SET007 Axioms: SET007+661.ax
%------------------------------------------------------------------------------
% File : SET007+661 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Correctness of the Fibonacci Sequence and the Euclide Algorithm
% Version : [Urb08] axioms.
% English : Justifying the Correctness of the Fibonacci Sequence and the
% Euclide Algorithm by Loop-Invariant
% 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 : scpinvar [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 38 ( 2 unt; 0 def)
% Number of atoms : 424 ( 117 equ)
% Maximal formula atoms : 46 ( 11 avg)
% Number of connectives : 406 ( 20 ~; 24 |; 199 &)
% ( 5 <=>; 158 =>; 0 <=; 0 <~>)
% Maximal formula depth : 41 ( 12 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 29 ( 28 usr; 0 prp; 1-3 aty)
% Number of functors : 89 ( 89 usr; 33 con; 0-4 aty)
% Number of variables : 126 ( 123 !; 3 ?)
% 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_scpinvar,axiom,
! [A,B,C] :
( ( v1_ami_3(A,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(A)
& v3_scmpds_4(A)
& m1_ami_1(A,k1_tarski(k4_numbers),k1_scmpds_2)
& m1_scmpds_2(B)
& v1_int_1(C) )
=> ( v1_relat_1(k3_scpinvar(B,C,A))
& v1_funct_1(k3_scpinvar(B,C,A))
& v1_finset_1(k3_scpinvar(B,C,A))
& v1_ami_3(k3_scpinvar(B,C,A),k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(k3_scpinvar(B,C,A))
& v3_scmpds_4(k3_scpinvar(B,C,A)) ) ) ).
fof(fc2_scpinvar,axiom,
! [A,B,C] :
( ( v1_ami_3(A,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(A)
& v3_scmpds_5(A)
& m1_ami_1(A,k1_tarski(k4_numbers),k1_scmpds_2)
& m1_scmpds_2(B)
& v1_int_1(C) )
=> ( v1_relat_1(k3_scpinvar(B,C,A))
& v1_funct_1(k3_scpinvar(B,C,A))
& v1_finset_1(k3_scpinvar(B,C,A))
& v1_ami_3(k3_scpinvar(B,C,A),k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(k3_scpinvar(B,C,A))
& v3_scmpds_5(k3_scpinvar(B,C,A)) ) ) ).
fof(t1_scpinvar,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_nat_1(A,B)
& r1_nat_1(A,C) )
=> r2_int_1(A,k6_xcmplx_0(B,C)) ) ) ) ) ).
fof(t2_scpinvar,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_nat_1(A,B)
<=> r2_int_1(A,B) ) ) ) ).
fof(t3_scpinvar,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k6_nat_1(A,B) = k6_nat_1(A,k1_int_2(k6_xcmplx_0(B,A))) ) ) ).
fof(t4_scpinvar,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> k3_int_2(A,B) = k3_int_2(A,k6_xcmplx_0(B,A)) ) ) ) ).
fof(t5_scpinvar,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))
=> ! [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(k6_scmpds_4(A,B),C),k2_scmpds_3(np__0)) = A
& k1_funct_1(k3_scmpds_4(k6_scmpds_4(A,B),C),k2_scmpds_3(np__1)) = B ) ) ) ) ).
fof(t6_scpinvar,axiom,
! [A] :
( m1_scmpds_2(A)
=> ! [B] :
( m1_scmpds_2(B)
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers)
& m2_relset_1(C,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers)
& ! [D] :
( m1_subset_1(D,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( k2_scmpds_2(D,A) = k2_scmpds_2(D,B)
=> k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,C,D) = np__0 )
& ( k2_scmpds_2(D,A) != k2_scmpds_2(D,B)
=> k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,C,D) = k4_square_1(k1_int_2(k2_scmpds_2(D,A)),k1_int_2(k2_scmpds_2(D,B))) ) ) ) ) ) ) ).
fof(t7_scpinvar,axiom,
! [A] :
( m1_scmpds_2(A)
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers)
& m2_relset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers)
& ! [C] :
( m1_subset_1(C,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( r1_xreal_0(np__0,k2_scmpds_2(C,A))
=> k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,B,C) = np__0 )
& ( ~ r1_xreal_0(np__0,k2_scmpds_2(C,A))
=> k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,B,C) = k4_xcmplx_0(k2_scmpds_2(C,A)) ) ) ) ) ) ).
