SET007 Axioms: SET007+660.ax
%------------------------------------------------------------------------------
% File : SET007+660 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Quick Sort on SCMPDS
% 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 : scpqsort [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 23 ( 2 unt; 0 def)
% Number of atoms : 300 ( 46 equ)
% Maximal formula atoms : 36 ( 13 avg)
% Number of connectives : 291 ( 14 ~; 17 |; 144 &)
% ( 1 <=>; 115 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 13 avg)
% Maximal term depth : 15 ( 2 avg)
% Number of predicates : 23 ( 22 usr; 0 prp; 1-3 aty)
% Number of functors : 61 ( 61 usr; 18 con; 0-4 aty)
% Number of variables : 98 ( 97 !; 1 ?)
% 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_scpqsort,axiom,
! [A,B,C,D] :
( ( 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)
& v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(B)
& v3_scmpds_4(B)
& m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2)
& m1_scmpds_2(C)
& v1_int_1(D) )
=> ( v1_relat_1(k3_scmpds_6(C,D,A,B))
& v1_funct_1(k3_scmpds_6(C,D,A,B))
& v1_finset_1(k3_scmpds_6(C,D,A,B))
& v1_ami_3(k3_scmpds_6(C,D,A,B),k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(k3_scmpds_6(C,D,A,B))
& v3_scmpds_4(k3_scmpds_6(C,D,A,B)) ) ) ).
fof(t1_scpqsort,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] :
( ( v1_ami_3(C,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(C)
& v3_scmpds_4(C)
& m1_ami_1(C,k1_tarski(k4_numbers),k1_scmpds_2) )
=> ! [D] :
( m1_scmpds_2(D)
=> ! [E] :
( m1_scmpds_2(E)
=> ! [F] :
( v1_int_1(F)
=> ( ( r1_scmpds_6(B,A)
& r2_scmpds_6(B,A) )
=> ( r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,D),F)),np__0)
| k2_scmpds_2(k9_scmpds_4(k3_scmpds_6(D,F,B,C),A),E) = k2_scmpds_2(k9_scmpds_4(B,A),E) ) ) ) ) ) ) ) ) ).
fof(t2_scpqsort,axiom,
! [A] :
( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ! [B] :
( m1_subset_1(B,k4_card_3(u5_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)
& v3_scmpds_4(C)
& v3_scmpds_5(C)
& m1_ami_1(C,k1_tarski(k4_numbers),k1_scmpds_2) )
=> ! [D] :
( m1_scmpds_2(D)
=> ! [E] :
( v1_int_1(E)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_scmpds_6(C,A)
& r2_scmpds_6(C,A)
& F = k1_nat_1(k2_scm_1(k1_tarski(k4_numbers),k1_scmpds_2,k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,A,k2_scmpds_4(k8_scmpds_4(C)))),np__2)
& B = k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,A,k2_scmpds_4(k8_scmpds_4(k3_scmpds_8(D,E,C))))),F) )
=> ( r1_xreal_0(k4_card_1(C),np__0)
| r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,D),E)),np__0)
| ( k7_relat_1(B,k2_ami_2) = k7_relat_1(k9_scmpds_4(C,A),k2_ami_2)
& k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,B,k2_scmpds_4(k8_scmpds_4(k3_scmpds_8(D,E,C)))) = B ) ) ) ) ) ) ) ) ) ).
fof(t3_scpqsort,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_subset_1(C,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( k7_relat_1(C,k2_ami_2) = k7_relat_1(A,k2_ami_2)
=> r2_scmpds_6(B,C) ) )
=> r1_scmpds_6(B,A) ) ) ) ).
fof(t4_scpqsort,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] :
( m2_subset_1(C,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))
=> ! [D] :
( m2_subset_1(D,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(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(A,B),C),D)) = np__4 ) ) ) ) ).
