SET007 Axioms: SET007+63.ax
%------------------------------------------------------------------------------
% File : SET007+63 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Introduction to Trees
% 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 128 ( 38 unt; 0 def)
% Number of atoms : 635 ( 56 equ)
% Maximal formula atoms : 16 ( 4 avg)
% Number of connectives : 618 ( 111 ~; 1 |; 284 &)
% ( 24 <=>; 198 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 32 ( 32 usr; 5 con; 0-3 aty)
% Number of variables : 210 ( 194 !; 16 ?)
% 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_1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_trees_1(A) ) ).
fof(rc2_trees_1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) ) ).
fof(fc1_trees_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A)
& m1_subset_1(B,A) )
=> ( ~ v1_xboole_0(k4_trees_1(A,B))
& v1_finset_1(k4_trees_1(A,B))
& v1_trees_1(k4_trees_1(A,B)) ) ) ).
fof(fc2_trees_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A)
& m1_subset_1(B,A)
& ~ v1_xboole_0(C)
& v1_finset_1(C)
& v1_trees_1(C) )
=> ( ~ v1_xboole_0(k5_trees_1(A,B,C))
& v1_finset_1(k5_trees_1(A,B,C))
& v1_trees_1(k5_trees_1(A,B,C)) ) ) ).
fof(fc3_trees_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> v1_finset_1(k1_trees_1(A)) ) ).
fof(rc3_trees_1,axiom,
? [A] : v2_trees_1(A) ).
fof(cc1_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m4_trees_1(B,A)
=> v1_finset_1(B) ) ) ).
fof(t1_trees_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [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) )
=> ( C = k7_relat_1(B,k2_finseq_1(A))
=> r1_xreal_0(k3_finseq_1(C),A) ) ) ) ) ).
fof(t2_trees_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [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) )
=> ( C = k7_relat_1(B,k2_finseq_1(A))
=> r1_xreal_0(k3_finseq_1(C),k3_finseq_1(B)) ) ) ) ) ).
fof(t3_trees_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [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) )
=> ~ ( C = k7_relat_1(B,k2_finseq_1(A))
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> B != k7_finseq_1(C,D) ) ) ) ) ) ).
fof(t4_trees_1,axiom,
! [A] : k1_xboole_0 != k9_finseq_1(A) ).
fof(t5_trees_1,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) )
=> ( ( A = k7_finseq_1(A,B)
| A = k7_finseq_1(B,A) )
=> B = k1_xboole_0 ) ) ) ).
fof(t6_trees_1,axiom,
! [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) )
=> ~ ( k7_finseq_1(B,C) = k9_finseq_1(A)
& ~ ( B = k9_finseq_1(A)
& C = k1_xboole_0 )
& ~ ( B = k1_xboole_0
& C = k9_finseq_1(A) ) ) ) ) ).
fof(d1_trees_1,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) )
=> ( r1_tarski(A,B)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k7_relat_1(B,k2_finseq_1(C)) ) ) ) ) ).
fof(t7_trees_1,axiom,
$true ).
fof(t8_trees_1,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) )
=> ( r1_tarski(A,B)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& B = k7_finseq_1(A,C) ) ) ) ) ).
fof(t9_trees_1,axiom,
$true ).
fof(t10_trees_1,axiom,
$true ).
fof(t11_trees_1,axiom,
$true ).
fof(t12_trees_1,axiom,
$true ).
fof(t13_trees_1,axiom,
$true ).
fof(t14_trees_1,axiom,
$true ).
fof(t15_trees_1,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) )
=> ( ( r1_tarski(A,B)
& k3_finseq_1(A) = k3_finseq_1(B) )
=> A = B ) ) ) ).
fof(t16_trees_1,axiom,
! [A,B] :
( r1_tarski(k9_finseq_1(A),k9_finseq_1(B))
<=> A = B ) ).
fof(t17_trees_1,axiom,
$true ).
fof(t18_trees_1,axiom,
$true ).
