SET007 Axioms: SET007+792.ax
%------------------------------------------------------------------------------
% File : SET007+792 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Calculus of 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 : binari_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 76 ( 3 unt; 0 def)
% Number of atoms : 284 ( 104 equ)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 208 ( 0 ~; 0 |; 43 &)
% ( 2 <=>; 163 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-2 aty)
% Number of functors : 15 ( 15 usr; 3 con; 0-2 aty)
% Number of variables : 160 ( 160 !; 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_binari_5,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> ( v4_ordinal2(k1_binari_5(A,B))
& v1_xcmplx_0(k1_binari_5(A,B))
& v2_margrel1(k1_binari_5(A,B))
& v1_xreal_0(k1_binari_5(A,B)) ) ) ).
fof(fc2_binari_5,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> ( v4_ordinal2(k3_binari_5(A,B))
& v1_xcmplx_0(k3_binari_5(A,B))
& v2_margrel1(k3_binari_5(A,B))
& v1_xreal_0(k3_binari_5(A,B)) ) ) ).
fof(fc3_binari_5,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> ( v4_ordinal2(k5_binari_5(A,B))
& v1_xcmplx_0(k5_binari_5(A,B))
& v2_margrel1(k5_binari_5(A,B))
& v1_xreal_0(k5_binari_5(A,B)) ) ) ).
fof(d1_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_binari_5(A,B) = k9_margrel1(k10_margrel1(A,B)) ) ) ).
fof(d2_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k3_binari_5(A,B) = k9_margrel1(k1_binarith(A,B)) ) ) ).
fof(d3_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k5_binari_5(A,B) = k9_margrel1(k2_binarith(A,B)) ) ) ).
fof(t1_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> k1_binari_5(k8_margrel1,A) = k9_margrel1(A) ) ).
fof(t2_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> k1_binari_5(k7_margrel1,A) = k8_margrel1 ) ).
fof(t3_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ( k1_binari_5(A,A) = k9_margrel1(A)
& k9_margrel1(k1_binari_5(A,A)) = A ) ) ).
fof(t4_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k9_margrel1(k1_binari_5(A,B)) = k10_margrel1(A,B) ) ) ).
fof(t5_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ( k1_binari_5(A,k9_margrel1(A)) = k8_margrel1
& k9_margrel1(k1_binari_5(A,k9_margrel1(A))) = k7_margrel1 ) ) ).
fof(t6_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_binari_5(A,k10_margrel1(B,C)) = k9_margrel1(k10_margrel1(k10_margrel1(A,B),C)) ) ) ) ).
fof(t7_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_binari_5(A,k10_margrel1(B,C)) = k1_binari_5(k10_margrel1(A,B),C) ) ) ) ).
fof(t8_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_binari_5(A,k1_binarith(B,C)) = k10_margrel1(k9_margrel1(k10_margrel1(A,B)),k9_margrel1(k10_margrel1(A,C))) ) ) ) ).
fof(t9_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_binari_5(A,k2_binarith(B,C)) = k2_bvfunc_1(k10_margrel1(A,B),k10_margrel1(A,C)) ) ) ) ).
fof(t10_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> k3_binari_5(k8_margrel1,A) = k7_margrel1 ) ).
fof(t11_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> k3_binari_5(k7_margrel1,A) = k9_margrel1(A) ) ).
fof(t12_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ( k3_binari_5(A,A) = k9_margrel1(A)
& k9_margrel1(k3_binari_5(A,A)) = A ) ) ).
fof(t13_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k9_margrel1(k3_binari_5(A,B)) = k1_binarith(A,B) ) ) ).
fof(t14_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ( k3_binari_5(A,k9_margrel1(A)) = k7_margrel1
& k9_margrel1(k3_binari_5(A,k9_margrel1(A))) = k8_margrel1 ) ) ).
fof(t15_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k3_binari_5(A,k10_margrel1(B,C)) = k1_binarith(k9_margrel1(k1_binarith(A,B)),k9_margrel1(k1_binarith(A,C))) ) ) ) ).
fof(t16_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k3_binari_5(A,k1_binarith(B,C)) = k9_margrel1(k1_binarith(k1_binarith(A,B),C)) ) ) ) ).
fof(t17_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> k5_binari_5(k8_margrel1,A) = A ) ).
fof(t18_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> k5_binari_5(k7_margrel1,A) = k9_margrel1(A) ) ).
fof(t19_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ( k5_binari_5(A,A) = k8_margrel1
& k9_margrel1(k5_binari_5(A,A)) = k7_margrel1 ) ) ).
fof(t20_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k9_margrel1(k5_binari_5(A,B)) = k2_binarith(A,B) ) ) ).
