SET007 Axioms: SET007+152.ax
%------------------------------------------------------------------------------
% File : SET007+152 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Koenig's Lemma
% 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_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 104 ( 6 unt; 0 def)
% Number of atoms : 665 ( 60 equ)
% Maximal formula atoms : 21 ( 6 avg)
% Number of connectives : 675 ( 114 ~; 2 |; 317 &)
% ( 21 <=>; 221 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-3 aty)
% Number of functors : 70 ( 70 usr; 24 con; 0-3 aty)
% Number of variables : 269 ( 233 !; 36 ?)
% 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_trees_2,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_trees_2(A) ) ).
fof(fc1_trees_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_trees_2(A)
& m1_subset_1(B,A) )
=> v1_finset_1(k1_trees_2(A,B)) ) ).
fof(rc2_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] :
( m1_trees_2(B,A)
& ~ v1_xboole_0(B) ) ) ).
fof(rc3_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] :
( m1_trees_2(B,A)
& v2_trees_2(B,A) ) ) ).
fof(cc1_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_2(B,A)
=> ( v2_trees_2(B,A)
=> ~ v1_xboole_0(B) ) ) ) ).
fof(rc4_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] :
( m1_trees_2(B,A)
& v1_finset_1(B) ) ) ).
fof(rc5_trees_2,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) ) ).
fof(fc2_trees_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ( ~ v1_xboole_0(k1_relat_1(A))
& v1_trees_1(k1_relat_1(A)) ) ) ).
fof(rc6_trees_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m3_trees_2(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) ) ) ).
fof(fc3_trees_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( v1_relat_1(k2_funcop_1(A,B))
& v1_funct_1(k2_funcop_1(A,B))
& v3_trees_2(k2_funcop_1(A,B)) ) ) ).
fof(fc4_trees_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A)
& m1_subset_1(B,A) )
=> v1_finset_1(k1_trees_2(A,B)) ) ).
fof(rc7_trees_2,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) ) ).
fof(rc8_trees_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m3_trees_2(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v3_trees_2(B) ) ) ).
fof(fc5_trees_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_finset_1(A) )
=> v1_finset_1(k1_relat_1(A)) ) ).
fof(rc9_trees_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& ~ v1_xboole_0(C) ) ) ).
fof(t1_trees_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( r1_tarski(A,C)
& r1_tarski(B,C) )
=> r3_xboole_0(A,B) ) ) ) ) ).
fof(t2_trees_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( r2_xboole_0(A,C)
& r1_tarski(B,C) )
=> r3_xboole_0(A,B) ) ) ) ) ).
fof(t3_trees_2,axiom,
$true ).
fof(t4_trees_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ~ ( k3_finseq_1(B) = k1_nat_1(A,np__1)
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
~ ( B = k7_finseq_1(C,k9_finseq_1(D))
& k3_finseq_1(C) = A ) ) ) ) ) ).
fof(t5_trees_2,axiom,
$true ).
fof(t6_trees_2,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> k1_trees_1(k7_finseq_1(B,k9_finseq_1(A))) = k2_xboole_0(k1_trees_1(B),k1_tarski(B)) ) ).
fof(t7_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ( ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_hidden(C,A)
<=> r2_hidden(C,B) ) )
=> A = B ) ) ) ).
fof(d1_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ( A = B
<=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_hidden(C,A)
<=> r2_hidden(C,B) ) ) ) ) ) ).
fof(t8_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ( r2_hidden(B,A)
=> A = k5_trees_1(A,B,k4_trees_1(A,B)) ) ) ) ).
fof(t9_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( ( r2_hidden(C,A)
& r2_hidden(D,A) )
=> ( r1_tarski(C,D)
| r2_hidden(D,k5_trees_1(A,C,B)) ) ) ) ) ) ) ).
fof(t10_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_trees_1(C) )
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ! [E] :
( m2_finseq_1(E,k5_numbers)
=> ( ( r2_hidden(D,A)
& r2_hidden(E,A) )
=> ( r3_xboole_0(D,E)
| k5_trees_1(k5_trees_1(A,D,B),E,C) = k5_trees_1(k5_trees_1(A,E,C),D,B) ) ) ) ) ) ) ) ).
fof(d2_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( v1_trees_2(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ! [C] :
( m1_trees_1(C,A)
=> ~ r2_hidden(k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,B)),A) ) ) ) ) ).
fof(d3_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( m1_trees_2(B,A)
<=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r3_xboole_0(C,D) ) ) ) ) ) ) ).
fof(t11_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_trees_2(B,A)
=> m4_trees_1(B,A) ) ) ).