fof(d1_scpinvar,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_scpinvar(A,B) = k3_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k6_scmpds_2(k2_scmp_gcd,np__0),k6_scmpds_2(k1_scmp_gcd(np__1),np__0)),k6_scmpds_2(k1_scmp_gcd(np__2),k4_xcmplx_0(A))),k6_scmpds_2(k1_scmp_gcd(np__3),k1_nat_1(B,np__1))),k2_scmpds_8(k2_scmp_gcd,np__2,k5_scmpds_4(k6_scmpds_4(k13_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__3),np__1,np__0),k12_scmpds_2(k2_scmp_gcd,np__2,np__1)),k12_scmpds_2(k2_scmp_gcd,np__3,np__1)))) ) ) ).
fof(t8_scpinvar,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_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)
& v3_scmpds_4(B)
& v3_scmpds_5(B)
& m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
=> ! [C] :
( m1_scmpds_2(C)
=> ! [D] :
( m1_scmpds_2(D)
=> ! [E] :
( m1_scmpds_2(E)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ! [I] :
( m1_trees_4(I,k1_numbers,k6_wsierp_1)
=> ( ( r1_scpisort(I,A,H)
& k3_finseq_1(I) = F
& k2_scmpds_2(A,D) = np__0
& k2_scmpds_2(A,C) = np__0
& k2_scmpds_2(A,k1_scmp_gcd(G)) = k4_xcmplx_0(F)
& k2_scmpds_2(A,E) = k1_nat_1(H,np__1)
& ! [J] :
( m1_subset_1(J,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( k2_scmpds_2(J,C) = np__0
& ! [K] :
( m2_subset_1(K,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(K,H)
=> k2_scmpds_2(J,k1_scmp_gcd(K)) = k2_scmpds_2(A,k1_scmp_gcd(K)) ) ) )
=> ( ! [K] :
( m1_trees_4(K,k1_numbers,k6_wsierp_1)
=> ~ ( r1_scpisort(K,A,H)
& k3_finseq_1(K) = k2_xcmplx_0(k2_scmpds_2(J,k1_scmp_gcd(G)),F)
& k2_scmpds_2(J,D) = k7_wsierp_1(K)
& k2_scmpds_2(J,E) = k1_nat_1(k1_nat_1(H,np__1),k3_finseq_1(K)) ) )
| r1_xreal_0(np__0,k2_scmpds_2(J,k1_scmp_gcd(G)))
| ( k2_scmpds_2(k9_scmpds_4(B,J),C) = np__0
& r1_scmpds_6(B,J)
& r2_scmpds_6(B,J)
& k2_scmpds_2(k9_scmpds_4(B,J),k1_scmp_gcd(G)) = k2_xcmplx_0(k2_scmpds_2(J,k1_scmp_gcd(G)),np__1)
& ? [K] :
( m1_trees_4(K,k1_numbers,k6_wsierp_1)
& r1_scpisort(K,A,H)
& k3_finseq_1(K) = k2_xcmplx_0(k2_xcmplx_0(k2_scmpds_2(J,k1_scmp_gcd(G)),F),np__1)
& k2_scmpds_2(k9_scmpds_4(B,J),E) = k1_nat_1(k1_nat_1(H,np__1),k3_finseq_1(K))
& k2_scmpds_2(k9_scmpds_4(B,J),D) = k7_wsierp_1(K) )
& ! [K] :
( m2_subset_1(K,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(K,H)
=> k2_scmpds_2(k9_scmpds_4(B,J),k1_scmp_gcd(K)) = k2_scmpds_2(A,k1_scmp_gcd(K)) ) ) ) ) ) ) )
=> ( r1_xreal_0(k4_card_1(B),np__0)
| ( k2_scmpds_2(k9_scmpds_4(k2_scmpds_8(C,G,B),A),D) = k7_wsierp_1(I)
& r1_scmpds_6(k2_scmpds_8(C,G,B),A)
& r2_scmpds_6(k2_scmpds_8(C,G,B),A) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t9_scpinvar,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)
=> ! [D] :
( m1_trees_4(D,k1_numbers,k6_wsierp_1)
=> ( ( r1_xreal_0(np__3,C)
& r1_scpisort(D,A,C)
& k3_finseq_1(D) = B )
=> ( k2_scmpds_2(k9_scmpds_4(k1_scpinvar(B,C),A),k1_scmp_gcd(np__1)) = k7_wsierp_1(D)
& v2_scmpds_4(k1_scpinvar(B,C)) ) ) ) ) ) ) ).
