SET007 Axioms: SET007+185.ax
%------------------------------------------------------------------------------
% File : SET007+185 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Subformula Tree of a Formula of the First Order Language
% 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 : qc_lang4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 100 ( 9 unt; 0 def)
% Number of atoms : 556 ( 81 equ)
% Maximal formula atoms : 14 ( 5 avg)
% Number of connectives : 512 ( 56 ~; 1 |; 184 &)
% ( 21 <=>; 250 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 36 ( 34 usr; 1 prp; 0-3 aty)
% Number of functors : 57 ( 57 usr; 11 con; 0-4 aty)
% Number of variables : 272 ( 249 !; 23 ?)
% 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_qc_lang4,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)
& v3_trees_9(B)
& v4_trees_9(B) ) ) ).
fof(t1_qc_lang4,axiom,
$true ).
fof(t2_qc_lang4,axiom,
$true ).
fof(t3_qc_lang4,axiom,
$true ).
fof(t4_qc_lang4,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_tarski(C,B) ) ) ) ).
fof(t5_qc_lang4,axiom,
$true ).
fof(t6_qc_lang4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [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] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(k1_nat_1(E,np__1),k3_finseq_1(B))
& C = k7_relat_1(B,k2_finseq_1(k1_nat_1(E,np__1)))
& D = k7_relat_1(B,k2_finseq_1(E))
& ! [F] :
( m1_subset_1(F,A)
=> C != k7_finseq_1(D,k12_finseq_1(A,F)) ) ) ) ) ) ) ) ).
fof(t7_qc_lang4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ~ ( r1_xreal_0(np__1,k3_finseq_1(B))
& C = k7_relat_1(B,k2_finseq_1(np__1))
& ! [D] :
( m1_subset_1(D,A)
=> C != k12_finseq_1(A,D) ) ) ) ) ) ).
fof(t8_qc_lang4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> k1_funct_1(A,B) = k1_funct_1(k5_trees_2(A,B),k1_xboole_0) ) ) ).
fof(t9_qc_lang4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> k2_trees_9(A,B) = k5_relat_1(k1_trees_9(k1_relat_1(A),B),A) ) ) ).
fof(t10_qc_lang4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> k1_relat_1(k5_relat_1(k1_trees_9(k1_relat_1(A),B),A)) = k4_finseq_1(k1_trees_9(k1_relat_1(A),B)) ) ) ).
fof(t11_qc_lang4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> k4_finseq_1(k2_trees_9(A,B)) = k4_finseq_1(k1_trees_9(k1_relat_1(A),B)) ) ) ).
fof(t12_qc_lang4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(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_relat_1(A))
<=> r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(k1_trees_9(k1_relat_1(A),B))) ) ) ) ) ).
fof(t13_qc_lang4,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) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(k7_finseq_1(B,k12_finseq_1(k5_numbers,C)),k1_relat_1(A))
=> k1_funct_1(A,k7_finseq_1(B,k12_finseq_1(k5_numbers,C))) = k1_funct_1(k2_trees_9(A,B),k1_nat_1(C,np__1)) ) ) ) ) ).
fof(t14_qc_lang4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> ! [C] :
( m1_trees_1(C,k1_relat_1(A))
=> ( r2_hidden(B,k1_trees_2(k1_relat_1(A),C))
=> r2_hidden(k1_funct_1(A,B),k2_relat_1(k2_trees_9(A,C))) ) ) ) ) ).
fof(t15_qc_lang4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> ! [C] :
~ ( r2_hidden(C,k2_relat_1(k2_trees_9(A,B)))
& ! [D] :
( m1_trees_1(D,k1_relat_1(A))
=> ~ ( C = k1_funct_1(A,D)
& r2_hidden(D,k1_trees_2(k1_relat_1(A),B)) ) ) ) ) ) ).
fof(t16_qc_lang4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ( v1_finset_1(k1_trees_1(B))
& m1_trees_2(k1_trees_1(B),A) ) ) ) ).
