SET007 Axioms: SET007+331.ax
%------------------------------------------------------------------------------
% File : SET007+331 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Context-Free Grammar - Part I
% 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 : lang1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 87 ( 9 unt; 0 def)
% Number of atoms : 369 ( 64 equ)
% Maximal formula atoms : 19 ( 4 avg)
% Number of connectives : 356 ( 74 ~; 0 |; 155 &)
% ( 15 <=>; 112 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 28 ( 26 usr; 1 prp; 0-3 aty)
% Number of functors : 52 ( 52 usr; 4 con; 0-6 aty)
% Number of variables : 194 ( 176 !; 18 ?)
% 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_lang1,axiom,
? [A] :
( l1_lang1(A)
& v1_lang1(A) ) ).
fof(rc2_lang1,axiom,
? [A] :
( l1_lang1(A)
& ~ v3_struct_0(A)
& v1_lang1(A) ) ).
fof(rc3_lang1,axiom,
? [A] :
( l2_lang1(A)
& v2_lang1(A) ) ).
fof(rc4_lang1,axiom,
? [A] :
( l2_lang1(A)
& ~ v3_struct_0(A) ) ).
fof(rc5_lang1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k3_finseq_2(A))
& v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B) ) ).
fof(fc1_lang1,axiom,
! [A] :
( ~ v3_struct_0(k10_lang1(A))
& v2_lang1(k10_lang1(A)) ) ).
fof(fc2_lang1,axiom,
! [A,B] :
( ~ v3_struct_0(k11_lang1(A,B))
& v2_lang1(k11_lang1(A,B)) ) ).
fof(fc3_lang1,axiom,
! [A,B] :
( ~ v3_struct_0(k12_lang1(A,B))
& v2_lang1(k12_lang1(A,B)) ) ).
fof(fc4_lang1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k13_lang1(A))
& v2_lang1(k13_lang1(A)) ) ) ).
fof(rc6_lang1,axiom,
? [A] :
( l2_lang1(A)
& ~ v3_struct_0(A)
& v3_lang1(A)
& v4_lang1(A) ) ).
fof(d1_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( r1_lang1(A,B,C)
<=> r2_hidden(k4_tarski(B,C),u1_lang1(A)) ) ) ) ) ).
fof(t1_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> k2_xboole_0(k5_lang1(A),k6_lang1(A)) = u1_struct_0(A) ) ).
fof(d4_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> ! [B] :
( m2_finseq_2(B,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ( r2_lang1(A,B,C)
<=> ? [D] :
( m2_finseq_2(D,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
& ? [E] :
( m2_finseq_2(E,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
& ? [F] :
( m2_finseq_2(F,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
& ? [G] :
( m1_subset_1(G,u1_struct_0(A))
& B = k1_lang1(u1_struct_0(A),k1_lang1(u1_struct_0(A),D,k3_lang1(u1_struct_0(A),G)),E)
& C = k1_lang1(u1_struct_0(A),k1_lang1(u1_struct_0(A),D,F),E)
& r1_lang1(A,G,F) ) ) ) ) ) ) ) ) ).
fof(t2_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ( r1_lang1(A,B,C)
=> r2_lang1(A,k1_lang1(u1_struct_0(A),k1_lang1(u1_struct_0(A),D,k3_lang1(u1_struct_0(A),B)),E),k1_lang1(u1_struct_0(A),k1_lang1(u1_struct_0(A),D,C),E)) ) ) ) ) ) ) ).
fof(t3_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ( r1_lang1(A,B,C)
=> r2_lang1(A,k3_lang1(u1_struct_0(A),B),C) ) ) ) ) ).
fof(t4_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ( r2_lang1(A,k3_lang1(u1_struct_0(A),B),C)
=> r1_lang1(A,B,C) ) ) ) ) ).
fof(t5_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> ! [B] :
( m2_finseq_2(B,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ( r2_lang1(A,C,D)
=> ( r2_lang1(A,k1_lang1(u1_struct_0(A),B,C),k1_lang1(u1_struct_0(A),B,D))
& r2_lang1(A,k1_lang1(u1_struct_0(A),C,B),k1_lang1(u1_struct_0(A),D,B)) ) ) ) ) ) ) ).
fof(d5_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> ! [B] :
( m2_finseq_2(B,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ( r3_lang1(A,B,C)
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& r1_xreal_0(np__1,k3_finseq_1(D))
& k1_funct_1(D,np__1) = B
& k1_funct_1(D,k3_finseq_1(D)) = C
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,E)
& ~ r1_xreal_0(k3_finseq_1(D),E)
& ! [F] :
( m2_finseq_2(F,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ! [G] :
( m2_finseq_2(G,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ~ ( k1_funct_1(D,E) = F
& k1_funct_1(D,k1_nat_1(E,np__1)) = G
& r2_lang1(A,F,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t6_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> ! [B] :
( m2_finseq_2(B,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> r3_lang1(A,B,B) ) ) ).
