SET007 Axioms: SET007+398.ax
%------------------------------------------------------------------------------
% File : SET007+398 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Free Many Sorted 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 : msafree [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 78 ( 4 unt; 0 def)
% Number of atoms : 606 ( 63 equ)
% Maximal formula atoms : 20 ( 7 avg)
% Number of connectives : 673 ( 145 ~; 0 |; 308 &)
% ( 27 <=>; 193 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 9 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 40 ( 38 usr; 1 prp; 0-4 aty)
% Number of functors : 69 ( 69 usr; 3 con; 0-5 aty)
% Number of variables : 257 ( 240 !; 17 ?)
% 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_msafree,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A) )
=> ( v1_relat_1(k2_card_3(B))
& v2_relat_1(k2_card_3(B))
& v1_funct_1(k2_card_3(B)) ) ) ).
fof(fc2_msafree,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A) )
=> ~ v1_xboole_0(k3_card_3(B)) ) ).
fof(fc3_msafree,axiom,
! [A] :
( ( ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ~ v1_xboole_0(u1_msualg_1(A)) ) ).
fof(fc4_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& m1_pboole(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k5_msafree(A,B))
& v1_lang1(k5_msafree(A,B)) ) ) ).
fof(fc5_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k5_msafree(A,B))
& v1_lang1(k5_msafree(A,B))
& v1_dtconstr(k5_msafree(A,B))
& v2_dtconstr(k5_msafree(A,B))
& v3_dtconstr(k5_msafree(A,B)) ) ) ).
fof(fc6_msafree,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)) )
=> ( ~ v1_xboole_0(k7_msafree(A,B,C))
& v3_trees_3(k7_msafree(A,B,C)) ) ) ).
fof(fc7_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ( v1_relat_1(k8_msafree(A,B))
& v2_relat_1(k8_msafree(A,B))
& ~ v3_relat_1(k8_msafree(A,B))
& v1_funct_1(k8_msafree(A,B)) ) ) ).
fof(fc8_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ( v4_msualg_1(k11_msafree(A,B),A)
& v5_msualg_1(k11_msafree(A,B),A) ) ) ).
fof(fc9_msafree,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)) )
=> ~ v1_xboole_0(k12_msafree(A,B,C)) ) ).
fof(rc1_msafree,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ? [B] :
( l3_msualg_1(B,A)
& v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& v2_msafree(B,A) ) ) ).
fof(rc2_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_msafree(B,A)
& l3_msualg_1(B,A) )
=> ? [C] :
( m1_msafree(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_msafree(C,A,B) ) ) ).
fof(t1_msafree,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k3_finseq_2(B))
& m2_relset_1(C,A,k3_finseq_2(B)) )
=> ! [D] :
( m1_pboole(D,B)
=> ! [E] :
( m2_finseq_2(E,B,k3_finseq_2(B))
=> ! [F] :
( ( r2_hidden(F,A)
& E = k1_funct_1(C,F) )
=> k1_funct_1(k7_pboole(A,k3_finseq_2(B),C,k6_pboole(B,D)),F) = k4_card_3(k5_relat_1(E,D)) ) ) ) ) ) ).
fof(d1_msafree,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m4_pboole(D,A,B)
=> ! [E] :
( m3_pboole(E,A,B,C)
=> ! [F] :
( m3_pboole(F,A,D,C)
=> ( F = k1_msafree(A,B,C,D,E)
<=> ! [G] :
( r2_hidden(G,A)
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k1_funct_1(B,G),k1_funct_1(C,G))
& m2_relset_1(H,k1_funct_1(B,G),k1_funct_1(C,G)) )
=> ( H = k1_funct_1(E,G)
=> k1_funct_1(F,G) = k8_relset_1(k1_funct_1(B,G),k1_funct_1(C,G),H,k1_funct_1(D,G)) ) ) ) ) ) ) ) ) ) ).
fof(d2_msafree,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( r2_hidden(C,A)
=> ! [D] :
( D = k2_msafree(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( r2_hidden(F,k1_funct_1(B,C))
& E = k4_tarski(F,C) ) ) ) ) ) ).
fof(d3_msafree,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( C = k3_msafree(A,B)
<=> ! [D] :
( r2_hidden(D,A)
=> k1_funct_1(C,D) = k2_msafree(A,B,D) ) ) ) ) ).
fof(t2_msafree,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( r2_hidden(C,A)
=> ( ~ ( k1_funct_1(B,C) != k1_xboole_0
& k1_funct_1(k3_msafree(A,B),C) = k1_xboole_0 )
& ~ ( k1_funct_1(k3_msafree(A,B),C) != k1_xboole_0
& k1_funct_1(B,C) = k1_xboole_0 ) ) ) ) ).