fof(t19_trees_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ( ( r3_xboole_0(A,B)
& k4_card_1(A) = k4_card_1(B) )
=> A = B ) ) ) ).
fof(t20_trees_1,axiom,
$true ).
fof(t21_trees_1,axiom,
$true ).
fof(t22_trees_1,axiom,
$true ).
fof(t23_trees_1,axiom,
! [A,B] :
( r3_xboole_0(k9_finseq_1(A),k9_finseq_1(B))
<=> A = B ) ).
fof(t24_trees_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ~ ( r2_xboole_0(A,B)
& r1_xreal_0(k4_card_1(B),k4_card_1(A)) ) ) ) ).
fof(t25_trees_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( ~ r2_xboole_0(B,k1_xboole_0)
& ~ r2_xboole_0(B,k6_finseq_1(A)) ) ) ) ).
fof(t26_trees_1,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) )
=> ~ ( r2_xboole_0(A,B)
& r2_xboole_0(B,A) ) ) ) ).
fof(t27_trees_1,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,B)
& r2_xboole_0(B,C) )
& ~ ( r2_xboole_0(A,B)
& r1_tarski(B,C) )
& ~ ( r1_tarski(A,B)
& r2_xboole_0(B,C) ) )
=> r2_xboole_0(A,C) ) ) ) ) ).
fof(t28_trees_1,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) )
=> ~ ( r1_tarski(A,B)
& r2_xboole_0(B,A) ) ) ) ).
fof(t29_trees_1,axiom,
$true ).
fof(t30_trees_1,axiom,
! [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(k7_finseq_1(B,k9_finseq_1(A)),C)
=> r2_xboole_0(B,C) ) ) ) ).
fof(t31_trees_1,axiom,
! [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(B,C)
=> r2_xboole_0(B,k7_finseq_1(C,k9_finseq_1(A))) ) ) ) ).
fof(t32_trees_1,axiom,
! [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(B,k7_finseq_1(C,k9_finseq_1(A)))
=> r1_tarski(B,C) ) ) ) ).
fof(t33_trees_1,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) )
=> ( ~ ( ~ r2_xboole_0(k1_xboole_0,A)
& k1_xboole_0 = A )
=> r2_xboole_0(B,k7_finseq_1(B,A)) ) ) ) ).
fof(d2_trees_1,axiom,
$true ).
fof(d3_trees_1,axiom,
$true ).
fof(d4_trees_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( B = k1_trees_1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& C = D
& r2_xboole_0(D,A) ) ) ) ) ).
fof(t34_trees_1,axiom,
$true ).
fof(t35_trees_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( r2_hidden(A,k1_trees_1(B))
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) ) ) ) ).
fof(t36_trees_1,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) )
=> ( r2_hidden(A,k1_trees_1(B))
<=> r2_xboole_0(A,B) ) ) ) ).
fof(t37_trees_1,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) )
=> ~ ( r2_hidden(A,k1_trees_1(B))
& r1_xreal_0(k3_finseq_1(B),k3_finseq_1(A)) ) ) ) ).
fof(t38_trees_1,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_hidden(k7_finseq_1(B,C),k1_trees_1(A))
=> r2_hidden(B,k1_trees_1(A)) ) ) ) ) ).
fof(t39_trees_1,axiom,
k1_trees_1(k1_xboole_0) = k1_xboole_0 ).
fof(t40_trees_1,axiom,
! [A] : k1_trees_1(k9_finseq_1(A)) = k1_tarski(k1_xboole_0) ).
fof(t41_trees_1,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) )
=> ( r1_tarski(A,B)
=> r1_tarski(k1_trees_1(A),k1_trees_1(B)) ) ) ) ).
fof(t42_trees_1,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_hidden(B,k1_trees_1(A))
& r2_hidden(C,k1_trees_1(A)) )
=> r3_xboole_0(B,C) ) ) ) ) ).
