SET007 Axioms: SET007+183.ax
%------------------------------------------------------------------------------
% File : SET007+183 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Subtrees
% 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 : trees_9 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 95 ( 3 unt; 0 def)
% Number of atoms : 589 ( 59 equ)
% Maximal formula atoms : 15 ( 6 avg)
% Number of connectives : 575 ( 81 ~; 1 |; 319 &)
% ( 28 <=>; 146 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 33 ( 31 usr; 1 prp; 0-4 aty)
% Number of functors : 55 ( 55 usr; 6 con; 0-3 aty)
% Number of variables : 207 ( 181 !; 26 ?)
% 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(cc1_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_trees_2(A) ) ) ).
fof(cc2_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v1_trees_9(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) ) ) ).
fof(rc1_trees_9,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A)
& v1_trees_9(A) ) ).
fof(rc2_trees_9,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A)
& ~ v1_trees_9(A) ) ).
fof(fc1_trees_9,axiom,
! [A] :
( v1_relat_1(k1_trees_4(A))
& v1_funct_1(k1_trees_4(A))
& v1_finset_1(k1_trees_4(A))
& v3_trees_2(k1_trees_4(A))
& v1_trees_9(k1_trees_4(A)) ) ).
fof(cc3_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_trees_2(A) )
=> ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v2_trees_9(A) ) ) ).
fof(rc3_trees_9,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A)
& v1_trees_2(A)
& v2_trees_9(A) ) ).
fof(cc4_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v3_trees_9(A) ) ) ).
fof(cc5_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v3_trees_9(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A) ) ) ).
fof(rc4_trees_9,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A)
& v3_trees_9(A)
& v4_trees_9(A) ) ).
fof(fc2_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v3_trees_9(A) )
=> ( ~ v1_xboole_0(k1_relat_1(A))
& v1_trees_1(k1_relat_1(A))
& v1_trees_2(k1_relat_1(A))
& v2_trees_9(k1_relat_1(A)) ) ) ).
fof(fc3_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A) )
=> ( ~ v1_xboole_0(k1_relat_1(A))
& v1_trees_1(k1_relat_1(A))
& v2_trees_9(k1_relat_1(A)) ) ) ).
fof(fc4_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v2_trees_9(A)
& m1_subset_1(B,A) )
=> v1_finset_1(k1_trees_2(A,B)) ) ).
fof(rc5_trees_9,axiom,
! [A] :
? [B] :
( m1_finseq_1(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_xboole_0(B)
& v1_finset_1(B)
& v1_finseq_1(B) ) ).
fof(fc5_trees_9,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A)
& m1_subset_1(B,k1_relat_1(A)) )
=> ( v1_relat_1(k5_trees_2(A,B))
& v1_funct_1(k5_trees_2(A,B))
& v1_finset_1(k5_trees_2(A,B))
& v3_trees_2(k5_trees_2(A,B))
& v3_trees_9(k5_trees_2(A,B))
& v4_trees_9(k5_trees_2(A,B)) ) ) ).
fof(cc6_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k5_trees_3(A))) )
=> ! [C] :
( m1_subset_1(C,B)
=> v1_finset_1(C) ) ) ).
fof(fc6_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ( ~ v1_xboole_0(k3_trees_9(A))
& v3_trees_3(k3_trees_9(A)) ) ) ).
fof(cc7_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m1_subset_1(B,k3_trees_9(A))
=> v1_finset_1(B) ) ) ).
fof(fc7_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ( v1_relat_1(k6_trees_9(A))
& ~ v1_xboole_0(k6_trees_9(A)) ) ) ).
fof(fc8_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_trees_3(A) )
=> ( ~ v1_xboole_0(k9_trees_9(A))
& v3_trees_3(k9_trees_9(A)) ) ) ).
fof(fc9_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ( ~ v1_xboole_0(k1_tarski(A))
& v1_finset_1(k1_tarski(A))
& v3_trees_3(k1_tarski(A)) ) ) ).
fof(rc6_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] :
( m1_subset_1(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_xboole_0(B)
& v1_finset_1(B)
& v1_finseq_1(B) ) ) ).