fof(d4_msafree,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B))
=> ( m1_msafree(C,A,B)
<=> r6_pboole(u1_struct_0(A),u4_msualg_1(A,k12_msualg_2(A,B,C)),u4_msualg_1(A,B)) ) ) ) ) ).
fof(t3_msafree,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B))
=> ( m1_msafree(C,A,B)
<=> k12_msualg_2(A,B,C) = B ) ) ) ) ).
fof(d5_msafree,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_msafree(C,A,B)
=> ( v1_msafree(C,A,B)
<=> ! [D] :
( ( v5_msualg_1(D,A)
& l3_msualg_1(D,A) )
=> ! [E] :
( m3_pboole(E,u1_struct_0(A),C,u4_msualg_1(A,D))
=> ? [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,D))
& r1_msualg_3(A,B,D,F)
& r6_pboole(u1_struct_0(A),k1_msafree(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,D),C,F),E) ) ) ) ) ) ) ) ).
fof(d6_msafree,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ( v2_msafree(B,A)
<=> ? [C] :
( m1_msafree(C,A,B)
& v1_msafree(C,A,B) ) ) ) ) ).
fof(t4_msafree,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> r1_xboole_0(k3_card_3(k3_msafree(u1_struct_0(A),B)),k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A)))) ) ) ).
fof(d7_msafree,axiom,
$true ).
fof(d8_msafree,axiom,
$true ).
fof(d9_msafree,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ! [C] :
( m2_relset_1(C,k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B))),k3_finseq_2(k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B)))))
=> ( C = k4_msafree(A,B)
<=> ! [D] :
( m1_subset_1(D,k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B))))
=> ! [E] :
( m2_finseq_2(E,k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B))),k3_finseq_2(k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B)))))
=> ( r2_hidden(k4_tarski(D,E),C)
<=> ( r2_hidden(D,k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))))
& ! [F] :
( m1_subset_1(F,u1_msualg_1(A))
=> ( k4_tarski(F,u1_struct_0(A)) = D
=> ( k3_finseq_1(E) = k3_finseq_1(k1_msualg_1(A,F))
& ! [G] :
( r2_hidden(G,k4_finseq_1(E))
=> ( ( r2_hidden(k1_funct_1(E,G),k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))))
=> ! [H] :
( m1_subset_1(H,u1_msualg_1(A))
=> ( k4_tarski(H,u1_struct_0(A)) = k1_funct_1(E,G)
=> k2_msualg_1(A,H) = k1_funct_1(k1_msualg_1(A,F),G) ) ) )
& ( r2_hidden(k1_funct_1(E,G),k3_card_3(k3_msafree(u1_struct_0(A),B)))
=> r2_hidden(k1_funct_1(E,G),k2_msafree(u1_struct_0(A),B,k1_funct_1(k1_msualg_1(A,F),G))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t5_msafree,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> ! [D] :
( m2_finseq_2(D,k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B))),k3_finseq_2(k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B)))))
=> ( r2_hidden(k4_tarski(k4_tarski(C,u1_struct_0(A)),D),k4_msafree(A,B))
<=> ( k3_finseq_1(D) = k3_finseq_1(k1_msualg_1(A,C))
& ! [E] :
( r2_hidden(E,k4_finseq_1(D))
=> ( ( r2_hidden(k1_funct_1(D,E),k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))))
=> ! [F] :
( m1_subset_1(F,u1_msualg_1(A))
=> ( k4_tarski(F,u1_struct_0(A)) = k1_funct_1(D,E)
=> k2_msualg_1(A,F) = k1_funct_1(k1_msualg_1(A,C),E) ) ) )
& ( r2_hidden(k1_funct_1(D,E),k3_card_3(k3_msafree(u1_struct_0(A),B)))
=> r2_hidden(k1_funct_1(D,E),k2_msafree(u1_struct_0(A),B,k1_funct_1(k1_msualg_1(A,C),E))) ) ) ) ) ) ) ) ) ) ).
fof(d10_msafree,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> k5_msafree(A,B) = g1_lang1(k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B))),k4_msafree(A,B)) ) ) ).
fof(t6_msafree,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)) )
=> ( k6_lang1(k5_msafree(A,B)) = k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A)))
& k5_lang1(k5_msafree(A,B)) = k3_card_3(k3_msafree(u1_struct_0(A),B)) ) ) ) ).
fof(t7_msafree,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] :
( r2_hidden(C,k6_dtconstr(k5_msafree(A,B)))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( r2_hidden(E,k1_funct_1(B,D))
& C = k4_tarski(E,D) ) ) ) ) ) ).