fof(d5_trees_1,axiom,
! [A] :
( v1_trees_1(A)
<=> ( r1_tarski(A,k13_finseq_1(k5_numbers))
& ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ( r2_hidden(B,A)
=> r1_tarski(k1_trees_1(B),A) ) )
& ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C)),A)
& r1_xreal_0(D,C) )
=> r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,D)),A) ) ) ) ) ) ) ).
fof(t43_trees_1,axiom,
$true ).
fof(t44_trees_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ( r2_hidden(A,B)
=> m2_finseq_1(A,k5_numbers) ) ) ).
fof(t45_trees_1,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) )
=> ( ( r2_hidden(B,A)
& r1_tarski(C,B) )
=> r2_hidden(C,A) ) ) ) ) ).
fof(t46_trees_1,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( r2_hidden(k7_finseq_1(A,C),B)
=> r2_hidden(A,B) ) ) ) ) ).
fof(t47_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( r2_hidden(k1_xboole_0,A)
& r2_hidden(k6_finseq_1(k5_numbers),A) ) ) ).
fof(t48_trees_1,axiom,
( ~ v1_xboole_0(k1_tarski(k1_xboole_0))
& v1_trees_1(k1_tarski(k1_xboole_0)) ) ).
fof(t49_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ( ~ v1_xboole_0(k2_xboole_0(A,B))
& v1_trees_1(k2_xboole_0(A,B)) ) ) ) ).
fof(t50_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ( ~ v1_xboole_0(k3_xboole_0(A,B))
& v1_trees_1(k3_xboole_0(A,B)) ) ) ) ).
fof(t51_trees_1,axiom,
$true ).
fof(t52_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_trees_1(B) )
=> ( ~ v1_xboole_0(k2_xboole_0(A,B))
& v1_finset_1(k2_xboole_0(A,B))
& v1_trees_1(k2_xboole_0(A,B)) ) ) ) ).
fof(t53_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_trees_1(B) )
=> ( ~ v1_xboole_0(k3_xboole_0(B,A))
& v1_finset_1(k3_xboole_0(B,A))
& v1_trees_1(k3_xboole_0(B,A))
& ~ v1_xboole_0(k3_xboole_0(A,B))
& v1_finset_1(k3_xboole_0(A,B))
& v1_trees_1(k3_xboole_0(A,B)) ) ) ) ).
fof(d6_trees_1,axiom,
$true ).
fof(t54_trees_1,axiom,
$true ).
fof(t55_trees_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,A)
=> r2_hidden(k12_finseq_1(k5_numbers,A),k2_trees_1(B)) ) ) ) ).
fof(t56_trees_1,axiom,
k2_trees_1(np__0) = k1_tarski(k1_xboole_0) ).
fof(t57_trees_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ~ ( r2_hidden(B,k2_trees_1(A))
& B != k1_xboole_0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& B = k12_finseq_1(k5_numbers,C) ) ) ) ) ) ).
fof(d8_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( B = k3_trees_1(A)
<=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_hidden(C,B)
<=> ( r2_hidden(C,A)
& ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ~ ( r2_hidden(D,A)
& r2_xboole_0(C,D) ) ) ) ) ) ) ) ) ).
fof(d9_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ( r2_hidden(B,A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_trees_1(C) )
=> ( C = k4_trees_1(A,B)
<=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( r2_hidden(D,C)
<=> r2_hidden(k8_finseq_1(k5_numbers,B,D),A) ) ) ) ) ) ) ) ).
fof(t58_trees_1,axiom,
$true ).
fof(t59_trees_1,axiom,
$true ).
fof(t60_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> k4_trees_1(A,k6_finseq_1(k5_numbers)) = A ) ).
fof(d10_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( k3_trees_1(A) != k1_xboole_0
=> ! [B] :
( m1_trees_1(B,A)
=> ( m2_trees_1(B,A)
<=> r2_hidden(B,k3_trees_1(A)) ) ) ) ) ).
fof(d11_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ( m3_trees_1(B,A)
<=> ? [C] :
( m1_trees_1(C,A)
& B = k4_trees_1(A,C) ) ) ) ) ).
