SET007 Axioms: SET007+164.ax
%------------------------------------------------------------------------------
% File : SET007+164 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Joining of Decorated 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_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 66 ( 4 unt; 0 def)
% Number of atoms : 493 ( 101 equ)
% Maximal formula atoms : 29 ( 7 avg)
% Number of connectives : 496 ( 69 ~; 1 |; 248 &)
% ( 8 <=>; 170 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-3 aty)
% Number of functors : 55 ( 55 usr; 8 con; 0-4 aty)
% Number of variables : 233 ( 226 !; 7 ?)
% 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_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v3_trees_3(A)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m1_finseq_1(C,B)
=> v6_trees_3(C) ) ) ).
fof(d1_trees_4,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) )
=> ( A = B
<=> ( k1_relat_1(A) = k1_relat_1(B)
& ! [C] :
( m1_trees_1(C,k1_relat_1(A))
=> k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) ) ) ).
fof(t1_trees_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_tarski(k2_trees_1(A),k2_trees_1(B))
=> r1_xreal_0(A,B) ) ) ) ).
fof(t2_trees_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k2_trees_1(A) = k2_trees_1(B)
=> A = B ) ) ) ).
fof(d2_trees_4,axiom,
! [A] : k1_trees_4(A) = k2_funcop_1(k2_trees_1(np__0),A) ).
fof(t3_trees_4,axiom,
! [A] :
( k1_relat_1(k1_trees_4(A)) = k2_trees_1(np__0)
& k1_funct_1(k1_trees_4(A),k1_xboole_0) = A ) ).
fof(t4_trees_4,axiom,
! [A,B] :
( k1_trees_4(A) = k1_trees_4(B)
=> A = B ) ).
fof(t5_trees_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ( k1_relat_1(A) = k2_trees_1(np__0)
=> A = k1_trees_4(k1_funct_1(A,k1_xboole_0)) ) ) ).
fof(t6_trees_4,axiom,
! [A] : k1_trees_4(A) = k1_tarski(k4_tarski(k1_xboole_0,A)) ).
fof(d3_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( C = k3_trees_4(A,B)
<=> ( k1_relat_1(C) = k2_trees_1(k3_finseq_1(B))
& k1_funct_1(C,k1_xboole_0) = A
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(B),D)
=> k1_funct_1(C,k12_finseq_1(k5_numbers,D)) = k1_funct_1(B,k1_nat_1(D,np__1)) ) ) ) ) ) ) ).
fof(t7_trees_4,axiom,
! [A,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) )
=> ( k3_trees_4(A,C) = k3_trees_4(B,D)
=> ( A = B
& C = D ) ) ) ) ).
fof(t8_trees_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,A)
=> k4_trees_1(k2_trees_1(B),k12_finseq_1(k5_numbers,A)) = k2_trees_1(np__0) ) ) ) ).
fof(t9_trees_4,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ~ r1_xreal_0(k3_finseq_1(C),B)
=> k5_trees_2(k3_trees_4(A,C),k12_finseq_1(k5_numbers,B)) = k1_trees_4(k1_funct_1(C,k1_nat_1(B,np__1))) ) ) ) ).
fof(d4_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( v6_trees_3(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( C = k4_trees_4(A,B)
<=> ( ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& v6_trees_3(D)
& B = D
& k1_relat_1(C) = k13_trees_3(k2_funct_6(D)) )
& k1_funct_1(C,k1_xboole_0) = A
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(B),D)
=> k5_trees_2(C,k12_finseq_1(k5_numbers,D)) = k1_funct_1(B,k1_nat_1(D,np__1)) ) ) ) ) ) ) ) ).
fof(d5_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> k5_trees_4(A,B) = k4_trees_4(A,k9_finseq_1(B)) ) ).
fof(d6_trees_4,axiom,
! [A,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) )
=> k6_trees_4(A,B,C) = k4_trees_4(A,k10_finseq_1(B,C)) ) ) ).
fof(t10_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v6_trees_3(B) )
=> k1_relat_1(k4_trees_4(A,B)) = k13_trees_3(k2_funct_6(B)) ) ).