fof(d11_msafree,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))
=> k6_msafree(A,B,C) = k4_tarski(C,u1_struct_0(A)) ) ) ) ).
fof(d13_msafree,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_pboole(C,u1_struct_0(A))
=> ( C = k8_msafree(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_funct_1(C,D) = k7_msafree(A,B,D) ) ) ) ) ) ).
fof(t8_msafree,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] :
( r2_hidden(D,k1_funct_1(k8_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),k8_msafree(A,B))),C))
=> m1_trees_4(D,k5_trees_3(u1_struct_0(k5_msafree(A,B))),k4_dtconstr(k5_msafree(A,B))) ) ) ) ) ).
fof(t9_msafree,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] :
( m1_trees_4(D,k5_trees_3(u1_struct_0(k5_msafree(A,B))),k4_dtconstr(k5_msafree(A,B)))
=> ( r2_hidden(D,k1_funct_1(k8_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),k8_msafree(A,B))),C))
<=> ( k4_finseq_1(D) = k4_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),k7_msafree(A,B,k4_finseq_4(k5_numbers,u1_struct_0(A),k1_msualg_1(A,C),E))) ) ) ) ) ) ) ) ) ).
fof(t10_msafree,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] :
( m1_trees_4(D,k5_trees_3(u1_struct_0(k5_msafree(A,B))),k4_dtconstr(k5_msafree(A,B)))
=> ( r1_lang1(k5_msafree(A,B),k6_msafree(A,B,C),k1_dtconstr(u1_struct_0(k5_msafree(A,B)),k5_trees_3(u1_struct_0(k5_msafree(A,B))),D))
<=> r2_hidden(D,k1_funct_1(k8_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),k8_msafree(A,B))),C)) ) ) ) ) ) ).
fof(t11_msafree,axiom,
$true ).
fof(t12_msafree,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)) )
=> k3_tarski(k2_relat_1(k8_msafree(A,B))) = k4_dtconstr(k5_msafree(A,B)) ) ) ).
fof(t13_msafree,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_struct_0(A))
=> ( C != D
=> r1_subset_1(k1_funct_1(k8_msafree(A,B),C),k1_funct_1(k8_msafree(A,B),D)) ) ) ) ) ) ).
fof(d14_msafree,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_funct_1(D)
& v1_funct_2(D,k1_funct_1(k8_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),k8_msafree(A,B))),C),k1_funct_1(k8_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),k8_msafree(A,B)),C))
& m2_relset_1(D,k1_funct_1(k8_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),k8_msafree(A,B))),C),k1_funct_1(k8_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),k8_msafree(A,B)),C)) )
=> ( D = k9_msafree(A,B,C)
<=> ! [E] :
( m1_trees_4(E,k5_trees_3(u1_struct_0(k5_msafree(A,B))),k4_dtconstr(k5_msafree(A,B)))
=> ( r1_lang1(k5_msafree(A,B),k6_msafree(A,B,C),k1_dtconstr(u1_struct_0(k5_msafree(A,B)),k5_trees_3(u1_struct_0(k5_msafree(A,B))),E))
=> k1_funct_1(D,E) = k12_trees_4(u1_struct_0(k5_msafree(A,B)),k6_msafree(A,B,C),E) ) ) ) ) ) ) ) ).
fof(d15_msafree,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] :
( m3_pboole(C,u1_msualg_1(A),k8_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),k8_msafree(A,B))),k8_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),k8_msafree(A,B)))
=> ( C = k10_msafree(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_msualg_1(A))
=> k1_msualg_3(u1_msualg_1(A),k8_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),k8_msafree(A,B))),k8_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),k8_msafree(A,B)),C,D) = k9_msafree(A,B,D) ) ) ) ) ) ).
fof(d16_msafree,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)) )
=> k11_msafree(A,B) = g3_msualg_1(A,k8_msafree(A,B),k10_msafree(A,B)) ) ) ).
fof(d17_msafree,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_zfmisc_1(k1_funct_1(k8_msafree(A,B),C)))
=> ( D = k12_msafree(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( r2_hidden(F,k1_funct_1(B,C))
& E = k1_trees_4(k4_tarski(F,C)) ) ) ) ) ) ) ) ).
fof(d18_msafree,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_msafree(C,A,k11_msafree(A,B))
=> ( C = k13_msafree(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_funct_1(C,D) = k12_msafree(A,B,D) ) ) ) ) ) ).
fof(t15_msafree,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)) )
=> v2_relat_1(k13_msafree(A,B)) ) ) ).
