SET007 Axioms: SET007+841.ax
%------------------------------------------------------------------------------
% File : SET007+841 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Hall Marriage Theorem
% 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 : hallmar1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 57 ( 1 unt; 0 def)
% Number of atoms : 338 ( 33 equ)
% Maximal formula atoms : 14 ( 5 avg)
% Number of connectives : 319 ( 38 ~; 6 |; 85 &)
% ( 11 <=>; 179 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-4 aty)
% Number of functors : 21 ( 21 usr; 6 con; 0-4 aty)
% Number of variables : 209 ( 199 !; 10 ?)
% 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_hallmar1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_finseq_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v2_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B) ) ) ).
fof(fc1_hallmar1,axiom,
! [A,B,C] :
( ( v1_finset_1(A)
& m1_finseq_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> v1_finset_1(k1_funct_1(B,C)) ) ).
fof(fc2_hallmar1,axiom,
! [A,B,C] :
( ( v1_finset_1(A)
& m1_finseq_1(B,k1_zfmisc_1(A)) )
=> v1_finset_1(k1_hallmar1(A,B,C)) ) ).
fof(cc1_hallmar1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& ~ v1_xboole_0(B)
& m1_finseq_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> ! [D] :
( m3_hallmar1(D,A,B,C)
=> ( ~ v1_xboole_0(D)
& v1_relat_1(D) ) ) ) ).
fof(cc2_hallmar1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& ~ v1_xboole_0(B)
& m1_finseq_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m4_hallmar1(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_relat_1(C) ) ) ) ).
fof(t1_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> k1_nat_1(k4_card_1(k2_xboole_0(A,B)),k4_card_1(k3_xboole_0(A,B))) = k1_nat_1(k4_card_1(A),k4_card_1(B)) ) ) ).
fof(t2_hallmar1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m2_finseq_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k4_finseq_1(B))
& k1_funct_1(B,C) = k1_xboole_0 ) ) ) ) ).
fof(d1_hallmar1,axiom,
! [A,B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C,D] :
( D = k1_hallmar1(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( r2_hidden(F,C)
& r2_hidden(F,k4_finseq_1(B))
& r2_hidden(E,k1_funct_1(B,F)) ) ) ) ) ).
fof(t3_hallmar1,axiom,
! [A,B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] : r1_tarski(k1_hallmar1(A,B,C),A) ) ).
fof(t4_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C,D] :
( r1_tarski(C,D)
=> r1_tarski(k1_hallmar1(A,B,C),k1_hallmar1(A,B,D)) ) ) ) ).
fof(t5_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> k1_hallmar1(A,B,k1_tarski(C)) = k1_funct_1(B,C) ) ) ) ) ).
fof(t6_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k4_finseq_1(B)) )
=> k1_hallmar1(A,B,k2_tarski(C,D)) = k2_xboole_0(k1_funct_1(B,C),k1_funct_1(B,D)) ) ) ) ) ) ).
fof(t7_hallmar1,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m2_finseq_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(D,A)
& r2_hidden(D,k4_finseq_1(C)) )
=> r1_tarski(k1_funct_1(C,D),k1_hallmar1(B,C,A)) ) ) ) ) ).
fof(t8_hallmar1,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_zfmisc_1(B))
=> ( ( r2_hidden(C,A)
& r2_hidden(C,k4_finseq_1(D)) )
=> k1_hallmar1(B,D,A) = k2_xboole_0(k1_hallmar1(B,D,k4_xboole_0(A,k1_tarski(C))),k1_funct_1(D,C)) ) ) ) ) ).
fof(t9_hallmar1,axiom,
! [A,B,C] :
( v1_finset_1(C)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_1(E,k1_zfmisc_1(C))
=> ( r2_hidden(D,k4_finseq_1(E))
=> k1_hallmar1(C,E,k2_xboole_0(k2_xboole_0(k1_tarski(D),A),B)) = k2_xboole_0(k1_funct_1(E,D),k1_hallmar1(C,E,k2_xboole_0(A,B))) ) ) ) ) ).
fof(t10_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D,E] :
( ( r2_hidden(D,k1_funct_1(B,C))
& r2_hidden(E,k1_funct_1(B,C)) )
=> ( D = E
| k2_xboole_0(k4_xboole_0(k1_funct_1(B,C),k1_tarski(D)),k4_xboole_0(k1_funct_1(B,C),k1_tarski(E))) = k1_funct_1(B,C) ) ) ) ) ) ).
fof(d2_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D,E] :
( m2_finseq_1(E,k1_zfmisc_1(A))
=> ( E = k2_hallmar1(A,B,C,D)
<=> ( k4_finseq_1(E) = k4_finseq_1(B)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k4_finseq_1(E))
=> ( ( C = F
=> k1_funct_1(E,F) = k4_xboole_0(k1_funct_1(B,F),k1_tarski(D)) )
& ( C != F
=> k1_funct_1(E,F) = k1_funct_1(B,F) ) ) ) ) ) ) ) ) ) ) ).