fof(t21_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ( k5_binari_5(A,k9_margrel1(A)) = k7_margrel1
& k9_margrel1(k5_binari_5(A,k9_margrel1(A))) = k8_margrel1 ) ) ).
fof(t22_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ( r1_xreal_0(A,k1_bvfunc_1(B,C))
<=> r1_xreal_0(k10_margrel1(A,B),C) ) ) ) ) ).
fof(t23_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k2_bvfunc_1(A,B) = k10_margrel1(k1_bvfunc_1(A,B),k1_bvfunc_1(B,A)) ) ) ).
fof(t24_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ( k2_bvfunc_1(A,B) = k8_margrel1
<=> ( k1_bvfunc_1(A,B) = k8_margrel1
& k1_bvfunc_1(B,A) = k8_margrel1 ) ) ) ) ).
fof(t25_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ( ( k1_bvfunc_1(A,B) = k8_margrel1
& k1_bvfunc_1(B,A) = k8_margrel1 )
=> A = B ) ) ) ).
fof(t26_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ( ( k1_bvfunc_1(A,B) = k8_margrel1
& k1_bvfunc_1(B,C) = k8_margrel1 )
=> k1_bvfunc_1(A,C) = k8_margrel1 ) ) ) ) ).
fof(t27_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ( ( k2_bvfunc_1(A,B) = k8_margrel1
& k2_bvfunc_1(B,C) = k8_margrel1 )
=> k2_bvfunc_1(A,C) = k8_margrel1 ) ) ) ) ).
fof(t28_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_bvfunc_1(A,B) = k1_bvfunc_1(k9_margrel1(B),k9_margrel1(A)) ) ) ).
fof(t29_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k2_bvfunc_1(A,B) = k2_bvfunc_1(k9_margrel1(A),k9_margrel1(B)) ) ) ).
fof(t30_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ! [D] :
( v2_margrel1(D)
=> ( ( k2_bvfunc_1(A,B) = k8_margrel1
& k2_bvfunc_1(C,D) = k8_margrel1 )
=> k2_bvfunc_1(k10_margrel1(A,C),k10_margrel1(B,D)) = k8_margrel1 ) ) ) ) ) ).
fof(t31_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ! [D] :
( v2_margrel1(D)
=> ( ( k2_bvfunc_1(A,B) = k8_margrel1
& k2_bvfunc_1(C,D) = k8_margrel1 )
=> k2_bvfunc_1(k1_bvfunc_1(A,C),k1_bvfunc_1(B,D)) = k8_margrel1 ) ) ) ) ) ).
fof(t32_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ! [D] :
( v2_margrel1(D)
=> ( ( k2_bvfunc_1(A,B) = k8_margrel1
& k2_bvfunc_1(C,D) = k8_margrel1 )
=> k2_bvfunc_1(k1_binarith(A,C),k1_binarith(B,D)) = k8_margrel1 ) ) ) ) ) ).
fof(t33_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ! [D] :
( v2_margrel1(D)
=> ( ( k2_bvfunc_1(A,B) = k8_margrel1
& k2_bvfunc_1(C,D) = k8_margrel1 )
=> k2_bvfunc_1(k2_bvfunc_1(A,C),k2_bvfunc_1(B,D)) = k8_margrel1 ) ) ) ) ) ).
fof(t34_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ( ( A = k8_margrel1
& k1_bvfunc_1(A,B) = k8_margrel1 )
=> B = k8_margrel1 ) ) ) ).
fof(t35_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ( A = k8_margrel1
=> k1_bvfunc_1(B,A) = k8_margrel1 ) ) ) ).
fof(t36_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ( k9_margrel1(A) = k8_margrel1
=> k1_bvfunc_1(A,B) = k8_margrel1 ) ) ) ).
fof(t37_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> k1_bvfunc_1(A,A) = k8_margrel1 ) ).
fof(t38_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ( ( k1_bvfunc_1(A,B) = k8_margrel1
& k1_bvfunc_1(A,k9_margrel1(B)) = k8_margrel1 )
=> k9_margrel1(A) = k8_margrel1 ) ) ) ).
fof(t39_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> k1_bvfunc_1(k1_bvfunc_1(k9_margrel1(A),A),A) = k8_margrel1 ) ).
fof(t40_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_bvfunc_1(k1_bvfunc_1(A,B),k1_bvfunc_1(k9_margrel1(k10_margrel1(B,C)),k9_margrel1(k10_margrel1(A,C)))) = k8_margrel1 ) ) ) ).
fof(t41_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_bvfunc_1(k1_bvfunc_1(A,B),k1_bvfunc_1(k1_bvfunc_1(B,C),k1_bvfunc_1(A,C))) = k8_margrel1 ) ) ) ).
