SET007 Axioms: SET007+515.ax
%------------------------------------------------------------------------------
% File : SET007+515 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Correctness of the Generic Algorithms of Brown and Henrici
% Version : [Urb08] axioms.
% English : The Correctness of the Generic Algorithms of Brown and Henrici
% Concerning Addition and Multiplication in Fraction Fields
% 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 : gcd_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 106 ( 4 unt; 0 def)
% Number of atoms : 1520 ( 158 equ)
% Maximal formula atoms : 44 ( 14 avg)
% Number of connectives : 1606 ( 192 ~; 21 |; 997 &)
% ( 26 <=>; 370 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 13 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 41 ( 39 usr; 1 prp; 0-4 aty)
% Number of functors : 19 ( 19 usr; 1 con; 0-6 aty)
% Number of variables : 363 ( 349 !; 14 ?)
% 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(cc1_gcd_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v6_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v8_vectsp_1(A) ) ) ) ).
fof(cc2_gcd_1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v4_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v7_vectsp_1(A) ) ) ) ).
fof(cc3_gcd_1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v5_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v7_vectsp_1(A) ) ) ) ).
fof(cc4_gcd_1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A) ) ) ) ).
fof(fc1_gcd_1,axiom,
( ~ v3_struct_0(k3_vectsp_1)
& v3_rlvect_1(k3_vectsp_1)
& v4_rlvect_1(k3_vectsp_1)
& v5_rlvect_1(k3_vectsp_1)
& v6_rlvect_1(k3_vectsp_1)
& v2_group_1(k3_vectsp_1)
& v4_group_1(k3_vectsp_1)
& v7_group_1(k3_vectsp_1)
& v3_vectsp_1(k3_vectsp_1)
& v4_vectsp_1(k3_vectsp_1)
& v5_vectsp_1(k3_vectsp_1)
& v6_vectsp_1(k3_vectsp_1)
& v7_vectsp_1(k3_vectsp_1)
& v8_vectsp_1(k3_vectsp_1)
& v9_vectsp_1(k3_vectsp_1)
& ~ v10_vectsp_1(k3_vectsp_1)
& v2_vectsp_2(k3_vectsp_1) ) ).
fof(rc1_gcd_1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_vectsp_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A) ) ).
fof(fc2_gcd_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ~ v1_xboole_0(k2_gcd_1(A,B)) ) ).
fof(fc3_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ~ v1_xboole_0(k3_gcd_1(A)) ) ).
fof(rc2_gcd_1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A) ) ).
fof(rc3_gcd_1,axiom,
? [A] :
( l1_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& v3_gcd_1(A) ) ).
fof(rc4_gcd_1,axiom,
? [A] :
( l2_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& v3_gcd_1(A) ) ).
fof(cc5_gcd_1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_group_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v7_group_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& v3_gcd_1(A) ) ) ) ).
fof(rc5_gcd_1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A) ) ).
fof(t1_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( B != k1_rlvect_1(A)
=> ( ( k10_group_1(A,B,C) = k10_group_1(A,B,D)
=> C = D )
& ( k10_group_1(A,C,B) = k10_group_1(A,D,B)
=> C = D ) ) ) ) ) ) ) ).
fof(d1_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_gcd_1(A,B,C)
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& C = k1_group_1(A,B,D) ) ) ) ) ) ).
fof(d2_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_gcd_1(B,A)
<=> r1_gcd_1(A,B,k2_group_1(A)) ) ) ) ).
fof(d3_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_gcd_1(A,B,C)
<=> ( r1_gcd_1(A,B,C)
& r1_gcd_1(A,C,B) ) ) ) ) ) ).
fof(d4_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_gcd_1(A,C,B)
=> ( C = k1_rlvect_1(A)
| ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( D = k1_gcd_1(A,B,C)
<=> k10_group_1(A,D,C) = B ) ) ) ) ) ) ) ).
fof(t2_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r1_gcd_1(A,B,C)
& r1_gcd_1(A,C,D) )
=> r1_gcd_1(A,B,D) ) ) ) ) ) ).
fof(t3_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r1_gcd_1(A,C,B)
& r1_gcd_1(A,E,D) )
=> r1_gcd_1(A,k10_group_1(A,C,E),k10_group_1(A,B,D)) ) ) ) ) ) ) ).
