SET007 Axioms: SET007+716.ax
%------------------------------------------------------------------------------
% File : SET007+716 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Dickson's Lemma
% 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 : dickson [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 99 ( 9 unt; 0 def)
% Number of atoms : 514 ( 37 equ)
% Maximal formula atoms : 17 ( 5 avg)
% Number of connectives : 492 ( 77 ~; 1 |; 204 &)
% ( 26 <=>; 184 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 51 ( 49 usr; 1 prp; 0-3 aty)
% Number of functors : 54 ( 54 usr; 10 con; 0-4 aty)
% Number of variables : 189 ( 176 !; 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_dickson,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_relat_1(k3_dickson(A))
& v5_relat_2(k3_dickson(A)) ) ) ).
fof(fc2_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ~ v3_struct_0(k5_dickson(A))
& v1_orders_2(k5_dickson(A)) ) ) ).
fof(fc3_dickson,axiom,
! [A] :
( ( v3_orders_2(A)
& l1_orders_2(A) )
=> ( v1_orders_2(k5_dickson(A))
& v3_orders_2(k5_dickson(A)) ) ) ).
fof(fc4_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_orders_2(k5_dickson(A))
& v4_orders_2(k5_dickson(A)) ) ) ).
fof(fc5_dickson,axiom,
! [A,B,C] :
( ( v1_yellow_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ( v1_relat_1(k7_relat_1(B,C))
& v1_funct_1(k7_relat_1(B,C))
& v1_yellow_1(k7_relat_1(B,C))
& v2_pralg_1(k7_relat_1(B,C)) ) ) ).
fof(fc6_dickson,axiom,
! [A] :
( ( v1_yellow_1(A)
& m1_pboole(A,k1_xboole_0) )
=> ( ~ v3_struct_0(k5_yellow_1(k1_xboole_0,A))
& v1_orders_2(k5_yellow_1(k1_xboole_0,A)) ) ) ).
fof(fc7_dickson,axiom,
! [A] :
( ( v1_yellow_1(A)
& m1_pboole(A,k1_xboole_0) )
=> ( v1_orders_2(k5_yellow_1(k1_xboole_0,A))
& v4_orders_2(k5_yellow_1(k1_xboole_0,A)) ) ) ).
fof(fc8_dickson,axiom,
! [A] :
( ( v1_yellow_1(A)
& m1_pboole(A,k1_xboole_0) )
=> ( v1_orders_2(k5_yellow_1(k1_xboole_0,A))
& v3_dickson(k5_yellow_1(k1_xboole_0,A)) ) ) ).
fof(fc9_dickson,axiom,
! [A] :
( ( v1_yellow_1(A)
& m1_pboole(A,k1_xboole_0) )
=> ( v1_orders_2(k5_yellow_1(k1_xboole_0,A))
& v4_dickson(k5_yellow_1(k1_xboole_0,A)) ) ) ).
fof(fc10_dickson,axiom,
( ~ v3_struct_0(k11_dickson)
& v16_waybel_0(k11_dickson) ) ).
fof(fc11_dickson,axiom,
( ~ v3_struct_0(k11_dickson)
& v4_dickson(k11_dickson) ) ).
fof(fc12_dickson,axiom,
( ~ v3_struct_0(k11_dickson)
& v3_dickson(k11_dickson) ) ).
fof(fc13_dickson,axiom,
( ~ v3_struct_0(k11_dickson)
& v4_orders_2(k11_dickson) ) ).
fof(fc14_dickson,axiom,
( ~ v3_struct_0(k11_dickson)
& v3_orders_2(k11_dickson) ) ).
fof(fc15_dickson,axiom,
( ~ v3_struct_0(k11_dickson)
& v1_wellfnd1(k11_dickson) ) ).
fof(fc16_dickson,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k5_yellow_1(A,k2_pre_circ(A,k11_dickson)))
& v1_orders_2(k5_yellow_1(A,k2_pre_circ(A,k11_dickson))) ) ) ).
fof(fc17_dickson,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_orders_2(k5_yellow_1(A,k2_pre_circ(A,k11_dickson)))
& v4_dickson(k5_yellow_1(A,k2_pre_circ(A,k11_dickson))) ) ) ).
fof(fc18_dickson,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_orders_2(k5_yellow_1(A,k2_pre_circ(A,k11_dickson)))
& v3_dickson(k5_yellow_1(A,k2_pre_circ(A,k11_dickson))) ) ) ).