fof(t11_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k1_funct_1(B,C)) )
=> k4_card_1(k1_funct_1(k2_hallmar1(A,B,C,D),C)) = k5_real_1(k4_card_1(k1_funct_1(B,C)),np__1) ) ) ) ) ).
fof(t12_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D,E] : k1_hallmar1(A,k2_hallmar1(A,B,C,D),k4_xboole_0(E,k1_tarski(C))) = k1_hallmar1(A,B,k4_xboole_0(E,k1_tarski(C))) ) ) ) ).
fof(t13_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D,E] :
( ~ r2_hidden(C,E)
=> k1_hallmar1(A,B,E) = k1_hallmar1(A,k2_hallmar1(A,B,C,D),E) ) ) ) ) ).
fof(t14_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D,E] :
( ( r2_hidden(C,k4_finseq_1(k2_hallmar1(A,B,C,D)))
& r1_tarski(E,k4_finseq_1(k2_hallmar1(A,B,C,D)))
& r2_hidden(C,E) )
=> k1_hallmar1(A,k2_hallmar1(A,B,C,D),E) = k2_xboole_0(k1_hallmar1(A,B,k4_xboole_0(E,k1_tarski(C))),k4_xboole_0(k1_funct_1(B,C),k1_tarski(D))) ) ) ) ) ).
fof(d3_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( r1_hallmar1(A,B,C)
<=> ? [D] :
( m2_finseq_1(D,A)
& D = C
& k4_finseq_1(B) = k4_finseq_1(D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(D))
=> r2_hidden(k1_funct_1(D,E),k1_funct_1(B,E)) ) )
& v2_funct_1(D) ) ) ) ) ).
fof(d4_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ( v1_hallmar1(B,A)
<=> ! [C] :
( v1_finset_1(C)
=> ( r1_tarski(C,k4_finseq_1(B))
=> r1_xreal_0(k4_card_1(C),k4_card_1(k1_hallmar1(A,B,C))) ) ) ) ) ) ).
fof(t15_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_finseq_1(B,k1_zfmisc_1(A)) )
=> ( v1_hallmar1(B,A)
=> v2_relat_1(B) ) ) ) ).
fof(t16_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& v1_hallmar1(B,A) )
=> r1_xreal_0(np__1,k4_card_1(k1_funct_1(B,C))) ) ) ) ) ).
fof(t17_hallmar1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_finseq_1(B,k1_zfmisc_1(A)) )
=> ~ ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> k4_card_1(k1_funct_1(B,C)) = np__1 ) )
& v1_hallmar1(B,A)
& ! [C] : ~ r1_hallmar1(A,B,C) ) ) ) ).
fof(t18_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ( ? [C] : r1_hallmar1(A,B,C)
=> v1_hallmar1(B,A) ) ) ) ).
fof(d5_hallmar1,axiom,
! [A,B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_zfmisc_1(A))
=> ( m1_hallmar1(D,A,B,C)
<=> ( k4_finseq_1(D) = k4_finseq_1(B)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(B))
=> ( E = C
| k1_funct_1(B,E) = k1_funct_1(D,E) ) ) )
& r1_tarski(k1_funct_1(D,C),k1_funct_1(B,C)) ) ) ) ) ) ).
fof(d6_hallmar1,axiom,
! [A,B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_finseq_1(C,k1_zfmisc_1(A))
=> ( m2_hallmar1(C,A,B)
<=> ( k4_finseq_1(C) = k4_finseq_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(B))
=> r1_tarski(k1_funct_1(C,D),k1_funct_1(B,D)) ) ) ) ) ) ) ).
fof(d7_hallmar1,axiom,
! [A,B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> ( k1_funct_1(B,C) = k1_xboole_0
| ! [D] :
( m2_hallmar1(D,A,B)
=> ( m3_hallmar1(D,A,B,C)
<=> k1_card_1(k1_funct_1(D,C)) = np__1 ) ) ) ) ) ) ).
fof(t19_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_hallmar1(D,A,B,C)
=> m2_hallmar1(D,A,B) ) ) ) ) ).
fof(t20_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k1_funct_1(B,C)) )
=> m1_hallmar1(k2_hallmar1(A,B,C,D),A,B,C) ) ) ) ) ).
fof(t21_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k1_funct_1(B,C)) )
=> m2_hallmar1(k2_hallmar1(A,B,C,D),A,B) ) ) ) ) ).
fof(t22_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_hallmar1(C,A,B)
=> ! [D] :
( m2_hallmar1(D,A,C)
=> m2_hallmar1(D,A,B) ) ) ) ) ).
fof(t23_hallmar1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m2_finseq_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m3_hallmar1(D,A,B,C)
=> ~ ( r2_hidden(C,k4_finseq_1(B))
& k1_funct_1(D,C) = k1_xboole_0 ) ) ) ) ) ).
