ) ) ) ) ) ) ).

tff(conj_thm_2Edft_2EDFT__FOLD,axiom,
    ! [A_27a: del,A_27b: del,V0G: $i] :
      ( mem(V0G,arr(A_27a,ty_2Elist_2Elist(A_27a)))
     => ! [V1f: $i] :
          ( mem(V1f,arr(A_27a,arr(A_27b,A_27b)))
         => ! [V2seen: $i] :
              ( mem(V2seen,ty_2Elist_2Elist(A_27a))
             => ! [V3to__visit: $i] :
                  ( mem(V3to__visit,ty_2Elist_2Elist(A_27a))
                 => ! [V4acc: $i] :
                      ( mem(V4acc,A_27b)
                     => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2EdirGraph_2EParents(A_27a,A_27a),V0G)))
                       => ( ap(ap(ap(ap(ap(c_2Edft_2EDFT(A_27b,A_27a),V0G),V1f),V2seen),V3to__visit),V4acc) = ap(ap(ap(c_2Elist_2EFOLDR(A_27a,A_27b),V1f),V4acc),ap(ap(ap(ap(ap(c_2Edft_2EDFT(ty_2Elist_2Elist(A_27a),A_27a),V0G),c_2Elist_2ECONS(A_27a)),V2seen),V3to__visit),c_2Elist_2ENIL(A_27a))) ) ) ) ) ) ) ) ).

tff(conj_thm_2Edft_2EDFT__ALL__DISTINCT,axiom,
    ! [A_27a: del,V0G: $i] :
      ( mem(V0G,arr(A_27a,ty_2Elist_2Elist(A_27a)))
     => ! [V1seen: $i] :
          ( mem(V1seen,ty_2Elist_2Elist(A_27a))
         => ! [V2to__visit: $i] :
              ( mem(V2to__visit,ty_2Elist_2Elist(A_27a))
             => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2EdirGraph_2EParents(A_27a,A_27a),V0G)))
               => p(ap(c_2Elist_2EALL__DISTINCT(A_27a),ap(ap(ap(ap(ap(c_2Edft_2EDFT(ty_2Elist_2Elist(A_27a),A_27a),V0G),c_2Elist_2ECONS(A_27a)),V1seen),V2to__visit),c_2Elist_2ENIL(A_27a)))) ) ) ) ) ).

tff(conj_thm_2Edft_2EDFT__REACH__1,axiom,
    ! [A_27a: del,V0G: $i] :
      ( mem(V0G,arr(A_27a,ty_2Elist_2Elist(A_27a)))
     => ! [V1f: $i] :
          ( mem(V1f,arr(A_27a,arr(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27a))))
         => ! [V2seen: $i] :
              ( mem(V2seen,ty_2Elist_2Elist(A_27a))
             => ! [V3to__visit: $i] :
                  ( mem(V3to__visit,ty_2Elist_2Elist(A_27a))
                 => ! [V4acc: $i] :
                      ( mem(V4acc,ty_2Elist_2Elist(A_27a))
                     => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2EdirGraph_2EParents(A_27a,A_27a),V0G)))
                          & ( V1f = c_2Elist_2ECONS(A_27a) ) )
                       => ! [V5x: $i] :
                            ( mem(V5x,A_27a)
                           => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(ap(ap(ap(c_2Edft_2EDFT(ty_2Elist_2Elist(A_27a),A_27a),V0G),V1f),V2seen),V3to__visit),V4acc))))
                             => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(ap(c_2EdirGraph_2EREACH__LIST(A_27a),V0G),V3to__visit)))
                                | p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Elist_2ELIST__TO__SET(A_27a),V4acc))) ) ) ) ) ) ) ) ) ) ).

tff(conj_thm_2Edft_2EDFT__REACH__2,axiom,
    ! [A_27a: del,V0G: $i] :
      ( mem(V0G,arr(A_27a,ty_2Elist_2Elist(A_27a)))
     => ! [V1f: $i] :
          ( mem(V1f,arr(A_27a,arr(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27a))))
         => ! [V2seen: $i] :
              ( mem(V2seen,ty_2Elist_2Elist(A_27a))
             => ! [V3to__visit: $i] :
                  ( mem(V3to__visit,ty_2Elist_2Elist(A_27a))
                 => ! [V4acc: $i] :
                      ( mem(V4acc,ty_2Elist_2Elist(A_27a))
                     => ! [V5x: $i] :
                          ( mem(V5x,A_27a)
                         => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2EdirGraph_2EParents(A_27a,A_27a),V0G)))
                              & ( V1f = c_2Elist_2ECONS(A_27a) )
                              & p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(ap(c_2EdirGraph_2EREACH__LIST(A_27a),ap(ap(c_2EdirGraph_2EEXCLUDE(A_27a,A_27a),V0G),ap(c_2Elist_2ELIST__TO__SET(A_27a),V2seen))),V3to__visit)))
                              & ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Elist_2ELIST__TO__SET(A_27a),V2seen))) )
                           => p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(ap(ap(ap(c_2Edft_2EDFT(ty_2Elist_2Elist(A_27a),A_27a),V0G),V1f),V2seen),V3to__visit),V4acc)))) ) ) ) ) ) ) ) ).

tff(conj_thm_2Edft_2EDFT__REACH__THM,axiom,
    ! [A_27a: del,V0G: $i] :
      ( mem(V0G,arr(A_27a,ty_2Elist_2Elist(A_27a)))
     => ! [V1to__visit: $i] :
          ( mem(V1to__visit,ty_2Elist_2Elist(A_27a))
         => ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2EdirGraph_2EParents(A_27a,A_27a),V0G)))
           => ! [V2x: $i] :
                ( mem(V2x,A_27a)
               => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(ap(c_2EdirGraph_2EREACH__LIST(A_27a),V0G),V1to__visit)))
                <=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(ap(ap(ap(c_2Edft_2EDFT(ty_2Elist_2Elist(A_27a),A_27a),V0G),c_2Elist_2ECONS(A_27a)),c_2Elist_2ENIL(A_27a)),V1to__visit),c_2Elist_2ENIL(A_27a))))) ) ) ) ) ) ).

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