SET007 Axioms: SET007+123.ax
%------------------------------------------------------------------------------
% File : SET007+123 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Monotonic and Continuous Real Function
% 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 : fcont_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 40 ( 4 unt; 0 def)
% Number of atoms : 222 ( 15 equ)
% Maximal formula atoms : 13 ( 5 avg)
% Number of connectives : 214 ( 32 ~; 16 |; 79 &)
% ( 2 <=>; 85 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 27 ( 26 usr; 0 prp; 1-3 aty)
% Number of functors : 27 ( 27 usr; 4 con; 0-4 aty)
% Number of variables : 75 ( 73 !; 2 ?)
% 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_fcont_3,axiom,
( v1_membered(k2_subset_1(k1_numbers))
& v2_membered(k2_subset_1(k1_numbers))
& v2_rcomp_1(k2_subset_1(k1_numbers))
& v3_rcomp_1(k2_subset_1(k1_numbers)) ) ).
fof(fc2_fcont_3,axiom,
( v1_xboole_0(k1_subset_1(k1_numbers))
& v1_membered(k1_subset_1(k1_numbers))
& v2_membered(k1_subset_1(k1_numbers))
& v3_membered(k1_subset_1(k1_numbers))
& v4_membered(k1_subset_1(k1_numbers))
& v5_membered(k1_subset_1(k1_numbers))
& v2_rcomp_1(k1_subset_1(k1_numbers))
& v3_rcomp_1(k1_subset_1(k1_numbers)) ) ).
fof(fc3_fcont_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_membered(k4_limfunc1(A))
& v2_membered(k4_limfunc1(A))
& v3_rcomp_1(k4_limfunc1(A)) ) ) ).
fof(fc4_fcont_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_membered(k12_prob_1(A))
& v2_membered(k12_prob_1(A))
& v3_rcomp_1(k12_prob_1(A)) ) ) ).
fof(fc5_fcont_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_membered(k3_limfunc1(A))
& v2_membered(k3_limfunc1(A))
& v2_rcomp_1(k3_limfunc1(A)) ) ) ).
fof(fc6_fcont_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_membered(k2_limfunc1(A))
& v2_membered(k2_limfunc1(A))
& v2_rcomp_1(k2_limfunc1(A)) ) ) ).
fof(t1_fcont_3,axiom,
v2_rcomp_1(k2_subset_1(k1_numbers)) ).
fof(t2_fcont_3,axiom,
v3_rcomp_1(k1_subset_1(k1_numbers)) ).
fof(t3_fcont_3,axiom,
v2_rcomp_1(k1_subset_1(k1_numbers)) ).
fof(t4_fcont_3,axiom,
v3_rcomp_1(k2_subset_1(k1_numbers)) ).
fof(t5_fcont_3,axiom,
! [A] :
( v1_xreal_0(A)
=> v2_rcomp_1(k3_limfunc1(A)) ) ).
fof(t6_fcont_3,axiom,
! [A] :
( v1_xreal_0(A)
=> v2_rcomp_1(k2_limfunc1(A)) ) ).
fof(t7_fcont_3,axiom,
! [A] :
( v1_xreal_0(A)
=> v3_rcomp_1(k4_limfunc1(A)) ) ).
fof(t8_fcont_3,axiom,
! [A] :
( v1_xreal_0(A)
=> v3_rcomp_1(k12_prob_1(A)) ) ).
fof(t9_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( ~ r1_xreal_0(C,np__0)
& r2_hidden(A,k2_rcomp_1(k6_xcmplx_0(B,C),k2_xcmplx_0(B,C))) )
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = k3_real_1(B,D)
& ~ r1_xreal_0(C,k18_complex1(D)) ) ) ) ) ) ).
fof(t10_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( ~ r1_xreal_0(C,np__0)
& r2_hidden(A,k2_rcomp_1(k6_xcmplx_0(B,C),k2_xcmplx_0(B,C))) )
<=> r2_hidden(k5_real_1(A,B),k2_rcomp_1(k4_xcmplx_0(C),C)) ) ) ) ) ).
fof(t11_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k2_limfunc1(A) = k2_xboole_0(k1_tarski(A),k12_prob_1(A)) ) ).
fof(t12_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k3_limfunc1(A) = k2_xboole_0(k1_tarski(A),k4_limfunc1(A)) ) ).
fof(t13_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( v1_xreal_0(C)
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,D) = k6_xcmplx_0(C,k6_real_1(A,k3_real_1(D,np__1))) )
=> ( v4_seq_2(B)
& k2_seq_2(B) = C ) ) ) ) ) ).
fof(t14_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( v1_xreal_0(C)
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,D) = k2_xcmplx_0(C,k6_real_1(A,k3_real_1(D,np__1))) )
=> ( v4_seq_2(B)
& k2_seq_2(B) = C ) ) ) ) ) ).
fof(t15_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ~ ( r1_fcont_1(C,A)
& k2_seq_1(k1_numbers,k1_numbers,C,A) != B
& ? [D] :
( m1_rcomp_1(D,A)
& r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,C)) )
& ! [D] :
( m1_rcomp_1(D,A)
=> ~ ( r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,C))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,D)
& k2_seq_1(k1_numbers,k1_numbers,C,E) = B ) ) ) ) ) ) ) ) ).
fof(t16_fcont_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_rfunct_2(B,A)
| r2_rfunct_2(B,A) )
=> v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,A)) ) ) ).