fof(t24_hallmar1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m2_finseq_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m3_hallmar1(E,A,B,C)
=> ! [F] :
( m3_hallmar1(F,A,E,D)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k4_finseq_1(B)) )
=> ( k1_funct_1(F,C) = k1_xboole_0
| k1_funct_1(E,D) = k1_xboole_0
| ( m3_hallmar1(F,A,B,D)
& m3_hallmar1(F,A,B,C) ) ) ) ) ) ) ) ) ) ).
fof(t25_hallmar1,axiom,
! [A,B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> m1_hallmar1(B,A,B,C) ) ) ).
fof(t26_hallmar1,axiom,
! [A,B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> m2_hallmar1(B,A,B) ) ).
fof(d8_hallmar1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ( v2_relat_1(B)
=> ! [C] :
( m2_hallmar1(C,A,B)
=> ( m4_hallmar1(C,A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(B))
=> k1_card_1(k1_funct_1(C,D)) = np__1 ) ) ) ) ) ) ) ).
fof(t27_hallmar1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_relat_1(B)
& m2_finseq_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( m4_hallmar1(C,A,B)
<=> ( k1_relat_1(C) = k4_finseq_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(B))
=> m3_hallmar1(C,A,B,D) ) ) ) ) ) ) ) ).
fof(t28_hallmar1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_finseq_1(B,k1_zfmisc_1(A)) )
=> ! [C,D] :
( m2_hallmar1(D,A,B)
=> ( r1_hallmar1(A,D,C)
=> r1_hallmar1(A,B,C) ) ) ) ) ).
fof(t29_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ( v1_hallmar1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,k4_card_1(k1_funct_1(B,C)))
& ! [D] :
~ ( r2_hidden(D,k1_funct_1(B,C))
& v1_hallmar1(k2_hallmar1(A,B,C,D),A) ) ) ) ) ) ) ).
fof(t30_hallmar1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k4_finseq_1(B))
& v1_hallmar1(B,A)
& ! [D] :
( m3_hallmar1(D,A,B,C)
=> ~ v1_hallmar1(D,A) ) ) ) ) ) ).
fof(t31_hallmar1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_finseq_1(B,k1_zfmisc_1(A)) )
=> ~ ( v1_hallmar1(B,A)
& ! [C] :
( m4_hallmar1(C,A,B)
=> ~ v1_hallmar1(C,A) ) ) ) ) ).
fof(t32_hallmar1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_finseq_1(B,k1_zfmisc_1(A)) )
=> ( ? [C] : r1_hallmar1(A,B,C)
<=> v1_hallmar1(B,A) ) ) ) ).
fof(s1_hallmar1,axiom,
( ( p1_s1_hallmar1(f1_s1_hallmar1)
& r1_xreal_0(np__2,f1_s1_hallmar1)
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,A)
& p1_s1_hallmar1(k1_nat_1(A,np__1)) )
=> ( r1_xreal_0(f1_s1_hallmar1,A)
| p1_s1_hallmar1(A) ) ) ) )
=> ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,f1_s1_hallmar1) )
=> p1_s1_hallmar1(A) ) ) ) ).
fof(s2_hallmar1,axiom,
( ( ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& ~ r1_xreal_0(A,np__1)
& p1_s2_hallmar1(A) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,A)
& p1_s2_hallmar1(k1_nat_1(A,np__1)) )
=> p1_s2_hallmar1(A) ) ) )
=> p1_s2_hallmar1(np__1) ) ).
fof(dt_m1_hallmar1,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> ! [D] :
( m1_hallmar1(D,A,B,C)
=> m2_finseq_1(D,k1_zfmisc_1(A)) ) ) ).
fof(existence_m1_hallmar1,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> ? [D] : m1_hallmar1(D,A,B,C) ) ).
fof(dt_m2_hallmar1,axiom,
! [A,B] :
( m1_finseq_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_hallmar1(C,A,B)
=> m2_finseq_1(C,k1_zfmisc_1(A)) ) ) ).
fof(existence_m2_hallmar1,axiom,
! [A,B] :
( m1_finseq_1(B,k1_zfmisc_1(A))
=> ? [C] : m2_hallmar1(C,A,B) ) ).
fof(dt_m3_hallmar1,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> ! [D] :
( m3_hallmar1(D,A,B,C)
=> m2_hallmar1(D,A,B) ) ) ).
fof(existence_m3_hallmar1,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> ? [D] : m3_hallmar1(D,A,B,C) ) ).
fof(dt_m4_hallmar1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m4_hallmar1(C,A,B)
=> m2_hallmar1(C,A,B) ) ) ).
fof(existence_m4_hallmar1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,k1_zfmisc_1(A)) )
=> ? [C] : m4_hallmar1(C,A,B) ) ).
fof(dt_k1_hallmar1,axiom,
$true ).
fof(dt_k2_hallmar1,axiom,
! [A,B,C,D] :
( ( v1_finset_1(A)
& m1_finseq_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> m2_finseq_1(k2_hallmar1(A,B,C,D),k1_zfmisc_1(A)) ) ).
%------------------------------------------------------------------------------