SET007 Axioms: SET007+421.ax
%------------------------------------------------------------------------------
% File : SET007+421 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A Scheme for Extensions of Homomorphisms of Many Sorted Algebras
% 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 : msafree1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 29 ( 1 unt; 0 def)
% Number of atoms : 223 ( 24 equ)
% Maximal formula atoms : 19 ( 7 avg)
% Number of connectives : 229 ( 35 ~; 0 |; 99 &)
% ( 6 <=>; 89 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 32 ( 31 usr; 0 prp; 1-4 aty)
% Number of functors : 56 ( 56 usr; 14 con; 0-6 aty)
% Number of variables : 96 ( 93 !; 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(fc1_msafree1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A) )
=> ( ~ v1_xboole_0(k2_relat_1(B))
& v1_setfam_1(k2_relat_1(B)) ) ) ).
fof(fc2_msafree1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_setfam_1(A) )
=> ~ v1_xboole_0(k3_tarski(A)) ) ).
fof(cc1_msafree1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v3_relat_1(B)
=> v1_prob_2(B) ) ) ).
fof(rc1_msafree1,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_prob_2(B) ) ).
fof(rc2_msafree1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ? [B] :
( l3_msualg_1(B,A)
& v5_msualg_1(B,A)
& v1_msafree1(B,A) ) ) ).
fof(fc3_msafree1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ( v4_msualg_1(k2_msafree1(A),A)
& v5_msualg_1(k2_msafree1(A),A)
& v1_msafree1(k2_msafree1(A),A) ) ) ).
fof(fc4_msafree1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& v1_msafree1(B,A)
& l3_msualg_1(B,A) )
=> ( v1_relat_1(u4_msualg_1(A,B))
& v1_funct_1(u4_msualg_1(A,B))
& v1_prob_2(u4_msualg_1(A,B)) ) ) ).
fof(fc5_msafree1,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))
& v1_prob_2(k8_msafree(A,B)) ) ) ).
fof(fc6_msafree1,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(fc7_msafree1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& m1_subset_1(B,u1_msualg_1(A))
& v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ( ~ v1_xboole_0(k3_msualg_1(A,B,C))
& v1_fraenkel(k3_msualg_1(A,B,C)) ) ) ).
fof(fc8_msafree1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& m1_subset_1(B,u1_msualg_1(A))
& v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ~ v1_xboole_0(k4_msualg_1(A,B,C)) ) ).
fof(fc9_msafree1,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(u4_msualg_1(A,k11_msafree(A,B)))
& v2_relat_1(u4_msualg_1(A,k11_msafree(A,B)))
& ~ v3_relat_1(u4_msualg_1(A,k11_msafree(A,B)))
& v1_funct_1(u4_msualg_1(A,k11_msafree(A,B)))
& v1_prob_2(u4_msualg_1(A,k11_msafree(A,B))) ) ) ).
fof(fc10_msafree1,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)
& v1_msafree1(k11_msafree(A,B),A) ) ) ).
fof(t1_msafree1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(B,k4_card_3(A))
=> r1_tarski(k2_relat_1(B),k3_card_3(A)) ) ) ) ).
fof(t2_msafree1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),k4_msafree(A,B))
=> ( r2_hidden(C,k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))))
& r2_hidden(D,k13_finseq_1(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(t3_msafree1,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] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ( 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(d1_msafree1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_prob_2(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k3_card_3(B),k3_card_3(C))
& m2_relset_1(E,k3_card_3(B),k3_card_3(C)) )
=> ( E = k1_msafree1(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( r2_hidden(G,k1_funct_1(B,F))
=> k1_funct_1(E,G) = k1_funct_1(k1_msualg_3(A,B,C,D,F),G) ) ) ) ) ) ) ) ) ).
fof(t4_msafree1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_prob_2(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( m3_pboole(E,A,B,C)
=> ( k1_msafree1(A,B,C,D) = k1_msafree1(A,B,C,E)
=> r6_pboole(A,D,E) ) ) ) ) ) ) ).