fof(t17_qc_lang4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> k2_trees_2(A,np__0) = k1_tarski(k1_xboole_0) ) ).
fof(t19_qc_lang4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v2_trees_9(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> v1_finset_1(k2_trees_2(A,B)) ) ) ).
fof(t20_qc_lang4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v2_trees_9(A) )
=> ( v1_finset_1(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& k2_trees_2(A,B) = k1_xboole_0 ) ) ) ).
fof(t21_qc_lang4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v2_trees_9(A) )
=> ~ ( ~ v1_finset_1(A)
& ! [B] :
( m1_trees_2(B,A)
=> v1_finset_1(B) ) ) ) ).
fof(t22_qc_lang4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v2_trees_9(A) )
=> ~ ( ~ v1_finset_1(A)
& ! [B] :
( ( v2_trees_2(B,A)
& m1_trees_2(B,A) )
=> v1_finset_1(B) ) ) ) ).
fof(t23_qc_lang4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_2(B,A)
=> ! [C] :
( m1_trees_1(C,A)
=> ~ ( r2_hidden(C,B)
& ~ v1_finset_1(B)
& ! [D] :
( m1_trees_1(D,A)
=> ~ ( r2_hidden(D,B)
& r2_xboole_0(C,D) ) ) ) ) ) ) ).
fof(t24_qc_lang4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( ( v2_trees_2(B,A)
& m1_trees_2(B,A) )
=> ! [C] :
( m1_trees_1(C,A)
=> ~ ( r2_hidden(C,B)
& ~ v1_finset_1(B)
& ! [D] :
( m1_trees_1(D,A)
=> ~ ( r2_hidden(D,B)
& r2_hidden(D,k1_trees_2(A,C)) ) ) ) ) ) ) ).
fof(t25_qc_lang4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k5_numbers)
& m2_relset_1(A,k5_numbers,k5_numbers) )
=> ~ ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k8_funct_2(k5_numbers,k5_numbers,A,k1_nat_1(B,np__1)),k8_funct_2(k5_numbers,k5_numbers,A,B)) )
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& r1_xreal_0(B,C)
& k8_funct_2(k5_numbers,k5_numbers,A,C) != k8_funct_2(k5_numbers,k5_numbers,A,B) ) ) ) ) ).
fof(t26_qc_lang4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> ! [C] :
( r2_hidden(C,k2_relat_1(B))
=> m1_subset_1(C,A) ) ) ) ).
fof(t27_qc_lang4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> ! [C] :
( r2_hidden(C,k1_relat_1(B))
=> m1_subset_1(k1_funct_1(B,C),A) ) ) ) ).
fof(t28_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( r2_qc_lang2(A,B)
=> r1_xreal_0(k3_finseq_1(k10_qc_lang1(A)),k3_finseq_1(k10_qc_lang1(B))) ) ) ) ).
fof(t29_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( ( r2_qc_lang2(A,B)
& k3_finseq_1(k10_qc_lang1(A)) = k3_finseq_1(k10_qc_lang1(B)) )
=> A = B ) ) ) ).
fof(d1_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( ( ( A = k11_qc_lang1
| v2_qc_lang1(A) )
=> k1_qc_lang4(A) = k6_finseq_1(k8_qc_lang1) )
& ( v3_qc_lang1(A)
=> k1_qc_lang4(A) = k12_finseq_1(k8_qc_lang1,k17_qc_lang1(A)) )
& ( v4_qc_lang1(A)
=> k1_qc_lang4(A) = k2_finseq_4(k8_qc_lang1,k18_qc_lang1(A),k19_qc_lang1(A)) )
& ~ ( A != k11_qc_lang1
& ~ v2_qc_lang1(A)
& ~ v3_qc_lang1(A)
& ~ v4_qc_lang1(A)
& k1_qc_lang4(A) != k12_finseq_1(k8_qc_lang1,k21_qc_lang1(A)) ) ) ) ).