fof(t11_trees_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v6_trees_3(C) )
=> ( r2_hidden(A,k1_relat_1(k4_trees_4(B,C)))
<=> ~ ( A != k1_xboole_0
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v3_trees_2(E) )
=> ! [F] :
( m1_trees_1(F,k1_relat_1(E))
=> ~ ( ~ r1_xreal_0(k3_finseq_1(C),D)
& E = k1_funct_1(C,k1_nat_1(D,np__1))
& A = k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,D),F) ) ) ) ) ) ) ) ).
fof(t12_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v6_trees_3(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v3_trees_2(D) )
=> ! [E] :
( m1_trees_1(E,k1_relat_1(D))
=> ( D = k1_funct_1(B,k1_nat_1(C,np__1))
=> ( r1_xreal_0(k3_finseq_1(B),C)
| k1_funct_1(k4_trees_4(A,B),k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,C),E)) = k1_funct_1(D,E) ) ) ) ) ) ) ).
fof(t13_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> k1_relat_1(k5_trees_4(A,B)) = k14_trees_3(k1_relat_1(B)) ) ).
fof(t14_trees_4,axiom,
! [A,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) )
=> k1_relat_1(k6_trees_4(A,B,C)) = k15_trees_3(k1_relat_1(B),k1_relat_1(C)) ) ) ).
fof(t15_trees_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v6_trees_3(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& v6_trees_3(D) )
=> ( k4_trees_4(A,C) = k4_trees_4(B,D)
=> ( A = B
& C = D ) ) ) ) ).
fof(t16_trees_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( k1_trees_4(A) = k3_trees_4(B,C)
=> ( A = B
& C = k1_xboole_0 ) ) ) ).
fof(t17_trees_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( k1_trees_4(A) = k4_trees_4(B,C)
& v6_trees_3(C) )
=> ( A = B
& C = k1_xboole_0 ) ) ) ).
fof(t18_trees_4,axiom,
! [A,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) )
=> ( ( k3_trees_4(A,C) = k4_trees_4(B,D)
& v6_trees_3(D) )
=> ( A = B
& k3_finseq_1(C) = k3_finseq_1(D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(C))
=> k1_funct_1(D,E) = k1_trees_4(k1_funct_1(C,E)) ) ) ) ) ) ) ).
fof(t19_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v6_trees_3(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ( r2_hidden(k7_finseq_1(k12_finseq_1(k5_numbers,C),D),k1_relat_1(k4_trees_4(A,B)))
=> k1_funct_1(k4_trees_4(A,B),k7_finseq_1(k12_finseq_1(k5_numbers,C),D)) = k5_funct_6(B,k1_nat_1(C,np__1),D) ) ) ) ) ).
fof(t20_trees_4,axiom,
! [A] :
( k3_trees_4(A,k1_xboole_0) = k1_trees_4(A)
& k4_trees_4(A,k1_xboole_0) = k1_trees_4(A) ) ).
fof(t21_trees_4,axiom,
! [A,B] : k3_trees_4(A,k9_finseq_1(B)) = k8_trees_2(k2_funcop_1(k2_trees_1(np__1),A),k12_finseq_1(k5_numbers,np__0),k1_trees_4(B)) ).
fof(t22_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> k4_trees_4(A,k9_finseq_1(B)) = k8_trees_2(k2_funcop_1(k2_trees_1(np__1),A),k12_finseq_1(k5_numbers,np__0),B) ) ).
fof(d7_trees_4,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,D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v3_trees_2(D) )
=> ( D = k13_trees_4(A,B,C)
<=> ( ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E) )
=> ( r2_hidden(E,k1_relat_1(D))
<=> ~ ( ~ r2_hidden(E,k1_relat_1(A))
& ! [F] :
( m1_trees_1(F,k1_relat_1(A))
=> ! [G] :
( m1_trees_1(G,k1_relat_1(B))
=> ~ ( r2_hidden(F,k3_trees_1(k1_relat_1(A)))
& k1_funct_1(A,F) = C
& E = k8_finseq_1(k5_numbers,F,G) ) ) ) ) ) )
& ! [E] :
( m1_trees_1(E,k1_relat_1(A))
=> ( ~ ( r2_hidden(E,k3_trees_1(k1_relat_1(A)))
& k1_funct_1(A,E) = C )
=> k1_funct_1(D,E) = k1_funct_1(A,E) ) )
& ! [E] :
( m1_trees_1(E,k1_relat_1(A))
=> ! [F] :
( m1_trees_1(F,k1_relat_1(B))
=> ( ( r2_hidden(E,k3_trees_1(k1_relat_1(A)))
& k1_funct_1(A,E) = C )
=> k1_funct_1(D,k8_finseq_1(k5_numbers,E,F)) = k1_funct_1(B,F) ) ) ) ) ) ) ) ) ).
