SET007 Axioms: SET007+867.ax
%------------------------------------------------------------------------------
% File : SET007+867 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Holder's Inequality and Minkowski'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 : holder_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 14 ( 0 unt; 0 def)
% Number of atoms : 212 ( 30 equ)
% Maximal formula atoms : 27 ( 15 avg)
% Number of connectives : 212 ( 14 ~; 6 |; 104 &)
% ( 0 <=>; 88 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 14 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-3 aty)
% Number of functors : 15 ( 15 usr; 4 con; 0-4 aty)
% Number of variables : 68 ( 66 !; 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_holder_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ v1_xboole_0(k3_limfunc1(A))
& v1_membered(k3_limfunc1(A))
& v2_membered(k3_limfunc1(A)) ) ) ).
fof(t1_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& ? [C] :
( m1_subset_1(C,k1_numbers)
& r1_xreal_0(np__0,C)
& k4_real_1(k4_power(C,A),k4_power(C,B)) != k4_power(C,k3_real_1(A,B)) ) ) ) ) ).
fof(t2_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& ? [C] :
( m1_subset_1(C,k1_numbers)
& r1_xreal_0(np__0,C)
& k4_power(k4_power(C,A),B) != k4_power(C,k4_real_1(A,B)) ) ) ) ) ).
fof(t3_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r1_xreal_0(np__0,B)
& r1_xreal_0(B,C) )
=> r1_xreal_0(k4_power(B,A),k4_power(C,A)) ) ) ) ) ) ).
fof(t4_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( k3_real_1(k6_real_1(np__1,A),k6_real_1(np__1,B)) = np__1
=> ( r1_xreal_0(A,np__1)
| r1_xreal_0(C,np__0)
| r1_xreal_0(D,np__0)
| ( r1_xreal_0(k4_real_1(C,D),k3_real_1(k6_real_1(k13_prepower(C,A),A),k6_real_1(k13_prepower(D,B),B)))
& ( k4_real_1(C,D) = k3_real_1(k6_real_1(k13_prepower(C,A),A),k6_real_1(k13_prepower(D,B),B))
=> k13_prepower(C,A) = k13_prepower(D,B) )
& ( k13_prepower(C,A) = k13_prepower(D,B)
=> k4_real_1(C,D) = k3_real_1(k6_real_1(k13_prepower(C,A),A),k6_real_1(k13_prepower(D,B),B)) ) ) ) ) ) ) ) ) ).
fof(t5_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( k3_real_1(k6_real_1(np__1,A),k6_real_1(np__1,B)) = np__1
& r1_xreal_0(np__0,C)
& r1_xreal_0(np__0,D) )
=> ( r1_xreal_0(A,np__1)
| ( r1_xreal_0(k4_real_1(C,D),k3_real_1(k6_real_1(k4_power(C,A),A),k6_real_1(k4_power(D,B),B)))
& ( k4_real_1(C,D) = k3_real_1(k6_real_1(k4_power(C,A),A),k6_real_1(k4_power(D,B),B))
=> k4_power(C,A) = k4_power(D,B) )
& ( k4_power(C,A) = k4_power(D,B)
=> k4_real_1(C,D) = k3_real_1(k6_real_1(k4_power(C,A),A),k6_real_1(k4_power(D,B),B)) ) ) ) ) ) ) ) ) ).
