SET007 Axioms: SET007+83.ax
%------------------------------------------------------------------------------
% File : SET007+83 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : sigma-Fields and Probability
% 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 : prob_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 132 ( 32 unt; 0 def)
% Number of atoms : 572 ( 54 equ)
% Maximal formula atoms : 17 ( 4 avg)
% Number of connectives : 487 ( 47 ~; 0 |; 199 &)
% ( 15 <=>; 226 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 29 ( 27 usr; 1 prp; 0-3 aty)
% Number of functors : 44 ( 44 usr; 8 con; 0-4 aty)
% Number of variables : 339 ( 320 !; 19 ?)
% 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(rc1_prob_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& ~ v1_xboole_0(B)
& v2_finsub_1(B)
& v1_prob_1(B,A) ) ).
fof(cc1_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ( v2_finsub_1(B)
& v1_prob_1(B,A) ) ) ).
fof(rc2_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ? [C] :
( m3_prob_1(C,A,B)
& v1_xboole_0(C)
& v1_funct_1(C)
& v1_membered(C)
& v2_membered(C)
& v3_membered(C)
& v4_membered(C)
& v5_membered(C) ) ) ).
fof(t1_prob_1,axiom,
$true ).
fof(t2_prob_1,axiom,
$true ).
fof(t3_prob_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> k2_seq_1(k5_numbers,k1_numbers,B,D) = A ) ) )
=> ( v4_seq_2(B)
& k2_seq_2(B) = A ) ) ) ) ).
fof(d1_prob_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v1_prob_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,B)
=> r2_hidden(k3_subset_1(A,C),B) ) ) ) ) ).
fof(t4_prob_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> m1_subset_1(k2_tarski(B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ).
fof(t5_prob_1,axiom,
$true ).
fof(t6_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v2_finsub_1(B)
& v1_prob_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
& r2_hidden(C,B) ) ) ).
fof(t7_prob_1,axiom,
$true ).
fof(t8_prob_1,axiom,
$true ).
fof(t9_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v2_finsub_1(B)
& v1_prob_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C,D] :
( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k2_xboole_0(C,D),B) ) ) ).
fof(t10_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v2_finsub_1(B)
& v1_prob_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> r2_hidden(k1_xboole_0,B) ) ).
fof(t11_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v2_finsub_1(B)
& v1_prob_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> r2_hidden(A,B) ) ).
fof(t12_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v2_finsub_1(B)
& v1_prob_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k6_subset_1(A,C,D),B) ) ) ) ) ).
fof(t13_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v2_finsub_1(B)
& v1_prob_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C,D] :
( r1_xboole_0(k4_xboole_0(C,D),D)
& ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k2_xboole_0(k4_xboole_0(C,D),D),B) ) ) ) ).
fof(t14_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(k2_tarski(k1_xboole_0,A))
& v2_finsub_1(k2_tarski(k1_xboole_0,A))
& v1_prob_1(k2_tarski(k1_xboole_0,A),A)
& m1_subset_1(k2_tarski(k1_xboole_0,A),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(t15_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(k1_zfmisc_1(A))
& v2_finsub_1(k1_zfmisc_1(A))
& v1_prob_1(k1_zfmisc_1(A),A)
& m1_subset_1(k1_zfmisc_1(A),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(t16_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v2_finsub_1(B)
& v1_prob_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( r1_tarski(k2_tarski(k1_xboole_0,A),B)
& r1_tarski(B,k1_zfmisc_1(A)) ) ) ).
fof(t17_prob_1,axiom,
$true ).
fof(t18_prob_1,axiom,
! [A] :
( ! [B] :
~ ( r2_hidden(B,k2_zfmisc_1(k5_numbers,k1_tarski(A)))
& ! [C,D] : k4_tarski(C,D) != B )
& ! [B,C,D] :
( ( r2_hidden(k4_tarski(B,C),k2_zfmisc_1(k5_numbers,k1_tarski(A)))
& r2_hidden(k4_tarski(B,D),k2_zfmisc_1(k5_numbers,k1_tarski(A))) )
=> C = D ) ) ).