fof(d2_scpinvar,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_scpinvar(A) = k3_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k6_scmpds_2(k2_scmp_gcd,np__0),k6_scmpds_2(k1_scmp_gcd(np__1),np__0)),k6_scmpds_2(k1_scmp_gcd(np__2),np__1)),k6_scmpds_2(k1_scmp_gcd(np__3),A)),k3_scmpds_8(k2_scmp_gcd,np__3,k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__4,np__2),k13_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__2,np__1)),k17_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__1,np__4)),k12_scmpds_2(k2_scmp_gcd,np__3,k4_xcmplx_0(np__1))))) ) ).
fof(t10_scpinvar,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_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)
& v3_scmpds_4(B)
& v3_scmpds_5(B)
& m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
=> ! [C] :
( m1_scmpds_2(C)
=> ! [D] :
( m1_scmpds_2(D)
=> ! [E] :
( m1_scmpds_2(E)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( ( k2_scmpds_2(A,C) = np__0
& k2_scmpds_2(A,D) = np__0
& k2_scmpds_2(A,E) = np__1
& k2_scmpds_2(A,k1_scmp_gcd(G)) = F
& ! [H] :
( m1_subset_1(H,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ( ( F = k2_xcmplx_0(k2_scmpds_2(H,k1_scmp_gcd(G)),I)
& k2_scmpds_2(H,D) = k3_pre_ff(I)
& k2_scmpds_2(H,E) = k3_pre_ff(k1_nat_1(I,np__1))
& k2_scmpds_2(H,C) = np__0 )
=> ( r1_xreal_0(k2_scmpds_2(H,k1_scmp_gcd(G)),np__0)
| ( k2_scmpds_2(k9_scmpds_4(B,H),C) = np__0
& r1_scmpds_6(B,H)
& r2_scmpds_6(B,H)
& k2_scmpds_2(k9_scmpds_4(B,H),k1_scmp_gcd(G)) = k6_xcmplx_0(k2_scmpds_2(H,k1_scmp_gcd(G)),np__1)
& k2_scmpds_2(k9_scmpds_4(B,H),D) = k3_pre_ff(k1_nat_1(I,np__1))
& k2_scmpds_2(k9_scmpds_4(B,H),E) = k3_pre_ff(k1_nat_1(k1_nat_1(I,np__1),np__1)) ) ) ) ) ) )
=> ( r1_xreal_0(k4_card_1(B),np__0)
| ( k2_scmpds_2(k9_scmpds_4(k3_scmpds_8(C,G,B),A),D) = k3_pre_ff(F)
& k2_scmpds_2(k9_scmpds_4(k3_scmpds_8(C,G,B),A),E) = k3_pre_ff(k1_nat_1(F,np__1))
& r1_scmpds_6(k3_scmpds_8(C,G,B),A)
& r2_scmpds_6(k3_scmpds_8(C,G,B),A) ) ) ) ) ) ) ) ) ) ) ).
fof(t11_scpinvar,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)
=> ( k2_scmpds_2(k9_scmpds_4(k2_scpinvar(B),A),k1_scmp_gcd(np__1)) = k3_pre_ff(B)
& k2_scmpds_2(k9_scmpds_4(k2_scpinvar(B),A),k1_scmp_gcd(np__2)) = k3_pre_ff(k1_nat_1(B,np__1))
& v2_scmpds_4(k2_scpinvar(B)) ) ) ) ).
fof(d3_scpinvar,axiom,
! [A] :
( m1_scmpds_2(A)
=> ! [B] :
( v1_int_1(B)
=> ! [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) )
=> k3_scpinvar(A,B,C) = k5_scmpds_4(k3_scmpds_4(k6_scmpds_4(k8_scmpds_2(A,B,np__2),k4_scmpds_2(k1_nat_1(k4_card_1(C),np__2))),C),k4_scmpds_2(k4_xcmplx_0(k1_nat_1(k4_card_1(C),np__2)))) ) ) ) ).
fof(t12_scpinvar,axiom,
! [A] :
( m1_scmpds_2(A)
=> ! [B] :
( v1_int_1(B)
=> ! [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) )
=> k4_card_1(k3_scpinvar(A,B,C)) = k1_nat_1(k4_card_1(C),np__3) ) ) ) ).