fof(t30_qc_lang4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k8_qc_lang1)
=> ( ( r2_hidden(A,k4_finseq_1(k1_qc_lang4(B)))
& C = k1_funct_1(k1_qc_lang4(B),A) )
=> r1_qc_lang2(C,B) ) ) ) ) ).
fof(d2_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( v1_funct_1(B)
& v1_finset_1(B)
& v3_trees_2(B)
& m3_trees_2(B,k8_qc_lang1) )
=> ( B = k2_qc_lang4(A)
<=> ( k1_funct_1(B,k1_xboole_0) = A
& ! [C] :
( m1_trees_1(C,k1_relat_1(B))
=> k2_trees_9(B,C) = k1_qc_lang4(k3_trees_2(k8_qc_lang1,B,C)) ) ) ) ) ) ).
fof(t32_qc_lang4,axiom,
$true ).
fof(t33_qc_lang4,axiom,
$true ).
fof(t34_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> r2_hidden(A,k2_relat_1(k2_qc_lang4(A))) ) ).
fof(t35_qc_lang4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(B)))
=> ~ ( r2_hidden(k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,A)),k1_relat_1(k2_qc_lang4(B)))
& ! [D] :
( m1_subset_1(D,k8_qc_lang1)
=> ~ ( D = k1_funct_1(k2_qc_lang4(B),k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,A)))
& r1_qc_lang2(D,k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),C)) ) ) ) ) ) ) ).
fof(t36_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(B)))
=> ( r1_qc_lang2(A,k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r2_hidden(k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D)),k1_relat_1(k2_qc_lang4(B)))
& A = k1_funct_1(k2_qc_lang4(B),k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D))) ) ) ) ) ) ).
fof(t37_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k8_qc_lang1)
=> ( ( r2_hidden(A,k2_relat_1(k2_qc_lang4(B)))
& r1_qc_lang2(C,A) )
=> r2_hidden(C,k2_relat_1(k2_qc_lang4(B))) ) ) ) ) ).
fof(t38_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k8_qc_lang1)
=> ( ( r2_hidden(A,k2_relat_1(k2_qc_lang4(B)))
& r2_qc_lang2(C,A) )
=> r2_hidden(C,k2_relat_1(k2_qc_lang4(B))) ) ) ) ) ).
fof(t39_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( r2_hidden(A,k2_relat_1(k2_qc_lang4(B)))
<=> r2_qc_lang2(A,B) ) ) ) ).
fof(t40_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> k2_relat_1(k2_qc_lang4(A)) = k15_qc_lang2(A) ) ).
fof(t41_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(A)))
=> ( r2_hidden(B,k1_trees_2(k1_relat_1(k2_qc_lang4(A)),C))
=> r1_qc_lang2(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B),k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),C)) ) ) ) ) ).
fof(t42_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(A)))
=> ( r1_tarski(B,C)
=> r2_qc_lang2(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),C),k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B)) ) ) ) ) ).
fof(t43_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(A)))
=> ~ ( r2_xboole_0(B,C)
& r1_xreal_0(k3_finseq_1(k10_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B))),k3_finseq_1(k10_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),C)))) ) ) ) ) ).
fof(t44_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(A)))
=> ~ ( r2_xboole_0(B,C)
& k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),C) = k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B) ) ) ) ) ).
fof(t45_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(A)))
=> ( r2_xboole_0(B,C)
=> r3_qc_lang2(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),C),k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B)) ) ) ) ) ).
fof(t46_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> ( k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B) = A
<=> B = k1_xboole_0 ) ) ) ).
fof(t47_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(A)))
=> ~ ( B != C
& k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B) = k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),C)
& r3_xboole_0(B,C) ) ) ) ) ).
fof(d3_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m4_trees_1(C,k1_relat_1(k2_qc_lang4(A)))
=> ( C = k3_qc_lang4(A,B)
<=> ! [D] :
( m1_trees_1(D,k1_relat_1(k2_qc_lang4(A)))
=> ( r2_hidden(D,C)
<=> k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),D) = B ) ) ) ) ) ) ).
