SET007 Axioms: SET007+159.ax
%------------------------------------------------------------------------------
% File : SET007+159 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Relations of Tolerance
% 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 : toler_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 92 ( 43 unt; 0 def)
% Number of atoms : 388 ( 16 equ)
% Maximal formula atoms : 17 ( 4 avg)
% Number of connectives : 304 ( 8 ~; 0 |; 199 &)
% ( 15 <=>; 82 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 19 ( 19 usr; 2 con; 0-6 aty)
% Number of variables : 161 ( 154 !; 7 ?)
% 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(fc1_toler_1,axiom,
( v1_relat_1(k1_xboole_0)
& v3_relat_1(k1_xboole_0)
& v1_relat_2(k1_xboole_0)
& v2_relat_2(k1_xboole_0)
& v3_relat_2(k1_xboole_0)
& v4_relat_2(k1_xboole_0)
& v5_relat_2(k1_xboole_0)
& v6_relat_2(k1_xboole_0)
& v7_relat_2(k1_xboole_0)
& v8_relat_2(k1_xboole_0)
& v1_xboole_0(k1_xboole_0) ) ).
fof(rc1_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> ? [C] :
( m1_toler_1(C,A,B)
& v1_toler_1(C,A,B) ) ) ).
fof(t1_toler_1,axiom,
k3_relat_1(k1_xboole_0) = k1_xboole_0 ).
fof(t2_toler_1,axiom,
v1_relat_2(k1_xboole_0) ).
fof(t3_toler_1,axiom,
v3_relat_2(k1_xboole_0) ).
fof(t4_toler_1,axiom,
v2_relat_2(k1_xboole_0) ).
fof(t5_toler_1,axiom,
v4_relat_2(k1_xboole_0) ).
fof(t6_toler_1,axiom,
v5_relat_2(k1_xboole_0) ).
fof(t7_toler_1,axiom,
v6_relat_2(k1_xboole_0) ).
fof(t8_toler_1,axiom,
v7_relat_2(k1_xboole_0) ).
fof(t9_toler_1,axiom,
v8_relat_2(k1_xboole_0) ).
fof(t10_toler_1,axiom,
$true ).
fof(t11_toler_1,axiom,
$true ).
fof(t12_toler_1,axiom,
$true ).
fof(t13_toler_1,axiom,
! [A] : k5_relset_1(A,A,k1_eqrel_1(A)) = A ).
fof(t14_toler_1,axiom,
$true ).
fof(t15_toler_1,axiom,
! [A,B,C] :
( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r2_hidden(k4_tarski(B,C),k1_eqrel_1(A)) ) ).
fof(t16_toler_1,axiom,
! [A,B,C] :
( ( r2_hidden(B,k3_relat_1(k1_eqrel_1(A)))
& r2_hidden(C,k3_relat_1(k1_eqrel_1(A))) )
=> r2_hidden(k4_tarski(B,C),k1_eqrel_1(A)) ) ).
fof(t17_toler_1,axiom,
$true ).
fof(t18_toler_1,axiom,
$true ).
fof(t19_toler_1,axiom,
! [A] : v7_relat_2(k1_eqrel_1(A)) ).
fof(t20_toler_1,axiom,
$true ).
fof(t21_toler_1,axiom,
! [A] : v6_relat_2(k1_eqrel_1(A)) ).
fof(t22_toler_1,axiom,
$true ).
fof(t23_toler_1,axiom,
$true ).
fof(t24_toler_1,axiom,
$true ).
fof(t25_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> k5_relset_1(A,A,B) = A ) ).
fof(t26_toler_1,axiom,
$true ).
fof(t27_toler_1,axiom,
! [A,B,C] :
( ( v1_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( r2_hidden(B,A)
<=> r2_hidden(k4_tarski(B,B),C) ) ) ).
fof(t28_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> r1_relat_2(B,A) ) ).
fof(t29_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> r3_relat_2(B,A) ) ).
fof(t30_toler_1,axiom,
$true ).
fof(t31_toler_1,axiom,
$true ).
fof(t32_toler_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,A,B)
=> ( v3_relat_2(D)
=> v3_relat_2(k1_toler_1(D,C)) ) ) ).
fof(t33_toler_1,axiom,
! [A,B,C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,B,B)
& m2_relset_1(C,B,B) )
=> ( r1_tarski(A,B)
=> ( v1_relat_2(k1_toler_1(C,A))
& v3_relat_2(k1_toler_1(C,A))
& v1_partfun1(k1_toler_1(C,A),A,A)
& m2_relset_1(k1_toler_1(C,A),A,A) ) ) ) ).
fof(d1_toler_1,axiom,
$true ).
fof(d2_toler_1,axiom,
$true ).
fof(d3_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m1_toler_1(C,A,B)
<=> ! [D,E] :
( ( r2_hidden(D,C)
& r2_hidden(E,C) )
=> r2_hidden(k4_tarski(D,E),B) ) ) ) ).
