SET007 Axioms: SET007+783.ax
%------------------------------------------------------------------------------
% File : SET007+783 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Definition of Convex Function and Jensen's Inequality
% 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 : convfun1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 2 unt; 0 def)
% Number of atoms : 351 ( 56 equ)
% Maximal formula atoms : 30 ( 10 avg)
% Number of connectives : 382 ( 66 ~; 1 |; 175 &)
% ( 11 <=>; 129 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 11 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 25 ( 24 usr; 0 prp; 1-3 aty)
% Number of functors : 53 ( 53 usr; 12 con; 0-6 aty)
% Number of variables : 122 ( 119 !; 3 ?)
% 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_convfun1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A)
& ~ v3_struct_0(B)
& l2_rlvect_1(B) )
=> ~ v3_struct_0(k3_convfun1(A,B)) ) ).
fof(fc2_convfun1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l2_rlvect_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& l2_rlvect_1(B) )
=> ( ~ v3_struct_0(k3_convfun1(A,B))
& v3_rlvect_1(k3_convfun1(A,B)) ) ) ).
fof(fc3_convfun1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& l2_rlvect_1(A)
& ~ v3_struct_0(B)
& v4_rlvect_1(B)
& l2_rlvect_1(B) )
=> ( ~ v3_struct_0(k3_convfun1(A,B))
& v4_rlvect_1(k3_convfun1(A,B)) ) ) ).
fof(fc4_convfun1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& l2_rlvect_1(A)
& ~ v3_struct_0(B)
& v5_rlvect_1(B)
& l2_rlvect_1(B) )
=> ( ~ v3_struct_0(k3_convfun1(A,B))
& v5_rlvect_1(k3_convfun1(A,B)) ) ) ).
fof(fc5_convfun1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_rlvect_1(A)
& l2_rlvect_1(A)
& ~ v3_struct_0(B)
& v6_rlvect_1(B)
& l2_rlvect_1(B) )
=> ( ~ v3_struct_0(k3_convfun1(A,B))
& v6_rlvect_1(k3_convfun1(A,B)) ) ) ).
fof(fc6_convfun1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A)
& ~ v3_struct_0(B)
& v7_rlvect_1(B)
& l2_rlvect_1(B) )
=> ( ~ v3_struct_0(k3_convfun1(A,B))
& v7_rlvect_1(k3_convfun1(A,B)) ) ) ).
fof(fc7_convfun1,axiom,
( ~ v3_struct_0(k4_convfun1)
& v3_rlvect_1(k4_convfun1)
& v4_rlvect_1(k4_convfun1)
& v5_rlvect_1(k4_convfun1)
& v6_rlvect_1(k4_convfun1)
& v7_rlvect_1(k4_convfun1) ) ).
fof(d1_convfun1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_rlvect_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
& m2_relset_1(C,k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))) )
=> ( C = k1_convfun1(A,B)
<=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(B))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(B))
=> ( ( D = k1_domain_1(u1_struct_0(A),u1_struct_0(B),F,H)
& E = k1_domain_1(u1_struct_0(A),u1_struct_0(B),G,I) )
=> k2_binop_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),C,D,E) = k1_domain_1(u1_struct_0(A),u1_struct_0(B),k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A),u1_rlvect_1(A),k1_domain_1(u1_struct_0(A),u1_struct_0(A),F,G)),k8_funct_2(k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)),u1_struct_0(B),u1_rlvect_1(B),k1_domain_1(u1_struct_0(B),u1_struct_0(B),H,I))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d2_convfun1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_rlvect_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k6_supinf_1,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
& m2_relset_1(C,k2_zfmisc_1(k6_supinf_1,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))) )
=> ( C = k2_convfun1(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ( E = k1_domain_1(u1_struct_0(A),u1_struct_0(B),F,G)
=> k8_funct_2(k2_zfmisc_1(k1_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),C,k1_domain_1(k1_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),D,E)) = k1_domain_1(u1_struct_0(A),u1_struct_0(B),k8_funct_2(k2_zfmisc_1(k1_numbers,u1_struct_0(A)),u1_struct_0(A),u2_rlvect_1(A),k1_domain_1(k1_numbers,u1_struct_0(A),D,F)),k8_funct_2(k2_zfmisc_1(k1_numbers,u1_struct_0(B)),u1_struct_0(B),u2_rlvect_1(B),k1_domain_1(k1_numbers,u1_struct_0(B),D,G))) ) ) ) ) ) ) ) ) ) ).