fof(t7_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> ! [B] :
( m2_finseq_2(B,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ( r2_lang1(A,B,C)
=> r3_lang1(A,B,C) ) ) ) ) ).
fof(t8_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> ! [B] :
( m2_finseq_2(B,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ! [C] :
( m2_finseq_2(C,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ! [D] :
( m2_finseq_2(D,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ( ( r3_lang1(A,C,B)
& r3_lang1(A,D,C) )
=> r3_lang1(A,D,B) ) ) ) ) ) ).
fof(t9_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lang1(A) )
=> ! [B] :
( m2_finseq_2(B,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ( r2_hidden(B,k7_lang1(A))
<=> ( r1_tarski(k2_relat_1(B),k5_lang1(A))
& r3_lang1(A,k3_lang1(u1_struct_0(A),u2_lang1(A)),B) ) ) ) ) ).
fof(d7_lang1,axiom,
! [A,B] :
( ( v2_lang1(B)
& l2_lang1(B) )
=> ( B = k10_lang1(A)
<=> ( u1_struct_0(B) = k1_tarski(A)
& u1_lang1(B) = k1_tarski(k4_tarski(A,k1_xboole_0)) ) ) ) ).
fof(d8_lang1,axiom,
! [A,B,C] :
( ( v2_lang1(C)
& l2_lang1(C) )
=> ( C = k11_lang1(A,B)
<=> ( u1_struct_0(C) = k2_tarski(A,B)
& u2_lang1(C) = A
& u1_lang1(C) = k1_tarski(k4_tarski(A,k9_finseq_1(B))) ) ) ) ).
fof(d9_lang1,axiom,
! [A,B,C] :
( ( v2_lang1(C)
& l2_lang1(C) )
=> ( C = k12_lang1(A,B)
<=> ( u1_struct_0(C) = k2_tarski(A,B)
& u2_lang1(C) = A
& u1_lang1(C) = k2_tarski(k4_tarski(A,k10_finseq_1(B,A)),k4_tarski(A,k1_xboole_0)) ) ) ) ).
fof(t10_lang1,axiom,
! [A] : k5_lang1(k10_lang1(A)) = k1_xboole_0 ).
fof(t11_lang1,axiom,
! [A] : k7_lang1(k10_lang1(A)) = k1_tarski(k1_xboole_0) ).
fof(t12_lang1,axiom,
! [A,B] :
( A != B
=> k5_lang1(k11_lang1(A,B)) = k1_tarski(B) ) ).
fof(t13_lang1,axiom,
! [A,B] :
( A != B
=> k7_lang1(k11_lang1(A,B)) = k1_tarski(k9_finseq_1(B)) ) ).
fof(t14_lang1,axiom,
! [A,B] :
( A != B
=> k5_lang1(k12_lang1(A,B)) = k1_tarski(B) ) ).
fof(t15_lang1,axiom,
! [A,B] :
( A != B
=> k7_lang1(k12_lang1(A,B)) = k3_finseq_2(k1_tarski(B)) ) ).
fof(t16_lang1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k5_lang1(k13_lang1(A)) = A ) ).
fof(t17_lang1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k7_lang1(k13_lang1(A)) = k3_finseq_2(A) ) ).
fof(d11_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lang1(A) )
=> ( v3_lang1(A)
<=> ( ~ v1_xboole_0(k7_lang1(A))
& r2_hidden(u2_lang1(A),k6_lang1(A))
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( r2_hidden(B,k5_lang1(A))
& ! [C] :
( m2_finseq_2(C,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ~ ( r2_hidden(C,k7_lang1(A))
& r2_hidden(B,k2_relat_1(C)) ) ) ) ) ) ) ) ).
fof(d12_lang1,axiom,
! [A] :
( l2_lang1(A)
=> ( v4_lang1(A)
<=> v1_finset_1(u1_lang1(A)) ) ) ).
fof(d13_lang1,axiom,
$true ).
fof(d14_lang1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k3_finseq_2(A),k3_finseq_2(B))
& m2_relset_1(D,k3_finseq_2(A),k3_finseq_2(B)) )
=> ( D = k16_lang1(A,B,C)
<=> ! [E] :
( m2_finseq_2(E,A,k3_finseq_2(A))
=> k8_funct_2(k3_finseq_2(A),k3_finseq_2(B),D,E) = k15_lang1(A,B,E,C) ) ) ) ) ) ) ).
fof(t18_lang1,axiom,
! [A] :
( v1_relat_1(A)
=> r1_tarski(A,k17_finseq_1(A)) ) ).
