SET007 Axioms: SET007+385.ax
%------------------------------------------------------------------------------
% File : SET007+385 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Homomorphisms of Algebras. Quotient Universal Algebra
% 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 : alg_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 52 ( 2 unt; 0 def)
% Number of atoms : 685 ( 41 equ)
% Maximal formula atoms : 24 ( 13 avg)
% Number of connectives : 719 ( 86 ~; 0 |; 435 &)
% ( 18 <=>; 180 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 33 ( 31 usr; 1 prp; 0-4 aty)
% Number of functors : 36 ( 36 usr; 2 con; 0-5 aty)
% Number of variables : 167 ( 164 !; 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(t1_alg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ( k4_finseq_1(k5_finseqop(A,B,C,D)) = k4_finseq_1(C)
& k3_finseq_1(k5_finseqop(A,B,C,D)) = k3_finseq_1(C)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(k5_finseqop(A,B,C,D)))
=> k1_funct_1(k5_finseqop(A,B,C,D),E) = k1_funct_1(D,k1_funct_1(C,E)) ) ) ) ) ) ) ) ).
fof(t2_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( B = u1_struct_0(A)
=> k3_unialg_2(A,B) = u1_unialg_1(A) ) ) ) ).
fof(t3_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> k5_finseqop(u1_struct_0(A),u1_struct_0(B),k6_finseq_1(u1_struct_0(A)),C) = k6_finseq_1(u1_struct_0(B)) ) ) ) ).
fof(t4_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> k5_finseqop(u1_struct_0(A),u1_struct_0(A),B,k6_partfun1(u1_struct_0(A))) = B ) ) ).
fof(t5_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_unialg_1(C)
& v7_unialg_1(C)
& v8_unialg_1(C)
& l1_unialg_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
=> ! [F] :
( m2_finseq_1(F,u1_struct_0(A))
=> k5_finseqop(u1_struct_0(B),u1_struct_0(C),k5_finseqop(u1_struct_0(A),u1_struct_0(B),F,D),E) = k5_finseqop(u1_struct_0(A),u1_struct_0(C),F,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) ) ) ) ) ) ) ).
fof(d1_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r1_alg_1(A,B,C)
<=> ( r1_unialg_2(A,B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(u1_unialg_1(A)))
=> ! [E] :
( m2_unialg_2(E,u1_struct_0(A),k1_unialg_2(A))
=> ! [F] :
( m2_unialg_2(F,u1_struct_0(B),k1_unialg_2(B))
=> ( ( E = k1_funct_1(u1_unialg_1(A),D)
& F = k1_funct_1(u1_unialg_1(B),D) )
=> ! [G] :
( m2_finseq_1(G,u1_struct_0(A))
=> ( r2_hidden(G,k4_relset_1(k3_finseq_2(u1_struct_0(A)),u1_struct_0(A),E))
=> k1_funct_1(C,k1_funct_1(E,G)) = k1_funct_1(F,k5_finseqop(u1_struct_0(A),u1_struct_0(B),G,C)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d2_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r2_alg_1(A,B,C)
<=> ( r1_alg_1(A,B,C)
& v2_funct_1(C) ) ) ) ) ) ).
fof(d3_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r3_alg_1(A,B,C)
<=> ( r1_alg_1(A,B,C)
& k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ).
fof(d4_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r4_alg_1(A,B,C)
<=> ( r2_alg_1(A,B,C)
& r3_alg_1(A,B,C) ) ) ) ) ) ).
fof(d5_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ( r5_alg_1(A,B)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& r4_alg_1(A,B,C) ) ) ) ) ).
fof(t6_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> r1_alg_1(A,A,k6_partfun1(u1_struct_0(A))) ) ).
fof(t7_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_unialg_1(C)
& v7_unialg_1(C)
& v8_unialg_1(C)
& l1_unialg_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
=> ( ( r1_alg_1(A,B,D)
& r1_alg_1(B,C,E) )
=> r1_alg_1(A,C,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) ) ) ) ) ) ) ).
fof(t8_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r4_alg_1(A,B,C)
<=> ( r1_alg_1(A,B,C)
& k2_relat_1(C) = u1_struct_0(B)
& v2_funct_1(C) ) ) ) ) ) ).
fof(t9_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r4_alg_1(A,B,C)
=> ( k4_relset_1(u1_struct_0(A),u1_struct_0(B),C) = u1_struct_0(A)
& k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ).
fof(t10_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
=> ( ( r4_alg_1(A,B,C)
& D = k2_funct_1(C) )
=> r1_alg_1(B,A,D) ) ) ) ) ) ).
fof(t11_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
=> ( ( r4_alg_1(A,B,C)
& D = k2_funct_1(C) )
=> r4_alg_1(B,A,D) ) ) ) ) ) ).
fof(t12_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_unialg_1(C)
& v7_unialg_1(C)
& v8_unialg_1(C)
& l1_unialg_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
=> ( ( r4_alg_1(A,B,D)
& r4_alg_1(B,C,E) )
=> r4_alg_1(A,C,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) ) ) ) ) ) ) ).
