SET007 Axioms: SET007+162.ax
%------------------------------------------------------------------------------
% File : SET007+162 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Introduction to Modal Propositional Logic
% 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 : modal_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 108 ( 19 unt; 0 def)
% Number of atoms : 504 ( 104 equ)
% Maximal formula atoms : 23 ( 4 avg)
% Number of connectives : 493 ( 97 ~; 1 |; 171 &)
% ( 11 <=>; 213 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 31 ( 29 usr; 1 prp; 0-3 aty)
% Number of functors : 60 ( 60 usr; 13 con; 0-4 aty)
% Number of variables : 212 ( 199 !; 13 ?)
% 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(fc1_modal_1,axiom,
~ v1_xboole_0(k3_modal_1) ).
fof(fc2_modal_1,axiom,
~ v1_xboole_0(k4_modal_1) ).
fof(cc1_modal_1,axiom,
! [A] :
( m1_subset_1(A,k6_modal_1)
=> v1_finset_1(A) ) ).
fof(rc1_modal_1,axiom,
? [A] :
( m1_subset_1(A,k6_modal_1)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A)
& v1_modal_1(A) ) ).
fof(rc2_modal_1,axiom,
? [A] :
( m1_subset_1(A,k6_modal_1)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A)
& v2_modal_1(A) ) ).
fof(rc3_modal_1,axiom,
? [A] :
( m1_subset_1(A,k6_modal_1)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A)
& v3_modal_1(A) ) ).
fof(rc4_modal_1,axiom,
? [A] :
( m1_subset_1(A,k6_modal_1)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A)
& v4_modal_1(A) ) ).
fof(d1_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> k1_modal_1(A) = k1_xboole_0 ) ).
fof(d2_modal_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> k2_modal_1(A,B) = k3_trees_2(A,B,k1_modal_1(k1_relat_1(B))) ) ) ).
fof(t1_modal_1,axiom,
$true ).
fof(t2_modal_1,axiom,
$true ).
fof(t3_modal_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ~ ( A != B
& r3_xboole_0(k12_finseq_1(k5_numbers,A),k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,B),C)) ) ) ) ) ).
fof(t4_modal_1,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ~ ( A != k1_xboole_0
& ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> A != k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,C),B) ) ) ) ) ).
fof(t5_modal_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ~ ( A != B
& r2_xboole_0(k12_finseq_1(k5_numbers,A),k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,B),C)) ) ) ) ) ).
fof(t6_modal_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ~ ( A != B
& r1_tarski(k12_finseq_1(k5_numbers,A),k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,B),C)) ) ) ) ) ).
fof(t7_modal_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ r2_xboole_0(k12_finseq_1(k5_numbers,A),k12_finseq_1(k5_numbers,B)) ) ) ).
fof(t8_modal_1,axiom,
$true ).
fof(t9_modal_1,axiom,
k2_trees_1(np__1) = k2_tarski(k1_xboole_0,k12_finseq_1(k5_numbers,np__0)) ).
fof(t10_modal_1,axiom,
k2_trees_1(np__2) = k1_enumset1(k1_xboole_0,k12_finseq_1(k5_numbers,np__0),k12_finseq_1(k5_numbers,np__1)) ).
fof(t11_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [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(k12_finseq_1(k5_numbers,C),A) )
=> r2_hidden(k12_finseq_1(k5_numbers,B),A) ) ) ) ) ).
fof(t12_modal_1,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( r2_xboole_0(k8_finseq_1(k5_numbers,A,B),k8_finseq_1(k5_numbers,A,C))
=> r2_xboole_0(B,C) ) ) ) ) ).
fof(t13_modal_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v3_trees_2(A)
& m3_trees_2(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) )
=> r2_hidden(A,k4_partfun1(k3_finseq_2(k5_numbers),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))) ) ).
fof(t14_modal_1,axiom,
$true ).
fof(t15_modal_1,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] :
( m1_trees_1(D,A)
=> ( k5_trees_1(A,D,B) = k5_trees_1(A,D,C)
=> B = C ) ) ) ) ) ).
fof(t16_modal_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v3_trees_2(D)
& m3_trees_2(D,A) )
=> ! [E] :
( m1_trees_1(E,k1_relat_1(B))
=> ( k8_trees_2(B,E,C) = k8_trees_2(B,E,D)
=> C = D ) ) ) ) ) ) ).