fof(t13_scpinvar,axiom,
! [A] :
( m1_scmpds_2(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_ami_3(D,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(D)
& m1_ami_1(D,k1_tarski(k4_numbers),k1_scmpds_2) )
=> ( ~ r1_xreal_0(k1_nat_1(k4_card_1(D),np__3),C)
<=> r2_hidden(k2_scmpds_3(C),k1_relat_1(k3_scpinvar(A,B,D))) ) ) ) ) ) ).
fof(t14_scpinvar,axiom,
! [A] :
( m1_scmpds_2(A)
=> ! [B] :
( v1_int_1(B)
=> ! [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) )
=> ( r2_hidden(k2_scmpds_3(np__0),k1_relat_1(k3_scpinvar(A,B,C)))
& r2_hidden(k2_scmpds_3(np__1),k1_relat_1(k3_scpinvar(A,B,C))) ) ) ) ) ).
fof(t15_scpinvar,axiom,
! [A] :
( m1_scmpds_2(A)
=> ! [B] :
( v1_int_1(B)
=> ! [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_scpinvar(A,B,C),k2_scmpds_3(np__0)) = k8_scmpds_2(A,B,np__2)
& k1_funct_1(k3_scpinvar(A,B,C),k2_scmpds_3(np__1)) = k4_scmpds_2(k1_nat_1(k4_card_1(C),np__2))
& k1_funct_1(k3_scpinvar(A,B,C),k2_scmpds_3(k1_nat_1(k4_card_1(C),np__2))) = k4_scmpds_2(k4_xcmplx_0(k1_nat_1(k4_card_1(C),np__2))) ) ) ) ) ).
fof(t16_scpinvar,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_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] :
( m1_scmpds_2(C)
=> ! [D] :
( v1_int_1(D)
=> ( k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),D)) = np__0
=> ( r1_scmpds_6(k3_scpinvar(C,D,B),A)
& r2_scmpds_6(k3_scpinvar(C,D,B),A) ) ) ) ) ) ) ).
fof(t17_scpinvar,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_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] :
( m1_scmpds_2(C)
=> ! [D] :
( m1_scmpds_2(D)
=> ! [E] :
( v1_int_1(E)
=> ( k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),E)) = np__0
=> k9_scmpds_4(k3_scpinvar(C,E,B),A) = k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,A,k12_ami_3(k1_tarski(k4_numbers),k1_scmpds_2,k2_scmpds_3(k1_nat_1(k4_card_1(B),np__3)))) ) ) ) ) ) ) ).
fof(t18_scpinvar,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_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] :
( m1_scmpds_2(C)
=> ! [D] :
( v1_int_1(D)
=> ( k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),D)) = np__0
=> k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k9_scmpds_4(k3_scpinvar(C,D,B),A)) = k2_scmpds_3(k1_nat_1(k4_card_1(B),np__3)) ) ) ) ) ) ).
fof(t19_scpinvar,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_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] :
( m1_scmpds_2(C)
=> ! [D] :
( m1_scmpds_2(D)
=> ! [E] :
( v1_int_1(E)
=> ( k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),E)) = np__0
=> k2_scmpds_2(k9_scmpds_4(k3_scpinvar(C,E,B),A),D) = k2_scmpds_2(A,D) ) ) ) ) ) ) ).
