SET007 Axioms: SET007+173.ax
%------------------------------------------------------------------------------
% File : SET007+173 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Binary Arithmetics
% 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 : binarith [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 87 ( 13 unt; 0 def)
% Number of atoms : 313 ( 69 equ)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 251 ( 25 ~; 2 |; 53 &)
% ( 5 <=>; 166 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-3 aty)
% Number of functors : 39 ( 39 usr; 9 con; 0-5 aty)
% Number of variables : 180 ( 180 !; 0 ?)
% 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_binarith,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> v2_margrel1(k1_binarith(A,B)) ) ).
fof(fc2_binarith,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> v2_margrel1(k2_binarith(A,B)) ) ).
fof(t1_binarith,axiom,
$true ).
fof(t2_binarith,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,C)
=> ! [E] :
( m2_finseq_2(E,C,k4_finseq_2(B,C))
=> ( r2_hidden(A,k2_finseq_1(B))
=> k4_finseq_4(k5_numbers,C,k8_finseq_1(C,E,k12_finseq_1(C,D)),A) = k4_finseq_4(k5_numbers,C,E,A) ) ) ) ) ) ) ).
fof(t3_binarith,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_2(D,B,k4_finseq_2(A,B))
=> k4_finseq_4(k5_numbers,B,k8_finseq_1(B,D,k12_finseq_1(B,C)),k23_binop_2(A,np__1)) = C ) ) ) ) ).
fof(t4_binarith,axiom,
$true ).
fof(t5_binarith,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(A,k2_finseq_1(B))
& v1_xboole_0(A) ) ) ) ).
fof(d1_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_binarith(A,B) = k9_margrel1(k10_margrel1(k9_margrel1(A),k9_margrel1(B))) ) ) ).
fof(d2_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k2_binarith(A,B) = k1_binarith(k10_margrel1(k9_margrel1(A),B),k10_margrel1(A,k9_margrel1(B))) ) ) ).
fof(t6_binarith,axiom,
$true ).
fof(t7_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> k1_binarith(A,k7_margrel1) = A ) ).
fof(t8_binarith,axiom,
$true ).
fof(t9_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k9_margrel1(k10_margrel1(A,B)) = k1_binarith(k9_margrel1(A),k9_margrel1(B)) ) ) ).
fof(t10_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k9_margrel1(k1_binarith(A,B)) = k10_margrel1(k9_margrel1(A),k9_margrel1(B)) ) ) ).
fof(t11_binarith,axiom,
$true ).
fof(t12_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k10_margrel1(A,B) = k9_margrel1(k1_binarith(k9_margrel1(A),k9_margrel1(B))) ) ) ).
fof(t13_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> k2_binarith(k8_margrel1,A) = k9_margrel1(A) ) ).
fof(t14_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> k2_binarith(k7_margrel1,A) = A ) ).
fof(t15_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> k2_binarith(A,A) = k7_margrel1 ) ).
fof(t16_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> k10_margrel1(A,A) = A ) ).
fof(t17_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> k2_binarith(A,k9_margrel1(A)) = k8_margrel1 ) ).
fof(t18_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> k1_binarith(A,k9_margrel1(A)) = k8_margrel1 ) ).
fof(t19_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> k1_binarith(A,k8_margrel1) = k8_margrel1 ) ).
fof(t20_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_binarith(k1_binarith(A,B),C) = k1_binarith(A,k1_binarith(B,C)) ) ) ) ).
fof(t21_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> k1_binarith(A,A) = A ) ).
fof(t22_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k10_margrel1(A,k1_binarith(B,C)) = k1_binarith(k10_margrel1(A,B),k10_margrel1(A,C)) ) ) ) ).
fof(t23_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_binarith(A,k10_margrel1(B,C)) = k10_margrel1(k1_binarith(A,B),k1_binarith(A,C)) ) ) ) ).
fof(t24_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_binarith(A,k10_margrel1(A,B)) = A ) ) ).