fof(t12_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> m4_trees_1(k1_trees_2(A,B),A) ) ) ).
fof(t13_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m4_trees_1(B,A)
=> ! [C] :
( m1_trees_2(C,A)
=> ? [D] :
( m1_trees_1(D,A)
& r1_tarski(k3_xboole_0(B,C),k1_tarski(D)) ) ) ) ) ).
fof(t14_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(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))
<=> r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C)),A) ) ) ) ) ).
fof(t15_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ( B = k1_xboole_0
=> k2_trees_2(A,np__1) = k1_trees_2(A,B) ) ) ) ).
fof(t18_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_trees_2(B,A)
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& B = k2_trees_2(A,C) ) ) ) ).
fof(t19_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ~ ( v1_trees_2(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( m1_trees_1(C,A)
& ! [D] :
( v1_finset_1(D)
=> ~ ( D = k1_trees_2(A,C)
& r1_xreal_0(k4_card_1(D),B) ) ) ) ) ) ) ).
fof(t20_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ( v1_trees_2(A)
=> v1_finset_1(k1_trees_2(A,B)) ) ) ) ).
fof(t21_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> m1_trees_2(k1_xboole_0,A) ) ).
fof(t22_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> m1_trees_2(k1_tarski(k1_xboole_0),A) ) ).
fof(d7_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_2(B,A)
=> ( v2_trees_2(B,A)
<=> ( ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_hidden(C,B)
=> r1_tarski(k1_trees_1(C),B) ) )
& ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ~ ( r2_hidden(C,A)
& ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( r2_hidden(D,B)
=> r2_xboole_0(D,C) ) ) ) ) ) ) ) ) ).
fof(t23_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( m1_trees_2(D,A)
=> ~ ( r2_hidden(B,D)
& r2_hidden(C,D)
& ~ r2_hidden(B,k1_trees_1(C))
& ~ r1_tarski(C,B) ) ) ) ) ) ).
fof(t24_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ! [E] :
( m1_trees_2(E,A)
=> ~ ( r2_hidden(B,E)
& r2_hidden(C,E)
& r1_tarski(D,C)
& ~ r2_hidden(B,k1_trees_1(D))
& ~ r1_tarski(D,B) ) ) ) ) ) ) ).
fof(t25_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_finset_1(C)
& m1_trees_2(C,A) )
=> ~ ( ~ r1_xreal_0(k4_card_1(C),B)
& ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ~ ( r2_hidden(D,C)
& r1_xreal_0(B,k3_finseq_1(D)) ) ) ) ) ) ) ).
fof(t27_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ! [D] :
( ( v2_trees_2(D,A)
& m1_trees_2(D,A) )
=> ( ( r1_tarski(B,C)
& r2_hidden(C,D) )
=> r2_hidden(B,D) ) ) ) ) ) ).
fof(t28_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( v2_trees_2(B,A)
& m1_trees_2(B,A) )
=> r2_hidden(k1_xboole_0,B) ) ) ).
fof(t29_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ! [D] :
( m1_trees_2(D,A)
=> ( ( r2_hidden(B,D)
& r2_hidden(C,D)
& r1_xreal_0(k3_finseq_1(B),k3_finseq_1(C)) )
=> r1_tarski(B,C) ) ) ) ) ) ).
fof(t30_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_2(B,A)
=> ? [C] :
( v2_trees_2(C,A)
& m1_trees_2(C,A)
& r1_tarski(B,C) ) ) ) ).
fof(t31_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ~ ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( v1_finset_1(C)
& m1_trees_2(C,A)
& k4_card_1(C) = B ) )
& ! [B] :
( m1_trees_1(B,A)
=> v1_finset_1(k1_trees_2(A,B)) )
& ! [B] :
( m1_trees_2(B,A)
=> v1_finset_1(B) ) ) ) ).
fof(t32_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_trees_2(A) )
=> ~ ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( v1_finset_1(C)
& m1_trees_2(C,A)
& k4_card_1(C) = B ) )
& ! [B] :
( m1_trees_2(B,A)
=> v1_finset_1(B) ) ) ) ).
fof(d8_trees_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v3_trees_2(A)
<=> ( ~ v1_xboole_0(k1_relat_1(A))
& v1_trees_1(k1_relat_1(A)) ) ) ) ).
fof(d9_trees_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( m3_trees_2(B,A)
<=> r1_tarski(k2_relat_1(B),A) ) ) ).
fof(t33_trees_2,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) )
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(A,C) = k1_funct_1(B,C) ) ) )
=> A = B ) ) ) ).