fof(t20_scpinvar,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_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)
& v3_scmpds_4(B)
& v3_scmpds_5(B)
& m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
=> ! [C] :
( m1_scmpds_2(C)
=> ! [D] :
( m1_scmpds_2(D)
=> ! [E] :
( m1_scmpds_2(E)
=> ! [F] :
( v1_int_1(F)
=> ! [G] :
( v1_int_1(G)
=> ( ( k2_scmpds_2(A,C) = G
& k2_scmpds_2(A,k3_scmpds_2(G,F)) = k6_xcmplx_0(k2_scmpds_2(A,D),k2_scmpds_2(A,E))
& ! [H] :
( m1_subset_1(H,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( k2_scmpds_2(H,C) = G
& k2_scmpds_2(H,k3_scmpds_2(G,F)) = k6_xcmplx_0(k2_scmpds_2(H,D),k2_scmpds_2(H,E)) )
=> ( r1_xreal_0(k2_scmpds_2(H,D),np__0)
| r1_xreal_0(k2_scmpds_2(H,E),np__0)
| k2_scmpds_2(H,D) = k2_scmpds_2(H,E)
| ( k2_scmpds_2(k9_scmpds_4(B,H),C) = G
& r1_scmpds_6(B,H)
& r2_scmpds_6(B,H)
& ( ~ r1_xreal_0(k2_scmpds_2(H,D),k2_scmpds_2(H,E))
=> ( k2_scmpds_2(k9_scmpds_4(B,H),D) = k6_xcmplx_0(k2_scmpds_2(H,D),k2_scmpds_2(H,E))
& k2_scmpds_2(k9_scmpds_4(B,H),E) = k2_scmpds_2(H,E) ) )
& ( r1_xreal_0(k2_scmpds_2(H,D),k2_scmpds_2(H,E))
=> ( k2_scmpds_2(k9_scmpds_4(B,H),E) = k6_xcmplx_0(k2_scmpds_2(H,E),k2_scmpds_2(H,D))
& k2_scmpds_2(k9_scmpds_4(B,H),D) = k2_scmpds_2(H,D) ) )
& k2_scmpds_2(k9_scmpds_4(B,H),k3_scmpds_2(G,F)) = k6_xcmplx_0(k2_scmpds_2(k9_scmpds_4(B,H),D),k2_scmpds_2(k9_scmpds_4(B,H),E)) ) ) ) ) )
=> ( r1_xreal_0(k4_card_1(B),np__0)
| r1_xreal_0(k2_scmpds_2(A,D),np__0)
| r1_xreal_0(k2_scmpds_2(A,E),np__0)
| ( r1_scmpds_6(k3_scpinvar(C,F,B),A)
& r2_scmpds_6(k3_scpinvar(C,F,B),A)
& ( k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),F)) != np__0
=> k9_scmpds_4(k3_scpinvar(C,F,B),A) = k9_scmpds_4(k3_scpinvar(C,F,B),k9_scmpds_4(B,A)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_scpinvar,axiom,
k4_scpinvar = k3_scmpds_4(k5_scmpds_4(k6_scmpds_4(k6_scmpds_2(k2_scmp_gcd,np__0),k17_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__3,np__1)),k14_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__3,np__2)),k3_scpinvar(k2_scmp_gcd,np__3,k5_scmpds_4(k5_scmpds_4(k3_scmpds_6(k2_scmp_gcd,np__3,k1_scmpds_4(k14_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__1,np__2)),k1_scmpds_4(k14_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__2,np__1))),k17_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__3,np__1)),k14_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__3,np__2)))) ).
fof(t21_scpinvar,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_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)
& v3_scmpds_4(B)
& v3_scmpds_5(B)
& m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
=> ! [C] :
( m1_scmpds_2(C)
=> ! [D] :
( m1_scmpds_2(D)
=> ! [E] :
( m1_scmpds_2(E)
=> ! [F] :
( v1_int_1(F)
=> ! [G] :
( v1_int_1(G)
=> ( ( k2_scmpds_2(A,C) = G
& k2_scmpds_2(A,k3_scmpds_2(G,F)) = k6_xcmplx_0(k2_scmpds_2(A,D),k2_scmpds_2(A,E))
& ! [H] :
( m1_subset_1(H,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( k2_scmpds_2(H,C) = G
& k2_scmpds_2(H,k3_scmpds_2(G,F)) = k6_xcmplx_0(k2_scmpds_2(H,D),k2_scmpds_2(H,E)) )
=> ( r1_xreal_0(k2_scmpds_2(H,D),np__0)
| r1_xreal_0(k2_scmpds_2(H,E),np__0)
| k2_scmpds_2(H,D) = k2_scmpds_2(H,E)
| ( k2_scmpds_2(k9_scmpds_4(B,H),C) = G
& r1_scmpds_6(B,H)
& r2_scmpds_6(B,H)
& ( ~ r1_xreal_0(k2_scmpds_2(H,D),k2_scmpds_2(H,E))
=> ( k2_scmpds_2(k9_scmpds_4(B,H),D) = k6_xcmplx_0(k2_scmpds_2(H,D),k2_scmpds_2(H,E))
& k2_scmpds_2(k9_scmpds_4(B,H),E) = k2_scmpds_2(H,E) ) )
& ( r1_xreal_0(k2_scmpds_2(H,D),k2_scmpds_2(H,E))
=> ( k2_scmpds_2(k9_scmpds_4(B,H),E) = k6_xcmplx_0(k2_scmpds_2(H,E),k2_scmpds_2(H,D))
& k2_scmpds_2(k9_scmpds_4(B,H),D) = k2_scmpds_2(H,D) ) )
& k2_scmpds_2(k9_scmpds_4(B,H),k3_scmpds_2(G,F)) = k6_xcmplx_0(k2_scmpds_2(k9_scmpds_4(B,H),D),k2_scmpds_2(k9_scmpds_4(B,H),E)) ) ) ) ) )
=> ( r1_xreal_0(k4_card_1(B),np__0)
| r1_xreal_0(k2_scmpds_2(A,D),np__0)
| r1_xreal_0(k2_scmpds_2(A,E),np__0)
| ( k2_scmpds_2(k9_scmpds_4(k3_scpinvar(C,F,B),A),D) = k3_int_2(k2_scmpds_2(A,D),k2_scmpds_2(A,E))
& k2_scmpds_2(k9_scmpds_4(k3_scpinvar(C,F,B),A),E) = k3_int_2(k2_scmpds_2(A,D),k2_scmpds_2(A,E)) ) ) ) ) ) ) ) ) ) ) ).
