SET007 Axioms: SET007+45.ax
%------------------------------------------------------------------------------
% File : SET007+45 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Properties of Real Numbers
% 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 : real_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 115 ( 92 unt; 0 def)
% Number of atoms : 176 ( 13 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 76 ( 15 ~; 1 |; 24 &)
% ( 2 <=>; 34 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 2 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-2 aty)
% Number of functors : 15 ( 15 usr; 2 con; 0-2 aty)
% Number of variables : 42 ( 41 !; 1 ?)
% 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(cc1_real_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( v1_xreal_0(A)
& v1_xcmplx_0(A) ) ) ).
fof(t1_real_1,axiom,
$true ).
fof(t2_real_1,axiom,
$true ).
fof(t3_real_1,axiom,
$true ).
fof(t4_real_1,axiom,
$true ).
fof(t5_real_1,axiom,
$true ).
fof(t6_real_1,axiom,
$true ).
fof(t7_real_1,axiom,
$true ).
fof(t8_real_1,axiom,
$true ).
fof(t9_real_1,axiom,
$true ).
fof(t10_real_1,axiom,
$true ).
fof(t11_real_1,axiom,
$true ).
fof(t12_real_1,axiom,
$true ).
fof(t13_real_1,axiom,
$true ).
fof(t14_real_1,axiom,
$true ).
fof(t15_real_1,axiom,
$true ).
fof(t16_real_1,axiom,
$true ).
fof(t17_real_1,axiom,
$true ).
fof(t18_real_1,axiom,
$true ).
fof(t19_real_1,axiom,
$true ).
fof(t20_real_1,axiom,
$true ).
fof(t21_real_1,axiom,
$true ).
fof(t22_real_1,axiom,
$true ).
fof(t23_real_1,axiom,
$true ).
fof(t24_real_1,axiom,
$true ).
fof(t25_real_1,axiom,
! [A] :
( v1_xreal_0(A)
=> k6_xcmplx_0(A,np__0) = A ) ).
fof(t26_real_1,axiom,
k1_real_1(np__0) = np__0 ).
fof(d1_real_1,axiom,
$true ).
fof(d2_real_1,axiom,
$true ).
fof(d3_real_1,axiom,
$true ).
fof(d4_real_1,axiom,
$true ).
fof(d5_real_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
<=> ~ ( r1_xreal_0(B,A)
& B != A ) ) ) ) ).
fof(t27_real_1,axiom,
$true ).
fof(t28_real_1,axiom,
$true ).
fof(t29_real_1,axiom,
$true ).
fof(t30_real_1,axiom,
$true ).
fof(t31_real_1,axiom,
$true ).
fof(t32_real_1,axiom,
$true ).
fof(t33_real_1,axiom,
$true ).
fof(t34_real_1,axiom,
$true ).
fof(t35_real_1,axiom,
$true ).
fof(t36_real_1,axiom,
$true ).
fof(t37_real_1,axiom,
$true ).
fof(t38_real_1,axiom,
$true ).
fof(t39_real_1,axiom,
$true ).
fof(t40_real_1,axiom,
$true ).
fof(t41_real_1,axiom,
$true ).
fof(t42_real_1,axiom,
$true ).
fof(t43_real_1,axiom,
$true ).
fof(t44_real_1,axiom,
$true ).
fof(t45_real_1,axiom,
$true ).
fof(t46_real_1,axiom,
$true ).
fof(t47_real_1,axiom,
$true ).
fof(t48_real_1,axiom,
$true ).
fof(t49_real_1,axiom,
$true ).
fof(t50_real_1,axiom,
$true ).
fof(t51_real_1,axiom,
$true ).
fof(t52_real_1,axiom,
$true ).
fof(t53_real_1,axiom,
$true ).
fof(t54_real_1,axiom,
$true ).
fof(t55_real_1,axiom,
$true ).
fof(t56_real_1,axiom,
$true ).
fof(t57_real_1,axiom,
$true ).
fof(t58_real_1,axiom,
$true ).
fof(t59_real_1,axiom,
$true ).
fof(t60_real_1,axiom,
$true ).
fof(t61_real_1,axiom,
$true ).