fof(t19_prob_1,axiom,
! [A] :
? [B] :
( v1_relat_1(B)
& v1_funct_1(B)
& k1_relat_1(B) = k5_numbers
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_funct_1(B,C) = A ) ) ).
fof(t20_prob_1,axiom,
$true ).
fof(t21_prob_1,axiom,
! [A] :
? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,k1_zfmisc_1(A),B,C) = A ) ) ).
fof(t22_prob_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(D,k5_numbers,k1_zfmisc_1(A))
& k8_funct_2(k5_numbers,k1_zfmisc_1(A),D,np__0) = B
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( E != np__0
=> k8_funct_2(k5_numbers,k1_zfmisc_1(A),D,E) = C ) ) ) ) ) ).
fof(t23_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> m1_subset_1(k3_tarski(k2_relat_1(B)),k1_zfmisc_1(A)) ) ).
fof(t24_prob_1,axiom,
$true ).
fof(t25_prob_1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
=> ( r2_hidden(B,k2_prob_1(A,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r2_hidden(B,k1_prob_1(A,C,D)) ) ) ) ).
fof(t26_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,D) = k3_subset_1(A,k1_prob_1(A,B,D)) ) ) ) ).
fof(d2_prob_1,axiom,
$true ).
fof(d3_prob_1,axiom,
$true ).
fof(d4_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
=> ( C = k3_prob_1(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,D) = k3_subset_1(A,k1_prob_1(A,B,D)) ) ) ) ) ).
fof(d5_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> k4_prob_1(A,B) = k3_subset_1(A,k2_prob_1(A,k3_prob_1(A,B))) ) ).
fof(t27_prob_1,axiom,
$true ).
fof(t28_prob_1,axiom,
$true ).
fof(t29_prob_1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
=> ( r2_hidden(B,k4_prob_1(A,C))
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(B,k1_prob_1(A,C,D)) ) ) ) ).
fof(t30_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( k1_prob_1(A,B,np__0) = C
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( E != np__0
=> k1_prob_1(A,B,E) = D ) ) )
=> k4_prob_1(A,B) = k5_subset_1(A,C,D) ) ) ) ) ).
fof(d6_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v2_prob_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> r1_tarski(k1_prob_1(A,B,D),k1_prob_1(A,B,C)) ) ) ) ) ) ).
fof(d7_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v3_prob_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> r1_tarski(k1_prob_1(A,B,C),k1_prob_1(A,B,D)) ) ) ) ) ) ).
fof(d8_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( m1_prob_1(B,A)
<=> ( ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k1_prob_1(A,C,D),B) )
=> r2_hidden(k4_prob_1(A,C),B) ) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,B)
=> r2_hidden(k3_subset_1(A,C),B) ) ) ) ) ) ).
fof(t31_prob_1,axiom,
$true ).
fof(t32_prob_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( m1_prob_1(B,A)
<=> ( r1_tarski(B,k1_zfmisc_1(A))
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k1_prob_1(A,C,D),B) )
=> r2_hidden(k4_prob_1(A,C),B) ) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,B)
=> r2_hidden(k3_subset_1(A,C),B) ) ) ) ) ) ).
fof(t33_prob_1,axiom,
$true ).
fof(t34_prob_1,axiom,
$true ).
fof(t35_prob_1,axiom,
! [A,B] :
( m1_prob_1(A,B)
=> ( ~ v1_xboole_0(A)
& v2_finsub_1(A)
& v1_prob_1(A,B)
& m1_subset_1(A,k1_zfmisc_1(k1_zfmisc_1(B))) ) ) ).
fof(t36_prob_1,axiom,
$true ).
fof(t37_prob_1,axiom,
$true ).
fof(t38_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
& r2_hidden(C,B) ) ) ).
fof(t39_prob_1,axiom,
$true ).
fof(t40_prob_1,axiom,
$true ).
fof(t41_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k4_subset_1(A,C,D),B) ) ) ) ) ).
fof(t42_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> r2_hidden(k1_xboole_0,B) ) ).
fof(t43_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> r2_hidden(A,B) ) ).
fof(t44_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k6_subset_1(A,C,D),B) ) ) ) ) ).
fof(d9_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
=> ( m2_prob_1(C,A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k1_prob_1(A,C,D),B) ) ) ) ) ).