fof(t5_scpqsort,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)
=> ( ( r1_xreal_0(k2_xcmplx_0(G,k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),F))),k2_scmpds_2(A,D))
& ! [H] :
( m1_subset_1(H,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( r1_xreal_0(k2_xcmplx_0(G,k2_scmpds_2(H,k3_scmpds_2(k2_scmpds_2(A,C),F))),k2_scmpds_2(H,D))
& k2_scmpds_2(H,E) = k2_scmpds_2(A,E)
& k2_scmpds_2(H,C) = k2_scmpds_2(A,C) )
=> ( r1_xreal_0(k2_scmpds_2(H,k3_scmpds_2(k2_scmpds_2(A,C),F)),np__0)
| ( k2_scmpds_2(k9_scmpds_4(B,H),C) = k2_scmpds_2(H,C)
& r1_scmpds_6(B,H)
& r2_scmpds_6(B,H)
& ~ r1_xreal_0(k2_scmpds_2(H,k3_scmpds_2(k2_scmpds_2(A,C),F)),k2_scmpds_2(k9_scmpds_4(B,H),k3_scmpds_2(k2_scmpds_2(A,C),F)))
& r1_xreal_0(k2_xcmplx_0(G,k2_scmpds_2(k9_scmpds_4(B,H),k3_scmpds_2(k2_scmpds_2(A,C),F))),k2_scmpds_2(k9_scmpds_4(B,H),D))
& k2_scmpds_2(k9_scmpds_4(B,H),E) = k2_scmpds_2(H,E) ) ) ) ) )
=> ( r1_xreal_0(k4_card_1(B),np__0)
| ( r1_scmpds_6(k3_scmpds_8(C,F,B),A)
& r2_scmpds_6(k3_scmpds_8(C,F,B),A)
& ( ~ r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),F)),np__0)
=> k9_scmpds_4(k3_scmpds_8(C,F,B),A) = k9_scmpds_4(k3_scmpds_8(C,F,B),k9_scmpds_4(B,A)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t6_scpqsort,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)
=> ( ( r1_xreal_0(G,k2_scmpds_2(A,D))
& ! [H] :
( m1_subset_1(H,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( r1_xreal_0(G,k2_scmpds_2(H,D))
& k2_scmpds_2(H,E) = k2_scmpds_2(A,E)
& k2_scmpds_2(H,C) = k2_scmpds_2(A,C) )
=> ( r1_xreal_0(k2_scmpds_2(H,k3_scmpds_2(k2_scmpds_2(A,C),F)),np__0)
| ( k2_scmpds_2(k9_scmpds_4(B,H),C) = k2_scmpds_2(H,C)
& r1_scmpds_6(B,H)
& r2_scmpds_6(B,H)
& ~ r1_xreal_0(k2_scmpds_2(H,k3_scmpds_2(k2_scmpds_2(A,C),F)),k2_scmpds_2(k9_scmpds_4(B,H),k3_scmpds_2(k2_scmpds_2(A,C),F)))
& r1_xreal_0(G,k2_scmpds_2(k9_scmpds_4(B,H),D))
& k2_scmpds_2(k9_scmpds_4(B,H),E) = k2_scmpds_2(H,E) ) ) ) ) )
=> ( r1_xreal_0(k4_card_1(B),np__0)
| ( r1_scmpds_6(k3_scmpds_8(C,F,B),A)
& r2_scmpds_6(k3_scmpds_8(C,F,B),A)
& ( ~ r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),F)),np__0)
=> k9_scmpds_4(k3_scmpds_8(C,F,B),A) = k9_scmpds_4(k3_scmpds_8(C,F,B),k9_scmpds_4(B,A)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t7_scpqsort,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] :
( m1_scmpds_2(F)
=> ! [G] :
( m1_scmpds_2(G)
=> ! [H] :
( v1_int_1(H)
=> ! [I] :
( v1_int_1(I)
=> ! [J] :
( v1_int_1(J)
=> ( ( k2_scmpds_2(A,G) = k2_xcmplx_0(k6_xcmplx_0(k2_scmpds_2(A,F),I),k2_scmpds_2(A,D))
& r1_xreal_0(J,k6_xcmplx_0(k2_scmpds_2(A,F),I))