fof(fc19_dickson,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_orders_2(k5_yellow_1(A,k2_pre_circ(A,k11_dickson)))
& v4_orders_2(k5_yellow_1(A,k2_pre_circ(A,k11_dickson))) ) ) ).
fof(t1_dickson,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( k1_relat_1(A) = k1_tarski(B)
=> A = k3_cqc_lang(B,k1_funct_1(A,B)) ) ) ).
fof(t2_dickson,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(A,k1_nat_1(A,np__1)) ) ).
fof(t3_dickson,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,A)
& m2_relset_1(B,k5_numbers,A)
& v2_funct_1(B) ) ) ).
fof(d1_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v1_dickson(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)) != k8_funct_2(k5_numbers,u1_struct_0(A),B,C)
& r2_hidden(k4_tarski(k8_funct_2(k5_numbers,u1_struct_0(A),B,C),k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1))),u1_orders_2(A)) ) ) ) ) ) ).
fof(d2_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v2_dickson(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r2_hidden(k4_tarski(k8_funct_2(k5_numbers,u1_struct_0(A),B,C),k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1))),u1_orders_2(A)) ) ) ) ) ).
fof(t4_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v2_dickson(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(D,C)
=> r1_orders_2(A,k2_normsp_1(A,B,C),k2_normsp_1(A,B,D)) ) ) ) ) ) ) ).
fof(t5_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v16_waybel_0(A)
<=> r7_relat_2(u1_orders_2(A),u1_struct_0(A)) ) ) ).
fof(t6_dickson,axiom,
$true ).
fof(t7_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B,C] :
( ( r1_xboole_0(k1_wellord1(u1_orders_2(A),C),B)
& r2_hidden(C,B) )
<=> r4_waybel_4(B,C,u1_orders_2(A)) ) ) ).
fof(t8_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C,D] :
( r4_waybel_4(k3_xboole_0(k1_wellord1(u1_orders_2(A),B),D),C,u1_orders_2(A))
=> r4_waybel_4(D,C,u1_orders_2(A)) ) ) ) ).
fof(d3_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( v3_dickson(A)
<=> ( v2_orders_2(A)
& v3_orders_2(A) ) ) ) ).
fof(d4_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( v3_dickson(A)
=> k1_dickson(A) = k2_eqrel_1(u1_struct_0(A),u1_orders_2(A),k6_relset_1(u1_struct_0(A),u1_struct_0(A),u1_orders_2(A))) ) ) ).
fof(t9_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( v3_dickson(A)
=> ( r2_hidden(B,k6_eqrel_1(u1_struct_0(A),k1_dickson(A),C))
<=> ( r1_orders_2(A,B,C)
& r1_orders_2(A,C,B) ) ) ) ) ) ) ).
fof(d5_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( m2_relset_1(B,k8_eqrel_1(u1_struct_0(A),k1_dickson(A)),k8_eqrel_1(u1_struct_0(A),k1_dickson(A)))
=> ( B = k2_dickson(A)
<=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),B)
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& C = k6_eqrel_1(u1_struct_0(A),k1_dickson(A),E)
& D = k6_eqrel_1(u1_struct_0(A),k1_dickson(A),F)
& r1_orders_2(A,E,F) ) ) ) ) ) ) ).
fof(t10_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( v3_dickson(A)
=> r2_orders_1(k2_dickson(A),k8_eqrel_1(u1_struct_0(A),k1_dickson(A))) ) ) ).
fof(t11_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v3_dickson(A)
& v16_waybel_0(A) )
=> r3_orders_1(k2_dickson(A),k8_eqrel_1(u1_struct_0(A),k1_dickson(A))) ) ) ).
fof(d6_dickson,axiom,
! [A] :
( v1_relat_1(A)
=> k3_dickson(A) = k4_xboole_0(A,k4_relat_1(A)) ) ).
fof(d7_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> k5_dickson(A) = g1_orders_2(u1_struct_0(A),k4_dickson(u1_struct_0(A),u1_orders_2(A))) ) ).
fof(t12_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_eqrel_1(u1_struct_0(A),k1_dickson(A),B) = k15_cqc_sim1(u1_struct_0(A),B) ) ) ).
fof(t13_dickson,axiom,
! [A] :
( v1_relat_1(A)
=> ( A = k3_dickson(A)
<=> v5_relat_2(A) ) ) ).
