SET007 Axioms: SET007+169.ax
%------------------------------------------------------------------------------
% File : SET007+169 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Replacement of Subtrees in a Tree
% 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_a [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 3 unt; 0 def)
% Number of atoms : 323 ( 43 equ)
% Maximal formula atoms : 25 ( 9 avg)
% Number of connectives : 349 ( 61 ~; 0 |; 170 &)
% ( 10 <=>; 108 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-2 aty)
% Number of functors : 23 ( 23 usr; 2 con; 0-4 aty)
% Number of variables : 131 ( 119 !; 12 ?)
% 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_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ? [B] :
( m4_trees_1(B,A)
& ~ v1_xboole_0(B)
& v2_trees_1(B) ) ) ).
fof(t1_trees_a,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) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ( k7_finseq_1(A,B) = k7_finseq_1(D,C)
=> r3_xboole_0(A,D) ) ) ) ) ) ).
fof(d1_trees_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( m4_trees_1(C,A)
=> ( C != k1_xboole_0
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_trees_1(D) )
=> ( D = k1_trees_a(A,B,C)
<=> ! [E] :
( m2_finseq_1(E,k5_numbers)
=> ( r2_hidden(E,D)
<=> ~ ( ~ ( r2_hidden(E,A)
& ! [F] :
( m2_finseq_1(F,k5_numbers)
=> ~ ( r2_hidden(F,C)
& r2_xboole_0(F,E) ) ) )
& ! [F] :
( m2_finseq_1(F,k5_numbers)
=> ! [G] :
( m2_finseq_1(G,k5_numbers)
=> ~ ( r2_hidden(F,C)
& r2_hidden(G,B)
& E = k8_finseq_1(k5_numbers,F,G) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_trees_a,axiom,
$true ).
fof(t9_trees_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( m4_trees_1(C,B)
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( r2_hidden(D,C)
=> A = k4_trees_1(k1_trees_a(B,A,C),D) ) ) ) ) ) ).
fof(t10_trees_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( m1_trees_1(C,A)
=> k1_trees_a(A,B,k2_trees_a(A,C)) = k5_trees_1(A,C,B) ) ) ) ).
fof(d2_trees_a,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m4_trees_1(B,k1_relat_1(A))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( B != k1_xboole_0
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v3_trees_2(D) )
=> ( D = k3_trees_a(A,B,C)
<=> ( k1_relat_1(D) = k1_trees_a(k1_relat_1(A),k1_relat_1(C),B)
& ! [E] :
( m2_finseq_1(E,k5_numbers)
=> ~ ( r2_hidden(E,k1_trees_a(k1_relat_1(A),k1_relat_1(C),B))
& ? [F] :
( m2_finseq_1(F,k5_numbers)
& r2_hidden(F,B)
& ~ ( ~ r1_tarski(F,E)
& k1_funct_1(D,E) = k1_funct_1(A,E) ) )
& ! [F] :
( m2_finseq_1(F,k5_numbers)
=> ! [G] :
( m2_finseq_1(G,k5_numbers)
=> ~ ( r2_hidden(F,B)
& r2_hidden(G,k1_relat_1(C))
& E = k8_finseq_1(k5_numbers,F,G)
& k1_funct_1(D,E) = k1_funct_1(C,G) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t11_trees_a,axiom,
$true ).
fof(t12_trees_a,axiom,
$true ).
fof(t13_trees_a,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) )
=> ! [C] :
( m4_trees_1(C,k1_relat_1(A))
=> ( C != k1_xboole_0
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ~ ( r2_hidden(D,k1_relat_1(k3_trees_a(A,C,B)))
& ? [E] :
( m2_finseq_1(E,k5_numbers)
& r2_hidden(E,C)
& ~ ( ~ r1_tarski(E,D)
& k1_funct_1(k3_trees_a(A,C,B),D) = k1_funct_1(A,D) ) )
& ! [E] :
( m2_finseq_1(E,k5_numbers)
=> ! [F] :
( m2_finseq_1(F,k5_numbers)
=> ~ ( r2_hidden(E,C)
& r2_hidden(F,k1_relat_1(B))
& D = k8_finseq_1(k5_numbers,E,F)
& k1_funct_1(k3_trees_a(A,C,B),D) = k1_funct_1(B,F) ) ) ) ) ) ) ) ) ) ).