fof(t45_prob_1,axiom,
$true ).
fof(t46_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> r2_hidden(k2_prob_1(A,C),B) ) ) ).
fof(d10_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( m3_prob_1(C,A,B)
<=> r2_hidden(C,B) ) ) ) ).
fof(t47_prob_1,axiom,
$true ).
fof(t48_prob_1,axiom,
! [A,B,C] :
( m1_prob_1(C,A)
=> ( r2_hidden(B,C)
=> m3_prob_1(B,A,C) ) ) ).
fof(t49_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> m3_prob_1(k5_subset_1(A,C,D),A,B) ) ) ) ).
fof(t50_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> m3_prob_1(k3_subset_1(A,C),A,B) ) ) ).
fof(t51_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> m3_prob_1(k4_subset_1(A,C,D),A,B) ) ) ) ).
fof(t52_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> m3_prob_1(k1_xboole_0,A,B) ) ).
fof(t53_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> m3_prob_1(A,A,B) ) ).
fof(t54_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> m3_prob_1(k6_subset_1(A,C,D),A,B) ) ) ) ).
fof(d11_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> k5_prob_1(A,B) = A ) ).
fof(t55_prob_1,axiom,
$true ).
fof(t56_prob_1,axiom,
$true ).
fof(t57_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ! [C] :
( m1_prob_1(C,A)
=> ( m2_prob_1(B,A,C)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> m3_prob_1(k1_prob_1(A,B,D),A,C) ) ) ) ) ) ).
fof(t58_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ! [C] :
( m1_prob_1(C,A)
=> ( m2_prob_1(B,A,C)
=> m3_prob_1(k2_prob_1(A,B),A,C) ) ) ) ) ).
fof(t59_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_prob_1(C,A)
=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& k1_relat_1(D) = C
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( r2_hidden(E,C)
=> ( ( r2_hidden(B,E)
=> k1_funct_1(D,E) = np__1 )
& ( ~ r2_hidden(B,E)
=> k1_funct_1(D,E) = np__0 ) ) ) ) ) ) ) ).
fof(t60_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_prob_1(C,A)
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,C,k1_numbers)
& m2_relset_1(D,C,k1_numbers)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( r2_hidden(E,C)
=> ( ( r2_hidden(B,E)
=> k2_seq_1(C,k1_numbers,D,E) = np__1 )
& ( ~ r2_hidden(B,E)
=> k2_seq_1(C,k1_numbers,D,E) = np__0 ) ) ) ) ) ) ) ).
fof(t61_prob_1,axiom,
$true ).
fof(t62_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,k1_numbers)
& m2_relset_1(D,B,k1_numbers) )
=> ( v1_funct_1(k5_relat_1(C,D))
& v1_funct_2(k5_relat_1(C,D),k5_numbers,k1_numbers)
& m2_relset_1(k5_relat_1(C,D),k5_numbers,k1_numbers) ) ) ) ) ) ).
fof(d12_prob_1,axiom,
$true ).
fof(d13_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ( m4_prob_1(C,A,B)
<=> ( ! [D] :
( m3_prob_1(D,A,B)
=> r1_xreal_0(np__0,k10_prob_1(A,B,C,D)) )
& k2_seq_1(B,k1_numbers,C,A) = np__1
& ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m3_prob_1(E,A,B)
=> ( r1_xboole_0(D,E)
=> k10_prob_1(A,B,C,k7_prob_1(A,B,D,E)) = k2_xcmplx_0(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)) ) ) )
& ! [D] :
( m2_prob_1(D,A,B)
=> ( v2_prob_1(D,A)
=> ( v4_seq_2(k9_prob_1(A,B,D,C))
& k2_seq_2(k9_prob_1(A,B,D,C)) = k2_seq_1(B,k1_numbers,C,k4_prob_1(A,D)) ) ) ) ) ) ) ) ) ).
fof(t63_prob_1,axiom,
$true ).
fof(t64_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> k2_seq_1(B,k1_numbers,C,k1_xboole_0) = np__0 ) ) ) ).
fof(t65_prob_1,axiom,
$true ).
fof(t66_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> k10_prob_1(A,B,C,k5_prob_1(A,B)) = np__1 ) ) ) ).