fof(d10_trees_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> k4_trees_2(A) = k9_relat_1(A,k3_trees_1(k1_relat_1(A))) ) ).
fof(d11_trees_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( C = k5_trees_2(A,B)
<=> ( k1_relat_1(C) = k4_trees_1(k1_relat_1(A),B)
& ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( r2_hidden(D,k4_trees_1(k1_relat_1(A),B))
=> k1_funct_1(C,D) = k1_funct_1(A,k8_finseq_1(k5_numbers,B,D)) ) ) ) ) ) ) ) ).
fof(t34_trees_2,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ( r2_hidden(A,k1_relat_1(B))
=> r1_tarski(k2_relat_1(k5_trees_2(B,A)),k2_relat_1(B)) ) ) ) ).
fof(d12_trees_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( r2_hidden(B,k1_relat_1(A))
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v3_trees_2(D) )
=> ( D = k8_trees_2(A,B,C)
<=> ( k1_relat_1(D) = k5_trees_1(k1_relat_1(A),B,k1_relat_1(C))
& ! [E] :
( m2_finseq_1(E,k5_numbers)
=> ~ ( r2_hidden(E,k5_trees_1(k1_relat_1(A),B,k1_relat_1(C)))
& ~ ( ~ r1_tarski(B,E)
& k1_funct_1(D,E) = k1_funct_1(A,E) )
& ! [F] :
( m2_finseq_1(F,k5_numbers)
=> ~ ( r2_hidden(F,k1_relat_1(C))
& E = k8_finseq_1(k5_numbers,B,F)
& k1_funct_1(D,E) = k1_funct_1(C,F) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t35_trees_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ! [B] :
( r2_hidden(B,A)
=> ( ~ v1_xboole_0(B)
& v1_trees_1(B) ) )
=> ( ~ v1_xboole_0(k3_tarski(A))
& v1_trees_1(k3_tarski(A)) ) ) ) ).
fof(t36_trees_2,axiom,
! [A] :
( ( ! [B] :
( r2_hidden(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B) ) )
& v6_ordinal1(A) )
=> ( v1_relat_1(k3_tarski(A))
& v1_funct_1(k3_tarski(A)) ) ) ).
fof(t37_trees_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ( ! [B] :
( r2_hidden(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) ) )
& v6_ordinal1(A) )
=> ( v1_relat_1(k3_tarski(A))
& v1_funct_1(k3_tarski(A))
& v3_trees_2(k3_tarski(A)) ) ) ) ).
fof(t38_trees_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( ( ! [C] :
( r2_hidden(C,A)
=> ( v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,B) ) )
& v6_ordinal1(A) )
=> ( v1_funct_1(k3_tarski(A))
& v3_trees_2(k3_tarski(A))
& m3_trees_2(k3_tarski(A),B) ) ) ) ) ).
fof(d13_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> k10_trees_2(A,B) = k4_card_1(k1_trees_2(A,B)) ) ) ).
fof(s1_trees_2,axiom,
( ( p1_s1_trees_2(k1_xboole_0)
& ! [A] :
( m1_trees_1(A,f1_s1_trees_2)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( p1_s1_trees_2(A)
& r2_hidden(k8_finseq_1(k5_numbers,A,k12_finseq_1(k5_numbers,B)),f1_s1_trees_2) )
=> p1_s1_trees_2(k8_finseq_1(k5_numbers,A,k12_finseq_1(k5_numbers,B))) ) ) ) )
=> ! [A] :
( m1_trees_1(A,f1_s1_trees_2)
=> p1_s1_trees_2(A) ) ) ).
fof(s4_trees_2,axiom,
( ! [A] :
~ ( r2_hidden(A,f1_s4_trees_2)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ p1_s4_trees_2(A,B) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s4_trees_2
& ! [B] :
~ ( r2_hidden(B,f1_s4_trees_2)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( k1_funct_1(A,B) = C
& p1_s4_trees_2(B,C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( p1_s4_trees_2(B,D)
=> r1_xreal_0(C,D) ) ) ) ) ) ) ) ).
fof(s5_trees_2,axiom,
( ( r2_hidden(f2_s5_trees_2,f1_s5_trees_2)
& p1_s5_trees_2(f2_s5_trees_2)
& ! [A] :
~ ( r2_hidden(A,f1_s5_trees_2)
& p1_s5_trees_2(A)
& ! [B] :
~ ( r2_hidden(B,f1_s5_trees_2)
& p2_s5_trees_2(A,B)
& p1_s5_trees_2(B) ) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = k5_numbers
& r1_tarski(k2_relat_1(A),f1_s5_trees_2)
& k1_funct_1(A,np__0) = f2_s5_trees_2
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( p2_s5_trees_2(k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1)))
& p1_s5_trees_2(k1_funct_1(A,B)) ) ) ) ) ).