& ! [K] :
( m1_subset_1(K,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
=> ( ( k2_scmpds_2(K,G) = k2_xcmplx_0(k6_xcmplx_0(k2_scmpds_2(K,F),I),k2_scmpds_2(K,D))
& r1_xreal_0(J,k6_xcmplx_0(k2_scmpds_2(K,F),I))
& k2_scmpds_2(K,E) = k2_scmpds_2(A,E)
& k2_scmpds_2(K,C) = k2_scmpds_2(A,C) )
=> ( r1_xreal_0(k2_scmpds_2(K,k3_scmpds_2(k2_scmpds_2(A,C),H)),np__0)
| ( k2_scmpds_2(k9_scmpds_4(B,K),C) = k2_scmpds_2(K,C)
& r1_scmpds_6(B,K)
& r2_scmpds_6(B,K)
& ~ r1_xreal_0(k2_scmpds_2(K,k3_scmpds_2(k2_scmpds_2(A,C),H)),k2_scmpds_2(k9_scmpds_4(B,K),k3_scmpds_2(k2_scmpds_2(A,C),H)))
& k2_scmpds_2(k9_scmpds_4(B,K),G) = k2_xcmplx_0(k6_xcmplx_0(k2_scmpds_2(k9_scmpds_4(B,K),F),I),k2_scmpds_2(k9_scmpds_4(B,K),D))
& r1_xreal_0(J,k6_xcmplx_0(k2_scmpds_2(k9_scmpds_4(B,K),F),I))
& k2_scmpds_2(k9_scmpds_4(B,K),E) = k2_scmpds_2(K,E) ) ) ) ) )
=> ( r1_xreal_0(k4_card_1(B),np__0)
| ( r1_scmpds_6(k3_scmpds_8(C,H,B),A)
& r2_scmpds_6(k3_scmpds_8(C,H,B),A)
& ( ~ r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),H)),np__0)
=> k9_scmpds_4(k3_scmpds_8(C,H,B),A) = k9_scmpds_4(k3_scmpds_8(C,H,B),k9_scmpds_4(B,A)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_scpqsort,axiom,
! [A] :
( m2_finseq_1(A,k4_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(B,D)
& r1_xreal_0(D,E)
& C = k6_xcmplx_0(D,np__1)
& r1_sfmastr3(A,B,C)
& r1_sfmastr3(A,k1_nat_1(D,np__1),E)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,F)
=> ( r1_xreal_0(D,F)
| r1_xreal_0(k1_funct_1(A,F),k1_funct_1(A,D)) ) ) )
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(F,E)
=> ( r1_xreal_0(F,D)
| r1_xreal_0(k1_funct_1(A,D),k1_funct_1(A,F)) ) ) ) )
=> r1_sfmastr3(A,B,E) ) ) ) ) ) ) ).
fof(t9_scpqsort,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
~ ( r2_hidden(C,k4_finseq_1(B))
& r1_rfinseq(A,B)
& ! [D] :
~ ( r2_hidden(D,k4_finseq_1(B))
& k1_funct_1(A,C) = k1_funct_1(B,D) ) ) ) ) ).
fof(t10_scpqsort,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( r1_rfinseq(A,B)
<=> r1_rfinseq(k7_finseq_1(C,A),k7_finseq_1(C,B)) ) ) ) ) ).
fof(t11_scpqsort,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_rfinseq(A,B)
& r1_xreal_0(C,D)
& r1_xreal_0(D,k3_finseq_1(A))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,F)
& r1_xreal_0(F,C) )
=> k1_funct_1(A,F) = k1_funct_1(B,F) ) )
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(F,k3_finseq_1(A))
=> ( r1_xreal_0(F,D)
| k1_funct_1(A,F) = k1_funct_1(B,F) ) ) )
& ~ r1_xreal_0(E,C)
& r1_xreal_0(E,D)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(F,C)
& r1_xreal_0(F,D)
& k1_funct_1(A,E) = k1_funct_1(B,F) ) ) ) ) ) ) ) ) ).