fof(t23_trees_4,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] :
( ~ ( r2_hidden(C,k2_relat_1(A))
& r2_hidden(C,k4_trees_2(A)) )
=> k13_trees_4(A,B,C) = A ) ) ) ).
fof(t24_trees_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,k2_zfmisc_1(A,B)) )
=> ( k1_relat_1(k10_trees_3(A,B,C)) = k1_relat_1(C)
& k1_relat_1(k11_trees_3(A,B,C)) = k1_relat_1(C) ) ) ) ) ).
fof(t25_trees_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ( k10_trees_3(A,B,k2_trees_4(k2_zfmisc_1(A,B),k1_domain_1(A,B,C,D))) = k2_trees_4(A,C)
& k11_trees_3(A,B,k2_trees_4(k2_zfmisc_1(A,B),k1_domain_1(A,B,C,D))) = k2_trees_4(B,D) ) ) ) ) ) ).
fof(t26_trees_4,axiom,
! [A,B] : k13_funct_3(k1_trees_4(A),k1_trees_4(B)) = k1_trees_4(k4_tarski(A,B)) ).
fof(t27_trees_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m4_trees_3(E,k2_zfmisc_1(A,B)) )
=> ! [F] :
( ( ~ v1_xboole_0(F)
& m4_trees_3(F,A) )
=> ! [G] :
( m2_finseq_1(G,E)
=> ! [H] :
( m2_finseq_1(H,F)
=> ( ( k4_finseq_1(H) = k4_finseq_1(G)
& ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ( r2_hidden(I,k4_finseq_1(G))
=> ! [J] :
( ( v1_funct_1(J)
& v3_trees_2(J)
& m3_trees_2(J,k2_zfmisc_1(A,B)) )
=> ( J = k1_funct_1(G,I)
=> k1_funct_1(H,I) = k10_trees_3(A,B,J) ) ) ) ) )
=> k10_trees_3(A,B,k8_trees_4(k2_zfmisc_1(A,B),E,k1_domain_1(A,B,C,D),G)) = k8_trees_4(A,F,C,H) ) ) ) ) ) ) ) ) ) ).
fof(t28_trees_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m4_trees_3(E,k2_zfmisc_1(A,B)) )
=> ! [F] :
( ( ~ v1_xboole_0(F)
& m4_trees_3(F,B) )
=> ! [G] :
( m2_finseq_1(G,E)
=> ! [H] :
( m2_finseq_1(H,F)
=> ( ( k4_finseq_1(H) = k4_finseq_1(G)
& ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ( r2_hidden(I,k4_finseq_1(G))
=> ! [J] :
( ( v1_funct_1(J)
& v3_trees_2(J)
& m3_trees_2(J,k2_zfmisc_1(A,B)) )
=> ( J = k1_funct_1(G,I)
=> k1_funct_1(H,I) = k11_trees_3(A,B,J) ) ) ) ) )
=> k11_trees_3(A,B,k8_trees_4(k2_zfmisc_1(A,B),E,k1_domain_1(A,B,C,D),G)) = k8_trees_4(B,F,D,H) ) ) ) ) ) ) ) ) ) ).
fof(t29_trees_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m4_trees_3(E,k2_zfmisc_1(A,B)) )
=> ! [F] :
( m2_finseq_1(F,E)
=> ? [G] :
( m2_finseq_1(G,k4_trees_3(A))
& k4_finseq_1(G) = k4_finseq_1(F)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(H,k4_finseq_1(F))
& ! [I] :
( m5_trees_3(I,k2_zfmisc_1(A,B),E)
=> ~ ( I = k1_funct_1(F,H)
& k1_funct_1(G,H) = k10_trees_3(A,B,I) ) ) ) )
& k10_trees_3(A,B,k8_trees_4(k2_zfmisc_1(A,B),E,k1_domain_1(A,B,C,D),F)) = k8_trees_4(A,k4_trees_3(A),C,G) ) ) ) ) ) ) ) ).