fof(t4_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r3_gcd_1(A,B,C)
& r3_gcd_1(A,C,D) )
=> r3_gcd_1(A,B,D) ) ) ) ) ) ).
fof(t5_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_gcd_1(A,B,C)
=> r1_gcd_1(A,k1_group_1(A,D,B),k1_group_1(A,D,C)) ) ) ) ) ) ).
fof(t6_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_gcd_1(A,B,k1_group_1(A,B,C))
& ( v7_group_1(A)
=> r1_gcd_1(A,C,k1_group_1(A,B,C)) ) ) ) ) ) ).
fof(t7_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_gcd_1(A,B,C)
=> r1_gcd_1(A,B,k1_group_1(A,C,D)) ) ) ) ) ) ).
fof(t8_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_gcd_1(A,C,B)
=> ( C = k1_rlvect_1(A)
| ( k1_gcd_1(A,B,C) = k1_rlvect_1(A)
<=> B = k1_rlvect_1(A) ) ) ) ) ) ) ).
fof(t9_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B != k1_rlvect_1(A)
=> k1_gcd_1(A,B,B) = k2_group_1(A) ) ) ) ).
fof(t10_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_gcd_1(A,B,k2_group_1(A)) = B ) ) ).
fof(t11_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( D != k1_rlvect_1(A)
=> ( ( ( r2_gcd_1(A,D,k10_group_1(A,B,C))
& r2_gcd_1(A,D,B) )
=> k1_gcd_1(A,k10_group_1(A,B,C),D) = k10_group_1(A,k1_gcd_1(A,B,D),C) )
& ( ( r2_gcd_1(A,D,k10_group_1(A,B,C))
& r2_gcd_1(A,D,C) )
=> k1_gcd_1(A,k10_group_1(A,B,C),D) = k10_group_1(A,B,k1_gcd_1(A,C,D)) ) ) ) ) ) ) ) ).
fof(t12_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_gcd_1(A,D,B)
& r2_gcd_1(A,D,C)
& r2_gcd_1(A,D,k4_rlvect_1(A,B,C)) )
=> ( D = k1_rlvect_1(A)
| k4_rlvect_1(A,k1_gcd_1(A,B,D),k1_gcd_1(A,C,D)) = k1_gcd_1(A,k4_rlvect_1(A,B,C),D) ) ) ) ) ) ) ).
fof(t13_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_gcd_1(A,D,B)
& r2_gcd_1(A,D,C) )
=> ( D = k1_rlvect_1(A)
| ( k1_gcd_1(A,B,D) = k1_gcd_1(A,C,D)
<=> B = C ) ) ) ) ) ) ) ).
fof(t14_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r2_gcd_1(A,C,B)
& r2_gcd_1(A,E,D) )
=> ( C = k1_rlvect_1(A)
| E = k1_rlvect_1(A)
| k10_group_1(A,k1_gcd_1(A,B,C),k1_gcd_1(A,D,E)) = k1_gcd_1(A,k10_group_1(A,B,D),k10_group_1(A,C,E)) ) ) ) ) ) ) ) ).
fof(t15_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_gcd_1(A,k10_group_1(A,B,C),k10_group_1(A,B,D))
=> ( B = k1_rlvect_1(A)
| r2_gcd_1(A,C,D) ) ) ) ) ) ) ).
fof(t16_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r4_gcd_1(A,B,k1_rlvect_1(A))
=> B = k1_rlvect_1(A) ) ) ) ).
fof(t17_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k10_group_1(A,B,C) = B
=> ( B = k1_rlvect_1(A)
| C = k2_group_1(A) ) ) ) ) ) ).
fof(t18_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r4_gcd_1(A,B,C)
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& v1_gcd_1(D,A)
& k10_group_1(A,B,D) = C ) ) ) ) ) ).
fof(t19_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r4_gcd_1(A,k10_group_1(A,D,B),k10_group_1(A,D,C))
=> ( D = k1_rlvect_1(A)
| r4_gcd_1(A,B,C) ) ) ) ) ) ) ).
fof(d5_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = k2_gcd_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,C)
<=> r3_gcd_1(A,D,B) ) ) ) ) ) ) ).
