SET007 Axioms: SET007+238.ax


%------------------------------------------------------------------------------
% File     : SET007+238 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Translations in Affine Planes
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : translac [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   17 (   2 unit)
%            Number of atoms       :  254 (  21 equality)
%            Maximal formula depth :   24 (  13 average)
%            Number of connectives :  286 (  49 ~  ;   5  |; 155  &)
%                                         (   2 <=>;  75 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   1 propositional; 0-5 arity)
%            Number of functors    :    4 (   0 constant; 1-4 arity)
%            Number of variables   :   69 (   0 singleton;  64 !;   5 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(d1_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ( v1_translac(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ( ( r2_analoaf(A,B,C,D,E)
                            & r2_analoaf(A,B,D,C,E)
                            & r2_analoaf(A,B,E,C,D) )
                         => r1_aff_1(A,B,C,D) ) ) ) ) ) ) ) )).

fof(t1_translac,axiom,(
    $true )).

fof(t2_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ( ? [B] :
            ( m1_subset_1(B,u1_struct_0(A))
            & ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & r1_aff_1(A,B,C,D)
                    & B != C
                    & B != D
                    & C != D ) ) )
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( B != C
                    & ! [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                       => ~ ( r1_aff_1(A,B,C,D)
                            & B != D
                            & C != D ) ) ) ) ) ) ) )).

fof(t3_translac,axiom,(
    $true )).

fof(t4_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ~ ( v1_translac(A)
                          & r2_analoaf(A,B,C,D,E)
                          & r2_analoaf(A,B,D,C,E)
                          & ~ r1_aff_1(A,B,C,D)
                          & ! [F] :
                              ( m1_subset_1(F,u1_struct_0(A))
                             => ~ ( r1_aff_1(A,C,D,F)
                                  & r1_aff_1(A,B,E,F) ) ) ) ) ) ) ) ) )).

fof(t5_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(A),u1_struct_0(A))
                        & v3_funct_2(E,u1_struct_0(A),u1_struct_0(A))
                        & m2_relset_1(E,u1_struct_0(A),u1_struct_0(A)) )
                     => ( ( v7_transgeo(E,A)
                          & r2_analoaf(A,B,k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,B),C,D)
                          & r2_analoaf(A,B,C,k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,B),D) )
                       => ( r1_aff_1(A,B,k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,B),C)
                          | D = k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,C) ) ) ) ) ) ) ) )).

fof(t6_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v11_aff_2(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ( ( r2_analoaf(A,B,C,D,F)
                                    & r2_analoaf(A,B,C,E,G)
                                    & r2_analoaf(A,B,D,C,F)
                                    & r2_analoaf(A,B,E,C,G) )
                                 => ( r1_aff_1(A,B,C,D)
                                    | r1_aff_1(A,B,C,E)
                                    | r2_analoaf(A,D,E,F,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t7_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ? [C] :
              ( v1_funct_1(C)
              & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
              & v3_funct_2(C,u1_struct_0(A),u1_struct_0(A))
              & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A))
              & v7_transgeo(C,A)
              & k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,B) = B ) ) ) )).

fof(t8_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ( ( ! [F] :
                              ( m1_subset_1(F,u1_struct_0(A))
                             => ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(A))
                                 => ! [H] :
                                      ( m1_subset_1(H,u1_struct_0(A))
                                     => ~ ( F != G
                                          & r1_aff_1(A,F,G,H)
                                          & H != F
                                          & H != G ) ) ) )
                          & r2_analoaf(A,B,C,D,E)
                          & r2_analoaf(A,B,D,C,E) )
                       => ( r1_aff_1(A,B,C,D)
                          | r2_analoaf(A,B,E,C,D) ) ) ) ) ) ) ) )).

fof(t9_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( v11_aff_2(A)
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                        & v3_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                        & m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
                     => ~ ( v7_transgeo(D,A)
                          & k8_funct_2(u1_struct_0(A),u1_struct_0(A),D,B) = C ) ) ) ) ) ) )).

fof(t10_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                    & v3_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(A))
                    & v7_transgeo(D,A)
                    & k8_funct_2(u1_struct_0(A),u1_struct_0(A),D,B) = C ) ) )
       => v11_aff_2(A) ) ) )).

fof(t11_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                & v3_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                    & v3_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
                 => ( ( v7_transgeo(C,A)
                      & v7_transgeo(D,A) )
                   => ( r1_aff_1(A,B,k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,B),k8_funct_2(u1_struct_0(A),u1_struct_0(A),D,B))
                      | k1_transgeo(u1_struct_0(A),D,C) = k1_transgeo(u1_struct_0(A),C,D) ) ) ) ) ) ) )).

fof(t12_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & v3_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                & v3_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
             => ( ( v11_aff_2(A)
                  & v7_transgeo(B,A)
                  & v7_transgeo(C,A) )
               => k1_transgeo(u1_struct_0(A),C,B) = k1_transgeo(u1_struct_0(A),B,C) ) ) ) ) )).

fof(t13_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                & v3_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                    & v3_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
                 => ( ( v7_transgeo(C,A)
                      & v7_transgeo(D,A)
                      & r2_analoaf(A,B,k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,B),B,k8_funct_2(u1_struct_0(A),u1_struct_0(A),D,B)) )
                   => r2_analoaf(A,B,k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,B),B,k8_funct_2(u1_struct_0(A),u1_struct_0(A),k1_transgeo(u1_struct_0(A),D,C),B)) ) ) ) ) ) )).

fof(t14_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & v3_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ~ ( v1_translac(A)
              & v11_aff_2(A)
              & v7_transgeo(B,A)
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                    & v3_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                    & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
                 => ~ ( v7_transgeo(C,A)
                      & k1_transgeo(u1_struct_0(A),C,C) = B ) ) ) ) ) )).

fof(t15_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & v3_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( ( v1_translac(A)
              & v7_transgeo(B,A)
              & k1_transgeo(u1_struct_0(A),B,B) = k6_partfun1(u1_struct_0(A)) )
           => B = k6_partfun1(u1_struct_0(A)) ) ) ) )).

fof(t16_translac,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & v3_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                & v3_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                    & v3_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
                 => ( ( v11_aff_2(A)
                      & v1_translac(A)
                      & v7_transgeo(B,A)
                      & v7_transgeo(C,A)
                      & v7_transgeo(D,A)
                      & B = k1_transgeo(u1_struct_0(A),C,C)
                      & B = k1_transgeo(u1_struct_0(A),D,D) )
                   => C = D ) ) ) ) ) )).
%------------------------------------------------------------------------------