fof(t34_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> m1_toler_1(k1_xboole_0,A,B) ) ).
fof(d4_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m1_toler_1(C,A,B)
=> ( v1_toler_1(C,A,B)
<=> ! [D] :
~ ( ~ r2_hidden(D,C)
& r2_hidden(D,A)
& ! [E] :
~ ( r2_hidden(E,C)
& ~ r2_hidden(k4_tarski(D,E),B) ) ) ) ) ) ).
fof(t35_toler_1,axiom,
$true ).
fof(t36_toler_1,axiom,
$true ).
fof(t37_toler_1,axiom,
$true ).
fof(t38_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ( ( v1_toler_1(k1_xboole_0,A,B)
& m1_toler_1(k1_xboole_0,A,B) )
=> B = k1_xboole_0 ) ) ).
fof(t39_toler_1,axiom,
( v1_relat_2(k1_xboole_0)
& v3_relat_2(k1_xboole_0)
& v1_partfun1(k1_xboole_0,k1_xboole_0,k1_xboole_0)
& m2_relset_1(k1_xboole_0,k1_xboole_0,k1_xboole_0) ) ).
fof(t40_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),B)
=> m1_toler_1(k2_tarski(C,D),A,B) ) ) ).
fof(t41_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( r2_hidden(C,A)
=> m1_toler_1(k1_tarski(C),A,B) ) ) ).
fof(t42_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C,D] :
( ( m1_toler_1(C,A,B)
& m1_toler_1(D,A,B) )
=> m1_toler_1(k3_xboole_0(C,D),A,B) ) ) ).
fof(t43_toler_1,axiom,
! [A,B,C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,B,B)
& m2_relset_1(C,B,B) )
=> ( m1_toler_1(A,B,C)
=> r1_tarski(A,B) ) ) ).
fof(t44_toler_1,axiom,
$true ).
fof(t45_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m1_toler_1(C,A,B)
=> ? [D] :
( v1_toler_1(D,A,B)
& m1_toler_1(D,A,B)
& r1_tarski(C,D) ) ) ) ).
fof(t46_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C,D] :
~ ( r2_hidden(k4_tarski(C,D),B)
& ! [E] :
( ( v1_toler_1(E,A,B)
& m1_toler_1(E,A,B) )
=> ~ ( r2_hidden(C,E)
& r2_hidden(D,E) ) ) ) ) ).
fof(t47_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
~ ( r2_hidden(C,A)
& ! [D] :
( ( v1_toler_1(D,A,B)
& m1_toler_1(D,A,B) )
=> ~ r2_hidden(C,D) ) ) ) ).
fof(t48_toler_1,axiom,
$true ).
fof(t49_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> r1_tarski(B,k1_eqrel_1(A)) ) ).
fof(t50_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> r1_tarski(k6_partfun1(A),B) ) ).
fof(t51_toler_1,axiom,
! [A] :
? [B] :
( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,k3_tarski(A),k3_tarski(A))
& m2_relset_1(B,k3_tarski(A),k3_tarski(A))
& ! [C] :
( r2_hidden(C,A)
=> m1_toler_1(C,k3_tarski(A),B) ) ) ).
fof(t52_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,k3_tarski(A),k3_tarski(A))
& m2_relset_1(B,k3_tarski(A),k3_tarski(A)) )
=> ! [C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,k3_tarski(A),k3_tarski(A))
& m2_relset_1(C,k3_tarski(A),k3_tarski(A)) )
=> ( ( ! [D,E] :
( r2_hidden(k4_tarski(D,E),B)
<=> ? [F] :
( r2_hidden(F,A)
& r2_hidden(D,F)
& r2_hidden(E,F) ) )
& ! [D,E] :
( r2_hidden(k4_tarski(D,E),C)
<=> ? [F] :
( r2_hidden(F,A)
& r2_hidden(D,F)
& r2_hidden(E,F) ) ) )
=> B = C ) ) ) ).
fof(t53_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( ! [D] :
( ( v1_toler_1(D,A,B)
& m1_toler_1(D,A,B) )
<=> ( v1_toler_1(D,A,C)
& m1_toler_1(D,A,C) ) )
=> B = C ) ) ) ).
fof(t54_toler_1,axiom,
! [A,B,C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( r2_hidden(D,k6_eqrel_1(A,C,B))
<=> r2_hidden(k4_tarski(B,D),C) ) ) ).
fof(t55_toler_1,axiom,
$true ).
fof(t56_toler_1,axiom,
$true ).
fof(t57_toler_1,axiom,
$true ).
fof(t58_toler_1,axiom,
! [A,B,C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( ! [E] :
( r2_hidden(E,D)
<=> ( r2_hidden(B,E)
& v1_toler_1(E,A,C)
& m1_toler_1(E,A,C) ) )
=> k6_eqrel_1(A,C,B) = k3_tarski(D) ) ) ).