fof(t17_modal_1,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)
=> ! [D] :
( m1_trees_1(D,k5_trees_1(A,C,B))
=> ! [E] :
( m1_trees_1(E,A)
=> ( ( D = E
& r2_xboole_0(E,C) )
=> k1_trees_2(k5_trees_1(A,C,B),D) = k1_trees_2(A,E) ) ) ) ) ) ) ) ).
fof(t18_modal_1,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)
=> ! [D] :
( m1_trees_1(D,k5_trees_1(A,C,B))
=> ! [E] :
( m1_trees_1(E,A)
=> ( D = E
=> ( r3_xboole_0(C,E)
| k1_trees_2(k5_trees_1(A,C,B),D) = k1_trees_2(A,E) ) ) ) ) ) ) ) ) ).
fof(t19_modal_1,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)
=> ! [D] :
( m1_trees_1(D,k5_trees_1(A,C,B))
=> ! [E] :
( m1_trees_1(E,B)
=> ( D = k8_finseq_1(k5_numbers,C,E)
=> r2_tarski(k1_trees_2(k5_trees_1(A,C,B),D),k1_trees_2(B,E)) ) ) ) ) ) ) ) ).
fof(t20_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ( r2_hidden(B,A)
=> ! [C] :
( m1_trees_1(C,A)
=> ! [D] :
( m1_trees_1(D,k4_trees_1(A,B))
=> ( C = k8_finseq_1(k5_numbers,B,D)
=> r2_tarski(k1_trees_2(A,C),k1_trees_2(k4_trees_1(A,B),D)) ) ) ) ) ) ) ).
fof(t21_modal_1,axiom,
$true ).
fof(t22_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ( k10_trees_2(A,k1_modal_1(A)) = np__0
=> ( k4_card_1(A) = np__1
& A = k1_tarski(k1_xboole_0) ) ) ) ).
fof(t23_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ( k10_trees_2(A,k1_modal_1(A)) = np__1
=> k1_trees_2(A,k1_modal_1(A)) = k6_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k12_finseq_1(k5_numbers,np__0)) ) ) ).
fof(t24_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ( k10_trees_2(A,k1_modal_1(A)) = np__2
=> k1_trees_2(A,k1_modal_1(A)) = k7_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k12_finseq_1(k5_numbers,np__0),k12_finseq_1(k5_numbers,np__1)) ) ) ).
fof(t26_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ~ ( B != k1_modal_1(A)
& r1_xreal_0(k4_card_1(A),k4_card_1(k4_trees_1(A,B))) ) ) ) ).
fof(t27_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ( k1_trees_2(A,k1_modal_1(A)) = k6_domain_1(A,B)
=> A = k5_trees_1(k2_trees_1(np__1),k12_finseq_1(k5_numbers,np__0),k4_trees_1(A,B)) ) ) ) ).
fof(t28_modal_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_finset_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> ! [C] :
( m1_trees_1(C,k1_relat_1(B))
=> ( ( k1_trees_2(k1_relat_1(B),k1_modal_1(k1_relat_1(B))) = k6_domain_1(k1_relat_1(B),C)
& v1_finset_1(k1_relat_1(B)) )
=> B = k8_trees_2(k9_trees_2(A,k2_trees_1(np__1),k2_modal_1(A,B)),k12_finseq_1(k5_numbers,np__0),k7_trees_2(A,B,C)) ) ) ) ) ).
fof(t29_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ! [C] :
( m1_trees_1(C,A)
=> ( ( v1_finset_1(A)
& B = k12_finseq_1(k5_numbers,np__0)
& C = k12_finseq_1(k5_numbers,np__1)
& k1_trees_2(A,k1_modal_1(A)) = k7_domain_1(A,B,C) )
=> A = k5_trees_1(k5_trees_1(k2_trees_1(np__2),k12_finseq_1(k5_numbers,np__0),k4_trees_1(A,B)),k12_finseq_1(k5_numbers,np__1),k4_trees_1(A,C)) ) ) ) ) ).