fof(t6_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( k3_real_1(k6_real_1(np__1,A),k6_real_1(np__1,B)) = np__1
=> ( r1_xreal_0(A,np__1)
| ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k5_numbers,k1_numbers)
& m2_relset_1(F,k5_numbers,k1_numbers) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k5_numbers,k1_numbers)
& m2_relset_1(G,k5_numbers,k1_numbers) )
=> ( ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( k2_seq_1(k5_numbers,k1_numbers,E,H) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,C,H)),A)
& k2_seq_1(k5_numbers,k1_numbers,F,H) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,D,H)),B)
& k2_seq_1(k5_numbers,k1_numbers,G,H) = k18_complex1(k4_real_1(k2_seq_1(k5_numbers,k1_numbers,C,H),k2_seq_1(k5_numbers,k1_numbers,D,H))) ) )
=> ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(G),H),k4_real_1(k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(E),H),k6_real_1(np__1,A)),k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(F),H),k6_real_1(np__1,B)))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t7_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(A,np__1)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k5_numbers,k1_numbers)
& m2_relset_1(F,k5_numbers,k1_numbers) )
=> ( ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( k2_seq_1(k5_numbers,k1_numbers,D,G) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,B,G)),A)
& k2_seq_1(k5_numbers,k1_numbers,E,G) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,C,G)),A)
& k2_seq_1(k5_numbers,k1_numbers,F,G) = k4_power(k18_complex1(k3_real_1(k2_seq_1(k5_numbers,k1_numbers,B,G),k2_seq_1(k5_numbers,k1_numbers,C,G))),A) ) )
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> r1_xreal_0(k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(F),G),k6_real_1(np__1,A)),k3_real_1(k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(D),G),k6_real_1(np__1,A)),k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(E),G),k6_real_1(np__1,A)))) ) ) ) ) ) ) ) ) ) ).
fof(t8_holder_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,B,C)) )
& v4_seq_2(B)
& v3_seqm_3(A) )
=> ( v4_seq_2(A)
& r1_xreal_0(k2_seq_2(A),k2_seq_2(B)) ) ) ) ) ).
fof(t9_holder_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,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_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),k3_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D))) )
& v4_seq_2(B)
& v4_seq_2(C)
& v3_seqm_3(A) )
=> ( v4_seq_2(A)
& r1_xreal_0(k2_seq_2(A),k3_real_1(k2_seq_2(B),k2_seq_2(C))) ) ) ) ) ) ).
fof(t10_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,D)) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_power(k2_seq_1(k5_numbers,k1_numbers,B,D),A) ) )
=> ( v4_seq_2(C)
& k2_seq_2(C) = k4_power(k2_seq_2(B),A) ) ) ) ) ) ) ).
fof(t11_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v1_series_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,D)) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),D),A) ) )
=> ( v4_seq_2(C)
& k2_seq_2(C) = k4_power(k2_series_1(B),A)
& v3_seqm_3(C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,D),k4_power(k2_series_1(B),A)) ) ) ) ) ) ) ) ).
fof(t12_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( k3_real_1(k6_real_1(np__1,A),k6_real_1(np__1,B)) = np__1
=> ( r1_xreal_0(A,np__1)
| ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k5_numbers,k1_numbers)
& m2_relset_1(F,k5_numbers,k1_numbers) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k5_numbers,k1_numbers)
& m2_relset_1(G,k5_numbers,k1_numbers) )
=> ( ( ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( k2_seq_1(k5_numbers,k1_numbers,E,H) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,C,H)),A)
& k2_seq_1(k5_numbers,k1_numbers,F,H) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,D,H)),B)
& k2_seq_1(k5_numbers,k1_numbers,G,H) = k18_complex1(k4_real_1(k2_seq_1(k5_numbers,k1_numbers,C,H),k2_seq_1(k5_numbers,k1_numbers,D,H))) ) )
& v1_series_1(E)
& v1_series_1(F) )
=> ( v1_series_1(G)
& r1_xreal_0(k2_series_1(G),k4_real_1(k4_power(k2_series_1(E),k6_real_1(np__1,A)),k4_power(k2_series_1(F),k6_real_1(np__1,B)))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t13_holder_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(A,np__1)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k5_numbers,k1_numbers)
& m2_relset_1(F,k5_numbers,k1_numbers) )
=> ( ( ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( k2_seq_1(k5_numbers,k1_numbers,D,G) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,B,G)),A)
& k2_seq_1(k5_numbers,k1_numbers,E,G) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,C,G)),A)
& k2_seq_1(k5_numbers,k1_numbers,F,G) = k4_power(k18_complex1(k3_real_1(k2_seq_1(k5_numbers,k1_numbers,B,G),k2_seq_1(k5_numbers,k1_numbers,C,G))),A) ) )
& v1_series_1(D)
& v1_series_1(E) )
=> ( v1_series_1(F)
& r1_xreal_0(k4_power(k2_series_1(F),k6_real_1(np__1,A)),k3_real_1(k4_power(k2_series_1(D),k6_real_1(np__1,A)),k4_power(k2_series_1(E),k6_real_1(np__1,A)))) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------