fof(t14_dickson,axiom,
! [A] :
( v1_relat_1(A)
=> ( v8_relat_2(A)
=> v8_relat_2(k3_dickson(A)) ) ) ).
fof(t15_dickson,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( v4_relat_2(A)
=> ( r2_hidden(k4_tarski(B,C),k3_dickson(A))
<=> ( r2_hidden(k4_tarski(B,C),A)
& B != C ) ) ) ) ).
fof(t16_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_wellfnd1(A)
=> v1_wellfnd1(k5_dickson(A)) ) ) ).
fof(t17_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( v1_wellfnd1(k5_dickson(A))
& v4_orders_2(A) )
=> v1_wellfnd1(A) ) ) ).
fof(t18_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B,C] :
( m1_subset_1(C,u1_struct_0(k5_dickson(A)))
=> ( r4_waybel_4(B,C,u1_orders_2(k5_dickson(A)))
<=> ( r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(D,B)
& r2_hidden(k4_tarski(D,C),u1_orders_2(A)) )
=> r2_hidden(k4_tarski(C,D),u1_orders_2(A)) ) ) ) ) ) ) ).
fof(t19_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( v3_dickson(A)
& v4_orders_2(B)
& v1_wellfnd1(k5_dickson(B))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r1_orders_2(A,D,E)
=> r1_orders_2(B,k1_waybel_0(A,B,C,D),k1_waybel_0(A,B,C,E)) )
& ( k1_waybel_0(A,B,C,D) = k1_waybel_0(A,B,C,E)
=> r2_hidden(k7_yellow_3(A,A,D,E),k1_dickson(A)) ) ) ) ) )
=> v1_wellfnd1(k5_dickson(A)) ) ) ) ) ).
fof(d8_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( C = k6_dickson(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k5_dickson(A)))
& r4_waybel_4(B,E,u1_orders_2(k5_dickson(A)))
& D = k5_subset_1(u1_struct_0(A),k6_eqrel_1(u1_struct_0(A),k1_dickson(A),E),B) ) ) ) ) ) ) ).
fof(t20_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v3_dickson(A)
& r2_hidden(C,k6_dickson(A,B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_dickson(A)))
=> ( r2_hidden(D,C)
=> r4_waybel_4(B,D,u1_orders_2(k5_dickson(A))) ) ) ) ) ) ).
fof(t21_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v1_wellfnd1(k5_dickson(A))
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B != k1_xboole_0
& ! [C] : ~ r2_hidden(C,k6_dickson(A,B)) ) ) ) ) ).
fof(t22_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_dickson(A)))
=> ~ ( r4_waybel_4(B,C,u1_orders_2(k5_dickson(A)))
& v1_xboole_0(k6_dickson(A,B)) ) ) ) ) ).
fof(t23_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
~ ( v3_dickson(A)
& r2_hidden(C,k6_dickson(A,B))
& v1_xboole_0(C) ) ) ) ).
fof(t24_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v3_dickson(A)
=> ( ( v16_waybel_0(A)
& v1_wellfnd1(k5_dickson(A)) )
<=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k1_card_1(k6_dickson(A,B)) = np__1 ) ) ) ) ).
fof(t25_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ( r2_wellord1(u1_orders_2(A),u1_struct_0(A))
<=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k1_card_1(k6_dickson(A,B)) = np__1 ) ) ) ).
fof(d9_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( r1_dickson(A,B,C)
<=> ( r1_tarski(C,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(D,B)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,C)
& r1_orders_2(A,E,D) ) ) ) ) ) ) ) ) ).
fof(t26_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> r1_dickson(A,k1_subset_1(u1_struct_0(A)),k1_xboole_0) ) ).
fof(t27_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
~ ( r1_dickson(A,B,C)
& v1_xboole_0(C) ) ) ) ).
fof(d10_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( v4_dickson(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ? [C] :
( r1_dickson(A,B,C)
& v1_finset_1(C) ) ) ) ) ).
fof(t28_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v1_wellfnd1(k5_dickson(A))
& v16_waybel_0(A) )
=> v4_dickson(A) ) ) ).
fof(t29_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ( ( r1_tarski(u1_orders_2(A),u1_orders_2(B))
& v4_dickson(A)
& u1_struct_0(A) = u1_struct_0(B) )
=> v4_dickson(B) ) ) ) ).