fof(t42_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ( k1_bvfunc_1(A,B) = k8_margrel1
=> k1_bvfunc_1(k1_bvfunc_1(B,C),k1_bvfunc_1(A,C)) = k8_margrel1 ) ) ) ) ).
fof(t43_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_bvfunc_1(A,k1_bvfunc_1(B,A)) = k8_margrel1 ) ) ).
fof(t44_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_bvfunc_1(k1_bvfunc_1(k1_bvfunc_1(A,B),C),k1_bvfunc_1(B,C)) = k8_margrel1 ) ) ) ).
fof(t45_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_bvfunc_1(A,k1_bvfunc_1(k1_bvfunc_1(A,B),B)) = k8_margrel1 ) ) ).
fof(t46_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_bvfunc_1(k1_bvfunc_1(A,k1_bvfunc_1(B,C)),k1_bvfunc_1(B,k1_bvfunc_1(A,C))) = k8_margrel1 ) ) ) ).
fof(t47_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_bvfunc_1(k1_bvfunc_1(A,B),k1_bvfunc_1(k1_bvfunc_1(C,A),k1_bvfunc_1(C,B))) = k8_margrel1 ) ) ) ).
fof(t48_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_bvfunc_1(k1_bvfunc_1(A,k1_bvfunc_1(A,B)),k1_bvfunc_1(A,B)) = k8_margrel1 ) ) ).
fof(t49_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> k1_bvfunc_1(k1_bvfunc_1(A,k1_bvfunc_1(B,C)),k1_bvfunc_1(k1_bvfunc_1(A,B),k1_bvfunc_1(A,C))) = k8_margrel1 ) ) ) ).
fof(t50_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ( A = k8_margrel1
=> k1_bvfunc_1(k1_bvfunc_1(A,B),B) = k8_margrel1 ) ) ) ).
fof(t51_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ( k1_bvfunc_1(A,k1_bvfunc_1(B,C)) = k8_margrel1
=> k1_bvfunc_1(B,k1_bvfunc_1(A,C)) = k8_margrel1 ) ) ) ) ).
fof(t52_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ( ( k1_bvfunc_1(A,k1_bvfunc_1(B,C)) = k8_margrel1
& B = k8_margrel1 )
=> k1_bvfunc_1(A,C) = k8_margrel1 ) ) ) ) ).
fof(t53_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ( ( k1_bvfunc_1(A,k1_bvfunc_1(B,C)) = k8_margrel1
& B = k8_margrel1
& A = k8_margrel1 )
=> C = k8_margrel1 ) ) ) ) ).
fof(t54_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ( k1_bvfunc_1(A,k1_bvfunc_1(A,B)) = k8_margrel1
=> k1_bvfunc_1(A,B) = k8_margrel1 ) ) ) ).
fof(t55_binari_5,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ! [C] :
( v2_margrel1(C)
=> ( k1_bvfunc_1(A,k1_bvfunc_1(B,C)) = k8_margrel1
=> k1_bvfunc_1(k1_bvfunc_1(A,B),k1_bvfunc_1(A,C)) = k8_margrel1 ) ) ) ) ).
fof(dt_k1_binari_5,axiom,
$true ).
fof(commutativity_k1_binari_5,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> k1_binari_5(A,B) = k1_binari_5(B,A) ) ).
fof(dt_k2_binari_5,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> m1_subset_1(k2_binari_5(A,B),k6_margrel1) ) ).
fof(commutativity_k2_binari_5,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> k2_binari_5(A,B) = k2_binari_5(B,A) ) ).
fof(redefinition_k2_binari_5,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> k2_binari_5(A,B) = k1_binari_5(A,B) ) ).
fof(dt_k3_binari_5,axiom,
$true ).
fof(commutativity_k3_binari_5,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> k3_binari_5(A,B) = k3_binari_5(B,A) ) ).
fof(dt_k4_binari_5,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> m1_subset_1(k4_binari_5(A,B),k6_margrel1) ) ).
fof(commutativity_k4_binari_5,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> k4_binari_5(A,B) = k4_binari_5(B,A) ) ).
fof(redefinition_k4_binari_5,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> k4_binari_5(A,B) = k3_binari_5(A,B) ) ).
fof(dt_k5_binari_5,axiom,
$true ).
fof(commutativity_k5_binari_5,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> k5_binari_5(A,B) = k5_binari_5(B,A) ) ).
fof(dt_k6_binari_5,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> m1_subset_1(k6_binari_5(A,B),k6_margrel1) ) ).
fof(commutativity_k6_binari_5,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> k6_binari_5(A,B) = k6_binari_5(B,A) ) ).
fof(redefinition_k6_binari_5,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_margrel1)
& m1_subset_1(B,k6_margrel1) )
=> k6_binari_5(A,B) = k5_binari_5(A,B) ) ).
%------------------------------------------------------------------------------