fof(t67_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m4_prob_1(D,A,B)
=> k2_xcmplx_0(k10_prob_1(A,B,D,k8_prob_1(A,B,k5_prob_1(A,B),C)),k10_prob_1(A,B,D,C)) = np__1 ) ) ) ) ).
fof(t68_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m4_prob_1(D,A,B)
=> k10_prob_1(A,B,D,k8_prob_1(A,B,k5_prob_1(A,B),C)) = k6_xcmplx_0(np__1,k10_prob_1(A,B,D,C)) ) ) ) ) ).
fof(t69_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m4_prob_1(E,A,B)
=> ( r1_tarski(C,D)
=> k10_prob_1(A,B,E,k8_prob_1(A,B,D,C)) = k6_xcmplx_0(k10_prob_1(A,B,E,D),k10_prob_1(A,B,E,C)) ) ) ) ) ) ) ).
fof(t70_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m4_prob_1(E,A,B)
=> ( r1_tarski(C,D)
=> r1_xreal_0(k10_prob_1(A,B,E,C),k10_prob_1(A,B,E,D)) ) ) ) ) ) ) ).
fof(t71_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m4_prob_1(D,A,B)
=> r1_xreal_0(k10_prob_1(A,B,D,C),np__1) ) ) ) ) ).
fof(t72_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m4_prob_1(E,A,B)
=> k10_prob_1(A,B,E,k7_prob_1(A,B,C,D)) = k2_xcmplx_0(k10_prob_1(A,B,E,C),k10_prob_1(A,B,E,k8_prob_1(A,B,D,C))) ) ) ) ) ) ).
fof(t73_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m4_prob_1(E,A,B)
=> k10_prob_1(A,B,E,k7_prob_1(A,B,C,D)) = k2_xcmplx_0(k10_prob_1(A,B,E,C),k10_prob_1(A,B,E,k8_prob_1(A,B,D,k6_prob_1(A,B,C,D)))) ) ) ) ) ) ).
fof(t74_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m4_prob_1(E,A,B)
=> k10_prob_1(A,B,E,k7_prob_1(A,B,C,D)) = k6_xcmplx_0(k2_xcmplx_0(k10_prob_1(A,B,E,C),k10_prob_1(A,B,E,D)),k10_prob_1(A,B,E,k6_prob_1(A,B,C,D))) ) ) ) ) ) ).
fof(t75_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m4_prob_1(E,A,B)
=> r1_xreal_0(k10_prob_1(A,B,E,k7_prob_1(A,B,C,D)),k2_xcmplx_0(k10_prob_1(A,B,E,C),k10_prob_1(A,B,E,D))) ) ) ) ) ) ).
fof(t76_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m1_prob_1(k1_zfmisc_1(A),A) ) ).
fof(d14_prob_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_prob_1(C,A)
=> ( C = k11_prob_1(A,B)
<=> ( r1_tarski(B,C)
& ! [D] :
( ( r1_tarski(B,D)
& m1_prob_1(D,A) )
=> r1_tarski(C,D) ) ) ) ) ) ) ).
fof(d17_prob_1,axiom,
k14_prob_1 = k11_prob_1(k1_numbers,k13_prob_1) ).
fof(dt_m1_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ).
fof(existence_m1_prob_1,axiom,
! [A] :
? [B] : m1_prob_1(B,A) ).
fof(dt_m2_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) ) ) ) ).
fof(existence_m2_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ? [C] : m2_prob_1(C,A,B) ) ).
fof(dt_m3_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(A)) ) ) ).
fof(existence_m3_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ? [C] : m3_prob_1(C,A,B) ) ).
fof(dt_m4_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A) )
=> ! [C] :
( m4_prob_1(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) ) ) ) ).
fof(existence_m4_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A) )
=> ? [C] : m4_prob_1(C,A,B) ) ).
fof(dt_k1_prob_1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k1_prob_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(redefinition_k1_prob_1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> k1_prob_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k2_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> m1_subset_1(k2_prob_1(A,B),k1_zfmisc_1(A)) ) ).
fof(redefinition_k2_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> k2_prob_1(A,B) = k3_card_3(B) ) ).