fof(d1_scpqsort,axiom,
k1_scpqsort = k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k3_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__5,np__4),k14_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__5,np__2)),k17_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__3,np__2)),k12_scmpds_2(k2_scmp_gcd,np__3,np__1)),k3_scmpds_8(k2_scmp_gcd,np__5,k3_scmpds_4(k3_scmpds_4(k3_scmpds_8(k2_scmp_gcd,np__5,k3_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__7,np__5),k12_scmpds_2(k2_scmp_gcd,np__5,k4_xcmplx_0(np__1))),k17_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__4),np__6,np__0)),k14_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__2),np__6,np__0)),k3_scmpds_6(k2_scmp_gcd,np__6,k6_scmpds_4(k12_scmpds_2(k2_scmp_gcd,np__4,k4_xcmplx_0(np__1)),k12_scmpds_2(k2_scmp_gcd,np__7,k4_xcmplx_0(np__1))),k1_scmpds_4(k11_scmpds_2(k2_scmp_gcd,np__5,np__0))))),k3_scmpds_8(k2_scmp_gcd,np__7,k3_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__5,np__7),k12_scmpds_2(k2_scmp_gcd,np__7,k4_xcmplx_0(np__1))),k17_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__2),np__6,np__0)),k14_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__3),np__6,np__0)),k3_scmpds_6(k2_scmp_gcd,np__6,k6_scmpds_4(k12_scmpds_2(k2_scmp_gcd,np__3,np__1),k12_scmpds_2(k2_scmp_gcd,np__5,k4_xcmplx_0(np__1))),k1_scmpds_4(k11_scmpds_2(k2_scmp_gcd,np__7,np__0)))))),k7_scmpds_6(k2_scmp_gcd,np__5,k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__4),np__6,np__0),k17_scmpds_2(k1_scmp_gcd(np__4),k1_scmp_gcd(np__3),np__0,np__0)),k17_scmpds_2(k1_scmp_gcd(np__3),k2_scmp_gcd,np__0,np__6)),k12_scmpds_2(k2_scmp_gcd,np__5,k4_xcmplx_0(np__2))),k12_scmpds_2(k2_scmp_gcd,np__3,np__1)),k12_scmpds_2(k2_scmp_gcd,np__4,k4_xcmplx_0(np__1))))))),k17_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__4),np__6,np__0)),k17_scmpds_2(k1_scmp_gcd(np__4),k1_scmp_gcd(np__2),np__0,np__0)),k17_scmpds_2(k1_scmp_gcd(np__2),k2_scmp_gcd,np__0,np__6)) ).
fof(d2_scpqsort,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_scpqsort(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(k3_scmp_gcd,np__1)),k11_scmpds_2(k3_scmp_gcd,k1_nat_1(B,A),k1_nat_1(B,np__1))),k11_scmpds_2(k3_scmp_gcd,k1_nat_1(k1_nat_1(B,A),np__1),k1_nat_1(B,A))),k3_scmpds_8(k2_scmp_gcd,np__1,k3_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k3_scmp_gcd,np__2,k1_nat_1(k1_nat_1(B,A),np__1)),k14_scmpds_2(k2_scmp_gcd,k3_scmp_gcd,np__2,k1_nat_1(B,A))),k3_scmpds_6(k2_scmp_gcd,np__2,k3_scmpds_4(k3_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k3_scmp_gcd,np__2,k1_nat_1(B,A)),k17_scmpds_2(k2_scmp_gcd,k3_scmp_gcd,np__4,k1_nat_1(k1_nat_1(B,A),np__1))),k1_scpqsort),k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,k1_nat_1(k1_nat_1(B,A),np__3),k1_nat_1(k1_nat_1(B,A),np__1)),k17_scmpds_2(k3_scmp_gcd,k2_scmp_gcd,k1_nat_1(k1_nat_1(B,A),np__1),np__4)),k17_scmpds_2(k3_scmp_gcd,k2_scmp_gcd,k1_nat_1(k1_nat_1(B,A),np__2),np__4)),k12_scmpds_2(k3_scmp_gcd,k1_nat_1(k1_nat_1(B,A),np__1),k4_xcmplx_0(np__1))),k12_scmpds_2(k3_scmp_gcd,k1_nat_1(k1_nat_1(B,A),np__2),np__1)),k12_scmpds_2(k2_scmp_gcd,np__1,np__2))),k1_scmpds_4(k12_scmpds_2(k2_scmp_gcd,np__1,k4_xcmplx_0(np__2))))))) ) ) ).
fof(t12_scpqsort,axiom,
k4_card_1(k1_scpqsort) = np__38 ).
fof(t13_scpqsort,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)
=> ( ( k2_scmpds_2(A,k2_scmp_gcd) = np__0
& k2_scmpds_2(A,k1_scmp_gcd(np__2)) = B
& r1_xreal_0(k1_nat_1(C,np__1),B)
& r1_xreal_0(np__7,C) )
=> ( r1_xreal_0(k6_xcmplx_0(k2_scmpds_2(A,k1_scmp_gcd(np__4)),k2_scmpds_2(A,k1_scmp_gcd(np__2))),np__0)
| ( r1_scmpds_6(k1_scpqsort,A)
& r2_scmpds_6(k1_scpqsort,A) ) ) ) ) ) ) ).