fof(t30_modal_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> ! [C] :
( m1_trees_1(C,k1_relat_1(B))
=> ! [D] :
( m1_trees_1(D,k1_relat_1(B))
=> ( ( v1_finset_1(k1_relat_1(B))
& C = k12_finseq_1(k5_numbers,np__0)
& D = k12_finseq_1(k5_numbers,np__1)
& k1_trees_2(k1_relat_1(B),k1_modal_1(k1_relat_1(B))) = k7_domain_1(k1_relat_1(B),C,D) )
=> B = k8_trees_2(k8_trees_2(k9_trees_2(A,k2_trees_1(np__2),k2_modal_1(A,B)),k12_finseq_1(k5_numbers,np__0),k7_trees_2(A,B,C)),k12_finseq_1(k5_numbers,np__1),k7_trees_2(A,B,D)) ) ) ) ) ) ).
fof(d3_modal_1,axiom,
k3_modal_1 = k12_mcart_1(k5_numbers,k1_numbers,k6_domain_1(k5_numbers,np__3),k5_numbers) ).
fof(d4_modal_1,axiom,
k4_modal_1 = k12_mcart_1(k5_numbers,k1_numbers,k8_domain_1(k5_numbers,np__0,np__1,np__2),k5_numbers) ).
fof(t31_modal_1,axiom,
r1_subset_1(k4_modal_1,k3_modal_1) ).
fof(d5_modal_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( m1_modal_1(B,A)
<=> ! [C] :
( r2_hidden(C,B)
=> ( v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,A) ) ) ) ) ) ).
fof(d6_modal_1,axiom,
! [A] :
( m1_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ( A = k6_modal_1
<=> ( ! [B] :
( ( v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) )
=> ( r2_hidden(B,A)
=> v1_finset_1(B) ) )
& ! [B] :
( ( v1_funct_1(B)
& v1_finset_1(B)
& v3_trees_2(B)
& m3_trees_2(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) )
=> ( r2_hidden(B,A)
<=> ! [C] :
( m1_trees_1(C,k1_relat_1(B))
=> ( r1_xreal_0(k5_modal_1(k1_relat_1(B),C),np__2)
& ~ ( k5_modal_1(k1_relat_1(B),C) = np__0
& k3_trees_2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),B,C) != k1_domain_1(k5_numbers,k5_numbers,np__0,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k3_trees_2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),B,C) != k1_domain_1(k5_numbers,k5_numbers,np__3,D) ) )
& ~ ( k5_modal_1(k1_relat_1(B),C) = np__1
& k3_trees_2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),B,C) != k1_domain_1(k5_numbers,k5_numbers,np__1,np__0)
& k3_trees_2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),B,C) != k1_domain_1(k5_numbers,k5_numbers,np__1,np__1) )
& ( k5_modal_1(k1_relat_1(B),C) = np__2
=> k3_trees_2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),B,C) = k1_domain_1(k5_numbers,k5_numbers,np__2,np__0) ) ) ) ) ) ) ) ) ).
fof(d7_modal_1,axiom,
! [A] :
( m1_subset_1(A,k4_modal_1)
=> k8_modal_1(A) = k1_mcart_1(A) ) ).
fof(d8_modal_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,A) )
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ( r2_hidden(D,k1_relat_1(B))
=> k9_modal_1(A,B,C,D) = k8_trees_2(B,D,C) ) ) ) ) ) ).
fof(t32_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> m2_modal_1(k8_trees_2(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(np__1),k1_domain_1(k5_numbers,k5_numbers,np__1,np__0)),k12_finseq_1(k5_numbers,np__0),A),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(t33_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> m2_modal_1(k8_trees_2(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(np__1),k1_domain_1(k5_numbers,k5_numbers,np__1,np__1)),k12_finseq_1(k5_numbers,np__0),A),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(t34_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> m2_modal_1(k8_trees_2(k8_trees_2(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(np__2),k1_domain_1(k5_numbers,k5_numbers,np__2,np__0)),k12_finseq_1(k5_numbers,np__0),A),k12_finseq_1(k5_numbers,np__1),B),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ) ).
fof(d9_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> k10_modal_1(A) = k8_trees_2(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(np__1),k1_domain_1(k5_numbers,k5_numbers,np__1,np__0)),k12_finseq_1(k5_numbers,np__0),A) ) ).