fof(d11_dickson,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( k1_relat_1(A) = k5_numbers
& r2_hidden(B,k2_relat_1(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k7_dickson(A,B)
<=> ( k1_funct_1(A,C) = B
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( k1_funct_1(A,D) = B
=> r1_xreal_0(C,D) ) ) ) ) ) ) ) ).
fof(d12_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C,D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& ~ r1_xreal_0(E,D)
& k2_normsp_1(A,B,E) = C )
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( E = k8_dickson(A,B,C,D)
<=> ( k2_normsp_1(A,B,E) = C
& ~ r1_xreal_0(E,D)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( k2_normsp_1(A,B,F) = C
=> ( r1_xreal_0(F,D)
| r1_xreal_0(E,F) ) ) ) ) ) ) ) ) ) ) ).
fof(t30_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v3_dickson(A)
& v4_dickson(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& ~ r1_xreal_0(D,C)
& r1_orders_2(A,k2_normsp_1(A,B,C),k2_normsp_1(A,B,D)) ) ) ) ) ) ).
fof(t31_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_dickson(A)))
=> ( ( v3_dickson(A)
& r2_hidden(C,B)
& r1_tarski(k3_xboole_0(k1_wellord1(u1_orders_2(A),C),B),k6_eqrel_1(u1_struct_0(A),k1_dickson(A),C)) )
=> r4_waybel_4(B,C,u1_orders_2(k5_dickson(A))) ) ) ) ) ).
fof(t32_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v3_dickson(A)
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& ~ r1_xreal_0(D,C)
& r1_orders_2(A,k2_normsp_1(A,B,C),k2_normsp_1(A,B,D)) ) ) ) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_finset_1(k6_dickson(A,B))
& ~ v1_xboole_0(k6_dickson(A,B)) ) ) ) ) ).
fof(t33_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v3_dickson(A)
& ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_finset_1(k6_dickson(A,B))
& ~ v1_xboole_0(k6_dickson(A,B)) ) ) )
=> v4_dickson(A) ) ) ).
fof(t34_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v3_dickson(A)
& v4_dickson(A) )
=> v1_wellfnd1(k5_dickson(A)) ) ) ).
fof(t35_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v4_dickson(A)
& ! [C] :
~ ( r1_dickson(A,B,C)
& ! [D] :
( r1_dickson(A,B,D)
=> r1_tarski(C,D) ) ) ) ) ) ).
fof(d13_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_dickson(A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> ( C = k9_dickson(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> r1_dickson(A,B,D) ) ) ) ) ) ) ).
fof(t36_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ~ ( v4_dickson(A)
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ~ ( m1_bhsp_3(C,A,B)
& v2_dickson(C,A) ) ) ) ) ) ).
fof(t37_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( v3_struct_0(A)
=> v4_dickson(A) ) ) ).
fof(t38_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ( ( v4_dickson(A)
& v4_dickson(B)
& v3_dickson(A)
& v3_dickson(B) )
=> ( v3_dickson(k3_yellow_3(A,B))
& v4_dickson(k3_yellow_3(A,B)) ) ) ) ) ).
fof(t39_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ( ( r5_waybel_1(A,B)
& v4_dickson(A)
& v3_dickson(A) )
=> ( v3_dickson(B)
& v4_dickson(B) ) ) ) ) ).
fof(t40_dickson,axiom,
! [A] :
( ( v1_yellow_1(A)
& m1_pboole(A,np__1) )
=> ! [B] :
( m1_subset_1(B,np__1)
=> r5_waybel_1(k4_yellow_1(np__1,A,B),k5_yellow_1(np__1,A)) ) ) ).
fof(t41_dickson,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( v1_yellow_1(B)
& m1_pboole(B,A) )
=> ( ~ v3_struct_0(k5_yellow_1(A,B))
<=> v4_waybel_3(B) ) ) ) ).
fof(t42_dickson,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( v1_yellow_1(B)
& m1_pboole(B,k1_nat_1(A,np__1)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_nat_1(A,np__1)))
=> ! [D] :
( m1_subset_1(D,k1_nat_1(A,np__1))
=> ( ( C = A
& D = A )
=> r5_waybel_1(k3_yellow_3(k5_yellow_1(C,k3_pre_circ(k1_nat_1(A,np__1),B,C)),k4_yellow_1(k1_nat_1(A,np__1),B,D)),k5_yellow_1(k1_nat_1(A,np__1),B)) ) ) ) ) ) ).