fof(t22_scpinvar,axiom,
k4_card_1(k4_scpinvar) = np__12 ).
fof(t23_scpinvar,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( k2_scmpds_2(A,k1_scmp_gcd(np__1)) = B
& k2_scmpds_2(A,k1_scmp_gcd(np__2)) = C )
=> ( r1_xreal_0(B,np__0)
| r1_xreal_0(C,np__0)
| ( k2_scmpds_2(k9_scmpds_4(k4_scpinvar,A),k1_scmp_gcd(np__1)) = k3_int_2(B,C)
& k2_scmpds_2(k9_scmpds_4(k4_scpinvar,A),k1_scmp_gcd(np__2)) = k3_int_2(B,C)
& r1_scmpds_6(k4_scpinvar,A)
& r2_scmpds_6(k4_scpinvar,A) ) ) ) ) ) ) ).
fof(s1_scpinvar,axiom,
( ( ~ r1_xreal_0(k4_card_1(f3_s1_scpinvar),np__0)
& ! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( p1_s1_scpinvar(k1_scmpds_8(A))
=> ( f1_s1_scpinvar(k1_scmpds_8(A)) = np__0
<=> r1_xreal_0(np__0,k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f2_s1_scpinvar,f4_s1_scpinvar),f5_s1_scpinvar))) ) ) )
& p1_s1_scpinvar(k1_scmpds_8(f2_s1_scpinvar))
& ! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( p1_s1_scpinvar(k1_scmpds_8(A))
& k2_scmpds_2(A,f4_s1_scpinvar) = k2_scmpds_2(f2_s1_scpinvar,f4_s1_scpinvar) )
=> ( r1_xreal_0(np__0,k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f2_s1_scpinvar,f4_s1_scpinvar),f5_s1_scpinvar)))
| ( k2_scmpds_2(k9_scmpds_4(f3_s1_scpinvar,A),f4_s1_scpinvar) = k2_scmpds_2(A,f4_s1_scpinvar)
& r1_scmpds_6(f3_s1_scpinvar,A)
& r2_scmpds_6(f3_s1_scpinvar,A)
& ~ r1_xreal_0(f1_s1_scpinvar(k1_scmpds_8(A)),f1_s1_scpinvar(k1_scmpds_8(k9_scmpds_4(f3_s1_scpinvar,A))))
& p1_s1_scpinvar(k1_scmpds_8(k9_scmpds_4(f3_s1_scpinvar,A))) ) ) ) ) )
=> ( f1_s1_scpinvar(k1_scmpds_8(k9_scmpds_4(k2_scmpds_8(f4_s1_scpinvar,f5_s1_scpinvar,f3_s1_scpinvar),f2_s1_scpinvar))) = np__0
& p1_s1_scpinvar(k1_scmpds_8(k9_scmpds_4(k2_scmpds_8(f4_s1_scpinvar,f5_s1_scpinvar,f3_s1_scpinvar),f2_s1_scpinvar))) ) ) ).