fof(t1_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> k5_trees_2(A,k6_finseq_1(k5_numbers)) = A ) ).
fof(t2_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_hidden(k8_finseq_1(k5_numbers,B,C),A)
=> k4_trees_1(A,k8_finseq_1(k5_numbers,B,C)) = k4_trees_1(k4_trees_1(A,B),C) ) ) ) ) ).
fof(t3_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_hidden(k8_finseq_1(k5_numbers,B,C),k1_relat_1(A))
=> k5_trees_2(A,k8_finseq_1(k5_numbers,B,C)) = k5_trees_2(k5_trees_2(A,B),C) ) ) ) ) ).
fof(d1_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ( v1_trees_9(A)
<=> k1_relat_1(A) = k2_trees_1(np__0) ) ) ).
fof(t4_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ( v1_trees_9(A)
<=> r2_hidden(k1_xboole_0,k3_trees_1(k1_relat_1(A))) ) ) ).
fof(t5_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ( k4_trees_1(A,B) = k2_trees_1(np__0)
<=> r2_hidden(B,k3_trees_1(A)) ) ) ) ).
fof(t6_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> ( v1_trees_9(k5_trees_2(A,B))
<=> r2_hidden(B,k3_trees_1(k1_relat_1(A))) ) ) ) ).
fof(d2_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( v2_trees_9(A)
<=> ! [B] :
( m1_trees_1(B,A)
=> v1_finset_1(k1_trees_2(A,B)) ) ) ) ).
fof(d3_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ( v3_trees_9(A)
<=> v1_trees_2(k1_relat_1(A)) ) ) ).
fof(d4_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ( v4_trees_9(A)
<=> v2_trees_9(k1_relat_1(A)) ) ) ).
fof(t7_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v2_trees_9(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C)),k1_trees_2(A,B))
<=> ~ r1_xreal_0(k4_card_1(k1_trees_2(A,B)),C) ) ) ) ) ).
fof(d5_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v2_trees_9(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ! [C] :
( ( v2_funct_1(C)
& m2_finseq_1(C,A) )
=> ( C = k1_trees_9(A,B)
<=> ( k3_finseq_1(C) = k4_card_1(k1_trees_2(A,B))
& k2_relat_1(C) = k1_trees_2(A,B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(C),D)
=> k1_funct_1(C,k1_nat_1(D,np__1)) = k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,D)) ) ) ) ) ) ) ) ).
fof(d6_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( r2_hidden(B,k1_relat_1(A))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( C = k2_trees_9(A,B)
<=> ? [D] :
( m1_trees_1(D,k1_relat_1(A))
& D = B
& C = k5_relat_1(k1_trees_9(k1_relat_1(A),D),A) ) ) ) ) ) ) ).
fof(t8_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A) )
=> ? [B,C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v6_trees_3(C)
& A = k4_trees_4(B,C) ) ) ).
fof(t9_trees_9,axiom,
$true ).
fof(t10_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ( A = k4_trees_1(A,B)
=> B = k1_xboole_0 ) ) ) ).
fof(t11_trees_9,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ( r2_hidden(A,k3_trees_9(B))
<=> ? [C] :
( m1_trees_1(C,k1_relat_1(B))
& A = k5_trees_2(B,C) ) ) ) ).
fof(t12_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> r2_hidden(A,k3_trees_9(A)) ) ).
fof(t13_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ( ( v1_finset_1(A)
& k3_trees_9(A) = k3_trees_9(B) )
=> A = B ) ) ) ).
fof(t14_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> r1_tarski(k3_trees_9(k5_trees_2(A,B)),k3_trees_9(A)) ) ) ).
fof(t15_trees_9,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ( r2_hidden(A,k6_trees_9(B))
<=> ? [C] :
( m1_trees_1(C,k1_relat_1(B))
& A = k4_tarski(C,k5_trees_2(B,C)) ) ) ) ).
fof(t16_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> r2_hidden(k4_tarski(k1_xboole_0,A),k6_trees_9(A)) ) ).
fof(t17_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ( k6_trees_9(A) = k6_trees_9(B)
=> A = B ) ) ) ).
