SET007 Axioms: SET007+423.ax
%------------------------------------------------------------------------------
% File : SET007+423 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Preliminaries to Circuits, II
% 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 : msafree2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 53 ( 0 unt; 0 def)
% Number of atoms : 392 ( 31 equ)
% Maximal formula atoms : 19 ( 7 avg)
% Number of connectives : 441 ( 102 ~; 0 |; 201 &)
% ( 12 <=>; 126 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 38 ( 37 usr; 0 prp; 1-4 aty)
% Number of functors : 41 ( 41 usr; 4 con; 0-5 aty)
% Number of variables : 135 ( 125 !; 10 ?)
% 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(rc1_msafree2,axiom,
? [A] :
( l1_msualg_1(A)
& ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v1_msafree2(A) ) ).
fof(fc1_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_msafree2(A)
& l1_msualg_1(A) )
=> ~ v1_xboole_0(k2_msafree2(A)) ) ).
fof(cc1_msafree2,axiom,
! [A] :
( l1_msualg_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_msualg_1(A) )
=> ( ~ v3_struct_0(A)
& v2_msafree2(A) ) ) ) ).
fof(rc2_msafree2,axiom,
? [A] :
( l1_msualg_1(A)
& ~ v3_struct_0(A)
& v1_msualg_1(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A) ) ).
fof(cc2_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ( ( v5_msualg_1(B,A)
& v4_msafree2(B,A) )
=> ( v5_msualg_1(B,A)
& v3_msafree2(B,A) ) ) ) ) ).
fof(rc3_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ? [B] :
( l3_msualg_1(B,A)
& v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& v3_msafree2(B,A)
& v4_msafree2(B,A) ) ) ).
fof(rc4_msafree2,axiom,
? [A] :
( l1_msualg_1(A)
& ~ v3_struct_0(A)
& v6_group_1(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A) ) ).
fof(cc3_msafree2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ! [D] :
( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
=> ( ~ v1_xboole_0(D)
& v1_relat_1(D)
& v1_funct_1(D)
& v1_finset_1(D) ) ) ) ).
fof(rc5_msafree2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ? [D] :
( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
& ~ v1_xboole_0(D)
& v1_relat_1(D)
& v1_funct_1(D)
& v1_finset_1(D) ) ) ).
fof(cc4_msafree2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ! [D] :
( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
=> ( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ~ v1_xboole_0(D)
& v1_relat_1(D)
& v1_funct_1(D)
& v1_finset_1(D)
& v3_trees_2(D) ) ) ) ) ).
fof(rc6_msafree2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ? [D] :
( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
& ~ v1_xboole_0(D)
& v1_relat_1(D)
& v1_funct_1(D)
& v1_finset_1(D)
& v3_trees_2(D) ) ) ).
fof(d2_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> k2_msafree2(A) = k4_xboole_0(u1_struct_0(A),k5_relset_1(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A))) ) ).
fof(d3_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> k3_msafree2(A) = k5_relset_1(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A)) ) ).
fof(t1_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_msualg_1(A)
& l1_msualg_1(A) )
=> k2_msafree2(A) = u1_struct_0(A) ) ).
fof(t2_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r2_hidden(B,k2_msafree2(A))
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> k2_msualg_1(A,C) != B ) ) ) ) ).
fof(t3_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> k4_subset_1(u1_struct_0(A),k2_msafree2(A),k3_msafree2(A)) = u1_struct_0(A) ) ).
fof(t4_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> r1_xboole_0(k2_msafree2(A),k3_msafree2(A)) ) ).
fof(t5_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> r1_tarski(k1_msafree2(A),k3_msafree2(A)) ) ).
fof(t6_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> r1_xboole_0(k2_msafree2(A),k1_msafree2(A)) ) ).
fof(d4_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ( v1_msafree2(A)
<=> k2_msafree2(A) != k1_xboole_0 ) ) ).