fof(s2_scpinvar,axiom,
( ( ~ r1_xreal_0(k4_card_1(f3_s2_scpinvar),np__0)
& ! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( p1_s2_scpinvar(k1_scmpds_8(A))
=> ( f1_s2_scpinvar(k1_scmpds_8(A)) = np__0
<=> r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f2_s2_scpinvar,f4_s2_scpinvar),f5_s2_scpinvar)),np__0) ) ) )
& p1_s2_scpinvar(k1_scmpds_8(f2_s2_scpinvar))
& ! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( p1_s2_scpinvar(k1_scmpds_8(A))
& k2_scmpds_2(A,f4_s2_scpinvar) = k2_scmpds_2(f2_s2_scpinvar,f4_s2_scpinvar) )
=> ( r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f2_s2_scpinvar,f4_s2_scpinvar),f5_s2_scpinvar)),np__0)
| ( k2_scmpds_2(k9_scmpds_4(f3_s2_scpinvar,A),f4_s2_scpinvar) = k2_scmpds_2(A,f4_s2_scpinvar)
& r1_scmpds_6(f3_s2_scpinvar,A)
& r2_scmpds_6(f3_s2_scpinvar,A)
& ~ r1_xreal_0(f1_s2_scpinvar(k1_scmpds_8(A)),f1_s2_scpinvar(k1_scmpds_8(k9_scmpds_4(f3_s2_scpinvar,A))))
& p1_s2_scpinvar(k1_scmpds_8(k9_scmpds_4(f3_s2_scpinvar,A))) ) ) ) ) )
=> ( f1_s2_scpinvar(k1_scmpds_8(k9_scmpds_4(k3_scmpds_8(f4_s2_scpinvar,f5_s2_scpinvar,f3_s2_scpinvar),f2_s2_scpinvar))) = np__0
& p1_s2_scpinvar(k1_scmpds_8(k9_scmpds_4(k3_scmpds_8(f4_s2_scpinvar,f5_s2_scpinvar,f3_s2_scpinvar),f2_s2_scpinvar))) ) ) ).
fof(s3_scpinvar,axiom,
( ( ~ r1_xreal_0(k4_card_1(f3_s3_scpinvar),np__0)
& ! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( p1_s3_scpinvar(k1_scmpds_8(A))
& f1_s3_scpinvar(k1_scmpds_8(A)) = np__0 )
=> k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f2_s3_scpinvar,f4_s3_scpinvar),f5_s3_scpinvar)) = np__0 ) )
& p1_s3_scpinvar(k1_scmpds_8(f2_s3_scpinvar))
& ! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( p1_s3_scpinvar(k1_scmpds_8(A))
& k2_scmpds_2(A,f4_s3_scpinvar) = k2_scmpds_2(f2_s3_scpinvar,f4_s3_scpinvar) )
=> ( k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f2_s3_scpinvar,f4_s3_scpinvar),f5_s3_scpinvar)) = np__0
| ( k2_scmpds_2(k9_scmpds_4(f3_s3_scpinvar,A),f4_s3_scpinvar) = k2_scmpds_2(A,f4_s3_scpinvar)
& r1_scmpds_6(f3_s3_scpinvar,A)
& r2_scmpds_6(f3_s3_scpinvar,A)
& ~ r1_xreal_0(f1_s3_scpinvar(k1_scmpds_8(A)),f1_s3_scpinvar(k1_scmpds_8(k9_scmpds_4(f3_s3_scpinvar,A))))
& p1_s3_scpinvar(k1_scmpds_8(k9_scmpds_4(f3_s3_scpinvar,A))) ) ) ) ) )
=> ( r1_scmpds_6(k3_scpinvar(f4_s3_scpinvar,f5_s3_scpinvar,f3_s3_scpinvar),f2_s3_scpinvar)
& r2_scmpds_6(k3_scpinvar(f4_s3_scpinvar,f5_s3_scpinvar,f3_s3_scpinvar),f2_s3_scpinvar) ) ) ).
