SET007 Axioms: SET007+38.ax
%------------------------------------------------------------------------------
% File : SET007+38 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Complex Numbers - Basic Definitions
% 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 : xcmplx_0 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 33 ( 8 unt; 0 def)
% Number of atoms : 103 ( 24 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 82 ( 12 ~; 0 |; 32 &)
% ( 7 <=>; 31 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-2 aty)
% Number of functors : 16 ( 16 usr; 5 con; 0-5 aty)
% Number of variables : 50 ( 40 !; 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(fc1_xcmplx_0,axiom,
v1_xcmplx_0(k1_xcmplx_0) ).
fof(rc1_xcmplx_0,axiom,
? [A] : v1_xcmplx_0(A) ).
fof(fc2_xcmplx_0,axiom,
! [A,B] :
( ( v1_xcmplx_0(A)
& v1_xcmplx_0(B) )
=> v1_xcmplx_0(k2_xcmplx_0(A,B)) ) ).
fof(fc3_xcmplx_0,axiom,
! [A,B] :
( ( v1_xcmplx_0(A)
& v1_xcmplx_0(B) )
=> v1_xcmplx_0(k3_xcmplx_0(A,B)) ) ).
fof(fc4_xcmplx_0,axiom,
! [A,B] :
( ( v1_xcmplx_0(A)
& v1_xcmplx_0(B) )
=> v1_xcmplx_0(k6_xcmplx_0(A,B)) ) ).
fof(fc5_xcmplx_0,axiom,
! [A,B] :
( ( v1_xcmplx_0(A)
& v1_xcmplx_0(B) )
=> v1_xcmplx_0(k7_xcmplx_0(A,B)) ) ).
fof(rc2_xcmplx_0,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_xcmplx_0(A) ) ).
fof(fc6_xcmplx_0,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_xcmplx_0(A) )
=> ( ~ v1_xboole_0(k4_xcmplx_0(A))
& v1_xcmplx_0(k4_xcmplx_0(A)) ) ) ).
fof(fc7_xcmplx_0,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_xcmplx_0(A) )
=> ( ~ v1_xboole_0(k5_xcmplx_0(A))
& v1_xcmplx_0(k5_xcmplx_0(A)) ) ) ).
fof(fc8_xcmplx_0,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_xcmplx_0(A)
& ~ v1_xboole_0(B)
& v1_xcmplx_0(B) )
=> ( ~ v1_xboole_0(k3_xcmplx_0(A,B))
& v1_xcmplx_0(k3_xcmplx_0(A,B)) ) ) ).
fof(fc9_xcmplx_0,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_xcmplx_0(A)
& ~ v1_xboole_0(B)
& v1_xcmplx_0(B) )
=> ( ~ v1_xboole_0(k7_xcmplx_0(A,B))
& v1_xcmplx_0(k7_xcmplx_0(A,B)) ) ) ).
fof(cc1_xcmplx_0,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> v1_xcmplx_0(A) ) ).
fof(cc2_xcmplx_0,axiom,
! [A] :
( v4_ordinal2(A)
=> v1_xcmplx_0(A) ) ).
fof(d1_xcmplx_0,axiom,
k1_xcmplx_0 = k5_funct_4(k1_numbers,np__0,np__1,np__0,np__1) ).
fof(d2_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
<=> r2_hidden(A,k2_numbers) ) ).
fof(d3_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( v1_xboole_0(A)
<=> A = np__0 ) ) ).
fof(d4_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ! [C] :
( C = k2_xcmplx_0(A,B)
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& ? [F] :
( m1_subset_1(F,k1_numbers)
& ? [G] :
( m1_subset_1(G,k1_numbers)
& A = k5_arytm_0(D,E)
& B = k5_arytm_0(F,G)
& C = k5_arytm_0(k1_arytm_0(D,F),k1_arytm_0(E,G)) ) ) ) ) ) ) ) ).
fof(d5_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ! [C] :
( C = k3_xcmplx_0(A,B)
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& ? [F] :
( m1_subset_1(F,k1_numbers)
& ? [G] :
( m1_subset_1(G,k1_numbers)
& A = k5_arytm_0(D,E)
& B = k5_arytm_0(F,G)
& C = k5_arytm_0(k1_arytm_0(k2_arytm_0(D,F),k3_arytm_0(k2_arytm_0(E,G))),k1_arytm_0(k2_arytm_0(D,G),k2_arytm_0(E,F))) ) ) ) ) ) ) ) ).
fof(d6_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( B = k4_xcmplx_0(A)
<=> k2_xcmplx_0(A,B) = np__0 ) ) ) ).
fof(d7_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( ( A != np__0
=> ( B = k5_xcmplx_0(A)
<=> k3_xcmplx_0(A,B) = np__1 ) )
& ( A = np__0
=> ( B = k5_xcmplx_0(A)
<=> B = np__0 ) ) ) ) ) ).
fof(d8_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k6_xcmplx_0(A,B) = k2_xcmplx_0(A,k4_xcmplx_0(B)) ) ) ).
fof(d9_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k7_xcmplx_0(A,B) = k3_xcmplx_0(A,k5_xcmplx_0(B)) ) ) ).
fof(dt_k1_xcmplx_0,axiom,
$true ).
fof(dt_k2_xcmplx_0,axiom,
$true ).
fof(commutativity_k2_xcmplx_0,axiom,
! [A,B] :
( ( v1_xcmplx_0(A)
& v1_xcmplx_0(B) )
=> k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ).
fof(dt_k3_xcmplx_0,axiom,
$true ).
fof(commutativity_k3_xcmplx_0,axiom,
! [A,B] :
( ( v1_xcmplx_0(A)
& v1_xcmplx_0(B) )
=> k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ).
fof(dt_k4_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> v1_xcmplx_0(k4_xcmplx_0(A)) ) ).
fof(involutiveness_k4_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ).
fof(dt_k5_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> v1_xcmplx_0(k5_xcmplx_0(A)) ) ).
fof(involutiveness_k5_xcmplx_0,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k5_xcmplx_0(k5_xcmplx_0(A)) = A ) ).
fof(dt_k6_xcmplx_0,axiom,
$true ).
fof(dt_k7_xcmplx_0,axiom,
$true ).
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