fof(t13_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> r5_alg_1(A,A) ) ).
fof(t14_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ( r5_alg_1(A,B)
=> r5_alg_1(B,A) ) ) ) ).
fof(t15_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_unialg_1(C)
& v7_unialg_1(C)
& v8_unialg_1(C)
& l1_unialg_1(C) )
=> ( ( r5_alg_1(A,B)
& r5_alg_1(B,C) )
=> r5_alg_1(A,C) ) ) ) ) ).
fof(d6_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r1_alg_1(A,B,C)
=> ! [D] :
( ( v3_unialg_1(D)
& m3_unialg_2(D,B) )
=> ( D = k1_alg_1(A,B,C)
<=> u1_struct_0(D) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(A)) ) ) ) ) ) ) ).
fof(t16_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r1_alg_1(A,B,C)
=> k2_relat_1(C) = u1_struct_0(k1_alg_1(A,B,C)) ) ) ) ) ).
fof(t17_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_unialg_1(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r1_alg_1(A,B,C)
=> ( r3_alg_1(A,B,C)
<=> k1_alg_1(A,B,C) = B ) ) ) ) ) ).
fof(d7_alg_1,axiom,
$true ).
fof(d8_alg_1,axiom,
$true ).
fof(d9_alg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ! [C] :
( m2_relset_1(C,k3_finseq_2(A),k3_finseq_2(A))
=> ( C = k2_alg_1(A,B)
<=> ! [D] :
( m2_finseq_1(D,A)
=> ! [E] :
( m2_finseq_1(E,A)
=> ( r2_hidden(k4_tarski(D,E),C)
<=> ( k3_finseq_1(D) = k3_finseq_1(E)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k4_finseq_1(D))
=> r2_hidden(k4_tarski(k1_funct_1(D,F),k1_funct_1(E,F)),B) ) ) ) ) ) ) ) ) ) ) ).
fof(t18_alg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k2_alg_1(A,k6_partfun1(A)) = k6_partfun1(k3_finseq_2(A)) ) ).
fof(d10_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( m1_alg_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_unialg_2(D,u1_struct_0(A),k1_unialg_2(A))
=> ( ( r2_hidden(C,k4_finseq_1(u1_unialg_1(A)))
& D = k1_funct_1(u1_unialg_1(A),C) )
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(A))
=> ! [F] :
( m2_finseq_1(F,u1_struct_0(A))
=> ( ( r2_hidden(E,k4_relset_1(k3_finseq_2(u1_struct_0(A)),u1_struct_0(A),D))
& r2_hidden(F,k4_relset_1(k3_finseq_2(u1_struct_0(A)),u1_struct_0(A),D))
& r2_hidden(k4_tarski(E,F),k2_alg_1(u1_struct_0(A),B)) )
=> r2_hidden(k4_tarski(k1_funct_1(D,E),k1_funct_1(D,F)),B) ) ) ) ) ) ) ) ) ) ).
fof(d11_alg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m2_finseq_1(C,k8_eqrel_1(A,B))
=> ! [D] :
( m2_finseq_1(D,A)
=> ( r6_alg_1(A,B,C,D)
<=> ( k3_finseq_1(D) = k3_finseq_1(C)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(D))
=> k6_eqrel_1(A,B,k1_funct_1(D,E)) = k1_funct_1(C,E) ) ) ) ) ) ) ) ) ).
fof(t19_alg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m2_finseq_1(C,k8_eqrel_1(A,B))
=> ? [D] :
( m2_finseq_1(D,A)
& r6_alg_1(A,B,C,D) ) ) ) ) ).
fof(d12_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_alg_1(B,A)
=> ! [C] :
( m2_unialg_2(C,u1_struct_0(A),k1_unialg_2(A))
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_funct_1(D)
& v1_unialg_1(D,k8_eqrel_1(u1_struct_0(A),B))
& v2_unialg_1(D,k8_eqrel_1(u1_struct_0(A),B))
& m2_relset_1(D,k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B)) )
=> ( D = k3_alg_1(A,B,C)
<=> ( k4_relset_1(k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B),D) = k4_finseq_2(k2_unialg_1(u1_struct_0(A),C),k8_eqrel_1(u1_struct_0(A),B))
& ! [E] :
( m2_finseq_1(E,k8_eqrel_1(u1_struct_0(A),B))
=> ( r2_hidden(E,k4_relset_1(k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B),D))
=> ! [F] :
( m2_finseq_1(F,u1_struct_0(A))
=> ( r6_alg_1(u1_struct_0(A),B,E,F)
=> k1_funct_1(D,E) = k6_eqrel_1(u1_struct_0(A),B,k1_funct_1(C,F)) ) ) ) ) ) ) ) ) ) ) ).
fof(d13_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_alg_1(B,A)
=> ! [C] :
( m2_finseq_1(C,k4_partfun1(k13_finseq_1(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B)))
=> ( C = k4_alg_1(A,B)
<=> ( k3_finseq_1(C) = k3_finseq_1(u1_unialg_1(A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(C))
=> ! [E] :
( m2_unialg_2(E,u1_struct_0(A),k1_unialg_2(A))
=> ( k1_funct_1(u1_unialg_1(A),D) = E
=> k1_funct_1(C,D) = k3_alg_1(A,B,E) ) ) ) ) ) ) ) ) ) ).