fof(t18_trees_9,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ! [C] :
( r2_hidden(A,k7_trees_9(B,C))
<=> ? [D] :
( m1_trees_1(D,k1_relat_1(B))
& A = k5_trees_2(B,D)
& ~ ( r2_hidden(D,k3_trees_1(k1_relat_1(B)))
& ~ r2_hidden(k1_funct_1(B,D),C) ) ) ) ) ).
fof(t19_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( v1_xboole_0(k7_trees_9(A,B))
<=> ( v1_trees_9(A)
& ~ r2_hidden(k1_funct_1(A,k1_xboole_0),B) ) ) ) ).
fof(d10_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) )
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k7_trees_9(A,B),k3_finseq_2(k3_trees_9(A)))
& m2_relset_1(C,k7_trees_9(A,B),k3_finseq_2(k3_trees_9(A))) )
=> ( C = k8_trees_9(A,B)
<=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v3_trees_2(D) )
=> ( r2_hidden(D,k7_trees_9(A,B))
=> ! [E] :
( m2_finseq_1(E,k3_trees_9(A))
=> ( E = k1_funct_1(C,D)
=> D = k4_trees_4(k1_funct_1(D,k1_xboole_0),E) ) ) ) ) ) ) ) ).
fof(t20_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v3_trees_3(B) )
=> ( r2_hidden(A,k9_trees_9(B))
<=> ? [C] :
( m3_trees_3(C,B)
& ? [D] :
( m1_trees_1(D,k1_relat_1(C))
& A = k5_trees_2(C,D) ) ) ) ) ).
fof(t21_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_trees_3(B) )
=> ( r2_hidden(A,B)
=> r2_hidden(A,k9_trees_9(B)) ) ) ) ).
fof(t22_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_trees_3(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_trees_3(B) )
=> ( r1_tarski(A,B)
=> r1_tarski(k9_trees_9(A),k9_trees_9(B)) ) ) ) ).
fof(t23_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> k9_trees_9(k1_tarski(A)) = k3_trees_9(A) ) ).
fof(t25_trees_9,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& v3_trees_3(C) )
=> ( r2_hidden(A,k12_trees_9(C,B))
<=> ? [D] :
( m3_trees_3(D,C)
& ? [E] :
( m1_trees_1(E,k1_relat_1(D))
& A = k5_trees_2(D,E)
& ~ ( r2_hidden(E,k3_trees_1(k1_relat_1(D)))
& ~ r2_hidden(k1_funct_1(D,E),B) ) ) ) ) ) ).
fof(t26_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v3_trees_3(B) )
=> ( v1_xboole_0(k12_trees_9(B,A))
<=> ! [C] :
( m3_trees_3(C,B)
=> ( v1_trees_9(C)
& ~ r2_hidden(k1_funct_1(C,k1_xboole_0),A) ) ) ) ) ).
fof(t27_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] : k12_trees_9(k1_tarski(A),B) = k7_trees_9(A,B) ) ).
fof(d13_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_trees_3(A) )
=> ( ! [B] :
( m3_trees_3(B,A)
=> v1_finset_1(B) )
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k12_trees_9(A,B),k3_finseq_2(k9_trees_9(A)))
& m2_relset_1(C,k12_trees_9(A,B),k3_finseq_2(k9_trees_9(A))) )
=> ( C = k13_trees_9(A,B)
<=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v3_trees_2(D) )
=> ( r2_hidden(D,k12_trees_9(A,B))
=> ! [E] :
( m2_finseq_1(E,k9_trees_9(A))
=> ( E = k1_funct_1(C,D)
=> D = k4_trees_4(k1_funct_1(D,k1_xboole_0),E) ) ) ) ) ) ) ) ) ).
fof(t29_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> ( k3_finseq_1(k2_trees_9(A,B)) = k3_finseq_1(k1_trees_9(k1_relat_1(A),B))
& k4_finseq_1(k2_trees_9(A,B)) = k4_finseq_1(k1_trees_9(k1_relat_1(A),B)) ) ) ) ).
fof(t30_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v5_trees_3(A) )
=> ! [B] :
( ( v1_xboole_0(B)
& m1_trees_1(B,k13_trees_3(A)) )
=> k4_card_1(k1_trees_2(k13_trees_3(A),B)) = k3_finseq_1(A) ) ) ).