fof(d12_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_trees_1(C) )
=> ( r2_hidden(B,A)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_trees_1(D) )
=> ( D = k5_trees_1(A,B,C)
<=> ! [E] :
( m2_finseq_1(E,k5_numbers)
=> ( r2_hidden(E,D)
<=> ~ ( ~ ( r2_hidden(E,A)
& ~ r2_xboole_0(B,E) )
& ! [F] :
( m2_finseq_1(F,k5_numbers)
=> ~ ( r2_hidden(F,C)
& E = k8_finseq_1(k5_numbers,B,F) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t61_trees_1,axiom,
$true ).
fof(t62_trees_1,axiom,
$true ).
fof(t63_trees_1,axiom,
$true ).
fof(t65_trees_1,axiom,
$true ).
fof(t66_trees_1,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_trees_1(C) )
=> ( r2_hidden(A,B)
=> C = k4_trees_1(k5_trees_1(B,A,C),A) ) ) ) ) ).
fof(t67_trees_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> r2_tarski(k1_trees_1(A),k4_finseq_1(A)) ) ).
fof(t68_trees_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> k4_card_1(k1_trees_1(A)) = k3_finseq_1(A) ) ).
fof(d13_trees_1,axiom,
! [A] :
( v2_trees_1(A)
<=> ( ! [B] :
( r2_hidden(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) ) )
& ! [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_hidden(B,A)
& r2_hidden(C,A)
& B != C
& r3_xboole_0(B,C) ) ) ) ) ) ).
fof(t69_trees_1,axiom,
$true ).
fof(t70_trees_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> v2_trees_1(k1_tarski(A)) ) ).
fof(t71_trees_1,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) )
=> ( ~ r3_xboole_0(A,B)
=> v2_trees_1(k2_tarski(A,B)) ) ) ) ).
fof(d14_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( v2_trees_1(B)
=> ( m4_trees_1(B,A)
<=> r1_tarski(B,A) ) ) ) ).
fof(t72_trees_1,axiom,
$true ).
fof(t73_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( m4_trees_1(k1_xboole_0,A)
& m4_trees_1(k1_tarski(k1_xboole_0),A) ) ) ).
fof(t74_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> m4_trees_1(k1_tarski(B),A) ) ) ).
fof(t75_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ! [C] :
( m1_trees_1(C,A)
=> ( ~ r3_xboole_0(B,C)
=> m4_trees_1(k2_tarski(B,C),A) ) ) ) ) ).
fof(d15_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k6_trees_1(A)
<=> ( ? [C] :
( m2_finseq_1(C,k5_numbers)
& r2_hidden(C,A)
& k3_finseq_1(C) = B )
& ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_hidden(C,A)
=> r1_xreal_0(k3_finseq_1(C),B) ) ) ) ) ) ) ).
fof(d16_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k7_trees_1(A)
<=> ? [C] :
( m4_trees_1(C,A)
& B = k4_card_1(C)
& ! [D] :
( m4_trees_1(D,A)
=> r1_xreal_0(k4_card_1(D),k4_card_1(C)) ) ) ) ) ) ).
fof(t76_trees_1,axiom,
$true ).
fof(t77_trees_1,axiom,
$true ).
fof(t78_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> r1_xreal_0(np__1,k7_trees_1(A)) ) ).
fof(t79_trees_1,axiom,
k6_trees_1(k2_trees_1(np__0)) = np__0 ).
fof(t80_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ( k6_trees_1(A) = np__0
=> A = k2_trees_1(np__0) ) ) ).
fof(t81_trees_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k6_trees_1(k2_trees_1(k1_nat_1(A,np__1))) = np__1 ) ).
fof(t82_trees_1,axiom,
k7_trees_1(k2_trees_1(np__0)) = np__1 ).
fof(t83_trees_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k7_trees_1(k2_trees_1(k1_nat_1(A,np__1))) = k1_nat_1(A,np__1) ) ).
fof(t84_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> r1_xreal_0(k6_trees_1(k4_trees_1(A,B)),k6_trees_1(A)) ) ) ).
