:: Extended Natural Numbers and Counters
:: by Sebastian Koch
::
:: Received October 25, 2020
:: Copyright (c) 2020-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, RELAT_1, TARSKI, XBOOLE_0, FUNCT_1, MEMBERED, SUBSET_1,
XCMPLX_0, XXREAL_0, INT_1, ORDINAL1, FUNCOP_1, FUNCT_4, FUNCT_2,
ZFMISC_1, ORDINAL2, ARYTM_3, NAT_1, SETFAM_1, CARD_1, AOFA_I00, CARD_3,
AFINSQ_1, PARTFUN1, ORDINAL4, CARD_2, VALUED_0, REAL_1, FINSET_1,
FUNCT_7, XXREAL_2, PBOOLE, COUNTERS;
notations TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, SETFAM_1, RELAT_1,
FUNCT_1, ORDINAL1, RELSET_1, PARTFUN1, FUNCT_2, FUNCOP_1, FUNCT_4,
ORDINAL2, FINSET_1, CARD_1, PBOOLE, CARD_3, NUMBERS, XXREAL_0, XREAL_0,
INT_1, RAT_1, MEMBERED, VALUED_0, CARD_2, XXREAL_2, XXREAL_3, XCMPLX_0,
NAT_1, ORDINAL4, AFINSQ_1, AOFA_I00;
constructors XCMPLX_0, RAT_1, MEMBERED, FUNCOP_1, FUNCT_4, XXREAL_0, PARTFUN1,
NAT_1, SETFAM_1, RELSET_1, NUMBERS, XXREAL_3, FINSET_1, VALUED_0,
XXREAL_2, ENUMSET1, INT_1, CARD_1, PBOOLE, AOFA_I00, AFINSQ_1, CARD_2;
registrations MEMBERED, RELAT_1, FUNCOP_1, XREAL_0, RAT_1, ORDINAL1, INT_1,
FUNCT_1, XBOOLE_0, RELSET_1, FUNCT_2, SETFAM_1, SUBSET_1, PBOOLE, NAT_1,
NUMBERS, XXREAL_0, XXREAL_3, CARD_1, VALUED_0, XXREAL_2, FOMODEL0,
FINSET_1, AFINSQ_1, CAYLDICK, ORDINAL2, CARD_3, CATALG_1, ZFMISC_1,
CARD_2, CARD_5, ORDINAL4;
requirements SUBSET, BOOLE, NUMERALS, ARITHM, REAL;
theorems ZFMISC_1, NUMBERS, TARSKI, XXREAL_0, XBOOLE_0, XBOOLE_1, ORDINAL1,
FINSET_1, XXREAL_3, NAT_1, XREAL_0, SUBSET_1, XXREAL_2, MEMBERED, INT_1,
ENUMSET1, VALUED_0, RELAT_1, FUNCOP_1, FUNCT_4, SETFAM_1, FUNCT_1,
PARTFUN1, FUNCT_2, AFINSQ_1, CARD_1, ORDINAL2, ORDINAL3, ORDINAL4,
XTUPLE_0, CARD_2, CARD_3, SYSREL;
schemes NAT_1, SUBSET_1, FUNCT_1, ORDINAL2;
begin :: Extended natural numbers
definition
func ExtNAT -> Subset of ExtREAL equals
NAT \/ {+infty};
coherence
proof
now
let x be object;
assume x in NAT \/ {+infty};
then per cases by ZFMISC_1:136;
suppose x in NAT;
then x in REAL by NUMBERS:19;
hence x in ExtREAL by NUMBERS:31;
end;
suppose x = +infty;
then x in {+infty,-infty} by TARSKI:def 2;
hence x in ExtREAL by XXREAL_0:def 4, XBOOLE_0:def 3;
end;
end;
hence thesis by TARSKI:def 3;
end;
end;
theorem Th1:
NAT c< ExtNAT & ExtNAT c< ExtREAL
proof
A2: +infty in ExtNAT by ZFMISC_1:136;
not +infty in NAT;
hence NAT c< ExtNAT by A2, XBOOLE_1:7, XBOOLE_0:def 8;
-infty in {+infty,-infty} by TARSKI:def 2;
then A2: -infty in ExtREAL by XXREAL_0:def 4, XBOOLE_0:def 3;
not -infty in ExtNAT
proof
assume -infty in ExtNAT;
then per cases by ZFMISC_1:136;
suppose -infty in NAT;
hence contradiction;
end;
suppose -infty in {+infty};
hence contradiction by TARSKI:def 1;
end;
end;
hence thesis by A2, XBOOLE_0:def 8;
end;
registration
cluster ExtNAT -> non empty infinite;
coherence by Th1, FINSET_1:1, XBOOLE_0:def 8;
end;
