SET007 Axioms: SET007+772.ax
%------------------------------------------------------------------------------
% File : SET007+772 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Properties for Convex Combinations
% 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 : convex2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 25 ( 1 unt; 0 def)
% Number of atoms : 277 ( 22 equ)
% Maximal formula atoms : 18 ( 11 avg)
% Number of connectives : 304 ( 52 ~; 0 |; 166 &)
% ( 7 <=>; 79 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 12 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 28 ( 26 usr; 1 prp; 0-3 aty)
% Number of functors : 30 ( 30 usr; 5 con; 0-3 aty)
% Number of variables : 99 ( 91 !; 8 ?)
% 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(rc1_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ? [B] :
( m1_rlvect_2(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v1_seq_1(B)
& v2_convex1(B,A) ) ) ).
fof(rc2_convex2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] :
( m2_rlvect_2(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v1_seq_1(C)
& v2_convex1(C,A) ) ) ).
fof(t1_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_convex1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v1_convex1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v1_convex1(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ).
fof(t6_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_rlvect_2(D,A,C)
=> ( ( v2_convex1(D,A)
& r1_tarski(C,B) )
=> r2_hidden(k13_rlvect_2(A,D),B) ) ) )
<=> v1_convex1(B,A) ) ) ) ).
fof(d1_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( C = k1_convex2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> m2_rlvect_2(D,A,B) ) ) ) ) ).
fof(t7_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k2_convex1(A,B) != k1_xboole_0 ) ) ).
fof(t8_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(B,k3_convex1(A,B)) ) ) ).
fof(t9_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v2_convex1(B,A)
& m1_rlvect_2(B,A) )
=> ! [C] :
( ( v2_convex1(C,A)
& m1_rlvect_2(C,A) )
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ~ r1_xreal_0(np__1,D)
& ~ ( v2_convex1(k14_rlvect_2(A,k15_rlvect_2(A,D,B),k15_rlvect_2(A,k5_real_1(np__1,D),C)),A)
& m1_rlvect_2(k14_rlvect_2(A,k15_rlvect_2(A,D,B),k15_rlvect_2(A,k5_real_1(np__1,D),C)),A) ) ) ) ) ) ) ).
fof(t10_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v2_convex1(C,A)
& m2_rlvect_2(C,A,B) )
=> ! [D] :
( ( v2_convex1(D,A)
& m2_rlvect_2(D,A,B) )
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ~ r1_xreal_0(np__1,E)
& ~ ( v2_convex1(k14_rlvect_2(A,k15_rlvect_2(A,E,C),k15_rlvect_2(A,k5_real_1(np__1,E),D)),A)
& m2_rlvect_2(k14_rlvect_2(A,k15_rlvect_2(A,E,C),k15_rlvect_2(A,k5_real_1(np__1,E),D)),A,B) ) ) ) ) ) ) ) ).
fof(t11_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ? [B] :
( m1_rlvect_2(B,A)
& v2_convex1(B,A) ) ) ).
fof(t12_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] :
( m2_rlvect_2(C,A,B)
& v2_convex1(C,A) ) ) ) ).
fof(t13_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,A)
=> k3_rusub_4(A,k1_rlsub_2(A,B,C)) = k7_rusub_4(A,k3_rusub_4(A,B),k3_rusub_4(A,C)) ) ) ) ).
fof(t14_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlsub_1(B,A)
=> ! [C] :
( m1_rlsub_1(C,A)
=> k3_rusub_4(A,k2_rlsub_2(A,B,C)) = k5_subset_1(u1_struct_0(A),k3_rusub_4(A,B),k3_rusub_4(A,C)) ) ) ) ).
fof(t15_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v2_convex1(B,A)
& m1_rlvect_2(B,A) )
=> ! [C] :
( ( v2_convex1(C,A)
& m1_rlvect_2(C,A) )
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ~ r1_xreal_0(k4_real_1(D,E),np__0)
=> k10_rlvect_2(A,k14_rlvect_2(A,k15_rlvect_2(A,D,B),k15_rlvect_2(A,E,C))) = k2_rlvect_2(A,k10_rlvect_2(A,k15_rlvect_2(A,D,B)),k10_rlvect_2(A,k15_rlvect_2(A,E,C))) ) ) ) ) ) ) ).
fof(t16_convex2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_rfinseq(A,B)
=> ! [C,D] :
~ ( r2_hidden(C,k1_relat_1(A))
& r2_hidden(D,k1_relat_1(A))
& C != D
& ! [E,F] :
~ ( r2_hidden(E,k1_relat_1(B))
& r2_hidden(F,k1_relat_1(B))
& E != F
& k1_funct_1(A,C) = k1_funct_1(B,E)
& k1_funct_1(A,D) = k1_funct_1(B,F) ) ) ) ) ) ).
fof(t17_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_rlvect_2(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r1_tarski(C,k10_rlvect_2(A,B))
& ! [D] :
( m1_rlvect_2(D,A)
=> k10_rlvect_2(A,D) != C ) ) ) ) ) ).
fof(dt_k1_convex2,axiom,
$true ).
fof(t2_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( B = a_3_0_convex2(A,C,D)
=> v1_convex1(B,A) ) ) ) ) ) ).
fof(t3_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( B = a_3_1_convex2(A,C,D)
=> v1_convex1(B,A) ) ) ) ) ) ).
fof(t4_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( B = a_3_2_convex2(A,C,D)
=> v1_convex1(B,A) ) ) ) ) ) ).
fof(t5_convex2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( B = a_3_3_convex2(A,C,D)
=> v1_convex1(B,A) ) ) ) ) ) ).
fof(fraenkel_a_3_0_convex2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_bhsp_1(B)
& l1_bhsp_1(B)
& m2_finseq_1(C,u1_struct_0(B))
& m2_finseq_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_0_convex2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k5_subset_1(k5_numbers,k4_finseq_1(C),k4_finseq_1(D)))
& ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ~ ( G = k1_funct_1(C,F)
& r1_xreal_0(k1_bhsp_1(B,E,G),k1_funct_1(D,F)) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_1_convex2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_bhsp_1(B)
& l1_bhsp_1(B)
& m2_finseq_1(C,u1_struct_0(B))
& m2_finseq_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_1_convex2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k5_subset_1(k5_numbers,k4_finseq_1(C),k4_finseq_1(D)))
& ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ~ ( G = k1_funct_1(C,F)
& ~ r1_xreal_0(k1_funct_1(D,F),k1_bhsp_1(B,E,G)) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_2_convex2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_bhsp_1(B)
& l1_bhsp_1(B)
& m2_finseq_1(C,u1_struct_0(B))
& m2_finseq_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_2_convex2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k5_subset_1(k5_numbers,k4_finseq_1(C),k4_finseq_1(D)))
& ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ~ ( G = k1_funct_1(C,F)
& r1_xreal_0(k1_funct_1(D,F),k1_bhsp_1(B,E,G)) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_3_convex2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_bhsp_1(B)
& l1_bhsp_1(B)
& m2_finseq_1(C,u1_struct_0(B))
& m2_finseq_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_3_convex2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k5_subset_1(k5_numbers,k4_finseq_1(C),k4_finseq_1(D)))
& ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ~ ( G = k1_funct_1(C,F)
& ~ r1_xreal_0(k1_bhsp_1(B,E,G),k1_funct_1(D,F)) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------