fof(d2_msafree1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ( v1_msafree1(B,A)
<=> v1_prob_2(u4_msualg_1(A,B)) ) ) ) ).
fof(d3_msafree1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v4_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( B = k2_msafree1(A)
<=> ! [C] :
( r2_hidden(C,u1_struct_0(A))
=> k1_funct_1(u4_msualg_1(A,B),C) = k1_tarski(C) ) ) ) ) ).
fof(t5_msafree1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_msualg_1(A))
=> ! [C] :
( ( v5_msualg_1(C,A)
& v1_msafree1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( ( v5_msualg_1(D,A)
& l3_msualg_1(D,A) )
=> ! [E] :
( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
=> ! [F] :
( m1_subset_1(F,k3_msualg_1(A,B,C))
=> k5_relat_1(F,k1_msafree1(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D),E)) = k6_msualg_3(A,C,D,B,E,F) ) ) ) ) ) ) ).
fof(s1_msafree1,axiom,
( ( ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s1_msafree1))
=> ( r2_hidden(A,k5_lang1(f1_s1_msafree1))
=> k1_funct_1(f5_s1_msafree1,k2_trees_4(u1_struct_0(f1_s1_msafree1),A)) = f3_s1_msafree1(A) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s1_msafree1))
=> ! [B] :
( m1_trees_4(B,k5_trees_3(u1_struct_0(f1_s1_msafree1)),k4_dtconstr(f1_s1_msafree1))
=> ( r1_lang1(f1_s1_msafree1,A,k1_dtconstr(u1_struct_0(f1_s1_msafree1),k5_trees_3(u1_struct_0(f1_s1_msafree1)),B))
=> ! [C] :
( m2_finseq_1(C,f2_s1_msafree1)
=> ( C = k5_relat_1(B,f5_s1_msafree1)
=> k1_funct_1(f5_s1_msafree1,k12_trees_4(u1_struct_0(f1_s1_msafree1),A,B)) = f4_s1_msafree1(A,B,C) ) ) ) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s1_msafree1))
=> ( r2_hidden(A,k5_lang1(f1_s1_msafree1))
=> k1_funct_1(f6_s1_msafree1,k2_trees_4(u1_struct_0(f1_s1_msafree1),A)) = f3_s1_msafree1(A) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s1_msafree1))
=> ! [B] :
( m1_trees_4(B,k5_trees_3(u1_struct_0(f1_s1_msafree1)),k4_dtconstr(f1_s1_msafree1))
=> ( r1_lang1(f1_s1_msafree1,A,k1_dtconstr(u1_struct_0(f1_s1_msafree1),k5_trees_3(u1_struct_0(f1_s1_msafree1)),B))
=> ! [C] :
( m2_finseq_1(C,f2_s1_msafree1)
=> ( C = k5_relat_1(B,f6_s1_msafree1)
=> k1_funct_1(f6_s1_msafree1,k12_trees_4(u1_struct_0(f1_s1_msafree1),A,B)) = f4_s1_msafree1(A,B,C) ) ) ) ) ) )
=> f5_s1_msafree1 = f6_s1_msafree1 ) ).