fof(t20_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ r1_xboole_0(k2_gcd_1(A,B),k2_gcd_1(A,C))
=> k2_gcd_1(A,B) = k2_gcd_1(A,C) ) ) ) ) ).
fof(d6_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( B = k3_gcd_1(A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& C = k2_gcd_1(A,D) ) ) ) ) ) ) ).
fof(t21_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,k3_gcd_1(A))
& v1_xboole_0(B) ) ) ) ).
fof(d7_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( m1_gcd_1(B,A)
<=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_struct_0(D,A,B)
& r4_gcd_1(A,D,C) ) )
& ! [C] :
( m1_struct_0(C,A,B)
=> ! [D] :
( m1_struct_0(D,A,B)
=> ~ ( C != D
& r3_gcd_1(A,C,D) ) ) ) ) ) ) ) ).
fof(d8_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( m2_gcd_1(B,A)
<=> ( m1_gcd_1(B,A)
& r2_hidden(k2_group_1(A),B) ) ) ) ) ).
fof(t22_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ( r2_hidden(k2_group_1(A),B)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_struct_0(D,A,B)
& r4_gcd_1(A,D,C) ) )
& ! [C] :
( m1_struct_0(C,A,B)
=> ! [D] :
( m1_struct_0(D,A,B)
=> ~ ( C != D
& r3_gcd_1(A,C,D) ) ) ) ) ) ) ).
fof(t23_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_struct_0(C,A,B)
=> ! [D] :
( m1_struct_0(D,A,B)
=> ( r4_gcd_1(A,C,D)
=> C = D ) ) ) ) ) ).
fof(t24_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> m1_struct_0(k1_rlvect_1(A),A,B) ) ) ).
fof(d9_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( D = k4_gcd_1(A,B,C)
<=> ( r2_hidden(D,B)
& r4_gcd_1(A,D,C) ) ) ) ) ) ) ).
fof(t25_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ( k4_gcd_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A)
& k4_gcd_1(A,B,k2_group_1(A)) = k2_group_1(A) ) ) ) ).
fof(t26_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
<=> C = k4_gcd_1(A,B,C) ) ) ) ) ).
fof(d10_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ( v2_gcd_1(B,A)
<=> ! [C] :
( m1_struct_0(C,A,B)
=> ! [D] :
( m1_struct_0(D,A,B)
=> r2_hidden(k1_group_1(A,C,D),B) ) ) ) ) ) ).
fof(t27_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ( v2_gcd_1(B,A)
=> ! [C] :
( m1_struct_0(C,A,B)
=> ! [D] :
( m1_struct_0(D,A,B)
=> ( r2_gcd_1(A,D,C)
=> ( D = k1_rlvect_1(A)
| r2_hidden(k1_gcd_1(A,C,D),B) ) ) ) ) ) ) ) ).
fof(d11_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ( v3_gcd_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& r1_gcd_1(A,D,B)
& r1_gcd_1(A,D,C)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r1_gcd_1(A,E,B)
& r1_gcd_1(A,E,C) )
=> r1_gcd_1(A,E,D) ) ) ) ) ) ) ) ).
fof(d12_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v3_gcd_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( E = k5_gcd_1(A,B,C,D)
<=> ( r2_hidden(E,B)
& r2_gcd_1(A,E,C)
& r2_gcd_1(A,E,D)
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r2_gcd_1(A,F,C)
& r2_gcd_1(A,F,D) )
=> r2_gcd_1(A,F,E) ) ) ) ) ) ) ) ) ) ).
fof(t28_gcd_1,axiom,
$true ).
fof(t29_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_gcd_1(A,E,k5_gcd_1(A,B,C,D))
=> ( r2_gcd_1(A,E,C)
& r2_gcd_1(A,E,D) ) ) ) ) ) ) ) ).
fof(t30_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k5_gcd_1(A,B,C,D) = k5_gcd_1(A,B,D,C) ) ) ) ) ).
fof(t31_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k5_gcd_1(A,B,C,k1_rlvect_1(A)) = k4_gcd_1(A,B,C)
& k5_gcd_1(A,B,k1_rlvect_1(A),C) = k4_gcd_1(A,B,C) ) ) ) ) ).
fof(t32_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> k5_gcd_1(A,B,k1_rlvect_1(A),k1_rlvect_1(A)) = k1_rlvect_1(A) ) ) ).