fof(d10_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> k11_modal_1(A) = k8_trees_2(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(np__1),k1_domain_1(k5_numbers,k5_numbers,np__1,np__1)),k12_finseq_1(k5_numbers,np__0),A) ) ).
fof(d11_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> k12_modal_1(A,B) = k8_trees_2(k8_trees_2(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(np__2),k1_domain_1(k5_numbers,k5_numbers,np__2,np__0)),k12_finseq_1(k5_numbers,np__0),A),k12_finseq_1(k5_numbers,np__1),B) ) ) ).
fof(d12_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> k13_modal_1(A) = k10_modal_1(k11_modal_1(k10_modal_1(A))) ) ).
fof(d13_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> k14_modal_1(A,B) = k10_modal_1(k12_modal_1(k10_modal_1(A),k10_modal_1(B))) ) ) ).
fof(d14_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> k15_modal_1(A,B) = k10_modal_1(k12_modal_1(A,k10_modal_1(B))) ) ) ).
fof(t35_modal_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> m2_modal_1(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(np__0),k1_domain_1(k5_numbers,k5_numbers,np__3,A)),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(t36_modal_1,axiom,
m2_modal_1(k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(np__0),k1_domain_1(k5_numbers,k5_numbers,np__0,np__0)),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ).
fof(d15_modal_1,axiom,
! [A] :
( m1_subset_1(A,k3_modal_1)
=> k16_modal_1(A) = k9_trees_2(k3_modal_1,k2_trees_1(np__0),A) ) ).
fof(t37_modal_1,axiom,
! [A] :
( m1_subset_1(A,k3_modal_1)
=> ! [B] :
( m1_subset_1(B,k3_modal_1)
=> ( k16_modal_1(A) = k16_modal_1(B)
=> A = B ) ) ) ).
fof(t38_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( k10_modal_1(A) = k10_modal_1(B)
=> A = B ) ) ) ).
fof(t39_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( k11_modal_1(A) = k11_modal_1(B)
=> A = B ) ) ) ).
fof(t40_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [C] :
( m2_modal_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [D] :
( m2_modal_1(D,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( k12_modal_1(A,B) = k12_modal_1(C,D)
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(d16_modal_1,axiom,
k17_modal_1 = k9_trees_2(k2_zfmisc_1(k5_numbers,k5_numbers),k2_trees_1(np__0),k1_domain_1(k5_numbers,k5_numbers,np__0,np__0)) ).
fof(t41_modal_1,axiom,
$true ).
fof(t42_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ~ ( k4_card_1(k1_relat_1(A)) = np__1
& A != k17_modal_1
& ! [B] :
( m1_subset_1(B,k3_modal_1)
=> A != k16_modal_1(B) ) ) ) ).
fof(t43_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ~ ( r1_xreal_0(np__2,k4_card_1(k1_relat_1(A)))
& ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( A != k10_modal_1(B)
& A != k11_modal_1(B) ) )
& ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [C] :
( m2_modal_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> A != k12_modal_1(B,C) ) ) ) ) ).
fof(t44_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ~ r1_xreal_0(k4_card_1(k1_relat_1(k10_modal_1(A))),k4_card_1(k1_relat_1(A))) ) ).
fof(t45_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ~ r1_xreal_0(k4_card_1(k1_relat_1(k11_modal_1(A))),k4_card_1(k1_relat_1(A))) ) ).
fof(t46_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( ~ r1_xreal_0(k4_card_1(k1_relat_1(k12_modal_1(A,B))),k4_card_1(k1_relat_1(A)))
& ~ r1_xreal_0(k4_card_1(k1_relat_1(k12_modal_1(A,B))),k4_card_1(k1_relat_1(B))) ) ) ) ).
fof(d17_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( v1_modal_1(A)
<=> ? [B] :
( m1_subset_1(B,k3_modal_1)
& A = k16_modal_1(B) ) ) ) ).
fof(d18_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( v2_modal_1(A)
<=> ? [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
& A = k10_modal_1(B) ) ) ) ).
fof(d19_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( v3_modal_1(A)
<=> ? [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
& A = k11_modal_1(B) ) ) ) ).
fof(d20_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( v4_modal_1(A)
<=> ? [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
& ? [C] :
( m2_modal_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
& A = k12_modal_1(B,C) ) ) ) ) ).