fof(d3_convfun1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_rlvect_1(B) )
=> k3_convfun1(A,B) = g2_rlvect_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_domain_1(u1_struct_0(A),u1_struct_0(B),k1_rlvect_1(A),k1_rlvect_1(B)),k1_convfun1(A,B),k2_convfun1(A,B)) ) ) ).
fof(t1_convfun1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_rlvect_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k3_convfun1(A,B)))
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( E = k1_domain_1(u1_struct_0(A),u1_struct_0(B),C,D)
=> k3_rlvect_1(k3_convfun1(A,B),E,F) = k1_domain_1(u1_struct_0(A),u1_struct_0(B),k3_rlvect_1(A,C,F),k3_rlvect_1(B,D,F)) ) ) ) ) ) ) ) ).
fof(t2_convfun1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_rlvect_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k3_convfun1(A,B)))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(k3_convfun1(A,B)))
=> ( ( G = k1_domain_1(u1_struct_0(A),u1_struct_0(B),C,E)
& H = k1_domain_1(u1_struct_0(A),u1_struct_0(B),D,F) )
=> k2_rlvect_1(k3_convfun1(A,B),G,H) = k1_domain_1(u1_struct_0(A),u1_struct_0(B),k2_rlvect_1(A,C,D),k2_rlvect_1(B,E,F)) ) ) ) ) ) ) ) ) ) ).
fof(t3_convfun1,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] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& l2_rlvect_1(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(A))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(B))
=> ! [F] :
( m2_finseq_1(F,u1_struct_0(k3_convfun1(A,B)))
=> ( ( k3_finseq_1(D) = C
& k3_finseq_1(E) = C
& k3_finseq_1(F) = C
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(G,k2_finseq_1(C))
=> k1_funct_1(F,G) = k4_tarski(k1_funct_1(D,G),k1_funct_1(E,G)) ) ) )
=> k9_rlvect_1(k3_convfun1(A,B),F) = k1_domain_1(u1_struct_0(A),u1_struct_0(B),k9_rlvect_1(A,D),k9_rlvect_1(B,E)) ) ) ) ) ) ) ) ).
fof(d4_convfun1,axiom,
k4_convfun1 = g2_rlvect_1(k6_supinf_1,np__0,k33_binop_2,k35_binop_2) ).
fof(d5_convfun1,axiom,
! [A] :
( m2_finseq_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( B = k5_convfun1(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k3_supinf_1)
& m2_relset_1(C,k5_numbers,k3_supinf_1)
& B = k4_mesfunc1(k5_numbers,C,k3_finseq_1(A))
& k4_mesfunc1(k5_numbers,C,np__0) = k1_supinf_2
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(A),D)
=> k4_mesfunc1(k5_numbers,C,k1_nat_1(D,np__1)) = k2_supinf_2(k4_mesfunc1(k5_numbers,C,D),k4_mesfunc1(k5_numbers,A,k1_nat_1(D,np__1))) ) ) ) ) ) ) ).
fof(t4_convfun1,axiom,
k5_convfun1(k6_finseq_1(k3_supinf_1)) = k1_supinf_2 ).
fof(t5_convfun1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> k5_convfun1(k13_binarith(k3_supinf_1,A)) = A ) ).
fof(t6_convfun1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> k5_convfun1(k2_finseq_4(k3_supinf_1,A,B)) = k2_supinf_2(A,B) ) ) ).
fof(t7_convfun1,axiom,
! [A] :
( m2_finseq_1(A,k3_supinf_1)
=> ! [B] :
( m2_finseq_1(B,k3_supinf_1)
=> ~ ( ~ r2_hidden(k4_measure6,k5_relset_1(k5_numbers,k3_supinf_1,A))
& ~ r2_hidden(k4_measure6,k5_relset_1(k5_numbers,k3_supinf_1,B))
& k5_convfun1(k8_finseq_1(k3_supinf_1,A,B)) != k2_supinf_2(k5_convfun1(A),k5_convfun1(B)) ) ) ) ).
fof(t8_convfun1,axiom,
! [A] :
( m2_finseq_1(A,k3_supinf_1)
=> ! [B] :
( m2_finseq_1(B,k3_supinf_1)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k4_finseq_1(A),k4_finseq_1(A))
& v3_funct_2(C,k4_finseq_1(A),k4_finseq_1(A))
& m2_relset_1(C,k4_finseq_1(A),k4_finseq_1(A)) )
=> ( B = k1_partfun1(k4_finseq_1(A),k4_finseq_1(A),k5_numbers,k3_supinf_1,C,A)
=> ( r2_hidden(k4_measure6,k5_relset_1(k5_numbers,k3_supinf_1,A))
| k5_convfun1(A) = k5_convfun1(B) ) ) ) ) ) ).