fof(t14_trees_a,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( r2_hidden(A,k1_relat_1(B))
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ~ ( r2_hidden(D,k1_relat_1(k8_trees_2(B,A,C)))
& ~ ( ~ r1_tarski(A,D)
& k1_funct_1(k8_trees_2(B,A,C),D) = k1_funct_1(B,D) )
& ! [E] :
( m2_finseq_1(E,k5_numbers)
=> ~ ( r2_hidden(E,k1_relat_1(C))
& D = k8_finseq_1(k5_numbers,A,E)
& k1_funct_1(k8_trees_2(B,A,C),D) = k1_funct_1(C,E) ) ) ) ) ) ) ) ) ).
fof(t19_trees_a,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) )
=> ! [C] :
( m1_trees_1(C,k1_relat_1(A))
=> k3_trees_a(A,k2_trees_a(k1_relat_1(A),C),B) = k8_trees_2(A,C,B) ) ) ) ).
fof(d3_trees_a,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> ! [C] :
( m4_trees_1(C,k1_relat_1(B))
=> ! [D] :
( ( v1_funct_1(D)
& v3_trees_2(D)
& m3_trees_2(D,A) )
=> ( C != k1_xboole_0
=> k4_trees_a(A,B,C,D) = k3_trees_a(B,C,D) ) ) ) ) ) ).
fof(dt_k1_trees_a,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& ~ v1_xboole_0(B)
& v1_trees_1(B)
& m4_trees_1(C,A) )
=> ( ~ v1_xboole_0(k1_trees_a(A,B,C))
& v1_trees_1(k1_trees_a(A,B,C)) ) ) ).
fof(dt_k2_trees_a,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& m1_subset_1(B,A) )
=> ( ~ v1_xboole_0(k2_trees_a(A,B))
& m4_trees_1(k2_trees_a(A,B),A) ) ) ).
fof(redefinition_k2_trees_a,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& m1_subset_1(B,A) )
=> k2_trees_a(A,B) = k1_tarski(B) ) ).
fof(dt_k3_trees_a,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& m4_trees_1(B,k1_relat_1(A))
& v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( v1_relat_1(k3_trees_a(A,B,C))
& v1_funct_1(k3_trees_a(A,B,C))
& v3_trees_2(k3_trees_a(A,B,C)) ) ) ).
fof(dt_k4_trees_a,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A)
& m4_trees_1(C,k1_relat_1(B))
& v1_funct_1(D)
& v3_trees_2(D)
& m3_trees_2(D,A) )
=> ( v1_funct_1(k4_trees_a(A,B,C,D))
& v3_trees_2(k4_trees_a(A,B,C,D))
& m3_trees_2(k4_trees_a(A,B,C,D),A) ) ) ).
fof(t2_trees_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( m4_trees_1(C,A)
=> ( C != k1_xboole_0
=> k1_trees_a(A,B,C) = k2_xboole_0(a_2_0_trees_a(A,C),a_3_0_trees_a(A,B,C)) ) ) ) ) ).
fof(t3_trees_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m4_trees_1(B,A)
=> r1_tarski(a_2_1_trees_a(A,B),a_2_0_trees_a(A,B)) ) ) ).
fof(t4_trees_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m4_trees_1(B,A)
=> r1_tarski(B,a_2_0_trees_a(A,B)) ) ) ).
fof(t5_trees_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m4_trees_1(B,A)
=> k4_xboole_0(a_2_0_trees_a(A,B),a_2_1_trees_a(A,B)) = B ) ) ).
fof(t6_trees_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( m4_trees_1(C,A)
=> r1_tarski(C,a_3_0_trees_a(A,B,C)) ) ) ) ).
fof(t7_trees_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> ! [C] :
( m4_trees_1(C,A)
=> ( C != k1_xboole_0
=> k1_trees_a(A,B,C) = k2_xboole_0(a_2_1_trees_a(A,C),a_3_0_trees_a(A,B,C)) ) ) ) ) ).
fof(t15_trees_a,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) )
=> ! [C] :
( m4_trees_1(C,k1_relat_1(A))
=> ( C != k1_xboole_0
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( ( r2_hidden(D,k1_relat_1(k3_trees_a(A,C,B)))
& r2_hidden(D,a_2_2_trees_a(A,C)) )
=> k1_funct_1(k3_trees_a(A,C,B),D) = k1_funct_1(A,D) ) ) ) ) ) ) ).