fof(s6_trees_2,axiom,
( ! [A] :
( m2_finseq_1(A,k5_numbers)
=> ~ ( r2_hidden(A,f1_s6_trees_2)
& ! [B] : ~ p1_s6_trees_2(A,B) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& k1_relat_1(A) = f1_s6_trees_2
& ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ( r2_hidden(B,f1_s6_trees_2)
=> p1_s6_trees_2(B,k1_funct_1(A,B)) ) ) ) ) ).
fof(s7_trees_2,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& k1_relat_1(A) = f1_s7_trees_2
& ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ( r2_hidden(B,f1_s7_trees_2)
=> k1_funct_1(A,B) = f2_s7_trees_2(B) ) ) ) ).
fof(dt_m1_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_2(B,A)
=> m1_subset_1(B,k1_zfmisc_1(A)) ) ) ).
fof(existence_m1_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] : m1_trees_2(B,A) ) ).
fof(dt_m2_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_trees_2(B,A)
=> m1_subset_1(B,k1_zfmisc_1(A)) ) ) ).
fof(existence_m2_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] : m2_trees_2(B,A) ) ).
fof(dt_m3_trees_2,axiom,
! [A,B] :
( m3_trees_2(B,A)
=> v1_relat_1(B) ) ).
fof(existence_m3_trees_2,axiom,
! [A] :
? [B] : m3_trees_2(B,A) ).
fof(dt_k1_trees_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k1_trees_2(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_k2_trees_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& m1_subset_1(B,k5_numbers) )
=> m2_trees_2(k2_trees_2(A,B),A) ) ).
fof(dt_k3_trees_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A)
& m1_subset_1(C,k1_relat_1(B)) )
=> m1_subset_1(k3_trees_2(A,B,C),A) ) ).
fof(redefinition_k3_trees_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A)
& m1_subset_1(C,k1_relat_1(B)) )
=> k3_trees_2(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k4_trees_2,axiom,
$true ).
fof(dt_k5_trees_2,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& m1_finseq_1(B,k5_numbers) )
=> ( v1_relat_1(k5_trees_2(A,B))
& v1_funct_1(k5_trees_2(A,B))
& v3_trees_2(k5_trees_2(A,B)) ) ) ).
fof(dt_k6_trees_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> m1_subset_1(k6_trees_2(A,B),k1_zfmisc_1(A)) ) ).
fof(redefinition_k6_trees_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> k6_trees_2(A,B) = k4_trees_2(B) ) ).
fof(dt_k7_trees_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A)
& m1_subset_1(C,k1_relat_1(B)) )
=> ( v1_funct_1(k7_trees_2(A,B,C))
& v3_trees_2(k7_trees_2(A,B,C))
& m3_trees_2(k7_trees_2(A,B,C),A) ) ) ).
fof(redefinition_k7_trees_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A)
& m1_subset_1(C,k1_relat_1(B)) )
=> k7_trees_2(A,B,C) = k5_trees_2(B,C) ) ).
fof(dt_k8_trees_2,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& m1_finseq_1(B,k5_numbers)
& v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( v1_relat_1(k8_trees_2(A,B,C))
& v1_funct_1(k8_trees_2(A,B,C))
& v3_trees_2(k8_trees_2(A,B,C)) ) ) ).
fof(dt_k9_trees_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_trees_1(B)
& m1_subset_1(C,A) )
=> ( v1_funct_1(k9_trees_2(A,B,C))
& v3_trees_2(k9_trees_2(A,B,C))
& m3_trees_2(k9_trees_2(A,B,C),A) ) ) ).
fof(redefinition_k9_trees_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_trees_1(B)
& m1_subset_1(C,A) )
=> k9_trees_2(A,B,C) = k2_funcop_1(B,C) ) ).
fof(dt_k10_trees_2,axiom,
$true ).
fof(d4_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( m2_trees_2(B,A)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& B = a_2_0_trees_2(A,C) ) ) ) ) ).
fof(d5_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> k1_trees_2(A,B) = a_2_1_trees_2(A,B) ) ) ).
fof(d6_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_trees_2(A,B) = a_2_0_trees_2(A,B) ) ) ).
fof(t16_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> A = k3_tarski(a_1_0_trees_2(A)) ) ).
fof(t17_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> A = k3_tarski(a_1_1_trees_2(A)) ) ).
fof(t26_trees_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_2(B,A)
=> m1_trees_2(a_2_2_trees_2(A,B),A) ) ) ).
