SET007 Axioms: SET007+610.ax
%------------------------------------------------------------------------------
% File : SET007+610 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Recursive Euclide 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 : scmp_gcd [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 27 ( 7 unt; 0 def)
% Number of atoms : 133 ( 56 equ)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 117 ( 11 ~; 3 |; 54 &)
% ( 1 <=>; 48 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Maximal term depth : 16 ( 2 avg)
% Number of predicates : 12 ( 11 usr; 0 prp; 1-3 aty)
% Number of functors : 74 ( 74 usr; 25 con; 0-4 aty)
% Number of variables : 38 ( 38 !; 0 ?)
% 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_scmp_gcd,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k6_nat_1(B,A) = k6_nat_1(A,k4_nat_1(B,A)) ) ) ) ).
fof(t2_scmp_gcd,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(B,np__0)
| k3_int_2(A,B) = k3_int_2(B,k6_int_1(A,B)) ) ) ) ) ).
fof(t3_scmp_gcd,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ( k2_scmpds_3(A) = B
=> k2_scmpds_3(k1_nat_1(A,np__2)) = k1_nat_1(k2_nat_1(np__2,k3_nat_1(k1_int_2(B),np__2)),np__4) ) ) ) ).
fof(d1_scmp_gcd,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_scmp_gcd(A) = k15_ami_3(A) ) ).
fof(t4_scmp_gcd,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A != B
& k1_scmp_gcd(A) = k1_scmp_gcd(B) ) ) ) ).
fof(t5_scmp_gcd,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k3_scmpds_2(A,B) = k1_scmp_gcd(k1_nat_1(A,B)) ) ) ).
fof(t6_scmp_gcd,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ! [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_scmpds_2,A) = k2_scmpds_3(k1_nat_1(B,C))
=> k19_scmpds_2(A,k4_xcmplx_0(C)) = k2_scmpds_3(B) ) ) ) ) ).
fof(d2_scmp_gcd,axiom,
k2_scmp_gcd = k1_scmp_gcd(np__0) ).
fof(d3_scmp_gcd,axiom,
k3_scmp_gcd = k1_scmp_gcd(np__1) ).
fof(t7_scmp_gcd,axiom,
k2_scmp_gcd != k3_scmp_gcd ).
fof(t8_scmp_gcd,axiom,
! [A] :
( m2_subset_1(A,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))
=> ! [B] :
( ( v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(B)
& m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
=> k4_card_1(k5_scmpds_4(B,A)) = k1_nat_1(k4_card_1(B),np__1) ) ) ).
fof(t9_scmp_gcd,axiom,
! [A] :
( m2_subset_1(A,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))
=> ! [B] :
( m2_subset_1(B,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))
=> k4_card_1(k6_scmpds_4(A,B)) = np__2 ) ) ).
fof(t10_scmp_gcd,axiom,
! [A] :
( m2_subset_1(A,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))
=> ! [B] :
( ( v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(B)
& m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
=> ( k1_funct_1(k5_scmpds_4(B,A),k2_scmpds_3(k4_card_1(B))) = A
& r2_hidden(k2_scmpds_3(k4_card_1(B)),k1_relat_1(k5_scmpds_4(B,A))) ) ) ) ).
fof(t11_scmp_gcd,axiom,
! [A] :
( m2_subset_1(A,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))
=> ! [B] :
( ( v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(B)
& m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
=> ! [C] :
( ( v1_ami_3(C,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(C)
& m1_ami_1(C,k1_tarski(k4_numbers),k1_scmpds_2) )
=> k1_funct_1(k3_scmpds_4(k5_scmpds_4(B,A),C),k2_scmpds_3(k4_card_1(B))) = A ) ) ) ).
fof(d4_scmp_gcd,axiom,
k4_scmp_gcd = k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k6_scmpds_2(k2_scmp_gcd,np__0),k6_scmpds_2(k3_scmp_gcd,np__7)),k7_scmpds_2(k3_scmp_gcd,k27_scmpds_1)),k4_scmpds_2(np__2)),k5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k9_scmpds_2(k3_scmp_gcd,np__3,np__9)),k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__6,np__3)),k16_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__2,np__3)),k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__7,np__3)),k17_scmpds_2(k3_scmp_gcd,k2_scmp_gcd,k1_nat_1(np__4,k26_scmpds_1),np__1)),k12_scmpds_2(k2_scmp_gcd,np__1,np__4)),k7_scmpds_2(k3_scmp_gcd,k27_scmpds_1)),k4_scmpds_2(k4_xcmplx_0(np__7))),k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__2,np__6)),k5_scmpds_2(k3_scmp_gcd)) ).
fof(t12_scmp_gcd,axiom,
k4_card_1(k4_scmp_gcd) = np__15 ).
fof(t13_scmp_gcd,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(np__15,A)
<=> r2_hidden(k2_scmpds_3(A),k1_relat_1(k4_scmp_gcd)) ) ) ).