fof(t85_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ~ ( B != k1_xboole_0
& r1_xreal_0(k6_trees_1(A),k6_trees_1(k4_trees_1(A,B))) ) ) ) ).
fof(s1_trees_1,axiom,
( ! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(k12_finseq_1(k5_numbers,B),A)
=> p1_s1_trees_1(k4_trees_1(A,k12_finseq_1(k5_numbers,B))) ) )
=> p1_s1_trees_1(A) ) )
=> ! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> p1_s1_trees_1(A) ) ) ).
fof(dt_m1_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> m2_finseq_1(B,k5_numbers) ) ) ).
fof(existence_m1_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] : m1_trees_1(B,A) ) ).
fof(redefinition_m1_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
<=> m1_subset_1(B,A) ) ) ).
fof(dt_m2_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_trees_1(B,A)
=> m1_trees_1(B,A) ) ) ).
fof(existence_m2_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] : m2_trees_1(B,A) ) ).
fof(dt_m3_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m3_trees_1(B,A)
=> ( ~ v1_xboole_0(B)
& v1_trees_1(B) ) ) ) ).
fof(existence_m3_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] : m3_trees_1(B,A) ) ).
fof(dt_m4_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m4_trees_1(B,A)
=> v2_trees_1(B) ) ) ).
fof(existence_m4_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] : m4_trees_1(B,A) ) ).
fof(dt_k1_trees_1,axiom,
$true ).
fof(dt_k2_trees_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v1_xboole_0(k2_trees_1(A))
& v1_finset_1(k2_trees_1(A))
& v1_trees_1(k2_trees_1(A)) ) ) ).
fof(dt_k3_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> m1_subset_1(k3_trees_1(A),k1_zfmisc_1(A)) ) ).
fof(dt_k4_trees_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& m1_finseq_1(B,k5_numbers) )
=> ( ~ v1_xboole_0(k4_trees_1(A,B))
& v1_trees_1(k4_trees_1(A,B)) ) ) ).
fof(dt_k5_trees_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& m1_finseq_1(B,k5_numbers)
& ~ v1_xboole_0(C)
& v1_trees_1(C) )
=> ( ~ v1_xboole_0(k5_trees_1(A,B,C))
& v1_trees_1(k5_trees_1(A,B,C)) ) ) ).
fof(dt_k6_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> m2_subset_1(k6_trees_1(A),k1_numbers,k5_numbers) ) ).
fof(dt_k7_trees_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> m2_subset_1(k7_trees_1(A),k1_numbers,k5_numbers) ) ).
fof(d7_trees_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_trees_1(A) = k2_xboole_0(a_1_0_trees_1(A),k1_tarski(k1_xboole_0)) ) ).
fof(t64_trees_1,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_trees_1(C) )
=> ( r2_hidden(A,B)
=> k5_trees_1(B,A,C) = k2_xboole_0(a_2_0_trees_1(A,B),a_2_1_trees_1(A,C)) ) ) ) ) ).
fof(fraenkel_a_1_0_trees_1,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_0_trees_1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k12_finseq_1(k5_numbers,C)
& ~ r1_xreal_0(B,C) ) ) ) ).
fof(fraenkel_a_2_0_trees_1,axiom,
! [A,B,C] :
( ( m2_finseq_1(B,k5_numbers)
& ~ v1_xboole_0(C)
& v1_trees_1(C) )
=> ( r2_hidden(A,a_2_0_trees_1(B,C))
<=> ? [D] :
( m1_trees_1(D,C)
& A = D
& ~ r2_xboole_0(B,D) ) ) ) ).
fof(fraenkel_a_2_1_trees_1,axiom,
! [A,B,C] :
( ( m2_finseq_1(B,k5_numbers)
& ~ v1_xboole_0(C)
& v1_trees_1(C) )
=> ( r2_hidden(A,a_2_1_trees_1(B,C))
<=> ? [D] :
( m1_trees_1(D,C)
& A = k8_finseq_1(k5_numbers,B,D)
& D = D ) ) ) ).
%------------------------------------------------------------------------------