fof(t31_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) )
=> ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v6_trees_3(C) )
=> ( A = k4_trees_4(B,C)
=> ! [D] :
( ( v1_xboole_0(D)
& m1_trees_1(D,k1_relat_1(A)) )
=> k2_trees_9(A,D) = k16_trees_3(C) ) ) ) ) ).
fof(t32_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k5_trees_2(A,B)))
=> k2_trees_9(A,k8_finseq_1(k5_numbers,B,C)) = k2_trees_9(k5_trees_2(A,B),C) ) ) ) ).
fof(s1_trees_9,axiom,
( ( ! [A] :
~ ( r2_hidden(A,f2_s1_trees_9)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> A != f1_s1_trees_9(B) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(f1_s1_trees_9(B),f2_s1_trees_9)
=> ( r1_xreal_0(B,A)
| r2_hidden(f1_s1_trees_9(A),f2_s1_trees_9) ) ) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( f1_s1_trees_9(A) = f1_s1_trees_9(B)
=> A = B ) ) ) )
=> ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r2_hidden(f1_s1_trees_9(A),f2_s1_trees_9)
<=> ~ r1_xreal_0(k4_card_1(f2_s1_trees_9),A) ) ) ) ).
fof(dt_m1_trees_9,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m4_trees_3(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ! [D] :
( m1_trees_9(D,A,B,C)
=> m5_trees_3(D,A,B) ) ) ).
fof(existence_m1_trees_9,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m4_trees_3(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ? [D] : m1_trees_9(D,A,B,C) ) ).
fof(redefinition_m1_trees_9,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m4_trees_3(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ! [D] :
( m1_trees_9(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(dt_k1_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v2_trees_9(A)
& m1_subset_1(B,A) )
=> ( v2_funct_1(k1_trees_9(A,B))
& m2_finseq_1(k1_trees_9(A,B),A) ) ) ).
fof(dt_k2_trees_9,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( v1_relat_1(k2_trees_9(A,B))
& v1_funct_1(k2_trees_9(A,B))
& v1_finseq_1(k2_trees_9(A,B)) ) ) ).
fof(dt_k3_trees_9,axiom,
$true ).
fof(dt_k4_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> ( ~ v1_xboole_0(k4_trees_9(A,B))
& m1_subset_1(k4_trees_9(A,B),k1_zfmisc_1(k4_trees_3(A))) ) ) ).
fof(redefinition_k4_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> k4_trees_9(A,B) = k3_trees_9(B) ) ).
fof(dt_k5_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_finset_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> ( ~ v1_xboole_0(k5_trees_9(A,B))
& m1_subset_1(k5_trees_9(A,B),k1_zfmisc_1(k5_trees_3(A))) ) ) ).
fof(redefinition_k5_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_finset_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> k5_trees_9(A,B) = k3_trees_9(B) ) ).
fof(dt_k6_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> m1_subset_1(k6_trees_9(A),k1_zfmisc_1(k2_zfmisc_1(k1_relat_1(A),k3_trees_9(A)))) ) ).
fof(dt_k7_trees_9,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> m1_subset_1(k7_trees_9(A,B),k1_zfmisc_1(k3_trees_9(A))) ) ).
fof(dt_k8_trees_9,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) )
=> ( v1_funct_1(k8_trees_9(A,B))
& v1_funct_2(k8_trees_9(A,B),k7_trees_9(A,B),k3_finseq_2(k3_trees_9(A)))
& m2_relset_1(k8_trees_9(A,B),k7_trees_9(A,B),k3_finseq_2(k3_trees_9(A))) ) ) ).
fof(dt_k9_trees_9,axiom,
$true ).
fof(dt_k10_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k4_trees_3(A))) )
=> ( ~ v1_xboole_0(k10_trees_9(A,B))
& m1_subset_1(k10_trees_9(A,B),k1_zfmisc_1(k4_trees_3(A))) ) ) ).
fof(redefinition_k10_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k4_trees_3(A))) )
=> k10_trees_9(A,B) = k9_trees_9(B) ) ).