fof(d7_convfun1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k3_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k3_supinf_1) )
=> ( v1_convfun1(B,A)
<=> v1_convex1(k6_convfun1(A,B),k3_convfun1(A,k4_convfun1)) ) ) ) ).
fof(t9_convfun1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k3_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k3_supinf_1) )
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_mesfunc1(u1_struct_0(A),B,C) != k4_measure6 )
=> ( v1_convfun1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ~ r1_xreal_0(np__1,E)
& ~ r1_supinf_1(k4_mesfunc1(u1_struct_0(A),B,k2_rlvect_1(A,k3_rlvect_1(A,C,E),k3_rlvect_1(A,D,k10_binop_2(np__1,E)))),k2_supinf_2(k2_extreal1(k1_measure6(E),k4_mesfunc1(u1_struct_0(A),B,C)),k2_extreal1(k1_measure6(k10_binop_2(np__1,E)),k4_mesfunc1(u1_struct_0(A),B,D)))) ) ) ) ) ) ) ) ) ).
fof(t10_convfun1,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_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k3_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k3_supinf_1) )
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_mesfunc1(u1_struct_0(A),B,C) != k4_measure6 )
=> ( v1_convfun1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r1_xreal_0(np__0,E)
& r1_xreal_0(E,np__1) )
=> r1_supinf_1(k4_mesfunc1(u1_struct_0(A),B,k4_rlvect_1(A,k3_rlvect_1(A,C,E),k3_rlvect_1(A,D,k10_binop_2(np__1,E)))),k2_supinf_2(k2_extreal1(k1_measure6(E),k4_mesfunc1(u1_struct_0(A),B,C)),k2_extreal1(k1_measure6(k10_binop_2(np__1,E)),k4_mesfunc1(u1_struct_0(A),B,D)))) ) ) ) ) ) ) ) ) ).
fof(t11_convfun1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k4_convfun1),k3_supinf_1)
& m2_relset_1(B,u1_struct_0(k4_convfun1),k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_convfun1)))
=> ( ( r1_tarski(C,k4_relset_1(k6_supinf_1,k6_supinf_1,A))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r2_hidden(D,C)
=> k4_mesfunc1(u1_struct_0(k4_convfun1),B,D) = k2_seq_1(k6_supinf_1,k6_supinf_1,A,D) )
& ( ~ r2_hidden(D,C)
=> k4_mesfunc1(u1_struct_0(k4_convfun1),B,D) = k5_measure6 ) ) ) )
=> ( v1_convfun1(B,k4_convfun1)
<=> ( r2_rfunct_3(A,C)
& v1_convex1(C,k4_convfun1) ) ) ) ) ) ) ).
fof(t12_convfun1,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)))
=> ( v1_convex1(B,A)
<=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ! [D] :
( m2_finseq_1(D,k6_supinf_1)
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(A))
=> ! [F] :
( m2_finseq_1(F,u1_struct_0(A))
=> ( ( k3_finseq_1(D) = C
& k3_finseq_1(E) = C
& k3_finseq_1(F) = C
& k15_rvsum_1(D) = np__1
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(G,k2_finseq_1(C))
=> ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k6_supinf_1,D,G),np__0)
& k1_funct_1(F,G) = k3_rlvect_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),E,G),k2_seq_1(k5_numbers,k6_supinf_1,D,G))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(A),E,G),B) ) ) ) )
=> r2_hidden(k9_rlvect_1(A,F),B) ) ) ) ) ) ) ) ) ).
fof(t13_convfun1,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_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k3_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k3_supinf_1) )
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_mesfunc1(u1_struct_0(A),B,C) != k4_measure6 )
=> ( v1_convfun1(B,A)
<=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ! [D] :
( m2_finseq_1(D,k6_supinf_1)
=> ! [E] :
( m2_finseq_1(E,k3_supinf_1)
=> ! [F] :
( m2_finseq_1(F,u1_struct_0(A))
=> ! [G] :
( m2_finseq_1(G,u1_struct_0(A))
=> ( ( k3_finseq_1(D) = C
& k3_finseq_1(E) = C
& k3_finseq_1(F) = C
& k3_finseq_1(G) = C
& k15_rvsum_1(D) = np__1
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k2_finseq_1(C))
=> ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k6_supinf_1,D,H),np__0)
& k1_funct_1(G,H) = k3_rlvect_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),F,H),k2_seq_1(k5_numbers,k6_supinf_1,D,H))
& k4_mesfunc1(k5_numbers,E,H) = k2_extreal1(k1_measure6(k2_seq_1(k5_numbers,k6_supinf_1,D,H)),k4_mesfunc1(u1_struct_0(A),B,k4_finseq_4(k5_numbers,u1_struct_0(A),F,H))) ) ) ) )
=> r1_supinf_1(k4_mesfunc1(u1_struct_0(A),B,k9_rlvect_1(A,G)),k5_convfun1(E)) ) ) ) ) ) ) ) ) ) ) ).