fof(t25_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k10_margrel1(A,k1_binarith(A,B)) = A ) ) ).
fof(t26_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_binarith(A,k10_margrel1(k9_margrel1(A),B)) = k1_binarith(A,B) ) ) ).
fof(t27_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k10_margrel1(A,k1_binarith(k9_margrel1(A),B)) = k10_margrel1(A,B) ) ) ).
fof(t28_binarith,axiom,
$true ).
fof(t29_binarith,axiom,
$true ).
fof(t30_binarith,axiom,
$true ).
fof(t31_binarith,axiom,
$true ).
fof(t32_binarith,axiom,
$true ).
fof(t33_binarith,axiom,
k4_binarith(k8_margrel1,k7_margrel1) = k8_margrel1 ).
fof(t34_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k2_binarith(k2_binarith(A,B),C) = k2_binarith(A,k2_binarith(B,C)) ) ) ) ).
fof(t35_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k2_binarith(A,k10_margrel1(k9_margrel1(A),B)) = k1_binarith(A,B) ) ) ).
fof(t36_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_binarith(A,k2_binarith(A,B)) = k1_binarith(A,B) ) ) ).
fof(t37_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_binarith(A,k2_binarith(k9_margrel1(A),B)) = k1_binarith(A,k9_margrel1(B)) ) ) ).
fof(t38_binarith,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k10_margrel1(A,k2_binarith(B,C)) = k2_binarith(k10_margrel1(A,B),k10_margrel1(A,C)) ) ) ) ).
fof(d3_binarith,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ( r1_xreal_0(np__0,k6_xcmplx_0(A,B))
=> k5_binarith(A,B) = k6_xcmplx_0(A,B) )
& ( ~ r1_xreal_0(np__0,k6_xcmplx_0(A,B))
=> k5_binarith(A,B) = np__0 ) ) ) ) ).
fof(t39_binarith,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> k5_binarith(k2_xcmplx_0(A,B),B) = A ) ) ).
fof(d4_binarith,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( C = k6_binarith(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_finseq_1(A))
=> k4_finseq_4(k5_numbers,k6_margrel1,C,D) = k11_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,D)) ) ) ) ) ) ) ).
fof(d5_binarith,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [D] :
( m2_finseq_2(D,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( D = k7_binarith(A,B,C)
<=> ( k4_finseq_4(k5_numbers,k6_margrel1,D,np__1) = k7_margrel1
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,E)
=> ( r1_xreal_0(A,E)
| k4_finseq_4(k5_numbers,k6_margrel1,D,k23_binop_2(E,np__1)) = k3_binarith(k3_binarith(k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,E),k4_finseq_4(k5_numbers,k6_margrel1,C,E)),k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,E),k4_finseq_4(k5_numbers,k6_margrel1,D,E))),k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,C,E),k4_finseq_4(k5_numbers,k6_margrel1,D,E))) ) ) ) ) ) ) ) ) ) ).
fof(d6_binarith,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ( C = k8_binarith(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_finseq_1(A))
=> k4_finseq_4(k5_numbers,k5_numbers,C,D) = k2_cqc_lang(k5_numbers,k4_finseq_4(k5_numbers,k6_margrel1,B,D),k7_margrel1,np__0,k3_series_1(np__2,k5_binarith(D,np__1))) ) ) ) ) ) ) ).
fof(d7_binarith,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> k9_binarith(A,B) = k2_finsop_1(k5_numbers,k8_binarith(A,B),k47_binop_2) ) ) ).
fof(d8_binarith,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [D] :
( m2_finseq_2(D,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( D = k10_binarith(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k4_finseq_4(k5_numbers,k6_margrel1,D,E) = k4_binarith(k4_binarith(k4_finseq_4(k5_numbers,k6_margrel1,B,E),k4_finseq_4(k5_numbers,k6_margrel1,C,E)),k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,C),E)) ) ) ) ) ) ) ) ).