fof(t14_scpqsort,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] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_1(E,k4_numbers)
=> ! [F] :
( m2_finseq_1(F,k4_numbers)
=> ( ( k2_scmpds_2(A,k2_scmp_gcd) = np__0
& k2_scmpds_2(A,k1_scmp_gcd(np__2)) = B
& r1_xreal_0(k1_nat_1(C,np__1),B)
& r1_xreal_0(k2_scmpds_2(A,k1_scmp_gcd(np__4)),k1_nat_1(C,D))
& r1_xreal_0(np__7,C)
& r1_scpisort(E,A,C)
& k3_finseq_1(E) = D
& r1_scpisort(F,k9_scmpds_4(k1_scpqsort,A),C)
& k3_finseq_1(F) = D )
=> ( r1_xreal_0(k6_xcmplx_0(k2_scmpds_2(A,k1_scmp_gcd(np__4)),k2_scmpds_2(A,k1_scmp_gcd(np__2))),np__0)
| ( k2_scmpds_2(k9_scmpds_4(k1_scpqsort,A),k2_scmp_gcd) = np__0
& k2_scmpds_2(k9_scmpds_4(k1_scpqsort,A),k1_scmp_gcd(np__1)) = k2_scmpds_2(A,k1_scmp_gcd(np__1))
& r1_rfinseq(E,F)
& ? [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
& G = k2_scmpds_2(k9_scmpds_4(k1_scpqsort,A),k1_scmp_gcd(np__4))
& r1_xreal_0(B,G)
& r1_xreal_0(G,k2_scmpds_2(A,k1_scmp_gcd(np__4)))
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,H)
=> ( r1_xreal_0(G,H)
| r1_xreal_0(k2_scmpds_2(k9_scmpds_4(k1_scpqsort,A),k1_scmp_gcd(H)),k2_scmpds_2(k9_scmpds_4(k1_scpqsort,A),k1_scmp_gcd(G))) ) ) )
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r1_xreal_0(H,k2_scmpds_2(A,k1_scmp_gcd(np__4)))
=> ( r1_xreal_0(H,G)
| r1_xreal_0(k2_scmpds_2(k9_scmpds_4(k1_scpqsort,A),k1_scmp_gcd(G)),k2_scmpds_2(k9_scmpds_4(k1_scpqsort,A),k1_scmp_gcd(H))) ) ) )
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k1_nat_1(C,np__1),H)
=> ( ( r1_xreal_0(k2_scmpds_2(A,k1_scmp_gcd(np__2)),H)
& r1_xreal_0(H,k2_scmpds_2(A,k1_scmp_gcd(np__4))) )
| k2_scmpds_2(k9_scmpds_4(k1_scpqsort,A),k1_scmp_gcd(H)) = k2_scmpds_2(A,k1_scmp_gcd(H)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_scpqsort,axiom,
( v3_scmpds_5(k1_scpqsort)
& v3_scmpds_4(k1_scpqsort) ) ).
fof(t16_scpqsort,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k4_card_1(k2_scpqsort(A,B)) = np__57 ) ) ).
fof(t17_scpqsort,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__7,A)
=> v2_scmpds_4(k2_scpqsort(B,A)) ) ) ) ).
fof(t18_scpqsort,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)
=> ~ ( r1_xreal_0(np__7,B)
& ! [D] :
( m2_finseq_1(D,k4_numbers)
=> ! [E] :
( m2_finseq_1(E,k4_numbers)
=> ~ ( k3_finseq_1(D) = C
& r1_scpisort(D,A,B)
& k3_finseq_1(E) = C
& r1_scpisort(E,k9_scmpds_4(k2_scpqsort(C,B),A),B)
& r1_rfinseq(D,E)
& r1_sfmastr3(E,np__1,C) ) ) ) ) ) ) ) ).
fof(dt_k1_scpqsort,axiom,
( v1_ami_3(k1_scpqsort,k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(k1_scpqsort)
& m1_ami_1(k1_scpqsort,k1_tarski(k4_numbers),k1_scmpds_2) ) ).
fof(dt_k2_scpqsort,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_ami_3(k2_scpqsort(A,B),k1_tarski(k4_numbers),k1_scmpds_2)
& v1_scmpds_3(k2_scpqsort(A,B))
& m1_ami_1(k2_scpqsort(A,B),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ).
%------------------------------------------------------------------------------