SET007 Axioms: SET007+238.ax
%------------------------------------------------------------------------------
% File : SET007+238 : TPTP v8.2.0. 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 unt; 0 def)
% Number of atoms : 254 ( 21 equ)
% Maximal formula atoms : 25 ( 14 avg)
% Number of connectives : 286 ( 49 ~; 5 |; 155 &)
% ( 2 <=>; 75 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 13 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 17 ( 15 usr; 1 prp; 0-5 aty)
% Number of functors : 4 ( 4 usr; 0 con; 1-4 aty)
% Number of variables : 69 ( 64 !; 5 ?)
% 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 ) ) ) ) ) ).
%------------------------------------------------------------------------------