fof(s4_scpinvar,axiom,
( ( ~ r1_xreal_0(k4_card_1(f3_s4_scpinvar),np__0)
& k2_scmpds_2(f2_s4_scpinvar,k3_scmpds_2(k2_scmpds_2(f2_s4_scpinvar,f4_s4_scpinvar),f5_s4_scpinvar)) != np__0
& ! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( p1_s4_scpinvar(k1_scmpds_8(A))
& f1_s4_scpinvar(k1_scmpds_8(A)) = np__0 )
=> k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f2_s4_scpinvar,f4_s4_scpinvar),f5_s4_scpinvar)) = np__0 ) )
& p1_s4_scpinvar(k1_scmpds_8(f2_s4_scpinvar))
& ! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( p1_s4_scpinvar(k1_scmpds_8(A))
& k2_scmpds_2(A,f4_s4_scpinvar) = k2_scmpds_2(f2_s4_scpinvar,f4_s4_scpinvar) )
=> ( k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f2_s4_scpinvar,f4_s4_scpinvar),f5_s4_scpinvar)) = np__0
| ( k2_scmpds_2(k9_scmpds_4(f3_s4_scpinvar,A),f4_s4_scpinvar) = k2_scmpds_2(A,f4_s4_scpinvar)
& r1_scmpds_6(f3_s4_scpinvar,A)
& r2_scmpds_6(f3_s4_scpinvar,A)
& ~ r1_xreal_0(f1_s4_scpinvar(k1_scmpds_8(A)),f1_s4_scpinvar(k1_scmpds_8(k9_scmpds_4(f3_s4_scpinvar,A))))
& p1_s4_scpinvar(k1_scmpds_8(k9_scmpds_4(f3_s4_scpinvar,A))) ) ) ) ) )
=> k9_scmpds_4(k3_scpinvar(f4_s4_scpinvar,f5_s4_scpinvar,f3_s4_scpinvar),f2_s4_scpinvar) = k9_scmpds_4(k3_scpinvar(f4_s4_scpinvar,f5_s4_scpinvar,f3_s4_scpinvar),k9_scmpds_4(f3_s4_scpinvar,f2_s4_scpinvar)) ) ).
fof(s5_scpinvar,axiom,
( ( ~ r1_xreal_0(k4_card_1(f3_s5_scpinvar),np__0)
& ! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( p1_s5_scpinvar(k1_scmpds_8(A))
=> ( f1_s5_scpinvar(k1_scmpds_8(A)) = np__0
<=> k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f2_s5_scpinvar,f4_s5_scpinvar),f5_s5_scpinvar)) = np__0 ) ) )
& p1_s5_scpinvar(k1_scmpds_8(f2_s5_scpinvar))
& ! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( p1_s5_scpinvar(k1_scmpds_8(A))
& k2_scmpds_2(A,f4_s5_scpinvar) = k2_scmpds_2(f2_s5_scpinvar,f4_s5_scpinvar) )
=> ( k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f2_s5_scpinvar,f4_s5_scpinvar),f5_s5_scpinvar)) = np__0
| ( k2_scmpds_2(k9_scmpds_4(f3_s5_scpinvar,A),f4_s5_scpinvar) = k2_scmpds_2(A,f4_s5_scpinvar)
& r1_scmpds_6(f3_s5_scpinvar,A)
& r2_scmpds_6(f3_s5_scpinvar,A)
& ~ r1_xreal_0(f1_s5_scpinvar(k1_scmpds_8(A)),f1_s5_scpinvar(k1_scmpds_8(k9_scmpds_4(f3_s5_scpinvar,A))))
& p1_s5_scpinvar(k1_scmpds_8(k9_scmpds_4(f3_s5_scpinvar,A))) ) ) ) ) )
=> ( f1_s5_scpinvar(k1_scmpds_8(k9_scmpds_4(k3_scpinvar(f4_s5_scpinvar,f5_s5_scpinvar,f3_s5_scpinvar),f2_s5_scpinvar))) = np__0
& p1_s5_scpinvar(k1_scmpds_8(k9_scmpds_4(k3_scpinvar(f4_s5_scpinvar,f5_s5_scpinvar,f3_s5_scpinvar),f2_s5_scpinvar))) ) ) ).
fof(dt_k1_scpinvar,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_ami_3(k1_scpinvar(A,B),k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(k1_scpinvar(A,B))
& m1_ami_1(k1_scpinvar(A,B),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ).
fof(dt_k2_scpinvar,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_ami_3(k2_scpinvar(A),k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(k2_scpinvar(A))
& m1_ami_1(k2_scpinvar(A),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ).
fof(dt_k3_scpinvar,axiom,
! [A,B,C] :
( ( m1_scmpds_2(A)
& v1_int_1(B)
& 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) )
=> ( v1_ami_3(k3_scpinvar(A,B,C),k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(k3_scpinvar(A,B,C))
& m1_ami_1(k3_scpinvar(A,B,C),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ).
fof(dt_k4_scpinvar,axiom,
( v1_ami_3(k4_scpinvar,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(k4_scpinvar)
& m1_ami_1(k4_scpinvar,k1_tarski(k4_numbers),k1_scmpds_2) ) ).
%------------------------------------------------------------------------------