fof(t14_convfun1,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_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k3_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k3_supinf_1) )
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_mesfunc1(u1_struct_0(A),B,C) != k4_measure6 )
=> ( v1_convfun1(B,A)
<=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ! [D] :
( m2_finseq_1(D,k6_supinf_1)
=> ! [E] :
( m2_finseq_1(E,k3_supinf_1)
=> ! [F] :
( m2_finseq_1(F,u1_struct_0(A))
=> ! [G] :
( m2_finseq_1(G,u1_struct_0(A))
=> ( ( k3_finseq_1(D) = C
& k3_finseq_1(E) = C
& k3_finseq_1(F) = C
& k3_finseq_1(G) = C
& k15_rvsum_1(D) = np__1
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k2_finseq_1(C))
=> ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k6_supinf_1,D,H))
& k1_funct_1(G,H) = k3_rlvect_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),F,H),k2_seq_1(k5_numbers,k6_supinf_1,D,H))
& k4_mesfunc1(k5_numbers,E,H) = k2_extreal1(k1_measure6(k2_seq_1(k5_numbers,k6_supinf_1,D,H)),k4_mesfunc1(u1_struct_0(A),B,k4_finseq_4(k5_numbers,u1_struct_0(A),F,H))) ) ) ) )
=> r1_supinf_1(k4_mesfunc1(u1_struct_0(A),B,k9_rlvect_1(A,G)),k5_convfun1(E)) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_convfun1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A)
& ~ v3_struct_0(B)
& l2_rlvect_1(B) )
=> ( v1_funct_1(k1_convfun1(A,B))
& v1_funct_2(k1_convfun1(A,B),k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
& m2_relset_1(k1_convfun1(A,B),k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))) ) ) ).
fof(dt_k2_convfun1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A)
& ~ v3_struct_0(B)
& l2_rlvect_1(B) )
=> ( v1_funct_1(k2_convfun1(A,B))
& v1_funct_2(k2_convfun1(A,B),k2_zfmisc_1(k6_supinf_1,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
& m2_relset_1(k2_convfun1(A,B),k2_zfmisc_1(k6_supinf_1,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))) ) ) ).
fof(dt_k3_convfun1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A)
& ~ v3_struct_0(B)
& l2_rlvect_1(B) )
=> l2_rlvect_1(k3_convfun1(A,B)) ) ).
fof(dt_k4_convfun1,axiom,
( ~ v3_struct_0(k4_convfun1)
& l2_rlvect_1(k4_convfun1) ) ).
fof(dt_k5_convfun1,axiom,
! [A] :
( m1_finseq_1(A,k3_supinf_1)
=> m1_subset_1(k5_convfun1(A),k3_supinf_1) ) ).
fof(dt_k6_convfun1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k3_supinf_1)
& m1_relset_1(B,u1_struct_0(A),k3_supinf_1) )
=> m1_subset_1(k6_convfun1(A,B),k1_zfmisc_1(u1_struct_0(k3_convfun1(A,k4_convfun1)))) ) ).
fof(d6_convfun1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k3_supinf_1)
& m2_relset_1(B,u1_struct_0(A),k3_supinf_1) )
=> k6_convfun1(A,B) = a_2_0_convfun1(A,B) ) ) ).
fof(fraenkel_a_2_0_convfun1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l2_rlvect_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),k3_supinf_1)
& m2_relset_1(C,u1_struct_0(B),k3_supinf_1) )
=> ( r2_hidden(A,a_2_0_convfun1(B,C))
<=> ? [D,E] :
( m1_subset_1(D,u1_struct_0(B))
& m2_subset_1(E,k3_supinf_1,k6_supinf_1)
& A = k1_domain_1(u1_struct_0(B),k6_supinf_1,D,E)
& r1_supinf_1(k4_mesfunc1(u1_struct_0(B),C,D),k1_measure6(E)) ) ) ) ).
%------------------------------------------------------------------------------