fof(dt_k3_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v1_funct_1(k3_prob_1(A,B))
& v1_funct_2(k3_prob_1(A,B),k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(k3_prob_1(A,B),k5_numbers,k1_zfmisc_1(A)) ) ) ).
fof(involutiveness_k3_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> k3_prob_1(A,k3_prob_1(A,B)) = B ) ).
fof(dt_k4_prob_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> m1_subset_1(k4_prob_1(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_k5_prob_1,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> m3_prob_1(k5_prob_1(A,B),A,B) ) ).
fof(dt_k6_prob_1,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m3_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> m3_prob_1(k6_prob_1(A,B,C,D),A,B) ) ).
fof(commutativity_k6_prob_1,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m3_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> k6_prob_1(A,B,C,D) = k6_prob_1(A,B,D,C) ) ).
fof(idempotence_k6_prob_1,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m3_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> k6_prob_1(A,B,C,C) = C ) ).
fof(redefinition_k6_prob_1,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m3_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> k6_prob_1(A,B,C,D) = k3_xboole_0(C,D) ) ).
fof(dt_k7_prob_1,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m3_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> m3_prob_1(k7_prob_1(A,B,C,D),A,B) ) ).
fof(commutativity_k7_prob_1,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m3_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> k7_prob_1(A,B,C,D) = k7_prob_1(A,B,D,C) ) ).
fof(idempotence_k7_prob_1,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m3_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> k7_prob_1(A,B,C,C) = C ) ).
fof(redefinition_k7_prob_1,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m3_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> k7_prob_1(A,B,C,D) = k2_xboole_0(C,D) ) ).
fof(dt_k8_prob_1,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m3_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> m3_prob_1(k8_prob_1(A,B,C,D),A,B) ) ).
fof(redefinition_k8_prob_1,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m3_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> k8_prob_1(A,B,C,D) = k4_xboole_0(C,D) ) ).
fof(dt_k9_prob_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m2_prob_1(C,A,B)
& v1_funct_1(D)
& v1_funct_2(D,B,k1_numbers)
& m1_relset_1(D,B,k1_numbers) )
=> ( v1_funct_1(k9_prob_1(A,B,C,D))
& v1_funct_2(k9_prob_1(A,B,C,D),k5_numbers,k1_numbers)
& m2_relset_1(k9_prob_1(A,B,C,D),k5_numbers,k1_numbers) ) ) ).
fof(redefinition_k9_prob_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m2_prob_1(C,A,B)
& v1_funct_1(D)
& v1_funct_2(D,B,k1_numbers)
& m1_relset_1(D,B,k1_numbers) )
=> k9_prob_1(A,B,C,D) = k5_relat_1(C,D) ) ).
fof(dt_k10_prob_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m1_relset_1(C,B,k1_numbers)
& m3_prob_1(D,A,B) )
=> m1_subset_1(k10_prob_1(A,B,C,D),k1_numbers) ) ).
fof(redefinition_k10_prob_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m1_relset_1(C,B,k1_numbers)
& m3_prob_1(D,A,B) )
=> k10_prob_1(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k11_prob_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> m1_prob_1(k11_prob_1(A,B),A) ) ).
fof(dt_k12_prob_1,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k12_prob_1(A),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k13_prob_1,axiom,
m1_subset_1(k13_prob_1,k1_zfmisc_1(k1_zfmisc_1(k1_numbers))) ).
fof(dt_k14_prob_1,axiom,
m1_prob_1(k14_prob_1,k1_numbers) ).
fof(d15_prob_1,axiom,
! [A] :
( v1_xreal_0(A)
=> k12_prob_1(A) = a_1_0_prob_1(A) ) ).
fof(d16_prob_1,axiom,
k13_prob_1 = a_0_0_prob_1 ).
fof(fraenkel_a_1_0_prob_1,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ( r2_hidden(A,a_1_0_prob_1(B))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& A = C
& ~ r1_xreal_0(B,C) ) ) ) ).
fof(fraenkel_a_0_0_prob_1,axiom,
! [A] :
( r2_hidden(A,a_0_0_prob_1)
<=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
& A = B
& ? [C] :
( v1_xreal_0(C)
& B = k12_prob_1(C) ) ) ) ).
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