fof(d5_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_pboole(C,k2_msafree2(A))
=> ( m1_msafree2(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,k2_msafree2(A))
=> r2_hidden(k1_funct_1(C,D),k1_funct_1(u4_msualg_1(A,B),D)) ) ) ) ) ) ) ).
fof(d6_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ( v2_msafree2(A)
<=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ( B = A
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(B))
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ( k2_msualg_1(B,C) = k2_msualg_1(B,D)
=> C = D ) ) ) ) ) ) ) ).
fof(d7_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r2_hidden(B,k4_msafree2(A))
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> ( C = k5_msafree2(A,B)
<=> k2_msualg_1(A,C) = B ) ) ) ) ) ).
fof(t7_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ( ( k3_finseq_1(D) = k3_finseq_1(k1_msualg_1(A,C))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(D))
=> r2_hidden(k1_funct_1(D,E),k1_funct_1(u4_msualg_1(A,B),k4_finseq_4(k5_numbers,u1_struct_0(A),k1_msualg_1(A,C),E))) ) ) )
=> r2_hidden(D,k3_msualg_1(A,C,B)) ) ) ) ) ) ).
fof(d8_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> k6_msafree2(A,B) = k11_msafree(A,u4_msualg_1(A,B)) ) ) ).
fof(t8_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( v1_msafree(k13_msafree(A,u4_msualg_1(A,B)),A,k6_msafree2(A,B))
& m1_msafree(k13_msafree(A,u4_msualg_1(A,B)),A,k6_msafree2(A,B)) ) ) ) ).
fof(d9_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m3_pboole(C,u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,B))
=> ( C = k7_msafree2(A,B)
<=> ( r1_msualg_3(A,k6_msafree2(A,B),B,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E,F] :
( ( r2_hidden(F,k7_msafree(A,u4_msualg_1(A,B),D))
& F = k1_trees_4(k4_tarski(E,D))
& r2_hidden(E,k1_funct_1(u4_msualg_1(A,B),D)) )
=> k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,B),C,D),F) = E ) ) ) ) ) ) ) ).
fof(t9_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> m1_msafree(u4_msualg_1(A,B),A,B) ) ) ).
fof(d10_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ( ( ~ v2_msualg_1(A)
=> ( v3_msafree2(B,A)
<=> ! [C] :
( ( ~ v3_struct_0(C)
& ~ v2_msualg_1(C)
& l1_msualg_1(C) )
=> ( C = A
=> ! [D] :
( l3_msualg_1(D,C)
=> ~ ( D = B
& ! [E] :
( m1_msafree(E,C,D)
=> ~ v1_pre_circ(E,u1_struct_0(C)) ) ) ) ) ) ) )
& ( v2_msualg_1(A)
=> ( v3_msafree2(B,A)
<=> v1_pre_circ(u4_msualg_1(A,B),u1_struct_0(A)) ) ) ) ) ) ).
fof(d11_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ( v4_msafree2(B,A)
<=> v1_pre_circ(u4_msualg_1(A,B),u1_struct_0(A)) ) ) ) ).
fof(d12_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v4_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( B = k8_msafree2(A)
<=> r6_pboole(u1_struct_0(A),u4_msualg_1(A,B),k2_pre_circ(u1_struct_0(A),k1_tarski(np__0))) ) ) ) ).
fof(d13_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ( v5_msafree2(A)
<=> ! [B] :
( ( v5_msualg_1(B,A)
& v3_msafree2(B,A)
& l3_msualg_1(B,A) )
=> v4_msafree2(B,A) ) ) ) ).
fof(t10_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
=> ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finset_1(D)
& v3_trees_2(D) ) ) ) ) ) ).
fof(t11_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& v1_pre_circ(B,u1_struct_0(A))
& m1_pboole(B,u1_struct_0(A)) )
=> v3_msafree2(k11_msafree(A,B),A) ) ) ).
fof(t12_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),C))
=> ~ ( r2_hidden(C,k2_msafree2(A))
& ! [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,B),C))
=> D != k1_trees_4(k4_tarski(E,C)) ) ) ) ) ) ) ).