definition
let x be object;
attr x is ext-natural means
:Def1:
x in ExtNAT;
end;
registration
cluster +infty -> ext-natural;
coherence by ZFMISC_1:136;
cluster ext-natural -> ext-real for object;
coherence;
cluster natural -> ext-natural for object;
coherence
proof
let x be object;
assume x is natural;
then x in NAT by ORDINAL1:def 12;
hence thesis by Th1, XBOOLE_0:def 8, TARSKI:def 3;
end;
cluster finite ext-natural -> natural for set;
coherence
proof
let x be set;
assume x is finite ext-natural;
then x in ExtNAT & not x in {+infty} by XXREAL_0:def 2, TARSKI:def 1;
then x in NAT by XBOOLE_0:def 3;
hence thesis;
end;
end;
registration
cluster zero ext-natural for object;
existence
proof
take the zero natural object;
thus thesis;
end;
cluster non zero ext-natural for object;
existence
proof
take the non zero natural object;
thus thesis;
end;
cluster ext-natural for number;
existence
proof
take the natural number;
thus thesis;
end;
cluster -> ext-natural for Element of ExtNAT;
coherence;
end;
definition
mode ExtNat is ext-natural ExtReal;
end;
registration
let x be ExtNat;
reduce In(x,ExtNAT) to x;
reducibility by Def1, SUBSET_1:def 8;
end;
registration
sethood of ExtNat
proof
take ExtNAT;
thus thesis by Def1;
end;
end;
theorem Th3:
for x being object holds x is ExtNat iff x is Nat or x = +infty
proof
let x be object;
hereby
assume x is ExtNat;
then x in NAT or x = +infty by Def1, ZFMISC_1:136;
hence x is Nat or x = +infty;
end;
assume x is Nat or x = +infty;
then x in NAT or x in {+infty} by ORDINAL1:def 12, TARSKI:def 1;
then x in ExtNAT by XBOOLE_0:def 3;
hence thesis;
end;
registration
cluster zero -> ext-natural for object;
coherence;
cluster ext-natural -> non negative for ExtReal;
coherence by Th3;
end;
registration
cluster -> non negative for ExtNat;
coherence;
cluster non zero -> positive for ExtNat;
coherence;
end;
reserve N,M,K for ExtNat;
registration
let N,M;
cluster min(N,M) -> ext-natural;
coherence by XXREAL_0:def 9;
cluster max(N,M) -> ext-natural;
coherence by XXREAL_0:def 10;
cluster N + M -> ext-natural;
coherence
proof
per cases by Th3;
suppose N is Nat;
then reconsider n = N as Nat;
per cases by Th3;
suppose M is Nat;
then reconsider k = M as Nat;
n + k is Nat;
hence thesis;
end;
suppose M = +infty;
hence thesis by XXREAL_3:def 2;
end;
end;
suppose N = +infty;
hence thesis by XXREAL_3:def 2;
end;
end;
cluster N * M -> ext-natural;
coherence
proof
per cases by Th3;
suppose N is Nat;
then reconsider n = N as Nat;
per cases by Th3;
suppose M is Nat;
then reconsider k = M as Nat;
n * k is Nat;
hence thesis;
end;
suppose A1: M = +infty;
N is positive or N is zero;
hence thesis by A1, XXREAL_3:def 5;
end;
end;
suppose A1: N = +infty;
M is positive or M is zero;
hence thesis by A1, XXREAL_3:def 5;
end;
end;
end;
theorem
0 <= N;
theorem