fof(t14_scmp_gcd,axiom,
( k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__0)) = k6_scmpds_2(k2_scmp_gcd,np__0)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__1)) = k6_scmpds_2(k3_scmp_gcd,np__7)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__2)) = k7_scmpds_2(k3_scmp_gcd,k27_scmpds_1)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__3)) = k4_scmpds_2(np__2)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__4)) = k5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__5)) = k9_scmpds_2(k3_scmp_gcd,np__3,np__9)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__6)) = k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__6,np__3)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__7)) = k16_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__2,np__3)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__8)) = k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__7,np__3)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__9)) = k17_scmpds_2(k3_scmp_gcd,k2_scmp_gcd,k1_nat_1(np__4,k26_scmpds_1),np__1)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__10)) = k12_scmpds_2(k2_scmp_gcd,np__1,np__4)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__11)) = k7_scmpds_2(k3_scmp_gcd,k27_scmpds_1)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__12)) = k4_scmpds_2(k4_xcmplx_0(np__7))
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__13)) = k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__2,np__6)
& k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__14)) = k5_scmpds_2(k3_scmp_gcd) ) ).
fof(t15_scmp_gcd,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( r1_tarski(k2_scmpds_4(k4_scmp_gcd),A)
=> ( k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4)) = k2_scmpds_3(np__5)
& k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4),k2_scmp_gcd) = np__0
& k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4),k3_scmp_gcd) = np__7
& k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4),k1_scmp_gcd(k1_nat_1(np__7,k27_scmpds_1))) = k2_scmpds_3(np__2)
& k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4),k1_scmp_gcd(np__9)) = k2_scmpds_2(A,k1_scmp_gcd(np__9))
& k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4),k1_scmp_gcd(np__10)) = k2_scmpds_2(A,k1_scmp_gcd(np__10)) ) ) ) ).
fof(t16_scmp_gcd,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ~ ( r1_tarski(k4_scmp_gcd,A)
& k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A) = k2_scmpds_3(np__5)
& ~ r1_xreal_0(k2_scmpds_2(A,k3_scmp_gcd),np__0)
& k2_scmpds_2(A,k2_scmp_gcd) = np__0
& r1_xreal_0(np__0,k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__3)))
& r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__3)),k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)))
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( k8_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B)) = k5_scmpds_2(k3_scmp_gcd)
& k2_scmpds_2(A,k3_scmp_gcd) = k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k3_scmp_gcd)
& k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)) = k3_int_2(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)),k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__3)))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,k2_xcmplx_0(k2_scmpds_2(A,k3_scmp_gcd),np__1))
=> ( r1_xreal_0(C,np__1)
| k2_scmpds_2(A,k1_scmp_gcd(C)) = k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k1_scmp_gcd(C)) ) ) ) ) ) ) ) ).
fof(t17_scmp_gcd,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ~ ( r1_tarski(k4_scmp_gcd,A)
& k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A) = k2_scmpds_3(np__5)
& ~ r1_xreal_0(k2_scmpds_2(A,k3_scmp_gcd),np__0)
& k2_scmpds_2(A,k2_scmp_gcd) = np__0
& r1_xreal_0(np__0,k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__3)))
& r1_xreal_0(np__0,k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)))
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( k8_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B)) = k5_scmpds_2(k3_scmp_gcd)
& k2_scmpds_2(A,k3_scmp_gcd) = k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k3_scmp_gcd)
& k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)) = k3_int_2(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)),k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__3)))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,k2_xcmplx_0(k2_scmpds_2(A,k3_scmp_gcd),np__1))
=> ( r1_xreal_0(C,np__1)
| k2_scmpds_2(A,k1_scmp_gcd(C)) = k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k1_scmp_gcd(C)) ) ) ) ) ) ) ) ).
fof(t18_scmp_gcd,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( r1_tarski(k2_scmpds_4(k4_scmp_gcd),A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( k2_scmpds_2(A,k1_scmp_gcd(np__9)) = B
& k2_scmpds_2(A,k1_scmp_gcd(np__10)) = C
& r1_xreal_0(np__0,B)
& r1_xreal_0(np__0,C) )
=> k2_scmpds_2(k12_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),k1_scmp_gcd(np__9)) = k3_int_2(B,C) ) ) ) ) ) ).
fof(t19_scmp_gcd,axiom,
! [A] :
( m1_ami_1(A,k1_tarski(k4_numbers),k1_scmpds_2)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r1_xreal_0(np__0,C)
& r1_xreal_0(C,B)
& A = k4_funct_4(k1_scmp_gcd(np__9),k1_scmp_gcd(np__10),B,C) )
=> v11_ami_1(k17_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k2_scmpds_4(k4_scmp_gcd),A),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ) ) ).
fof(dt_k1_scmp_gcd,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_scmpds_2(k1_scmp_gcd(A)) ) ).
fof(dt_k2_scmp_gcd,axiom,
m1_scmpds_2(k2_scmp_gcd) ).
fof(dt_k3_scmp_gcd,axiom,
m1_scmpds_2(k3_scmp_gcd) ).
fof(dt_k4_scmp_gcd,axiom,
( v1_ami_3(k4_scmp_gcd,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(k4_scmp_gcd)
& m1_ami_1(k4_scmp_gcd,k1_tarski(k4_numbers),k1_scmpds_2) ) ).
%------------------------------------------------------------------------------