fof(t17_fcont_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v2_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_rfunct_2(B,A)
=> r1_rfunct_2(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,A)),k10_relset_1(k1_numbers,k1_numbers,B,A)) ) ) ).
fof(t18_fcont_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v2_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_rfunct_2(B,A)
=> r2_rfunct_2(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,A)),k10_relset_1(k1_numbers,k1_numbers,B,A)) ) ) ).
fof(t19_fcont_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& r5_rfunct_2(B,A) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> k10_relset_1(k1_numbers,k1_numbers,B,A) != k12_prob_1(C) )
| r2_fcont_1(B,A) ) ) ) ).
fof(t20_fcont_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& r5_rfunct_2(B,A) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> k10_relset_1(k1_numbers,k1_numbers,B,A) != k4_limfunc1(C) )
| r2_fcont_1(B,A) ) ) ) ).
fof(t21_fcont_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& r5_rfunct_2(B,A) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> k10_relset_1(k1_numbers,k1_numbers,B,A) != k2_limfunc1(C) )
| r2_fcont_1(B,A) ) ) ) ).
fof(t22_fcont_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& r5_rfunct_2(B,A) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> k10_relset_1(k1_numbers,k1_numbers,B,A) != k3_limfunc1(C) )
| r2_fcont_1(B,A) ) ) ) ).
fof(t23_fcont_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& r5_rfunct_2(B,A) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k10_relset_1(k1_numbers,k1_numbers,B,A) != k2_rcomp_1(C,D) ) )
| r2_fcont_1(B,A) ) ) ) ).
fof(t24_fcont_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& r5_rfunct_2(B,A)
& k10_relset_1(k1_numbers,k1_numbers,B,A) = k1_numbers )
=> r2_fcont_1(B,A) ) ) ).
fof(t25_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v2_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r1_tarski(k2_rcomp_1(A,B),k4_relset_1(k1_numbers,k1_numbers,C))
=> ( ( ~ r1_rfunct_2(C,k2_rcomp_1(A,B))
& ~ r2_rfunct_2(C,k2_rcomp_1(A,B)) )
| r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B))),k10_relset_1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B))) ) ) ) ) ) ).
fof(t26_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v2_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_tarski(k12_prob_1(A),k4_relset_1(k1_numbers,k1_numbers,B))
=> ( ( ~ r1_rfunct_2(B,k12_prob_1(A))
& ~ r2_rfunct_2(B,k12_prob_1(A)) )
| r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))),k10_relset_1(k1_numbers,k1_numbers,B,k12_prob_1(A))) ) ) ) ) ).
fof(t27_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v2_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_tarski(k4_limfunc1(A),k4_relset_1(k1_numbers,k1_numbers,B))
=> ( ( ~ r1_rfunct_2(B,k4_limfunc1(A))
& ~ r2_rfunct_2(B,k4_limfunc1(A)) )
| r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))),k10_relset_1(k1_numbers,k1_numbers,B,k4_limfunc1(A))) ) ) ) ) ).
fof(t28_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v2_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_tarski(k2_limfunc1(A),k4_relset_1(k1_numbers,k1_numbers,B))
=> ( ( ~ r1_rfunct_2(B,k2_limfunc1(A))
& ~ r2_rfunct_2(B,k2_limfunc1(A)) )
| r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k2_limfunc1(A))),k10_relset_1(k1_numbers,k1_numbers,B,k2_limfunc1(A))) ) ) ) ) ).
fof(t29_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v2_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_tarski(k3_limfunc1(A),k4_relset_1(k1_numbers,k1_numbers,B))
=> ( ( ~ r1_rfunct_2(B,k3_limfunc1(A))
& ~ r2_rfunct_2(B,k3_limfunc1(A)) )
| r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k3_limfunc1(A))),k10_relset_1(k1_numbers,k1_numbers,B,k3_limfunc1(A))) ) ) ) ) ).
fof(t30_fcont_3,axiom,
! [A] :
( ( v1_funct_1(A)
& v2_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v1_partfun1(A,k1_numbers,k1_numbers)
=> ( ( ~ r1_rfunct_2(A,k2_subset_1(k1_numbers))
& ~ r2_rfunct_2(A,k2_subset_1(k1_numbers)) )
| r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,A),k5_relset_1(k1_numbers,k1_numbers,A)) ) ) ) ).
fof(t31_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(C,k2_rcomp_1(A,B))
=> ( ( ~ r1_rfunct_2(C,k2_rcomp_1(A,B))
& ~ r2_rfunct_2(C,k2_rcomp_1(A,B)) )
| v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)))) ) ) ) ) ) ).
fof(t32_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(B,k12_prob_1(A))
=> ( ( ~ r1_rfunct_2(B,k12_prob_1(A))
& ~ r2_rfunct_2(B,k12_prob_1(A)) )
| v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A)))) ) ) ) ) ).
fof(t33_fcont_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(B,k4_limfunc1(A))
=> ( ( ~ r1_rfunct_2(B,k4_limfunc1(A))
& ~ r2_rfunct_2(B,k4_limfunc1(A)) )
| v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A)))) ) ) ) ) ).
fof(t34_fcont_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(A,k2_subset_1(k1_numbers))
=> ( ( ~ r1_rfunct_2(A,k2_subset_1(k1_numbers))
& ~ r2_rfunct_2(A,k2_subset_1(k1_numbers)) )
| v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,A)) ) ) ) ).
%------------------------------------------------------------------------------