fof(dt_k11_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k5_trees_3(A))) )
=> ( ~ v1_xboole_0(k11_trees_9(A,B))
& m1_subset_1(k11_trees_9(A,B),k1_zfmisc_1(k5_trees_3(A))) ) ) ).
fof(redefinition_k11_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k5_trees_3(A))) )
=> k11_trees_9(A,B) = k9_trees_9(B) ) ).
fof(dt_k12_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v3_trees_3(A) )
=> m1_subset_1(k12_trees_9(A,B),k1_zfmisc_1(k9_trees_9(A))) ) ).
fof(dt_k13_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v3_trees_3(A) )
=> ( v1_funct_1(k13_trees_9(A,B))
& v1_funct_2(k13_trees_9(A,B),k12_trees_9(A,B),k3_finseq_2(k9_trees_9(A)))
& m2_relset_1(k13_trees_9(A,B),k12_trees_9(A,B),k3_finseq_2(k9_trees_9(A))) ) ) ).
fof(d7_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> k3_trees_9(A) = a_1_0_trees_9(A) ) ).
fof(d8_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> k6_trees_9(A) = a_1_1_trees_9(A) ) ).
fof(d9_trees_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] : k7_trees_9(A,B) = a_2_0_trees_9(A,B) ) ).
fof(d11_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_trees_3(A) )
=> k9_trees_9(A) = a_1_2_trees_9(A) ) ).
fof(t24_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_trees_3(A) )
=> k9_trees_9(A) = k3_tarski(a_1_3_trees_9(A)) ) ).
fof(d12_trees_9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_trees_3(A) )
=> ! [B] : k12_trees_9(A,B) = a_2_1_trees_9(A,B) ) ).
fof(t28_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v3_trees_3(B) )
=> k12_trees_9(B,A) = k3_tarski(a_2_2_trees_9(A,B)) ) ).
fof(fraenkel_a_1_0_trees_9,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ( r2_hidden(A,a_1_0_trees_9(B))
<=> ? [C] :
( m1_trees_1(C,k1_relat_1(B))
& A = k5_trees_2(B,C) ) ) ) ).
fof(fraenkel_a_1_1_trees_9,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ( r2_hidden(A,a_1_1_trees_9(B))
<=> ? [C] :
( m1_trees_1(C,k1_relat_1(B))
& A = k4_tarski(C,k5_trees_2(B,C)) ) ) ) ).
fof(fraenkel_a_2_0_trees_9,axiom,
! [A,B,C] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ( r2_hidden(A,a_2_0_trees_9(B,C))
<=> ? [D] :
( m1_trees_1(D,k1_relat_1(B))
& A = k5_trees_2(B,D)
& ~ ( r2_hidden(D,k3_trees_1(k1_relat_1(B)))
& ~ r2_hidden(k1_funct_1(B,D),C) ) ) ) ) ).
fof(fraenkel_a_1_2_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v3_trees_3(B) )
=> ( r2_hidden(A,a_1_2_trees_9(B))
<=> ? [C,D] :
( m3_trees_3(C,B)
& m1_trees_1(D,k1_relat_1(C))
& A = k5_trees_2(C,D) ) ) ) ).
fof(fraenkel_a_1_3_trees_9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v3_trees_3(B) )
=> ( r2_hidden(A,a_1_3_trees_9(B))
<=> ? [C] :
( m3_trees_3(C,B)
& A = k3_trees_9(C) ) ) ) ).
fof(fraenkel_a_2_1_trees_9,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v3_trees_3(B) )
=> ( r2_hidden(A,a_2_1_trees_9(B,C))
<=> ? [D,E] :
( m3_trees_3(D,B)
& m1_trees_1(E,k1_relat_1(D))
& A = k5_trees_2(D,E)
& ~ ( r2_hidden(E,k3_trees_1(k1_relat_1(D)))
& ~ r2_hidden(k1_funct_1(D,E),C) ) ) ) ) ).
fof(fraenkel_a_2_2_trees_9,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& v3_trees_3(C) )
=> ( r2_hidden(A,a_2_2_trees_9(B,C))
<=> ? [D] :
( m3_trees_3(D,C)
& A = k7_trees_9(D,B) ) ) ) ).
%------------------------------------------------------------------------------