fof(t13_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& v6_trees_3(D) )
=> ( r2_hidden(k4_trees_4(k4_tarski(C,u1_struct_0(A)),D),k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),k2_msualg_1(A,C)))
=> k3_finseq_1(D) = k3_finseq_1(k1_msualg_1(A,C)) ) ) ) ) ) ).
fof(t14_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& v6_trees_3(D) )
=> ( r2_hidden(k4_trees_4(k4_tarski(C,u1_struct_0(A)),D),k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),k2_msualg_1(A,C)))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_relset_1(k5_numbers,u1_struct_0(A),k1_msualg_1(A,C)))
=> r2_hidden(k1_funct_1(D,E),k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),k1_funct_1(k1_msualg_1(A,C),E))) ) ) ) ) ) ) ) ).
fof(t15_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(A))
=> ! [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
=> ~ ( r2_hidden(C,k4_msafree2(A))
& k1_funct_1(E,k1_xboole_0) = k4_tarski(D,u1_struct_0(A))
& ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F)
& v1_finseq_1(F)
& v6_trees_3(F) )
=> ~ ( k3_finseq_1(F) = k3_finseq_1(k1_msualg_1(A,D))
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(G,k4_finseq_1(F))
=> r2_hidden(k1_funct_1(F,G),k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),k1_funct_1(k1_msualg_1(A,D),G))) ) ) ) ) ) ) ) ) ) ) ).
fof(d14_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( E = k9_msafree2(A,B,C,D)
<=> ? [F] :
( v1_relat_1(F)
& v1_funct_1(F)
& v1_finset_1(F)
& v3_trees_2(F)
& ? [G] :
( ~ v1_xboole_0(G)
& v1_finset_1(G)
& v1_trees_1(G)
& F = D
& G = k1_relat_1(F)
& E = k6_trees_1(G) ) ) ) ) ) ) ) ) ).
fof(dt_m1_msafree2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msafree2(C,A,B)
=> m1_pboole(C,k2_msafree2(A)) ) ) ).
fof(existence_m1_msafree2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ? [C] : m1_msafree2(C,A,B) ) ).
fof(dt_k1_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> m1_subset_1(k1_msafree2(A),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> m1_subset_1(k2_msafree2(A),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k3_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> m1_subset_1(k3_msafree2(A),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k4_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ( ~ v1_xboole_0(k4_msafree2(A))
& m1_subset_1(k4_msafree2(A),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(redefinition_k4_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> k4_msafree2(A) = k3_msafree2(A) ) ).
fof(dt_k5_msafree2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& l1_msualg_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k5_msafree2(A,B),u1_msualg_1(A)) ) ).
fof(dt_k6_msafree2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( v4_msualg_1(k6_msafree2(A,B),A)
& v5_msualg_1(k6_msafree2(A,B),A)
& v2_msafree(k6_msafree2(A,B),A)
& l3_msualg_1(k6_msafree2(A,B),A) ) ) ).
fof(dt_k7_msafree2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> m3_pboole(k7_msafree2(A,B),u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,B)) ) ).
fof(dt_k8_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ( v4_msualg_1(k8_msafree2(A),A)
& l3_msualg_1(k8_msafree2(A),A) ) ) ).
fof(dt_k9_msafree2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C)) )
=> m2_subset_1(k9_msafree2(A,B,C,D),k1_numbers,k5_numbers) ) ).
fof(d1_msafree2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ( ( ~ v2_msualg_1(A)
=> k1_msafree2(A) = a_1_0_msafree2(A) )
& ( v2_msualg_1(A)
=> k1_msafree2(A) = k1_xboole_0 ) ) ) ).
fof(fraenkel_a_1_0_msafree2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B) )
=> ( r2_hidden(A,a_1_0_msafree2(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& A = C
& v1_msualg_2(C,B) ) ) ) ).
%------------------------------------------------------------------------------