fof(d15_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lang1(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),B)
& m2_relset_1(C,u1_struct_0(A),B) )
=> k18_lang1(A,B,C) = g2_lang1(B,k8_funct_2(u1_struct_0(A),B,C,u2_lang1(A)),k7_relset_1(B,k3_finseq_2(u1_struct_0(A)),k3_finseq_2(u1_struct_0(A)),k3_finseq_2(B),k7_relset_1(B,u1_struct_0(A),u1_struct_0(A),k3_finseq_2(u1_struct_0(A)),k6_relset_1(u1_struct_0(A),B,C),u1_lang1(A)),k16_lang1(u1_struct_0(A),B,C))) ) ) ) ).
fof(dt_l1_lang1,axiom,
! [A] :
( l1_lang1(A)
=> l1_struct_0(A) ) ).
fof(existence_l1_lang1,axiom,
? [A] : l1_lang1(A) ).
fof(dt_l2_lang1,axiom,
! [A] :
( l2_lang1(A)
=> l1_lang1(A) ) ).
fof(existence_l2_lang1,axiom,
? [A] : l2_lang1(A) ).
fof(abstractness_v1_lang1,axiom,
! [A] :
( l1_lang1(A)
=> ( v1_lang1(A)
=> A = g1_lang1(u1_struct_0(A),u1_lang1(A)) ) ) ).
fof(abstractness_v2_lang1,axiom,
! [A] :
( l2_lang1(A)
=> ( v2_lang1(A)
=> A = g2_lang1(u1_struct_0(A),u2_lang1(A),u1_lang1(A)) ) ) ).
fof(dt_k1_lang1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k3_finseq_2(A))
& m1_subset_1(C,k3_finseq_2(A)) )
=> m2_finseq_2(k1_lang1(A,B,C),A,k3_finseq_2(A)) ) ).
fof(redefinition_k1_lang1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k3_finseq_2(A))
& m1_subset_1(C,k3_finseq_2(A)) )
=> k1_lang1(A,B,C) = k7_finseq_1(B,C) ) ).
fof(dt_k2_lang1,axiom,
! [A] :
( v1_xboole_0(k2_lang1(A))
& m2_finseq_2(k2_lang1(A),A,k3_finseq_2(A)) ) ).
fof(redefinition_k2_lang1,axiom,
! [A] : k2_lang1(A) = k6_finseq_1(A) ).
fof(dt_k3_lang1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m2_finseq_2(k3_lang1(A,B),A,k3_finseq_2(A)) ) ).
fof(redefinition_k3_lang1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> k3_lang1(A,B) = k5_finseq_1(B) ) ).
fof(dt_k4_lang1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m2_finseq_2(k4_lang1(A,B,C),A,k3_finseq_2(A)) ) ).
fof(redefinition_k4_lang1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k4_lang1(A,B,C) = k10_finseq_1(B,C) ) ).
fof(dt_k5_lang1,axiom,
$true ).
fof(dt_k6_lang1,axiom,
$true ).
fof(dt_k7_lang1,axiom,
$true ).
fof(dt_k8_lang1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k2_zfmisc_1(A,B)) )
=> m2_relset_1(k8_lang1(A,B,C),A,B) ) ).
fof(redefinition_k8_lang1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k2_zfmisc_1(A,B)) )
=> k8_lang1(A,B,C) = k1_tarski(C) ) ).
fof(dt_k9_lang1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> m2_relset_1(k9_lang1(A,B,C,D),A,B) ) ).
fof(commutativity_k9_lang1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> k9_lang1(A,B,C,D) = k9_lang1(A,B,D,C) ) ).
fof(redefinition_k9_lang1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> k9_lang1(A,B,C,D) = k2_tarski(C,D) ) ).
fof(dt_k10_lang1,axiom,
! [A] :
( v2_lang1(k10_lang1(A))
& l2_lang1(k10_lang1(A)) ) ).
fof(dt_k11_lang1,axiom,
! [A,B] :
( v2_lang1(k11_lang1(A,B))
& l2_lang1(k11_lang1(A,B)) ) ).
fof(dt_k12_lang1,axiom,
! [A,B] :
( v2_lang1(k12_lang1(A,B))
& l2_lang1(k12_lang1(A,B)) ) ).
fof(dt_k13_lang1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v2_lang1(k13_lang1(A))
& l2_lang1(k13_lang1(A)) ) ) ).
fof(dt_k14_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_lang1(A)
& l2_lang1(A) )
=> ( ~ v1_xboole_0(k14_lang1(A))
& m1_subset_1(k14_lang1(A),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(redefinition_k14_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_lang1(A)
& l2_lang1(A) )
=> k14_lang1(A) = k6_lang1(A) ) ).