fof(t30_trees_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m4_trees_3(E,k2_zfmisc_1(A,B)) )
=> ! [F] :
( m2_finseq_1(F,E)
=> ? [G] :
( m2_finseq_1(G,k4_trees_3(B))
& k4_finseq_1(G) = k4_finseq_1(F)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(H,k4_finseq_1(F))
& ! [I] :
( m5_trees_3(I,k2_zfmisc_1(A,B),E)
=> ~ ( I = k1_funct_1(F,H)
& k1_funct_1(G,H) = k11_trees_3(A,B,I) ) ) ) )
& k11_trees_3(A,B,k8_trees_4(k2_zfmisc_1(A,B),E,k1_domain_1(A,B,C,D),F)) = k8_trees_4(B,k4_trees_3(B),D,G) ) ) ) ) ) ) ) ).
fof(t31_trees_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( m2_finseq_1(E,k5_trees_3(k2_zfmisc_1(A,B)))
=> ? [F] :
( m2_finseq_1(F,k5_trees_3(A))
& k4_finseq_1(F) = k4_finseq_1(E)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(G,k4_finseq_1(E))
& ! [H] :
( m5_trees_3(H,k2_zfmisc_1(A,B),k5_trees_3(k2_zfmisc_1(A,B)))
=> ~ ( H = k1_funct_1(E,G)
& k1_funct_1(F,G) = k10_trees_3(A,B,H) ) ) ) )
& k10_trees_3(A,B,k12_trees_4(k2_zfmisc_1(A,B),k1_domain_1(A,B,C,D),E)) = k12_trees_4(A,C,F) ) ) ) ) ) ) ).
fof(t32_trees_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( m2_finseq_1(E,k5_trees_3(k2_zfmisc_1(A,B)))
=> ? [F] :
( m2_finseq_1(F,k5_trees_3(B))
& k4_finseq_1(F) = k4_finseq_1(E)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(G,k4_finseq_1(E))
& ! [H] :
( m5_trees_3(H,k2_zfmisc_1(A,B),k5_trees_3(k2_zfmisc_1(A,B)))
=> ~ ( H = k1_funct_1(E,G)
& k1_funct_1(F,G) = k11_trees_3(A,B,H) ) ) ) )
& k11_trees_3(A,B,k12_trees_4(k2_zfmisc_1(A,B),k1_domain_1(A,B,C,D),E)) = k12_trees_4(B,D,F) ) ) ) ) ) ) ).
fof(s1_trees_4,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_trees_1(A)
& ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( r2_hidden(B,A)
<=> ~ ( ~ r2_hidden(B,f1_s1_trees_4)
& ! [C] :
( m1_trees_1(C,f1_s1_trees_4)
=> ! [D] :
( m1_trees_1(D,f2_s1_trees_4)
=> ~ ( p1_s1_trees_4(C)
& B = k8_finseq_1(k5_numbers,C,D) ) ) ) ) ) ) ) ).
fof(dt_m1_trees_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m1_trees_4(C,A,B)
=> m2_finseq_1(C,A) ) ) ).
fof(existence_m1_trees_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ? [C] : m1_trees_4(C,A,B) ) ).
fof(redefinition_m1_trees_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m1_trees_4(C,A,B)
<=> m1_finseq_1(C,B) ) ) ).
fof(dt_k1_trees_4,axiom,
! [A] :
( v1_relat_1(k1_trees_4(A))
& v1_funct_1(k1_trees_4(A))
& v3_trees_2(k1_trees_4(A)) ) ).
fof(dt_k2_trees_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m5_trees_3(k2_trees_4(A,B),A,k5_trees_3(A)) ) ).
fof(redefinition_k2_trees_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> k2_trees_4(A,B) = k1_trees_4(B) ) ).
fof(dt_k3_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( v1_relat_1(k3_trees_4(A,B))
& v1_funct_1(k3_trees_4(A,B))
& v3_trees_2(k3_trees_4(A,B)) ) ) ).