fof(t48_qc_lang4,axiom,
$true ).
fof(t50_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( r2_qc_lang2(A,B)
<=> k3_qc_lang4(B,A) != k1_xboole_0 ) ) ) ).
fof(t51_qc_lang4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(B)))
=> ! [D] :
( m1_trees_1(D,k1_relat_1(k2_qc_lang4(B)))
=> ( ( C = k8_finseq_1(k5_numbers,D,k12_finseq_1(k5_numbers,A))
& v3_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),D)) )
=> ( k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),C) = k17_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),D))
& A = np__0 ) ) ) ) ) ) ).
fof(t52_qc_lang4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(B)))
=> ! [D] :
( m1_trees_1(D,k1_relat_1(k2_qc_lang4(B)))
=> ~ ( C = k8_finseq_1(k5_numbers,D,k12_finseq_1(k5_numbers,A))
& v4_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),D))
& ~ ( k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),C) = k18_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),D))
& A = np__0 )
& ~ ( k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),C) = k19_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),D))
& A = np__1 ) ) ) ) ) ) ).
fof(t53_qc_lang4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(B)))
=> ! [D] :
( m1_trees_1(D,k1_relat_1(k2_qc_lang4(B)))
=> ( ( C = k8_finseq_1(k5_numbers,D,k12_finseq_1(k5_numbers,A))
& v5_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),D)) )
=> ( k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),C) = k21_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),D))
& A = np__0 ) ) ) ) ) ) ).
fof(t54_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> ( v3_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B))
=> ( r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,np__0)),k1_relat_1(k2_qc_lang4(A)))
& k1_funct_1(k2_qc_lang4(A),k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,np__0))) = k17_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B)) ) ) ) ) ).
fof(t55_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> ( v4_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B))
=> ( r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,np__0)),k1_relat_1(k2_qc_lang4(A)))
& k1_funct_1(k2_qc_lang4(A),k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,np__0))) = k18_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B))
& r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,np__1)),k1_relat_1(k2_qc_lang4(A)))
& k1_funct_1(k2_qc_lang4(A),k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,np__1))) = k19_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B)) ) ) ) ) ).
fof(t56_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> ( v5_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B))
=> ( r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,np__0)),k1_relat_1(k2_qc_lang4(A)))
& k1_funct_1(k2_qc_lang4(A),k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,np__0))) = k21_qc_lang1(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B)) ) ) ) ) ).
fof(t57_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k8_qc_lang1)
=> ! [D] :
( m1_trees_1(D,k1_relat_1(k2_qc_lang4(A)))
=> ! [E] :
( m1_trees_1(E,k1_relat_1(k2_qc_lang4(B)))
=> ( ( r2_hidden(D,k3_qc_lang4(A,B))
& r2_hidden(E,k3_qc_lang4(B,C)) )
=> r2_hidden(k8_finseq_1(k5_numbers,D,E),k3_qc_lang4(A,C)) ) ) ) ) ) ) ).
fof(t58_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k8_qc_lang1)
=> ! [D] :
( m1_trees_1(D,k1_relat_1(k2_qc_lang4(A)))
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E) )
=> ( ( r2_hidden(D,k3_qc_lang4(A,B))
& r2_hidden(k7_finseq_1(D,E),k3_qc_lang4(A,C)) )
=> r2_hidden(E,k3_qc_lang4(B,C)) ) ) ) ) ) ) ).
fof(t60_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_trees_1(B,k1_relat_1(k2_qc_lang4(A)))
=> k7_trees_2(k8_qc_lang1,k2_qc_lang4(A),B) = k2_qc_lang4(k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),B)) ) ) ).
fof(t61_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(A)))
=> ( r2_hidden(C,k3_qc_lang4(A,B))
<=> k7_trees_2(k8_qc_lang1,k2_qc_lang4(A),C) = k2_qc_lang4(B) ) ) ) ) ).
fof(d4_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( m1_qc_lang4(B,A)
<=> r2_qc_lang2(B,A) ) ) ) ).