fof(t59_toler_1,axiom,
! [A,B,C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( ! [E] :
( r2_hidden(E,D)
<=> ( r2_hidden(B,E)
& m1_toler_1(E,A,C) ) )
=> k6_eqrel_1(A,C,B) = k3_tarski(D) ) ) ).
fof(d5_toler_1,axiom,
$true ).
fof(d6_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( C = k3_toler_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> m1_toler_1(D,A,B) ) ) ) ).
fof(d7_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( C = k4_toler_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( v1_toler_1(D,A,B)
& m1_toler_1(D,A,B) ) ) ) ) ).
fof(t60_toler_1,axiom,
$true ).
fof(t61_toler_1,axiom,
$true ).
fof(t62_toler_1,axiom,
$true ).
fof(t63_toler_1,axiom,
$true ).
fof(t64_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( r1_tarski(k4_toler_1(A,B),k4_toler_1(A,C))
=> r1_tarski(B,C) ) ) ) ).
fof(t65_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( k4_toler_1(A,B) = k4_toler_1(A,C)
=> B = C ) ) ) ).
fof(t66_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> k3_tarski(k4_toler_1(A,B)) = A ) ).
fof(t67_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> k3_tarski(k3_toler_1(A,B)) = A ) ).
fof(t68_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ( ! [C] :
( r2_hidden(C,A)
=> m1_toler_1(k6_eqrel_1(A,B,C),A,B) )
=> v8_relat_2(B) ) ) ).
fof(t69_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ( v8_relat_2(B)
=> ! [C] :
( r2_hidden(C,A)
=> ( v1_toler_1(k6_eqrel_1(A,B,C),A,B)
& m1_toler_1(k6_eqrel_1(A,B,C),A,B) ) ) ) ) ).
fof(t70_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C,D] :
( ( v1_toler_1(D,A,B)
& m1_toler_1(D,A,B) )
=> ( r2_hidden(C,D)
=> r1_tarski(D,k6_eqrel_1(A,B,C)) ) ) ) ).
fof(t71_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( r1_tarski(k3_toler_1(A,B),k3_toler_1(A,C))
<=> r1_tarski(B,C) ) ) ) ).
fof(t72_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> r1_tarski(k4_toler_1(A,B),k3_toler_1(A,B)) ) ).
fof(t73_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( ! [D] :
( r2_hidden(D,A)
=> r1_tarski(k6_eqrel_1(A,B,D),k6_eqrel_1(A,C,D)) )
=> r1_tarski(B,C) ) ) ) ).
fof(t74_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> r1_tarski(B,k7_relset_1(A,A,A,A,B,B)) ) ).
fof(s1_toler_1,axiom,
( ( ! [A] :
( r2_hidden(A,f1_s1_toler_1)
=> p1_s1_toler_1(A,A) )
& ! [A,B] :
( ( r2_hidden(A,f1_s1_toler_1)
& r2_hidden(B,f1_s1_toler_1)
& p1_s1_toler_1(A,B) )
=> p1_s1_toler_1(B,A) ) )
=> ? [A] :
( v1_relat_2(A)
& v3_relat_2(A)
& v1_partfun1(A,f1_s1_toler_1,f1_s1_toler_1)
& m2_relset_1(A,f1_s1_toler_1,f1_s1_toler_1)
& ! [B,C] :
( ( r2_hidden(B,f1_s1_toler_1)
& r2_hidden(C,f1_s1_toler_1) )
=> ( r2_hidden(k4_tarski(B,C),A)
<=> p1_s1_toler_1(B,C) ) ) ) ) ).
fof(dt_m1_toler_1,axiom,
$true ).
fof(existence_m1_toler_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> ? [C] : m1_toler_1(C,A,B) ) ).
fof(dt_k1_toler_1,axiom,
! [A,B] :
( v1_relat_1(A)
=> m2_relset_1(k1_toler_1(A,B),B,B) ) ).
fof(redefinition_k1_toler_1,axiom,
! [A,B] :
( v1_relat_1(A)
=> k1_toler_1(A,B) = k2_wellord1(A,B) ) ).
fof(dt_k2_toler_1,axiom,
! [A,B,C] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ( v1_relat_2(k2_toler_1(A,B,C))
& v3_relat_2(k2_toler_1(A,B,C))
& v1_partfun1(k2_toler_1(A,B,C),C,C)
& m2_relset_1(k2_toler_1(A,B,C),C,C) ) ) ).
fof(redefinition_k2_toler_1,axiom,
! [A,B,C] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k2_toler_1(A,B,C) = k2_wellord1(B,C) ) ).
fof(dt_k3_toler_1,axiom,
$true ).
fof(dt_k4_toler_1,axiom,
$true ).
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