fof(d9_binarith,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> k11_binarith(A,B,C) = k3_binarith(k3_binarith(k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,A),k4_finseq_4(k5_numbers,k6_margrel1,C,A)),k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,A),k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,C),A))),k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,C,A),k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,C),A))) ) ) ) ).
fof(d10_binarith,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( r1_binarith(A,B,C)
<=> k11_binarith(A,B,C) = k7_margrel1 ) ) ) ) ).
fof(t40_binarith,axiom,
! [A] :
( m2_finseq_2(A,k6_margrel1,k4_finseq_2(np__1,k6_margrel1))
=> ( A = k12_finseq_1(k6_margrel1,k7_margrel1)
| A = k12_finseq_1(k6_margrel1,k8_margrel1) ) ) ).
fof(t41_binarith,axiom,
! [A] :
( m2_finseq_2(A,k6_margrel1,k4_finseq_2(np__1,k6_margrel1))
=> ( A = k12_finseq_1(k6_margrel1,k7_margrel1)
=> k9_binarith(np__1,A) = np__0 ) ) ).
fof(t42_binarith,axiom,
! [A] :
( m2_finseq_2(A,k6_margrel1,k4_finseq_2(np__1,k6_margrel1))
=> ( A = k12_finseq_1(k6_margrel1,k8_margrel1)
=> k9_binarith(np__1,A) = np__1 ) ) ).
fof(t43_binarith,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k6_margrel1)
=> ! [E] :
( m1_subset_1(E,k6_margrel1)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k2_finseq_1(A))
=> k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(k23_binop_2(A,np__1),k12_binarith(A,np__1,k6_margrel1,B,k13_binarith(k6_margrel1,D)),k12_binarith(A,np__1,k6_margrel1,C,k13_binarith(k6_margrel1,E))),F) = k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,C),F) ) ) ) ) ) ) ) ).
fof(t44_binarith,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k6_margrel1)
=> ! [E] :
( m1_subset_1(E,k6_margrel1)
=> k11_binarith(A,B,C) = k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(k23_binop_2(A,np__1),k12_binarith(A,np__1,k6_margrel1,B,k13_binarith(k6_margrel1,D)),k12_binarith(A,np__1,k6_margrel1,C,k13_binarith(k6_margrel1,E))),k23_binop_2(A,np__1)) ) ) ) ) ) ).
fof(t45_binarith,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k6_margrel1)
=> ! [E] :
( m1_subset_1(E,k6_margrel1)
=> k10_binarith(k23_binop_2(A,np__1),k12_binarith(A,np__1,k6_margrel1,B,k13_binarith(k6_margrel1,D)),k12_binarith(A,np__1,k6_margrel1,C,k13_binarith(k6_margrel1,E))) = k12_binarith(A,np__1,k6_margrel1,k10_binarith(A,B,C),k13_binarith(k6_margrel1,k4_binarith(k4_binarith(D,E),k11_binarith(A,B,C)))) ) ) ) ) ) ).
fof(t46_binarith,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k6_margrel1)
=> k9_binarith(k23_binop_2(A,np__1),k12_binarith(A,np__1,k6_margrel1,B,k13_binarith(k6_margrel1,C))) = k23_binop_2(k9_binarith(A,B),k2_cqc_lang(k5_numbers,C,k7_margrel1,np__0,k3_series_1(np__2,A))) ) ) ) ).
fof(t47_binarith,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> k23_binop_2(k9_binarith(A,k10_binarith(A,B,C)),k2_cqc_lang(k5_numbers,k11_binarith(A,B,C),k7_margrel1,np__0,k3_series_1(np__2,A))) = k23_binop_2(k9_binarith(A,B),k9_binarith(A,C)) ) ) ) ).
fof(t48_binarith,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( r1_binarith(A,B,C)
=> k9_binarith(A,k10_binarith(A,B,C)) = k23_binop_2(k9_binarith(A,B),k9_binarith(A,C)) ) ) ) ) ).
fof(t49_binarith,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(A,C)
& k5_binarith(B,A) = k5_binarith(C,A) )
=> B = C ) ) ) ) ).