fof(d5_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> ! [C] :
( m1_trees_1(C,k1_relat_1(k2_qc_lang4(A)))
=> ( m2_qc_lang4(C,A,B)
<=> k3_trees_2(k8_qc_lang1,k2_qc_lang4(A),C) = B ) ) ) ) ).
fof(t64_qc_lang4,axiom,
$true ).
fof(t65_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> ! [C] :
( m2_qc_lang4(C,A,B)
=> ! [D] :
( m2_qc_lang4(D,A,B)
=> ~ ( C != D
& r3_xboole_0(C,D) ) ) ) ) ) ).
fof(d6_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> k4_qc_lang4(A,B) = k3_qc_lang4(A,B) ) ) ).
fof(t66_qc_lang4,axiom,
$true ).
fof(t67_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> ! [C] :
( m2_qc_lang4(C,A,B)
=> r2_hidden(C,k4_qc_lang4(A,B)) ) ) ) ).
fof(t69_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> ! [C] :
( m1_qc_lang4(C,A)
=> ! [D] :
( m2_qc_lang4(D,A,B)
=> ! [E] :
( m1_trees_1(E,k1_relat_1(k2_qc_lang4(B)))
=> ( r2_hidden(E,k3_qc_lang4(B,C))
=> m2_qc_lang4(k8_finseq_1(k5_numbers,D,E),A,C) ) ) ) ) ) ) ).
fof(t70_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> ! [C] :
( m1_qc_lang4(C,A)
=> ! [D] :
( m2_qc_lang4(D,A,C)
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E) )
=> ( m2_qc_lang4(k7_finseq_1(D,E),A,B)
=> r2_hidden(E,k3_qc_lang4(C,B)) ) ) ) ) ) ) ).
fof(t73_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> ! [C] :
( m1_qc_lang4(C,A)
=> ( ? [D] :
( m2_qc_lang4(D,A,B)
& ? [E] :
( m2_qc_lang4(E,A,C)
& r1_tarski(D,E) ) )
=> r2_qc_lang2(C,B) ) ) ) ) ).
fof(t74_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> ! [C] :
( m1_qc_lang4(C,A)
=> ( r2_qc_lang2(B,C)
=> ! [D] :
( m2_qc_lang4(D,A,C)
=> ? [E] :
( m2_qc_lang4(E,A,B)
& r1_tarski(D,E) ) ) ) ) ) ) ).
fof(s1_qc_lang4,axiom,
? [A] :
( v1_funct_1(A)
& v3_trees_2(A)
& v4_trees_9(A)
& m3_trees_2(A,f1_s1_qc_lang4)
& k1_funct_1(A,k1_xboole_0) = f2_s1_qc_lang4
& ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> ! [C] :
( m1_subset_1(C,f1_s1_qc_lang4)
=> ( C = k3_trees_2(f1_s1_qc_lang4,A,B)
=> k2_trees_9(A,B) = f3_s1_qc_lang4(C) ) ) ) ) ).
fof(s2_qc_lang4,axiom,
( ! [A] :
( m1_trees_1(A,k1_relat_1(f2_s2_qc_lang4))
=> ! [B] :
( m1_trees_1(B,k1_relat_1(f2_s2_qc_lang4))
=> ! [C] :
( m1_subset_1(C,f1_s2_qc_lang4)
=> ~ ( r2_hidden(B,k1_trees_2(k1_relat_1(f2_s2_qc_lang4),A))
& C = k3_trees_2(f1_s2_qc_lang4,f2_s2_qc_lang4,B)
& r1_xreal_0(f3_s2_qc_lang4(k3_trees_2(f1_s2_qc_lang4,f2_s2_qc_lang4,A)),f3_s2_qc_lang4(C)) ) ) ) )
=> v1_finset_1(f2_s2_qc_lang4) ) ).
fof(dt_m1_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> m1_subset_1(B,k8_qc_lang1) ) ) ).
fof(existence_m1_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ? [B] : m1_qc_lang4(B,A) ) ).