fof(s2_trees_2,axiom,
r1_tarski(k1_card_1(a_0_0_trees_2),k1_card_1(f2_s2_trees_2)) ).
fof(s3_trees_2,axiom,
( f3_s3_trees_2 = a_0_1_trees_2
=> r1_xreal_0(k4_card_1(f3_s3_trees_2),k4_card_1(f2_s3_trees_2)) ) ).
fof(s8_trees_2,axiom,
( ! [A] :
( m1_subset_1(A,f1_s8_trees_2)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(B,C)
& r2_hidden(C,f3_s8_trees_2(A)) )
=> r2_hidden(B,f3_s8_trees_2(A)) ) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v3_trees_2(A)
& m3_trees_2(A,f1_s8_trees_2)
& k1_funct_1(A,k1_xboole_0) = f2_s8_trees_2
& ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> ( k1_trees_2(k1_relat_1(A),B) = a_2_3_trees_2(A,B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,f3_s8_trees_2(k3_trees_2(f1_s8_trees_2,A,B)))
=> k1_funct_1(A,k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C))) = k1_funct_1(f4_s8_trees_2,k4_tarski(k3_trees_2(f1_s8_trees_2,A,B),C)) ) ) ) ) ) ) ).
fof(s9_trees_2,axiom,
? [A] :
( v1_funct_1(A)
& v3_trees_2(A)
& m3_trees_2(A,f1_s9_trees_2)
& k1_funct_1(A,k1_xboole_0) = f2_s9_trees_2
& ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> ( k1_trees_2(k1_relat_1(A),B) = a_2_4_trees_2(A,B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(f3_s9_trees_2(k3_trees_2(f1_s9_trees_2,A,B)),C)
=> k1_funct_1(A,k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C))) = k1_funct_1(f4_s9_trees_2,k4_tarski(k3_trees_2(f1_s9_trees_2,A,B),C)) ) ) ) ) ) ).
fof(fraenkel_a_2_0_trees_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_0_trees_2(B,C))
<=> ? [D] :
( m1_trees_1(D,B)
& A = D
& k3_finseq_1(D) = C ) ) ) ).
fof(fraenkel_a_2_1_trees_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B)
& m1_trees_1(C,B) )
=> ( r2_hidden(A,a_2_1_trees_2(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D))
& r2_hidden(k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D)),B) ) ) ) ).
fof(fraenkel_a_1_0_trees_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ( r2_hidden(A,a_1_0_trees_2(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k2_trees_2(B,C) ) ) ) ).
fof(fraenkel_a_1_1_trees_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_trees_1(B) )
=> ( r2_hidden(A,a_1_1_trees_2(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k2_trees_2(B,C)
& r1_xreal_0(C,k6_trees_1(B)) ) ) ) ).
fof(fraenkel_a_2_2_trees_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B)
& m1_trees_2(C,B) )
=> ( r2_hidden(A,a_2_2_trees_2(B,C))
<=> ? [D] :
( m1_trees_1(D,B)
& A = D
& ? [E] :
( m2_finseq_1(E,k5_numbers)
& r2_hidden(E,C)
& r1_tarski(D,E) ) ) ) ) ).
fof(fraenkel_a_0_0_trees_2,axiom,
! [A] :
( r2_hidden(A,a_0_0_trees_2)
<=> ? [B] :
( m1_subset_1(B,f1_s2_trees_2)
& A = f3_s2_trees_2(B)
& r2_hidden(B,f2_s2_trees_2) ) ) ).
fof(fraenkel_a_0_1_trees_2,axiom,
! [A] :
( r2_hidden(A,a_0_1_trees_2)
<=> ? [B] :
( m1_subset_1(B,f1_s3_trees_2)
& A = f4_s3_trees_2(B)
& r2_hidden(B,f2_s3_trees_2) ) ) ).
fof(fraenkel_a_2_3_trees_2,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,f1_s8_trees_2)
& m1_trees_1(C,k1_relat_1(B)) )
=> ( r2_hidden(A,a_2_3_trees_2(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D))
& r2_hidden(D,f3_s8_trees_2(k3_trees_2(f1_s8_trees_2,B,C))) ) ) ) ).
fof(fraenkel_a_2_4_trees_2,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,f1_s9_trees_2)
& m1_trees_1(C,k1_relat_1(B)) )
=> ( r2_hidden(A,a_2_4_trees_2(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D))
& ~ r1_xreal_0(f3_s9_trees_2(k3_trees_2(f1_s9_trees_2,B,C)),D) ) ) ) ).
%------------------------------------------------------------------------------