fof(dt_k4_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( v1_relat_1(k4_trees_4(A,B))
& v1_funct_1(k4_trees_4(A,B))
& v3_trees_2(k4_trees_4(A,B)) ) ) ).
fof(dt_k5_trees_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ( v1_relat_1(k5_trees_4(A,B))
& v1_funct_1(k5_trees_4(A,B))
& v3_trees_2(k5_trees_4(A,B)) ) ) ).
fof(dt_k6_trees_4,axiom,
! [A,B,C] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B)
& v1_relat_1(C)
& v1_funct_1(C)
& v3_trees_2(C) )
=> ( v1_relat_1(k6_trees_4(A,B,C))
& v1_funct_1(k6_trees_4(A,B,C))
& v3_trees_2(k6_trees_4(A,B,C)) ) ) ).
fof(dt_k7_trees_4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_finseq_1(C,A) )
=> ( v1_funct_1(k7_trees_4(A,B,C))
& v3_trees_2(k7_trees_4(A,B,C))
& m3_trees_2(k7_trees_4(A,B,C),A) ) ) ).
fof(redefinition_k7_trees_4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_finseq_1(C,A) )
=> k7_trees_4(A,B,C) = k3_trees_4(B,C) ) ).
fof(dt_k8_trees_4,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m4_trees_3(B,A)
& m1_subset_1(C,A)
& m1_finseq_1(D,B) )
=> ( v1_funct_1(k8_trees_4(A,B,C,D))
& v3_trees_2(k8_trees_4(A,B,C,D))
& m3_trees_2(k8_trees_4(A,B,C,D),A) ) ) ).
fof(redefinition_k8_trees_4,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m4_trees_3(B,A)
& m1_subset_1(C,A)
& m1_finseq_1(D,B) )
=> k8_trees_4(A,B,C,D) = k4_trees_4(C,D) ) ).
fof(dt_k9_trees_4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,A) )
=> ( v1_funct_1(k9_trees_4(A,B,C))
& v3_trees_2(k9_trees_4(A,B,C))
& m3_trees_2(k9_trees_4(A,B,C),A) ) ) ).
fof(redefinition_k9_trees_4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,A) )
=> k9_trees_4(A,B,C) = k5_trees_4(B,C) ) ).
fof(dt_k10_trees_4,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,A)
& v1_funct_1(D)
& v3_trees_2(D)
& m3_trees_2(D,A) )
=> ( v1_funct_1(k10_trees_4(A,B,C,D))
& v3_trees_2(k10_trees_4(A,B,C,D))
& m3_trees_2(k10_trees_4(A,B,C,D),A) ) ) ).
fof(redefinition_k10_trees_4,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,A)
& v1_funct_1(D)
& v3_trees_2(D)
& m3_trees_2(D,A) )
=> k10_trees_4(A,B,C,D) = k6_trees_4(B,C,D) ) ).
fof(dt_k11_trees_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,k5_trees_3(A)) )
=> m2_finseq_1(k11_trees_4(A,B),k2_trees_3) ) ).
fof(redefinition_k11_trees_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,k5_trees_3(A)) )
=> k11_trees_4(A,B) = k2_funct_6(B) ) ).
fof(dt_k12_trees_4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_finseq_1(C,k5_trees_3(A)) )
=> m5_trees_3(k12_trees_4(A,B,C),A,k5_trees_3(A)) ) ).
fof(redefinition_k12_trees_4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_finseq_1(C,k5_trees_3(A)) )
=> k12_trees_4(A,B,C) = k4_trees_4(B,C) ) ).
fof(dt_k13_trees_4,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v3_trees_2(B) )
=> ( v1_relat_1(k13_trees_4(A,B,C))
& v1_funct_1(k13_trees_4(A,B,C))
& v3_trees_2(k13_trees_4(A,B,C)) ) ) ).
fof(dt_k14_trees_4,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A)
& v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,A) )
=> ( v1_funct_1(k14_trees_4(A,B,C,D))
& v3_trees_2(k14_trees_4(A,B,C,D))
& m3_trees_2(k14_trees_4(A,B,C,D),A) ) ) ).
fof(redefinition_k14_trees_4,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A)
& v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,A) )
=> k14_trees_4(A,B,C,D) = k13_trees_4(B,C,D) ) ).
%------------------------------------------------------------------------------