fof(dt_k15_lang1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_finseq_1(C,A)
& v1_funct_1(D)
& v1_funct_2(D,A,B)
& m1_relset_1(D,A,B) )
=> m2_finseq_2(k15_lang1(A,B,C,D),B,k3_finseq_2(B)) ) ).
fof(redefinition_k15_lang1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_finseq_1(C,A)
& v1_funct_1(D)
& v1_funct_2(D,A,B)
& m1_relset_1(D,A,B) )
=> k15_lang1(A,B,C,D) = k5_relat_1(C,D) ) ).
fof(dt_k16_lang1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B) )
=> ( v1_funct_1(k16_lang1(A,B,C))
& v1_funct_2(k16_lang1(A,B,C),k3_finseq_2(A),k3_finseq_2(B))
& m2_relset_1(k16_lang1(A,B,C),k3_finseq_2(A),k3_finseq_2(B)) ) ) ).
fof(dt_k17_lang1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_relset_1(B,A,A) )
=> m2_relset_1(k17_lang1(A,B),A,A) ) ).
fof(redefinition_k17_lang1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_relset_1(B,A,A) )
=> k17_lang1(A,B) = k17_finseq_1(B) ) ).
fof(dt_k18_lang1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l2_lang1(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),B)
& m1_relset_1(C,u1_struct_0(A),B) )
=> ( v2_lang1(k18_lang1(A,B,C))
& l2_lang1(k18_lang1(A,B,C)) ) ) ).
fof(dt_u1_lang1,axiom,
! [A] :
( l1_lang1(A)
=> m2_relset_1(u1_lang1(A),u1_struct_0(A),k3_finseq_2(u1_struct_0(A))) ) ).
fof(dt_u2_lang1,axiom,
! [A] :
( l2_lang1(A)
=> m1_subset_1(u2_lang1(A),u1_struct_0(A)) ) ).
fof(dt_g1_lang1,axiom,
! [A,B] :
( m1_relset_1(B,A,k3_finseq_2(A))
=> ( v1_lang1(g1_lang1(A,B))
& l1_lang1(g1_lang1(A,B)) ) ) ).
fof(free_g1_lang1,axiom,
! [A,B] :
( m1_relset_1(B,A,k3_finseq_2(A))
=> ! [C,D] :
( g1_lang1(A,B) = g1_lang1(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(dt_g2_lang1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,A)
& m1_relset_1(C,A,k3_finseq_2(A)) )
=> ( v2_lang1(g2_lang1(A,B,C))
& l2_lang1(g2_lang1(A,B,C)) ) ) ).
fof(free_g2_lang1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,A)
& m1_relset_1(C,A,k3_finseq_2(A)) )
=> ! [D,E,F] :
( g2_lang1(A,B,C) = g2_lang1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(d2_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> k5_lang1(A) = a_1_0_lang1(A) ) ).
fof(d3_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lang1(A) )
=> k6_lang1(A) = a_1_1_lang1(A) ) ).
fof(d6_lang1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lang1(A) )
=> k7_lang1(A) = a_1_2_lang1(A) ) ).
fof(d10_lang1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_lang1(B)
& l2_lang1(B) )
=> ( B = k13_lang1(A)
<=> ( u1_struct_0(B) = k2_xboole_0(A,k1_tarski(A))
& u2_lang1(B) = A
& u1_lang1(B) = k2_xboole_0(a_1_3_lang1(A),k1_tarski(k4_tarski(A,k1_xboole_0))) ) ) ) ) ).
fof(fraenkel_a_1_0_lang1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_lang1(B) )
=> ( r2_hidden(A,a_1_0_lang1(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& A = C
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ~ r1_lang1(B,C,D) ) ) ) ) ).
fof(fraenkel_a_1_1_lang1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_lang1(B) )
=> ( r2_hidden(A,a_1_1_lang1(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& A = C
& ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& r1_lang1(B,C,D) ) ) ) ) ).
fof(fraenkel_a_1_2_lang1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_lang1(B) )
=> ( r2_hidden(A,a_1_2_lang1(B))
<=> ? [C] :
( m2_finseq_2(C,u1_struct_0(B),k3_finseq_2(u1_struct_0(B)))
& A = C
& r1_tarski(k2_relat_1(C),k5_lang1(B))
& r3_lang1(B,k3_lang1(u1_struct_0(B),u2_lang1(B)),C) ) ) ) ).
fof(fraenkel_a_1_3_lang1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_3_lang1(B))
<=> ? [C] :
( m1_subset_1(C,B)
& A = k4_tarski(B,k10_finseq_1(C,B))
& C = C ) ) ) ).
%------------------------------------------------------------------------------