fof(t50_binarith,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_xreal_0(A,B)
=> k5_binarith(B,A) = k6_xcmplx_0(B,A) ) ) ) ).
fof(t51_binarith,axiom,
! [A] :
( v4_ordinal2(A)
=> k5_binarith(A,A) = np__0 ) ).
fof(t52_binarith,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> r1_xreal_0(k5_binarith(A,B),A) ) ) ).
fof(t53_binarith,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_xreal_0(B,A)
=> k2_xcmplx_0(k5_binarith(A,B),B) = A ) ) ) ).
fof(s1_binarith,axiom,
( ( p1_s1_binarith(np__1)
& ! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ( p1_s1_binarith(A)
=> p1_s1_binarith(k23_binop_2(A,np__1)) ) ) )
=> ! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> p1_s1_binarith(A) ) ) ).
fof(dt_k1_binarith,axiom,
$true ).
fof(commutativity_k1_binarith,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> k1_binarith(A,B) = k1_binarith(B,A) ) ).
fof(dt_k2_binarith,axiom,
$true ).
fof(commutativity_k2_binarith,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> k2_binarith(A,B) = k2_binarith(B,A) ) ).
fof(dt_k3_binarith,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> m1_subset_1(k3_binarith(A,B),k6_margrel1) ) ).
fof(commutativity_k3_binarith,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> k3_binarith(A,B) = k3_binarith(B,A) ) ).
fof(redefinition_k3_binarith,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> k3_binarith(A,B) = k1_binarith(A,B) ) ).
fof(dt_k4_binarith,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> m1_subset_1(k4_binarith(A,B),k6_margrel1) ) ).
fof(commutativity_k4_binarith,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> k4_binarith(A,B) = k4_binarith(B,A) ) ).
fof(redefinition_k4_binarith,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> k4_binarith(A,B) = k2_binarith(A,B) ) ).
fof(dt_k5_binarith,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> m2_subset_1(k5_binarith(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k6_binarith,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k6_margrel1)) )
=> m2_finseq_2(k6_binarith(A,B),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) ).
fof(dt_k7_binarith,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k6_margrel1))
& m1_subset_1(C,k4_finseq_2(A,k6_margrel1)) )
=> m2_finseq_2(k7_binarith(A,B,C),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) ).
fof(dt_k8_binarith,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k6_margrel1)) )
=> m2_finseq_2(k8_binarith(A,B),k5_numbers,k4_finseq_2(A,k5_numbers)) ) ).
fof(dt_k9_binarith,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k6_margrel1)) )
=> m2_subset_1(k9_binarith(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k10_binarith,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k6_margrel1))
& m1_subset_1(C,k4_finseq_2(A,k6_margrel1)) )
=> m2_finseq_2(k10_binarith(A,B,C),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) ).
fof(dt_k11_binarith,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k6_margrel1))
& m1_subset_1(C,k4_finseq_2(A,k6_margrel1)) )
=> m1_subset_1(k11_binarith(A,B,C),k6_margrel1) ) ).
fof(dt_k12_binarith,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& ~ v1_xboole_0(C)
& m1_subset_1(D,k4_finseq_2(A,C))
& m1_subset_1(E,k4_finseq_2(B,C)) )
=> m2_finseq_2(k12_binarith(A,B,C,D,E),C,k4_finseq_2(k23_binop_2(A,B),C)) ) ).
fof(redefinition_k12_binarith,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& ~ v1_xboole_0(C)
& m1_subset_1(D,k4_finseq_2(A,C))
& m1_subset_1(E,k4_finseq_2(B,C)) )
=> k12_binarith(A,B,C,D,E) = k7_finseq_1(D,E) ) ).
fof(dt_k13_binarith,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m2_finseq_2(k13_binarith(A,B),A,k4_finseq_2(np__1,A)) ) ).
fof(redefinition_k13_binarith,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> k13_binarith(A,B) = k5_finseq_1(B) ) ).
%------------------------------------------------------------------------------