fof(t16_trees_a,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( r2_hidden(A,k1_relat_1(B))
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( ( r2_hidden(D,k1_relat_1(k8_trees_2(B,A,C)))
& r2_hidden(D,a_2_3_trees_a(A,B)) )
=> k1_funct_1(k8_trees_2(B,A,C),D) = k1_funct_1(B,D) ) ) ) ) ) ) ).
fof(t17_trees_a,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) )
=> ! [C] :
( m4_trees_1(C,k1_relat_1(A))
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ~ ( r2_hidden(D,k1_relat_1(k3_trees_a(A,C,B)))
& r2_hidden(D,a_3_1_trees_a(A,B,C))
& ! [E] :
( m1_trees_1(E,k1_relat_1(A))
=> ! [F] :
( m1_trees_1(F,k1_relat_1(B))
=> ~ ( D = k8_finseq_1(k5_numbers,E,F)
& r2_hidden(E,C)
& k1_funct_1(k3_trees_a(A,C,B),D) = k1_funct_1(B,F) ) ) ) ) ) ) ) ) ).
fof(t18_trees_a,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( r2_hidden(A,k1_relat_1(B))
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ~ ( r2_hidden(D,k1_relat_1(k8_trees_2(B,A,C)))
& r2_hidden(D,a_2_4_trees_a(A,C))
& ! [E] :
( m1_trees_1(E,k1_relat_1(C))
=> ~ ( D = k8_finseq_1(k5_numbers,A,E)
& k1_funct_1(k8_trees_2(B,A,C),D) = k1_funct_1(C,E) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_2_0_trees_a,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B)
& m4_trees_1(C,B) )
=> ( r2_hidden(A,a_2_0_trees_a(B,C))
<=> ? [D] :
( m1_trees_1(D,B)
& A = D
& ! [E] :
( m2_finseq_1(E,k5_numbers)
=> ~ ( r2_hidden(E,C)
& r2_xboole_0(E,D) ) ) ) ) ) ).
fof(fraenkel_a_3_0_trees_a,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B)
& ~ v1_xboole_0(C)
& v1_trees_1(C)
& m4_trees_1(D,B) )
=> ( r2_hidden(A,a_3_0_trees_a(B,C,D))
<=> ? [E,F] :
( m1_trees_1(E,B)
& m1_trees_1(F,C)
& A = k8_finseq_1(k5_numbers,E,F)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_2_1_trees_a,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B)
& m4_trees_1(C,B) )
=> ( r2_hidden(A,a_2_1_trees_a(B,C))
<=> ? [D] :
( m1_trees_1(D,B)
& A = D
& ! [E] :
( m2_finseq_1(E,k5_numbers)
=> ~ ( r2_hidden(E,C)
& r1_tarski(E,D) ) ) ) ) ) ).
fof(fraenkel_a_2_2_trees_a,axiom,
! [A,B,C] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B)
& m4_trees_1(C,k1_relat_1(B)) )
=> ( r2_hidden(A,a_2_2_trees_a(B,C))
<=> ? [D] :
( m1_trees_1(D,k1_relat_1(B))
& A = D
& ! [E] :
( m2_finseq_1(E,k5_numbers)
=> ~ ( r2_hidden(E,C)
& r1_tarski(E,D) ) ) ) ) ) ).
fof(fraenkel_a_2_3_trees_a,axiom,
! [A,B,C] :
( ( m2_finseq_1(B,k5_numbers)
& v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( r2_hidden(A,a_2_3_trees_a(B,C))
<=> ? [D] :
( m1_trees_1(D,k1_relat_1(C))
& A = D
& ~ r1_tarski(B,D) ) ) ) ).
fof(fraenkel_a_3_1_trees_a,axiom,
! [A,B,C,D] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B)
& v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C)
& m4_trees_1(D,k1_relat_1(B)) )
=> ( r2_hidden(A,a_3_1_trees_a(B,C,D))
<=> ? [E,F] :
( m1_trees_1(E,k1_relat_1(B))
& m1_trees_1(F,k1_relat_1(C))
& A = k8_finseq_1(k5_numbers,E,F)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_2_4_trees_a,axiom,
! [A,B,C] :
( ( m2_finseq_1(B,k5_numbers)
& v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( r2_hidden(A,a_2_4_trees_a(B,C))
<=> ? [D] :
( m1_trees_1(D,k1_relat_1(C))
& A = k8_finseq_1(k5_numbers,B,D)
& D = D ) ) ) ).
%------------------------------------------------------------------------------