fof(t62_real_1,axiom,
$true ).
fof(t63_real_1,axiom,
$true ).
fof(t64_real_1,axiom,
$true ).
fof(t65_real_1,axiom,
$true ).
fof(t66_real_1,axiom,
$true ).
fof(t67_real_1,axiom,
$true ).
fof(t68_real_1,axiom,
$true ).
fof(t69_real_1,axiom,
$true ).
fof(t70_real_1,axiom,
$true ).
fof(t71_real_1,axiom,
$true ).
fof(t72_real_1,axiom,
$true ).
fof(t73_real_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(A,np__0)
=> ( ~ ( ~ r1_xreal_0(C,B)
& r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A)) )
& ~ ( ~ r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A))
& r1_xreal_0(C,B) ) ) ) ) ) ) ).
fof(t74_real_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(np__0,A)
=> ( ~ ( ~ r1_xreal_0(C,B)
& r1_xreal_0(k7_xcmplx_0(B,A),k7_xcmplx_0(C,A)) )
& ~ ( ~ r1_xreal_0(k7_xcmplx_0(B,A),k7_xcmplx_0(C,A))
& r1_xreal_0(C,B) ) ) ) ) ) ) ).
fof(t75_real_1,axiom,
$true ).
fof(t76_real_1,axiom,
$true ).
fof(t77_real_1,axiom,
$true ).
fof(t78_real_1,axiom,
$true ).
fof(t79_real_1,axiom,
$true ).
fof(t80_real_1,axiom,
$true ).
fof(t81_real_1,axiom,
$true ).
fof(t82_real_1,axiom,
$true ).
fof(t83_real_1,axiom,
$true ).
fof(t84_real_1,axiom,
$true ).
fof(t85_real_1,axiom,
$true ).
fof(t86_real_1,axiom,
$true ).
fof(t87_real_1,axiom,
$true ).
fof(t88_real_1,axiom,
$true ).
fof(t89_real_1,axiom,
$true ).
fof(t90_real_1,axiom,
$true ).
fof(t91_real_1,axiom,
$true ).
fof(t92_real_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,D) )
=> r1_xreal_0(k6_xcmplx_0(A,D),k6_xcmplx_0(B,C)) )
& ~ ( ( ( ~ r1_xreal_0(B,A)
& r1_xreal_0(C,D) )
| ( r1_xreal_0(A,B)
& ~ r1_xreal_0(D,C) ) )
& r1_xreal_0(k6_xcmplx_0(B,C),k6_xcmplx_0(A,D)) ) ) ) ) ) ) ).
fof(s1_real_1,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(B,A)
<=> p1_s1_real_1(B) ) ) ) ).
fof(dt_k1_real_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> m1_subset_1(k1_real_1(A),k1_numbers) ) ).
fof(involutiveness_k1_real_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k1_real_1(k1_real_1(A)) = A ) ).
fof(redefinition_k1_real_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k1_real_1(A) = k4_xcmplx_0(A) ) ).
fof(dt_k2_real_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> m1_subset_1(k2_real_1(A),k1_numbers) ) ).
fof(involutiveness_k2_real_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k2_real_1(k2_real_1(A)) = A ) ).
fof(redefinition_k2_real_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k2_real_1(A) = k5_xcmplx_0(A) ) ).
fof(dt_k3_real_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k3_real_1(A,B),k1_numbers) ) ).
fof(commutativity_k3_real_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k3_real_1(A,B) = k3_real_1(B,A) ) ).
fof(redefinition_k3_real_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ).
fof(dt_k4_real_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k4_real_1(A,B),k1_numbers) ) ).
fof(commutativity_k4_real_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k4_real_1(A,B) = k4_real_1(B,A) ) ).
fof(redefinition_k4_real_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k4_real_1(A,B) = k3_xcmplx_0(A,B) ) ).
fof(dt_k5_real_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k5_real_1(A,B),k1_numbers) ) ).
fof(redefinition_k5_real_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ).
fof(dt_k6_real_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k6_real_1(A,B),k1_numbers) ) ).
fof(redefinition_k6_real_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k6_real_1(A,B) = k7_xcmplx_0(A,B) ) ).
%------------------------------------------------------------------------------