fof(t33_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k5_gcd_1(A,B,C,k2_group_1(A)) = k2_group_1(A)
& k5_gcd_1(A,B,k2_group_1(A),C) = k2_group_1(A) ) ) ) ) ).
fof(t34_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k5_gcd_1(A,B,C,D) = k1_rlvect_1(A)
<=> ( C = k1_rlvect_1(A)
& D = k1_rlvect_1(A) ) ) ) ) ) ) ).
fof(t35_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r4_gcd_1(A,D,E)
=> ( r4_gcd_1(A,k5_gcd_1(A,B,C,D),k5_gcd_1(A,B,C,E))
& r4_gcd_1(A,k5_gcd_1(A,B,D,C),k5_gcd_1(A,B,E,C)) ) ) ) ) ) ) ) ).
fof(t36_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k5_gcd_1(A,B,k5_gcd_1(A,B,C,D),E) = k5_gcd_1(A,B,C,k5_gcd_1(A,B,D,E)) ) ) ) ) ) ).
fof(t37_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> r4_gcd_1(A,k5_gcd_1(A,B,k10_group_1(A,C,E),k10_group_1(A,D,E)),k10_group_1(A,E,k5_gcd_1(A,B,C,D))) ) ) ) ) ) ).
fof(t38_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( k5_gcd_1(A,B,C,D) = k2_group_1(A)
=> k5_gcd_1(A,B,C,k10_group_1(A,D,E)) = k5_gcd_1(A,B,C,E) ) ) ) ) ) ) ).
fof(t39_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( E = k5_gcd_1(A,B,C,D)
=> ( E = k1_rlvect_1(A)
| k5_gcd_1(A,B,k1_gcd_1(A,C,E),k1_gcd_1(A,D,E)) = k2_group_1(A) ) ) ) ) ) ) ) ).
fof(t40_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k5_gcd_1(A,B,k4_rlvect_1(A,C,k10_group_1(A,D,E)),E) = k5_gcd_1(A,B,C,E) ) ) ) ) ) ).
fof(t41_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( k5_gcd_1(A,B,C,D) = k2_group_1(A)
& k5_gcd_1(A,B,E,F) = k2_group_1(A) )
=> ( D = k1_rlvect_1(A)
| F = k1_rlvect_1(A)
| k5_gcd_1(A,B,k4_rlvect_1(A,k10_group_1(A,C,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F))),k10_group_1(A,E,k1_gcd_1(A,D,k5_gcd_1(A,B,D,F)))),k10_group_1(A,D,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F)))) = k5_gcd_1(A,B,k4_rlvect_1(A,k10_group_1(A,C,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F))),k10_group_1(A,E,k1_gcd_1(A,D,k5_gcd_1(A,B,D,F)))),k5_gcd_1(A,B,D,F)) ) ) ) ) ) ) ) ) ).
fof(t42_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( k5_gcd_1(A,B,C,D) = k2_group_1(A)
& k5_gcd_1(A,B,E,F) = k2_group_1(A) )
=> ( D = k1_rlvect_1(A)
| F = k1_rlvect_1(A)
| k5_gcd_1(A,B,k10_group_1(A,k1_gcd_1(A,C,k5_gcd_1(A,B,C,F)),k1_gcd_1(A,E,k5_gcd_1(A,B,E,D))),k10_group_1(A,k1_gcd_1(A,D,k5_gcd_1(A,B,E,D)),k1_gcd_1(A,F,k5_gcd_1(A,B,C,F)))) = k2_group_1(A) ) ) ) ) ) ) ) ) ).
fof(d13_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v3_gcd_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r5_gcd_1(A,B,C,D)
<=> k5_gcd_1(A,B,C,D) = k2_group_1(A) ) ) ) ) ) ).
fof(t43_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m2_gcd_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r5_gcd_1(A,B,D,E)
<=> r5_gcd_1(A,C,D,E) ) ) ) ) ) ) ).
fof(d14_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v3_gcd_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r6_gcd_1(A,B,C)
<=> ? [D] :
( m2_gcd_1(D,A)
& k5_gcd_1(A,D,B,C) = k2_group_1(A) ) ) ) ) ) ).
fof(t44_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r7_gcd_1(A,C,D)
=> k5_gcd_1(A,B,C,D) = k2_group_1(A) ) ) ) ) ) ).