fof(dt_m2_qc_lang4,axiom,
! [A,B] :
( ( m1_subset_1(A,k8_qc_lang1)
& m1_qc_lang4(B,A) )
=> ! [C] :
( m2_qc_lang4(C,A,B)
=> m1_trees_1(C,k1_relat_1(k2_qc_lang4(A))) ) ) ).
fof(existence_m2_qc_lang4,axiom,
! [A,B] :
( ( m1_subset_1(A,k8_qc_lang1)
& m1_qc_lang4(B,A) )
=> ? [C] : m2_qc_lang4(C,A,B) ) ).
fof(dt_k1_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m2_finseq_1(k1_qc_lang4(A),k8_qc_lang1) ) ).
fof(dt_k2_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v1_funct_1(k2_qc_lang4(A))
& v1_finset_1(k2_qc_lang4(A))
& v3_trees_2(k2_qc_lang4(A))
& m3_trees_2(k2_qc_lang4(A),k8_qc_lang1) ) ) ).
fof(dt_k3_qc_lang4,axiom,
! [A,B] :
( ( m1_subset_1(A,k8_qc_lang1)
& m1_subset_1(B,k8_qc_lang1) )
=> m4_trees_1(k3_qc_lang4(A,B),k1_relat_1(k2_qc_lang4(A))) ) ).
fof(dt_k4_qc_lang4,axiom,
! [A,B] :
( ( m1_subset_1(A,k8_qc_lang1)
& m1_qc_lang4(B,A) )
=> ( ~ v1_xboole_0(k4_qc_lang4(A,B))
& m4_trees_1(k4_qc_lang4(A,B),k1_relat_1(k2_qc_lang4(A))) ) ) ).
fof(t18_qc_lang4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B) )
=> k2_trees_2(B,k1_nat_1(A,np__1)) = k3_tarski(a_2_0_qc_lang4(A,B)) ) ) ).
fof(t31_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> k2_relat_1(k1_qc_lang4(A)) = a_1_0_qc_lang4(A) ) ).
fof(t49_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k3_qc_lang4(A,B) = a_2_1_qc_lang4(A,B) ) ) ).
fof(t59_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k8_qc_lang1)
=> r1_tarski(a_3_0_qc_lang4(A,B,C),k3_qc_lang4(A,C)) ) ) ) ).
fof(t62_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k3_qc_lang4(A,B) = a_2_2_qc_lang4(A,B) ) ) ).
fof(t63_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k8_qc_lang1)
=> ! [D] :
( m1_trees_2(D,k1_relat_1(k2_qc_lang4(A)))
=> ~ ( r2_hidden(B,a_2_3_qc_lang4(A,D))
& r2_hidden(C,a_2_3_qc_lang4(A,D))
& ~ r2_qc_lang2(B,C)
& ~ r2_qc_lang2(C,B) ) ) ) ) ) ).
fof(t68_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> k4_qc_lang4(A,B) = a_2_4_qc_lang4(A,B) ) ) ).
fof(t71_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> ! [C] :
( m1_qc_lang4(C,A)
=> a_3_1_qc_lang4(A,B,C) = a_3_2_qc_lang4(A,B,C) ) ) ) ).
fof(t72_qc_lang4,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_qc_lang4(B,A)
=> ! [C] :
( m1_qc_lang4(C,A)
=> r1_tarski(a_3_1_qc_lang4(A,B,C),k4_qc_lang4(A,C)) ) ) ) ).
fof(fraenkel_a_2_0_qc_lang4,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& ~ v1_xboole_0(C)
& v1_trees_1(C) )
=> ( r2_hidden(A,a_2_0_qc_lang4(B,C))
<=> ? [D] :
( m1_trees_1(D,C)
& A = k1_trees_2(C,D)
& k3_finseq_1(D) = B ) ) ) ).
fof(fraenkel_a_1_0_qc_lang4,axiom,
! [A,B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( r2_hidden(A,a_1_0_qc_lang4(B))
<=> ? [C] :
( m1_subset_1(C,k8_qc_lang1)
& A = C
& r1_qc_lang2(C,B) ) ) ) ).