fof(s2_msafree1,axiom,
( ( ! [A] :
( m1_subset_1(A,u1_msualg_1(f1_s2_msafree1))
=> ! [B] :
( m1_subset_1(B,k3_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)))
=> ! [C] :
( m1_msafree1(C,k3_card_3(f3_s2_msafree1))
=> ( C = k5_relat_1(B,k1_msafree1(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f6_s2_msafree1))
=> k1_funct_1(k1_msualg_3(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f6_s2_msafree1,k2_msualg_1(f1_s2_msafree1,A)),k8_funct_2(k3_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),k4_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),k5_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),B)) = f5_s2_msafree1(A,B,C) ) ) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s2_msafree1))
=> ! [B] :
( r2_hidden(B,k12_msafree(f1_s2_msafree1,f2_s2_msafree1,A))
=> k1_funct_1(k1_msualg_3(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f6_s2_msafree1,A),B) = f4_s2_msafree1(B) ) )
& ! [A] :
( m1_subset_1(A,u1_msualg_1(f1_s2_msafree1))
=> ! [B] :
( m1_subset_1(B,k3_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)))
=> ! [C] :
( m1_msafree1(C,k3_card_3(f3_s2_msafree1))
=> ( C = k5_relat_1(B,k1_msafree1(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f7_s2_msafree1))
=> k1_funct_1(k1_msualg_3(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f7_s2_msafree1,k2_msualg_1(f1_s2_msafree1,A)),k8_funct_2(k3_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),k4_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),k5_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),B)) = f5_s2_msafree1(A,B,C) ) ) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s2_msafree1))
=> ! [B] :
( r2_hidden(B,k12_msafree(f1_s2_msafree1,f2_s2_msafree1,A))
=> k1_funct_1(k1_msualg_3(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f7_s2_msafree1,A),B) = f4_s2_msafree1(B) ) ) )
=> r6_pboole(u1_struct_0(f1_s2_msafree1),f6_s2_msafree1,f7_s2_msafree1) ) ).
fof(s3_msafree1,axiom,
( ( r1_msualg_3(f1_s3_msafree1,k11_msafree(f1_s3_msafree1,f2_s3_msafree1),f3_s3_msafree1,f4_s3_msafree1)
& ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s3_msafree1))
=> ! [B,C] :
( r2_hidden(C,k12_msafree(f1_s3_msafree1,f2_s3_msafree1,A))
=> ( k1_funct_1(k1_msualg_3(u1_struct_0(f1_s3_msafree1),u4_msualg_1(f1_s3_msafree1,k11_msafree(f1_s3_msafree1,f2_s3_msafree1)),u4_msualg_1(f1_s3_msafree1,f3_s3_msafree1),f4_s3_msafree1,A),C) = B
<=> p1_s3_msafree1(A,B,C) ) ) )
& r1_msualg_3(f1_s3_msafree1,k11_msafree(f1_s3_msafree1,f2_s3_msafree1),f3_s3_msafree1,f5_s3_msafree1)
& ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s3_msafree1))
=> ! [B,C] :
( r2_hidden(C,k12_msafree(f1_s3_msafree1,f2_s3_msafree1,A))
=> ( k1_funct_1(k1_msualg_3(u1_struct_0(f1_s3_msafree1),u4_msualg_1(f1_s3_msafree1,k11_msafree(f1_s3_msafree1,f2_s3_msafree1)),u4_msualg_1(f1_s3_msafree1,f3_s3_msafree1),f5_s3_msafree1,A),C) = B
<=> p1_s3_msafree1(A,B,C) ) ) ) )
=> r6_pboole(u1_struct_0(f1_s3_msafree1),f4_s3_msafree1,f5_s3_msafree1) ) ).
fof(dt_m1_msafree1,axiom,
! [A,B] :
( m1_msafree1(B,A)
=> m1_subset_1(B,k13_finseq_1(A)) ) ).
fof(existence_m1_msafree1,axiom,
! [A] :
? [B] : m1_msafree1(B,A) ).
fof(redefinition_m1_msafree1,axiom,
! [A,B] :
( m1_msafree1(B,A)
<=> m1_finseq_1(B,A) ) ).
fof(dt_k1_msafree1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_prob_2(B)
& m1_pboole(B,A)
& v2_relat_1(C)
& m1_pboole(C,A)
& m3_pboole(D,A,B,C) )
=> ( v1_funct_1(k1_msafree1(A,B,C,D))
& v1_funct_2(k1_msafree1(A,B,C,D),k3_card_3(B),k3_card_3(C))
& m2_relset_1(k1_msafree1(A,B,C,D),k3_card_3(B),k3_card_3(C)) ) ) ).
fof(dt_k2_msafree1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ( v4_msualg_1(k2_msafree1(A),A)
& l3_msualg_1(k2_msafree1(A),A) ) ) ).
%------------------------------------------------------------------------------