fof(t20_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_alg_1(B,A)
=> ( ~ v3_struct_0(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
& v3_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
& v6_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
& v7_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
& v8_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
& l1_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B))) ) ) ) ).
fof(d14_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_alg_1(B,A)
=> k5_alg_1(A,B) = g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)) ) ) ).
fof(d15_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_alg_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(k5_alg_1(A,B)))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(k5_alg_1(A,B))) )
=> ( C = k6_alg_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(k5_alg_1(A,B)),C,D) = k6_eqrel_1(u1_struct_0(A),B,D) ) ) ) ) ) ).
fof(t21_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_alg_1(B,A)
=> r1_alg_1(A,k5_alg_1(A,B),k6_alg_1(A,B)) ) ) ).
fof(t22_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_alg_1(B,A)
=> r3_alg_1(A,k5_alg_1(A,B),k6_alg_1(A,B)) ) ) ).
fof(d16_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r1_alg_1(A,B,C)
=> ! [D] :
( m1_alg_1(D,A)
=> ( D = k7_alg_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( r2_hidden(k4_tarski(E,F),D)
<=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,F) ) ) ) ) ) ) ) ) ) ).
fof(d17_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r1_alg_1(A,B,C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_alg_1(A,k7_alg_1(A,B,C))),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(k5_alg_1(A,k7_alg_1(A,B,C))),u1_struct_0(B)) )
=> ( D = k8_alg_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k1_funct_1(D,k6_eqrel_1(u1_struct_0(A),k7_alg_1(A,B,C),E)) = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) ) ) ) ) ) ) ) ).
fof(t23_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r1_alg_1(A,B,C)
=> ( r1_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B,k8_alg_1(A,B,C))
& r2_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B,k8_alg_1(A,B,C)) ) ) ) ) ) ).
fof(t24_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r3_alg_1(A,B,C)
=> r4_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B,k8_alg_1(A,B,C)) ) ) ) ) ).
fof(t25_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r3_alg_1(A,B,C)
=> r5_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B) ) ) ) ) ).
fof(dt_m1_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_alg_1(B,A)
=> ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) ) ) ) ).
fof(existence_m1_alg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ? [B] : m1_alg_1(B,A) ) ).
fof(dt_k1_alg_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A)
& ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_unialg_1(k1_alg_1(A,B,C))
& m3_unialg_2(k1_alg_1(A,B,C),B) ) ) ).
fof(dt_k2_alg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_relset_1(B,A,A) )
=> m2_relset_1(k2_alg_1(A,B),k3_finseq_2(A),k3_finseq_2(A)) ) ).
fof(dt_k3_alg_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A)
& m1_alg_1(B,A)
& m1_subset_1(C,k1_unialg_2(A)) )
=> ( ~ v1_xboole_0(k3_alg_1(A,B,C))
& v1_funct_1(k3_alg_1(A,B,C))
& v1_unialg_1(k3_alg_1(A,B,C),k8_eqrel_1(u1_struct_0(A),B))
& v2_unialg_1(k3_alg_1(A,B,C),k8_eqrel_1(u1_struct_0(A),B))
& m2_relset_1(k3_alg_1(A,B,C),k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B)) ) ) ).
fof(dt_k4_alg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A)
& m1_alg_1(B,A) )
=> m2_finseq_1(k4_alg_1(A,B),k4_partfun1(k13_finseq_1(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B))) ) ).
fof(dt_k5_alg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A)
& m1_alg_1(B,A) )
=> ( ~ v3_struct_0(k5_alg_1(A,B))
& v3_unialg_1(k5_alg_1(A,B))
& v6_unialg_1(k5_alg_1(A,B))
& v7_unialg_1(k5_alg_1(A,B))
& v8_unialg_1(k5_alg_1(A,B))
& l1_unialg_1(k5_alg_1(A,B)) ) ) ).
fof(dt_k6_alg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A)
& m1_alg_1(B,A) )
=> ( v1_funct_1(k6_alg_1(A,B))
& v1_funct_2(k6_alg_1(A,B),u1_struct_0(A),u1_struct_0(k5_alg_1(A,B)))
& m2_relset_1(k6_alg_1(A,B),u1_struct_0(A),u1_struct_0(k5_alg_1(A,B))) ) ) ).
fof(dt_k7_alg_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A)
& ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> m1_alg_1(k7_alg_1(A,B,C),A) ) ).
fof(dt_k8_alg_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A)
& ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_funct_1(k8_alg_1(A,B,C))
& v1_funct_2(k8_alg_1(A,B,C),u1_struct_0(k5_alg_1(A,k7_alg_1(A,B,C))),u1_struct_0(B))
& m2_relset_1(k8_alg_1(A,B,C),u1_struct_0(k5_alg_1(A,k7_alg_1(A,B,C))),u1_struct_0(B)) ) ) ).
%------------------------------------------------------------------------------