fof(t47_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ~ ( A != k17_modal_1
& ~ ( v1_modal_1(A)
& m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) )
& ~ ( v2_modal_1(A)
& m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) )
& ~ ( v3_modal_1(A)
& m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) )
& ~ ( v4_modal_1(A)
& m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ) ) ).
fof(t48_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ~ ( A != k17_modal_1
& ! [B] :
( m1_subset_1(B,k3_modal_1)
=> A != k16_modal_1(B) )
& ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> A != k10_modal_1(B) )
& ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> A != k11_modal_1(B) )
& ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [C] :
( m2_modal_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> A != k12_modal_1(B,C) ) ) ) ) ).
fof(t49_modal_1,axiom,
! [A] :
( m1_subset_1(A,k3_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [C] :
( m2_modal_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( k16_modal_1(A) != k10_modal_1(B)
& k16_modal_1(A) != k11_modal_1(B)
& k16_modal_1(A) != k12_modal_1(B,C) ) ) ) ) ).
fof(t50_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [C] :
( m2_modal_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( k10_modal_1(A) != k11_modal_1(B)
& k10_modal_1(A) != k12_modal_1(B,C) ) ) ) ) ).
fof(t51_modal_1,axiom,
! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [C] :
( m2_modal_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> k11_modal_1(A) != k12_modal_1(B,C) ) ) ) ).
fof(t52_modal_1,axiom,
! [A] :
( m1_subset_1(A,k3_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [C] :
( m2_modal_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( k17_modal_1 != k16_modal_1(A)
& k17_modal_1 != k10_modal_1(B)
& k17_modal_1 != k11_modal_1(B)
& k17_modal_1 != k12_modal_1(B,C) ) ) ) ) ).
fof(t53_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ( r2_hidden(B,k3_trees_1(A))
<=> ~ r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,np__0)),A) ) ) ) ).
fof(t54_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ( r2_hidden(B,k3_trees_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)),A) ) ) ) ) ).
fof(s1_modal_1,axiom,
( ( p1_s1_modal_1(k17_modal_1)
& ! [A] :
( m1_subset_1(A,k3_modal_1)
=> p1_s1_modal_1(k16_modal_1(A)) )
& ! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( p1_s1_modal_1(A)
=> p1_s1_modal_1(k10_modal_1(A)) ) )
& ! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( p1_s1_modal_1(A)
=> p1_s1_modal_1(k11_modal_1(A)) ) )
& ! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ( ( p1_s1_modal_1(A)
& p1_s1_modal_1(B) )
=> p1_s1_modal_1(k12_modal_1(A,B)) ) ) ) )
=> ! [A] :
( m2_modal_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> p1_s1_modal_1(A) ) ) ).
fof(s2_modal_1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k6_modal_1,f1_s2_modal_1)
& m2_relset_1(A,k6_modal_1,f1_s2_modal_1)
& k8_funct_2(k6_modal_1,f1_s2_modal_1,A,k17_modal_1) = f2_s2_modal_1
& ! [B] :
( m1_subset_1(B,k3_modal_1)
=> k8_funct_2(k6_modal_1,f1_s2_modal_1,A,k16_modal_1(B)) = f3_s2_modal_1(B) )
& ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> k8_funct_2(k6_modal_1,f1_s2_modal_1,A,k10_modal_1(B)) = f4_s2_modal_1(k8_funct_2(k6_modal_1,f1_s2_modal_1,A,B)) )
& ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> k8_funct_2(k6_modal_1,f1_s2_modal_1,A,k11_modal_1(B)) = f5_s2_modal_1(k8_funct_2(k6_modal_1,f1_s2_modal_1,A,B)) )
& ! [B] :
( m2_modal_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> ! [C] :
( m2_modal_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1)
=> k8_funct_2(k6_modal_1,f1_s2_modal_1,A,k12_modal_1(B,C)) = f6_s2_modal_1(k8_funct_2(k6_modal_1,f1_s2_modal_1,A,B),k8_funct_2(k6_modal_1,f1_s2_modal_1,A,C)) ) ) ) ).
fof(dt_m1_modal_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_modal_1(B,A)
=> ~ v1_xboole_0(B) ) ) ).
fof(existence_m1_modal_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] : m1_modal_1(B,A) ) ).