fof(d15_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v3_gcd_1(A)
& l2_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r8_gcd_1(A,B,C,D)
<=> ( k5_gcd_1(A,B,C,D) = k2_group_1(A)
& r2_hidden(D,B)
& D != k1_rlvect_1(A) ) ) ) ) ) ) ).
fof(d16_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r7_gcd_1(A,C,D)
& r7_gcd_1(A,E,F)
& D = k4_gcd_1(A,B,D)
& F = k4_gcd_1(A,B,F) )
=> ( ( C = k1_rlvect_1(A)
=> k6_gcd_1(A,B,C,D,E,F) = E )
& ( E = k1_rlvect_1(A)
=> k6_gcd_1(A,B,C,D,E,F) = C )
& ( k5_gcd_1(A,B,D,F) = k2_group_1(A)
=> k6_gcd_1(A,B,C,D,E,F) = k4_rlvect_1(A,k10_group_1(A,C,F),k10_group_1(A,D,E)) )
& ( k4_rlvect_1(A,k10_group_1(A,C,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F))),k10_group_1(A,E,k1_gcd_1(A,D,k5_gcd_1(A,B,D,F)))) = k1_rlvect_1(A)
=> k6_gcd_1(A,B,C,D,E,F) = k1_rlvect_1(A) )
& ~ ( C != k1_rlvect_1(A)
& E != k1_rlvect_1(A)
& k5_gcd_1(A,B,D,F) != k2_group_1(A)
& k4_rlvect_1(A,k10_group_1(A,C,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F))),k10_group_1(A,E,k1_gcd_1(A,D,k5_gcd_1(A,B,D,F)))) != k1_rlvect_1(A)
& k6_gcd_1(A,B,C,D,E,F) != k1_gcd_1(A,k4_rlvect_1(A,k10_group_1(A,C,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F))),k10_group_1(A,E,k1_gcd_1(A,D,k5_gcd_1(A,B,D,F)))),k5_gcd_1(A,B,k4_rlvect_1(A,k10_group_1(A,C,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F))),k10_group_1(A,E,k1_gcd_1(A,D,k5_gcd_1(A,B,D,F)))),k5_gcd_1(A,B,D,F))) ) ) ) ) ) ) ) ) ) ).
fof(d17_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r7_gcd_1(A,C,D)
& r7_gcd_1(A,E,F)
& D = k4_gcd_1(A,B,D)
& F = k4_gcd_1(A,B,F) )
=> ( ( C = k1_rlvect_1(A)
=> k7_gcd_1(A,B,C,D,E,F) = F )
& ( E = k1_rlvect_1(A)
=> k7_gcd_1(A,B,C,D,E,F) = D )
& ( k5_gcd_1(A,B,D,F) = k2_group_1(A)
=> k7_gcd_1(A,B,C,D,E,F) = k10_group_1(A,D,F) )
& ( k4_rlvect_1(A,k10_group_1(A,C,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F))),k10_group_1(A,E,k1_gcd_1(A,D,k5_gcd_1(A,B,D,F)))) = k1_rlvect_1(A)
=> k7_gcd_1(A,B,C,D,E,F) = k2_group_1(A) )
& ~ ( C != k1_rlvect_1(A)
& E != k1_rlvect_1(A)
& k5_gcd_1(A,B,D,F) != k2_group_1(A)
& k4_rlvect_1(A,k10_group_1(A,C,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F))),k10_group_1(A,E,k1_gcd_1(A,D,k5_gcd_1(A,B,D,F)))) != k1_rlvect_1(A)
& k7_gcd_1(A,B,C,D,E,F) != k1_gcd_1(A,k10_group_1(A,D,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F))),k5_gcd_1(A,B,k4_rlvect_1(A,k10_group_1(A,C,k1_gcd_1(A,F,k5_gcd_1(A,B,D,F))),k10_group_1(A,E,k1_gcd_1(A,D,k5_gcd_1(A,B,D,F)))),k5_gcd_1(A,B,D,F))) ) ) ) ) ) ) ) ) ) ).
fof(t45_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( v2_gcd_1(B,A)
& r8_gcd_1(A,B,C,D)
& r8_gcd_1(A,B,E,F) )
=> r8_gcd_1(A,B,k6_gcd_1(A,B,C,D,E,F),k7_gcd_1(A,B,C,D,E,F)) ) ) ) ) ) ) ) ).