fof(d19_msafree,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] :
( ( v1_funct_1(D)
& v1_funct_2(D,k12_msafree(A,B,C),k1_funct_1(B,C))
& m2_relset_1(D,k12_msafree(A,B,C),k1_funct_1(B,C)) )
=> ( D = k14_msafree(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(k5_msafree(A,B)))
=> ( r2_hidden(k2_trees_4(u1_struct_0(k5_msafree(A,B)),E),k12_msafree(A,B,C))
=> k1_funct_1(D,k2_trees_4(u1_struct_0(k5_msafree(A,B)),E)) = k1_mcart_1(E) ) ) ) ) ) ) ) ).
fof(d20_msafree,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] :
( m3_pboole(C,u1_struct_0(A),k13_msafree(A,B),B)
=> ( C = k15_msafree(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_msualg_3(u1_struct_0(A),k13_msafree(A,B),B,C,D) = k14_msafree(A,B,D) ) ) ) ) ) ).
fof(d21_msafree,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] :
( ( v2_relat_1(C)
& m1_pboole(C,u1_struct_0(A)) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),k13_msafree(A,B),C)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k5_msafree(A,B)))
=> ( r2_hidden(E,k6_dtconstr(k5_msafree(A,B)))
=> ! [F] :
( m1_subset_1(F,k3_card_3(C))
=> ( F = k16_msafree(A,B,C,D,E)
<=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ( G = k1_funct_1(D,k2_mcart_1(E))
=> F = k1_funct_1(G,k2_trees_4(u1_struct_0(k5_msafree(A,B)),E)) ) ) ) ) ) ) ) ) ) ) ).
fof(d22_msafree,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(k5_msafree(A,B)))
=> ( ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& r1_lang1(k5_msafree(A,B),C,D) )
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(A))
=> ( D = k17_msafree(A,B,C)
<=> k4_tarski(D,u1_struct_0(A)) = C ) ) ) ) ) ) ).
fof(d23_msafree,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_msualg_1(A))
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ( r2_hidden(D,k3_msualg_1(A,C,B))
=> k18_msafree(A,B,C,D) = k1_funct_1(k5_msualg_1(A,C,B),D) ) ) ) ) ) ).
fof(t17_msafree,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)) )
=> v1_msafree(k13_msafree(A,B),A,k11_msafree(A,B)) ) ) ).
fof(t18_msafree,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)) )
=> v2_msafree(k11_msafree(A,B),A) ) ) ).
fof(t19_msafree,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] :
( v4_msualg_1(C,A)
& v5_msualg_1(C,A)
& v2_msafree(C,A)
& l3_msualg_1(C,A)
& ? [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B))
& r2_msualg_3(A,C,B,D) ) ) ) ) ).
fof(t20_msafree,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ? [C] :
( v4_msualg_1(C,A)
& v5_msualg_1(C,A)
& v2_msafree(C,A)
& l3_msualg_1(C,A)
& ? [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B))
& r2_msualg_3(A,C,B,D)
& k7_msualg_3(A,C,B,D) = B ) ) ) ) ).
fof(dt_m1_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msafree(C,A,B)
=> m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B)) ) ) ).
fof(existence_m1_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& l3_msualg_1(B,A) )
=> ? [C] : m1_msafree(C,A,B) ) ).
fof(dt_k1_msafree,axiom,
! [A,B,C,D,E] :
( ( m1_pboole(B,A)
& m1_pboole(C,A)
& m4_pboole(D,A,B)
& m3_pboole(E,A,B,C) )
=> m3_pboole(k1_msafree(A,B,C,D,E),A,D,C) ) ).
fof(dt_k2_msafree,axiom,
$true ).
fof(dt_k3_msafree,axiom,
! [A,B] :
( m1_pboole(B,A)
=> m1_pboole(k3_msafree(A,B),A) ) ).
fof(redefinition_k3_msafree,axiom,
! [A,B] :
( m1_pboole(B,A)
=> k3_msafree(A,B) = k2_card_3(B) ) ).
fof(dt_k4_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& m1_pboole(B,u1_struct_0(A)) )
=> m2_relset_1(k4_msafree(A,B),k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B))),k3_finseq_2(k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B))))) ) ).
fof(dt_k5_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& m1_pboole(B,u1_struct_0(A)) )
=> l1_lang1(k5_msafree(A,B)) ) ).
fof(dt_k6_msafree,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_msualg_1(A)) )
=> m1_subset_1(k6_msafree(A,B,C),u1_struct_0(k5_msafree(A,B))) ) ).
fof(dt_k7_msafree,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)) )
=> m1_subset_1(k7_msafree(A,B,C),k1_zfmisc_1(k4_dtconstr(k5_msafree(A,B)))) ) ).