fof(dt_m2_modal_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_modal_1(B,A) )
=> ! [C] :
( m2_modal_1(C,A,B)
=> ( v1_funct_1(C)
& v3_trees_2(C)
& m3_trees_2(C,A) ) ) ) ).
fof(existence_m2_modal_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_modal_1(B,A) )
=> ? [C] : m2_modal_1(C,A,B) ) ).
fof(redefinition_m2_modal_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_modal_1(B,A) )
=> ! [C] :
( m2_modal_1(C,A,B)
<=> m1_subset_1(C,B) ) ) ).
fof(dt_k1_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> m1_trees_1(k1_modal_1(A),A) ) ).
fof(dt_k2_modal_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_trees_2(B)
& m3_trees_2(B,A) )
=> m1_subset_1(k2_modal_1(A,B),A) ) ).
fof(dt_k3_modal_1,axiom,
$true ).
fof(dt_k4_modal_1,axiom,
$true ).
fof(dt_k5_modal_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A)
& m1_subset_1(B,A) )
=> m2_subset_1(k5_modal_1(A,B),k1_numbers,k5_numbers) ) ).
fof(redefinition_k5_modal_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A)
& m1_subset_1(B,A) )
=> k5_modal_1(A,B) = k10_trees_2(A,B) ) ).
fof(dt_k6_modal_1,axiom,
m1_modal_1(k6_modal_1,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ).
fof(dt_k7_modal_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_modal_1)
& m1_subset_1(B,k1_relat_1(A)) )
=> m2_modal_1(k7_modal_1(A,B),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(redefinition_k7_modal_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_modal_1)
& m1_subset_1(B,k1_relat_1(A)) )
=> k7_modal_1(A,B) = k5_trees_2(A,B) ) ).
fof(dt_k8_modal_1,axiom,
! [A] :
( m1_subset_1(A,k4_modal_1)
=> m2_subset_1(k8_modal_1(A),k1_numbers,k5_numbers) ) ).
fof(dt_k9_modal_1,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)
& m1_finseq_1(D,k5_numbers) )
=> ( v1_funct_1(k9_modal_1(A,B,C,D))
& v3_trees_2(k9_modal_1(A,B,C,D))
& m3_trees_2(k9_modal_1(A,B,C,D),A) ) ) ).
fof(dt_k10_modal_1,axiom,
! [A] :
( m1_subset_1(A,k6_modal_1)
=> m2_modal_1(k10_modal_1(A),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(dt_k11_modal_1,axiom,
! [A] :
( m1_subset_1(A,k6_modal_1)
=> m2_modal_1(k11_modal_1(A),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(dt_k12_modal_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_modal_1)
& m1_subset_1(B,k6_modal_1) )
=> m2_modal_1(k12_modal_1(A,B),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(dt_k13_modal_1,axiom,
! [A] :
( m1_subset_1(A,k6_modal_1)
=> m2_modal_1(k13_modal_1(A),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(dt_k14_modal_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_modal_1)
& m1_subset_1(B,k6_modal_1) )
=> m2_modal_1(k14_modal_1(A,B),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(dt_k15_modal_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_modal_1)
& m1_subset_1(B,k6_modal_1) )
=> m2_modal_1(k15_modal_1(A,B),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(dt_k16_modal_1,axiom,
! [A] :
( m1_subset_1(A,k3_modal_1)
=> m2_modal_1(k16_modal_1(A),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ) ).
fof(dt_k17_modal_1,axiom,
m2_modal_1(k17_modal_1,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k6_modal_1) ).
fof(t25_modal_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ( B != k1_modal_1(A)
=> ( r2_tarski(k4_trees_1(A,B),a_2_0_modal_1(A,B))
& ~ r2_hidden(k1_modal_1(A),a_2_0_modal_1(A,B)) ) ) ) ) ).
fof(fraenkel_a_2_0_modal_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_trees_1(B)
& m1_trees_1(C,B) )
=> ( r2_hidden(A,a_2_0_modal_1(B,C))
<=> ? [D] :
( m2_finseq_2(D,k5_numbers,k3_finseq_2(k5_numbers))
& A = k8_finseq_1(k5_numbers,C,D)
& r2_hidden(k8_finseq_1(k5_numbers,C,D),B) ) ) ) ).
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