fof(t46_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( v2_gcd_1(B,A)
& r8_gcd_1(A,B,C,D)
& r8_gcd_1(A,B,E,F) )
=> k10_group_1(A,k6_gcd_1(A,B,C,D,E,F),k10_group_1(A,D,F)) = k10_group_1(A,k7_gcd_1(A,B,C,D,E,F),k4_rlvect_1(A,k10_group_1(A,C,F),k10_group_1(A,E,D))) ) ) ) ) ) ) ) ).
fof(d18_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( ( C = k1_rlvect_1(A)
| E = k1_rlvect_1(A) )
=> k8_gcd_1(A,B,C,D,E,F) = k1_rlvect_1(A) )
& ( ( D = k2_group_1(A)
& F = k2_group_1(A) )
=> k8_gcd_1(A,B,C,D,E,F) = k10_group_1(A,C,E) )
& ( D = k2_group_1(A)
=> ( F = k1_rlvect_1(A)
| k8_gcd_1(A,B,C,D,E,F) = k1_gcd_1(A,k10_group_1(A,C,E),k5_gcd_1(A,B,C,F)) ) )
& ( F = k2_group_1(A)
=> ( D = k1_rlvect_1(A)
| k8_gcd_1(A,B,C,D,E,F) = k1_gcd_1(A,k10_group_1(A,C,E),k5_gcd_1(A,B,E,D)) ) )
& ~ ( C != k1_rlvect_1(A)
& E != k1_rlvect_1(A)
& ~ ( D = k2_group_1(A)
& F = k2_group_1(A) )
& ~ ( F != k1_rlvect_1(A)
& D = k2_group_1(A) )
& ~ ( D != k1_rlvect_1(A)
& F = k2_group_1(A) )
& k8_gcd_1(A,B,C,D,E,F) != k10_group_1(A,k1_gcd_1(A,C,k5_gcd_1(A,B,C,F)),k1_gcd_1(A,E,k5_gcd_1(A,B,E,D))) ) ) ) ) ) ) ) ) ).
fof(d19_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r7_gcd_1(A,C,D)
& r7_gcd_1(A,E,F)
& D = k4_gcd_1(A,B,D)
& F = k4_gcd_1(A,B,F) )
=> ( ( ( C = k1_rlvect_1(A)
| E = k1_rlvect_1(A) )
=> k9_gcd_1(A,B,C,D,E,F) = k2_group_1(A) )
& ( ( D = k2_group_1(A)
& F = k2_group_1(A) )
=> k9_gcd_1(A,B,C,D,E,F) = k2_group_1(A) )
& ( D = k2_group_1(A)
=> ( F = k1_rlvect_1(A)
| k9_gcd_1(A,B,C,D,E,F) = k1_gcd_1(A,F,k5_gcd_1(A,B,C,F)) ) )
& ( F = k2_group_1(A)
=> ( D = k1_rlvect_1(A)
| k9_gcd_1(A,B,C,D,E,F) = k1_gcd_1(A,D,k5_gcd_1(A,B,E,D)) ) )
& ~ ( C != k1_rlvect_1(A)
& E != k1_rlvect_1(A)
& ~ ( D = k2_group_1(A)
& F = k2_group_1(A) )
& ~ ( F != k1_rlvect_1(A)
& D = k2_group_1(A) )
& ~ ( D != k1_rlvect_1(A)
& F = k2_group_1(A) )
& k9_gcd_1(A,B,C,D,E,F) != k10_group_1(A,k1_gcd_1(A,D,k5_gcd_1(A,B,E,D)),k1_gcd_1(A,F,k5_gcd_1(A,B,C,F))) ) ) ) ) ) ) ) ) ) ).
fof(t47_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( v2_gcd_1(B,A)
& r8_gcd_1(A,B,C,D)
& r8_gcd_1(A,B,E,F) )
=> r8_gcd_1(A,B,k8_gcd_1(A,B,C,D,E,F),k9_gcd_1(A,B,C,D,E,F)) ) ) ) ) ) ) ) ).