fof(fraenkel_a_2_1_qc_lang4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k8_qc_lang1)
& m1_subset_1(C,k8_qc_lang1) )
=> ( r2_hidden(A,a_2_1_qc_lang4(B,C))
<=> ? [D] :
( m1_trees_1(D,k1_relat_1(k2_qc_lang4(B)))
& A = D
& k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),D) = C ) ) ) ).
fof(fraenkel_a_3_0_qc_lang4,axiom,
! [A,B,C,D] :
( ( m1_subset_1(B,k8_qc_lang1)
& m1_subset_1(C,k8_qc_lang1)
& m1_subset_1(D,k8_qc_lang1) )
=> ( r2_hidden(A,a_3_0_qc_lang4(B,C,D))
<=> ? [E,F] :
( m1_trees_1(E,k1_relat_1(k2_qc_lang4(B)))
& m1_trees_1(F,k1_relat_1(k2_qc_lang4(C)))
& A = k8_finseq_1(k5_numbers,E,F)
& r2_hidden(E,k3_qc_lang4(B,C))
& r2_hidden(F,k3_qc_lang4(C,D)) ) ) ) ).
fof(fraenkel_a_2_2_qc_lang4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k8_qc_lang1)
& m1_subset_1(C,k8_qc_lang1) )
=> ( r2_hidden(A,a_2_2_qc_lang4(B,C))
<=> ? [D] :
( m1_trees_1(D,k1_relat_1(k2_qc_lang4(B)))
& A = D
& k7_trees_2(k8_qc_lang1,k2_qc_lang4(B),D) = k2_qc_lang4(C) ) ) ) ).
fof(fraenkel_a_2_3_qc_lang4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k8_qc_lang1)
& m1_trees_2(C,k1_relat_1(k2_qc_lang4(B))) )
=> ( r2_hidden(A,a_2_3_qc_lang4(B,C))
<=> ? [D] :
( m1_trees_1(D,k1_relat_1(k2_qc_lang4(B)))
& A = k3_trees_2(k8_qc_lang1,k2_qc_lang4(B),D)
& r2_hidden(D,C) ) ) ) ).
fof(fraenkel_a_2_4_qc_lang4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k8_qc_lang1)
& m1_qc_lang4(C,B) )
=> ( r2_hidden(A,a_2_4_qc_lang4(B,C))
<=> ? [D] :
( m2_qc_lang4(D,B,C)
& A = D
& D = D ) ) ) ).
fof(fraenkel_a_3_1_qc_lang4,axiom,
! [A,B,C,D] :
( ( m1_subset_1(B,k8_qc_lang1)
& m1_qc_lang4(C,B)
& m1_qc_lang4(D,B) )
=> ( r2_hidden(A,a_3_1_qc_lang4(B,C,D))
<=> ? [E,F] :
( m2_qc_lang4(E,B,C)
& m1_trees_1(F,k1_relat_1(k2_qc_lang4(C)))
& A = k8_finseq_1(k5_numbers,E,F)
& r2_hidden(F,k3_qc_lang4(C,D)) ) ) ) ).
fof(fraenkel_a_3_2_qc_lang4,axiom,
! [A,B,C,D] :
( ( m1_subset_1(B,k8_qc_lang1)
& m1_qc_lang4(C,B)
& m1_qc_lang4(D,B) )
=> ( r2_hidden(A,a_3_2_qc_lang4(B,C,D))
<=> ? [E,F] :
( m1_trees_1(E,k1_relat_1(k2_qc_lang4(B)))
& m1_trees_1(F,k1_relat_1(k2_qc_lang4(C)))
& A = k8_finseq_1(k5_numbers,E,F)
& r2_hidden(E,k3_qc_lang4(B,C))
& r2_hidden(F,k3_qc_lang4(C,D)) ) ) ) ).
%------------------------------------------------------------------------------