fof(dt_k8_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> m1_pboole(k8_msafree(A,B),u1_struct_0(A)) ) ).
fof(dt_k9_msafree,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_msualg_1(A)) )
=> ( v1_funct_1(k9_msafree(A,B,C))
& v1_funct_2(k9_msafree(A,B,C),k1_funct_1(k8_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),k8_msafree(A,B))),C),k1_funct_1(k8_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),k8_msafree(A,B)),C))
& m2_relset_1(k9_msafree(A,B,C),k1_funct_1(k8_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),k8_msafree(A,B))),C),k1_funct_1(k8_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),k8_msafree(A,B)),C)) ) ) ).
fof(dt_k10_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> m3_pboole(k10_msafree(A,B),u1_msualg_1(A),k8_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),k8_msafree(A,B))),k8_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),k8_msafree(A,B))) ) ).
fof(dt_k11_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> l3_msualg_1(k11_msafree(A,B),A) ) ).
fof(dt_k12_msafree,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)) )
=> m1_subset_1(k12_msafree(A,B,C),k1_zfmisc_1(k1_funct_1(k8_msafree(A,B),C))) ) ).
fof(dt_k13_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> m1_msafree(k13_msafree(A,B),A,k11_msafree(A,B)) ) ).
fof(dt_k14_msafree,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)) )
=> ( v1_funct_1(k14_msafree(A,B,C))
& v1_funct_2(k14_msafree(A,B,C),k12_msafree(A,B,C),k1_funct_1(B,C))
& m2_relset_1(k14_msafree(A,B,C),k12_msafree(A,B,C),k1_funct_1(B,C)) ) ) ).
fof(dt_k15_msafree,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> m3_pboole(k15_msafree(A,B),u1_struct_0(A),k13_msafree(A,B),B) ) ).
fof(dt_k16_msafree,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A))
& v2_relat_1(C)
& m1_pboole(C,u1_struct_0(A))
& m3_pboole(D,u1_struct_0(A),k13_msafree(A,B),C)
& m1_subset_1(E,u1_struct_0(k5_msafree(A,B))) )
=> m1_subset_1(k16_msafree(A,B,C,D,E),k3_card_3(C)) ) ).
fof(dt_k17_msafree,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(k5_msafree(A,B))) )
=> m1_subset_1(k17_msafree(A,B,C),u1_msualg_1(A)) ) ).
fof(dt_k18_msafree,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A)
& m1_subset_1(C,u1_msualg_1(A))
& v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> m1_subset_1(k18_msafree(A,B,C,D),k3_card_3(u4_msualg_1(A,B))) ) ).
fof(d12_msafree,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))
=> k7_msafree(A,B,C) = a_3_0_msafree(A,B,C) ) ) ) ).
fof(t14_msafree,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))
=> k12_msafree(A,B,C) = a_3_1_msafree(A,B,C) ) ) ) ).
fof(t16_msafree,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)) )
=> k3_tarski(k2_relat_1(k13_msafree(A,B))) = a_2_0_msafree(A,B) ) ) ).
fof(fraenkel_a_3_0_msafree,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& v2_relat_1(C)
& m1_pboole(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_msafree(B,C,D))
<=> ? [E] :
( m1_dtconstr(E,u1_struct_0(k5_msafree(B,C)),k5_trees_3(u1_struct_0(k5_msafree(B,C))),k4_dtconstr(k5_msafree(B,C)))
& A = E
& ~ ( ! [F] :
~ ( r2_hidden(F,k1_funct_1(C,D))
& E = k1_trees_4(k4_tarski(F,D)) )
& ! [F] :
( m1_subset_1(F,u1_msualg_1(B))
=> ~ ( k4_tarski(F,u1_struct_0(B)) = k1_funct_1(E,k1_xboole_0)
& k2_msualg_1(B,F) = D ) ) ) ) ) ) ).
fof(fraenkel_a_3_1_msafree,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& v2_relat_1(C)
& m1_pboole(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_1_msafree(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k5_msafree(B,C)))
& A = k2_trees_4(u1_struct_0(k5_msafree(B,C)),E)
& r2_hidden(E,k6_dtconstr(k5_msafree(B,C)))
& k2_mcart_1(E) = D ) ) ) ).
fof(fraenkel_a_2_0_msafree,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& v2_relat_1(C)
& m1_pboole(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_0_msafree(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(k5_msafree(B,C)))
& A = k2_trees_4(u1_struct_0(k5_msafree(B,C)),D)
& r2_hidden(D,k6_dtconstr(k5_msafree(B,C))) ) ) ) ).
%------------------------------------------------------------------------------