SET007 Axioms: SET007+379.ax
%------------------------------------------------------------------------------
% File : SET007+379 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Euclid's Algorithm
% 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 : ami_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 18 ( 4 unt; 0 def)
% Number of atoms : 103 ( 40 equ)
% Maximal formula atoms : 12 ( 5 avg)
% Number of connectives : 97 ( 12 ~; 2 |; 38 &)
% ( 3 <=>; 42 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 14 ( 12 usr; 1 prp; 0-4 aty)
% Number of functors : 47 ( 47 usr; 13 con; 0-5 aty)
% Number of variables : 30 ( 25 !; 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(t1_ami_4,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> r1_xreal_0(np__0,k5_int_1(A,B)) ) ) ) ).
fof(t2_ami_4,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> ( k4_nat_1(k1_int_2(A),k1_int_2(B)) = k6_int_1(A,B)
& k3_nat_1(k1_int_2(A),k1_int_2(B)) = k5_int_1(A,B) ) ) ) ) ).
fof(d1_ami_4,axiom,
k1_ami_4 = k17_ami_1(k1_tarski(k4_numbers),k1_ami_3,k14_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__0),k3_ami_3(k15_ami_3(np__2),k15_ami_3(np__1))),k17_ami_1(k1_tarski(k4_numbers),k1_ami_3,k14_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__1),k7_ami_3(k15_ami_3(np__0),k15_ami_3(np__1))),k17_ami_1(k1_tarski(k4_numbers),k1_ami_3,k14_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__2),k3_ami_3(k15_ami_3(np__0),k15_ami_3(np__2))),k17_ami_1(k1_tarski(k4_numbers),k1_ami_3,k14_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__3),k10_ami_3(k16_ami_3(np__0),k15_ami_3(np__1))),k14_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__4),k5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))))) ).
fof(t3_ami_4,axiom,
$true ).
fof(t4_ami_4,axiom,
k1_relat_1(k1_ami_4) = k3_enumset1(k16_ami_3(np__0),k16_ami_3(np__1),k16_ami_3(np__2),k16_ami_3(np__3),k16_ami_3(np__4)) ).
fof(t5_ami_4,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
=> ( r1_tarski(k1_ami_4,A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B)) = k16_ami_3(np__0)
=> ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1))) = k16_ami_3(np__1)
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__0)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__0))
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__1)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1))
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__2)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)) ) ) ) ) ) ).
fof(t6_ami_4,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
=> ( r1_tarski(k1_ami_4,A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B)) = k16_ami_3(np__1)
=> ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1))) = k16_ami_3(np__2)
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__0)) = k5_int_1(k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__0)),k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)))
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__1)) = k6_int_1(k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__0)),k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)))
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__2)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__2)) ) ) ) ) ) ).
fof(t7_ami_4,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
=> ( r1_tarski(k1_ami_4,A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B)) = k16_ami_3(np__2)
=> ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1))) = k16_ami_3(np__3)
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__0)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__2))
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__1)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1))
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__2)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__2)) ) ) ) ) ) ).
fof(t8_ami_4,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
=> ( r1_tarski(k1_ami_4,A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B)) = k16_ami_3(np__3)
=> ( ( ~ r1_xreal_0(k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)),np__0)
=> k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1))) = k16_ami_3(np__0) )
& ( r1_xreal_0(k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)),np__0)
=> k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1))) = k16_ami_3(np__4) )
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__0)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__0))
& k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__1)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)) ) ) ) ) ) ).
fof(t9_ami_4,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
=> ( r1_tarski(k1_ami_4,A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B)) = k16_ami_3(np__4)
=> k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,C)) = k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B) ) ) ) ) ) ).
fof(t10_ami_4,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
=> ( ( r1_ami_3(k1_tarski(k4_numbers),k1_ami_3,A,k16_ami_3(np__0))
& r1_tarski(k1_ami_4,A) )
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( k2_ami_3(A,k15_ami_3(np__0)) = B
& k2_ami_3(A,k15_ami_3(np__1)) = C )
=> ( r1_xreal_0(B,np__0)
| r1_xreal_0(C,np__0)
| k2_ami_3(k12_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k15_ami_3(np__0)) = k3_int_2(B,C) ) ) ) ) ) ) ).
fof(d2_ami_4,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k14_ami_1(k1_tarski(k4_numbers),k1_ami_3),k14_ami_1(k1_tarski(k4_numbers),k1_ami_3)) )
=> ( A = k2_ami_4
<=> ! [B] :
( m1_ami_1(B,k1_tarski(k4_numbers),k1_ami_3)
=> ! [C] :
( m1_ami_1(C,k1_tarski(k4_numbers),k1_ami_3)
=> ( r2_hidden(k4_tarski(B,C),A)
<=> ? [D] :
( v1_int_1(D)
& ? [E] :
( v1_int_1(E)
& ~ r1_xreal_0(D,np__0)
& ~ r1_xreal_0(E,np__0)
& B = k18_ami_3(k15_ami_3(np__0),k15_ami_3(np__1),D,E)
& C = k17_ami_3(k15_ami_3(np__0),k3_int_2(D,E)) ) ) ) ) ) ) ) ).
fof(t11_ami_4,axiom,
! [A] :
( r2_hidden(A,k1_relat_1(k2_ami_4))
<=> ? [B] :
( v1_int_1(B)
& ? [C] :
( v1_int_1(C)
& ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(C,np__0)
& A = k18_ami_3(k15_ami_3(np__0),k15_ami_3(np__1),B,C) ) ) ) ).
fof(t12_ami_4,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& k1_funct_1(k2_ami_4,k18_ami_3(k15_ami_3(np__0),k15_ami_3(np__1),A,B)) != k17_ami_3(k15_ami_3(np__0),k3_int_2(A,B)) ) ) ) ).
fof(t13_ami_4,axiom,
r1_ami_1(k1_tarski(k4_numbers),k1_ami_3,k17_ami_1(k1_tarski(k4_numbers),k1_ami_3,k12_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__0)),k1_ami_4),k2_ami_4) ).
fof(s1_ami_4,axiom,
( ( ~ r1_xreal_0(f4_s1_ami_4,np__0)
& ~ r1_xreal_0(f3_s1_ami_4,f4_s1_ami_4)
& f1_s1_ami_4(np__0) = f3_s1_ami_4
& f2_s1_ami_4(np__0) = f4_s1_ami_4
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(f2_s1_ami_4(A),np__0)
=> ( f1_s1_ami_4(k1_nat_1(A,np__1)) = f2_s1_ami_4(A)
& f2_s1_ami_4(k1_nat_1(A,np__1)) = k4_nat_1(f1_s1_ami_4(A),f2_s1_ami_4(A)) ) ) ) )
=> ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& f1_s1_ami_4(A) = k6_nat_1(f3_s1_ami_4,f4_s1_ami_4)
& f2_s1_ami_4(A) = np__0 ) ) ).
fof(dt_k1_ami_4,axiom,
( v1_ami_3(k1_ami_4,k1_tarski(k4_numbers),k1_ami_3)
& m1_ami_1(k1_ami_4,k1_tarski(k4_numbers),k1_ami_3) ) ).
fof(dt_k2_ami_4,axiom,
( v1_funct_1(k2_ami_4)
& m2_relset_1(k2_ami_4,k14_ami_1(k1_tarski(k4_numbers),k1_ami_3),k14_ami_1(k1_tarski(k4_numbers),k1_ami_3)) ) ).
%------------------------------------------------------------------------------