fof(t48_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( v2_gcd_1(B,A)
& r8_gcd_1(A,B,C,D)
& r8_gcd_1(A,B,E,F) )
=> k10_group_1(A,k8_gcd_1(A,B,C,D,E,F),k10_group_1(A,D,F)) = k10_group_1(A,k9_gcd_1(A,B,C,D,E,F),k10_group_1(A,C,E)) ) ) ) ) ) ) ) ).
fof(t49_gcd_1,axiom,
$true ).
fof(t50_gcd_1,axiom,
$true ).
fof(t51_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_group_1(A,k5_rlvect_1(A,B),C) = k5_rlvect_1(A,k1_group_1(A,B,C))
& k1_group_1(A,B,k5_rlvect_1(A,C)) = k5_rlvect_1(A,k1_group_1(A,B,C)) ) ) ) ) ).
fof(t52_gcd_1,axiom,
$true ).
fof(t53_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& C != k1_rlvect_1(A)
& k10_group_1(A,k4_vectsp_1(A,B),k4_vectsp_1(A,C)) != k4_vectsp_1(A,k10_group_1(A,C,B)) ) ) ) ) ).
fof(dt_m1_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_gcd_1(B,A)
=> ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(existence_m1_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ? [B] : m1_gcd_1(B,A) ) ).
fof(dt_m2_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m2_gcd_1(B,A)
=> ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(existence_m2_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ? [B] : m2_gcd_1(B,A) ) ).
fof(reflexivity_r2_gcd_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> r2_gcd_1(A,B,B) ) ).
fof(redefinition_r2_gcd_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r2_gcd_1(A,B,C)
<=> r1_gcd_1(A,B,C) ) ) ).
fof(symmetry_r3_gcd_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r3_gcd_1(A,B,C)
=> r3_gcd_1(A,C,B) ) ) ).
fof(symmetry_r4_gcd_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r4_gcd_1(A,B,C)
=> r4_gcd_1(A,C,B) ) ) ).
fof(reflexivity_r4_gcd_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> r4_gcd_1(A,B,B) ) ).
fof(redefinition_r4_gcd_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r4_gcd_1(A,B,C)
<=> r3_gcd_1(A,B,C) ) ) ).
fof(symmetry_r7_gcd_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r7_gcd_1(A,B,C)
=> r7_gcd_1(A,C,B) ) ) ).
fof(redefinition_r7_gcd_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r7_gcd_1(A,B,C)
<=> r6_gcd_1(A,B,C) ) ) ).
fof(dt_k1_gcd_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k1_gcd_1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k2_gcd_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k2_gcd_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k3_gcd_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> m1_subset_1(k3_gcd_1(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k4_gcd_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_vectsp_1(A)
& m2_gcd_1(B,A)
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k4_gcd_1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k5_gcd_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v3_gcd_1(A)
& l1_vectsp_1(A)
& m2_gcd_1(B,A)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(A)) )
=> m1_subset_1(k5_gcd_1(A,B,C,D),u1_struct_0(A)) ) ).
fof(dt_k6_gcd_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A)
& m2_gcd_1(B,A)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(A))
& m1_subset_1(E,u1_struct_0(A))
& m1_subset_1(F,u1_struct_0(A)) )
=> m1_subset_1(k6_gcd_1(A,B,C,D,E,F),u1_struct_0(A)) ) ).
fof(dt_k7_gcd_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A)
& m2_gcd_1(B,A)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(A))
& m1_subset_1(E,u1_struct_0(A))
& m1_subset_1(F,u1_struct_0(A)) )
=> m1_subset_1(k7_gcd_1(A,B,C,D,E,F),u1_struct_0(A)) ) ).
fof(dt_k8_gcd_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A)
& m2_gcd_1(B,A)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(A))
& m1_subset_1(E,u1_struct_0(A))
& m1_subset_1(F,u1_struct_0(A)) )
=> m1_subset_1(k8_gcd_1(A,B,C,D,E,F),u1_struct_0(A)) ) ).
fof(dt_k9_gcd_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A)
& m2_gcd_1(B,A)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(A))
& m1_subset_1(E,u1_struct_0(A))
& m1_subset_1(F,u1_struct_0(A)) )
=> m1_subset_1(k9_gcd_1(A,B,C,D,E,F),u1_struct_0(A)) ) ).
%------------------------------------------------------------------------------