fof(t43_dickson,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( v1_yellow_1(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v4_dickson(k4_yellow_1(A,B,C))
& v3_dickson(k4_yellow_1(A,B,C)) ) )
=> ( v3_dickson(k5_yellow_1(A,B))
& v4_dickson(k5_yellow_1(A,B)) ) ) ) ) ).
fof(t44_dickson,axiom,
r1_relat_2(k10_dickson,k5_numbers) ).
fof(t45_dickson,axiom,
r4_relat_2(k10_dickson,k5_numbers) ).
fof(t46_dickson,axiom,
r7_relat_2(k10_dickson,k5_numbers) ).
fof(t47_dickson,axiom,
r8_relat_2(k10_dickson,k5_numbers) ).
fof(d15_dickson,axiom,
k11_dickson = g1_orders_2(k5_numbers,k10_dickson) ).
fof(t48_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( v4_dickson(A)
& v3_dickson(A) )
=> ( v3_dickson(k3_yellow_3(A,k11_dickson))
& v4_dickson(k3_yellow_3(A,k11_dickson)) ) ) ) ).
fof(t49_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ( ( v4_dickson(A)
& v3_dickson(A)
& v3_dickson(B)
& r1_tarski(u1_orders_2(A),u1_orders_2(B))
& u1_struct_0(A) = u1_struct_0(B) )
=> v1_wellfnd1(k5_dickson(B)) ) ) ) ).
fof(t50_dickson,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v3_dickson(A)
=> ( v4_dickson(A)
<=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ( ( v3_dickson(B)
& r1_tarski(u1_orders_2(A),u1_orders_2(B))
& u1_struct_0(A) = u1_struct_0(B) )
=> v1_wellfnd1(k5_dickson(B)) ) ) ) ) ) ).
fof(dt_k1_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_partfun1(k1_dickson(A),u1_struct_0(A),u1_struct_0(A))
& v3_relat_2(k1_dickson(A))
& v8_relat_2(k1_dickson(A))
& m2_relset_1(k1_dickson(A),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(dt_k2_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> m2_relset_1(k2_dickson(A),k8_eqrel_1(u1_struct_0(A),k1_dickson(A)),k8_eqrel_1(u1_struct_0(A),k1_dickson(A))) ) ).
fof(dt_k3_dickson,axiom,
! [A] :
( v1_relat_1(A)
=> v1_relat_1(k3_dickson(A)) ) ).
fof(dt_k4_dickson,axiom,
! [A,B] :
( m1_relset_1(B,A,A)
=> m2_relset_1(k4_dickson(A,B),A,A) ) ).
fof(redefinition_k4_dickson,axiom,
! [A,B] :
( m1_relset_1(B,A,A)
=> k4_dickson(A,B) = k3_dickson(B) ) ).
fof(dt_k5_dickson,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_orders_2(k5_dickson(A))
& l1_orders_2(k5_dickson(A)) ) ) ).
fof(dt_k6_dickson,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k6_dickson(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k7_dickson,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> m2_subset_1(k7_dickson(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k8_dickson,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& m1_subset_1(D,k5_numbers) )
=> m2_subset_1(k8_dickson(A,B,C,D),k1_numbers,k5_numbers) ) ).
fof(dt_k9_dickson,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k9_dickson(A,B))
& m1_subset_1(k9_dickson(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ).
fof(dt_k10_dickson,axiom,
m2_relset_1(k10_dickson,k5_numbers,k5_numbers) ).
fof(dt_k11_dickson,axiom,
( ~ v3_struct_0(k11_dickson)
& l1_orders_2(k11_dickson) ) ).
fof(d14_dickson,axiom,
k10_dickson = a_0_0_dickson ).
fof(s1_dickson,axiom,
v1_finset_1(a_0_1_dickson) ).
fof(fraenkel_a_0_0_dickson,axiom,
! [A] :
( r2_hidden(A,a_0_0_dickson)
<=> ? [B,C] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_subset_1(C,k1_numbers,k5_numbers)
& A = k1_domain_1(k5_numbers,k5_numbers,B,C)
& r1_xreal_0(B,C) ) ) ).
fof(fraenkel_a_0_1_dickson,axiom,
! [A] :
( r2_hidden(A,a_0_1_dickson)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = f3_s1_dickson(B)
& ~ r1_xreal_0(B,f1_s1_dickson)
& r1_xreal_0(B,f2_s1_dickson)
& p1_s1_dickson(B) ) ) ).
%------------------------------------------------------------------------------