0 <> N implies 0 < N;
theorem
0 < N + 1;
theorem Th5:
M in NAT & N <= M implies N in NAT
proof
assume A1: M in NAT & N <= M;
0 <= N & 0 in REAL by XREAL_0:def 1;
then N in REAL by A1, NUMBERS:19, XXREAL_0:45;
then N is Nat by Th3;
hence thesis by ORDINAL1:def 12;
end;
theorem
N < M implies N in NAT
proof
assume A1: N < M;
0 <= N & 0 in REAL by XREAL_0:def 1;
then N in REAL by A1, XXREAL_0:46;
then N is Nat by Th3;
hence thesis by ORDINAL1:def 12;
end;
theorem
N <= M implies N * K <= M * K by XXREAL_3:71;
theorem
N = 0 or ex K st N = K + 1
proof
per cases by Th3;
suppose N is Nat;
then reconsider n = N as Nat;
assume N <> 0;
then consider k being Nat such that
A1: n = k + 1 by NAT_1:6;
reconsider K = k as ExtNat;
take K;
N = k + (1 qua ExtNat) by A1;
hence thesis;
end;
suppose A2: N = +infty;
assume N <> 0;
take +infty;
thus thesis by A2, XXREAL_3:def 2;
end;
end;
theorem
N + M = 0 implies N = 0 & M = 0;
registration
let M be ExtNat, N be non zero ExtNat;
cluster M + N -> non zero;
coherence;
cluster N + M -> non zero;
coherence;
end;
theorem Th10:
N <= M + 1 implies N <= M or N = M + 1
proof
assume A1: N <= M + 1;
per cases by Th3;
suppose M is Nat;
then reconsider m = M as Nat;
m + (1 qua ExtNat) = M + 1;
then A2: m + 1 in NAT & N <= m + 1 by A1;
then N in NAT by Th5;
then reconsider n = N as Nat;
n <= m or n = m + 1 by A2, NAT_1:8;
then N <= M or N = m + (1 qua ExtNat);
hence thesis;
end;
suppose M = +infty;
hence thesis by A1, XXREAL_3:def 2;
end;
end;
theorem
N <= M & M <= N + 1 implies N = M or M = N + 1
proof
assume A1: N <= M & M <= N + 1;
then M <= N or M = N + 1 by Th10;
hence thesis by A1, XXREAL_0:1;
end;
theorem
N <= M implies ex K st M = N + K
proof
assume A1: N <= M;
per cases by Th3;
suppose M is Nat;
then reconsider m = M as Nat;
m in NAT by ORDINAL1:def 12;
then N in NAT by A1, Th5;
then reconsider n = N as Nat;
consider k being Nat such that
A2: m = n + k by A1, NAT_1:10;
reconsider K = k as ExtNat;
take K;
m = (n qua ExtNat) + k by A2;
hence thesis;
end;
suppose A3: M = +infty;
take M;
thus thesis by A3, XXREAL_3:def 2;
end;
end;
theorem
N <= N + M
proof
0 + N <= M + N by XXREAL_3:35;
hence thesis by XXREAL_3:4;
end;
theorem
N <= M implies N <= M + K
proof
assume A1: N <= M;
N + 0 <= M + K by A1, XXREAL_3:36;
hence thesis by XXREAL_3:4;
end;
theorem Th4c:
N < 1 implies N = 0
proof
assume A1: N < 1;
then N in NAT by Th5;
then reconsider n = N as Nat;
n = 0 by A1, NAT_1:14;
hence thesis;
end;
theorem
N * M = 1 implies N = 1
proof
assume A1: N * M = 1;
N <> +infty & M <> +infty
proof
assume N = +infty or M = +infty;
then (N = +infty & (M is zero or M is positive)) or
(M = +infty & (N is zero or N is positive));
hence contradiction by A1, XXREAL_3:def 5;
end;
then reconsider n = N, m = M as Nat by Th3;
n * (m qua ExtNat) = 1 by A1;
hence thesis by NAT_1:15;
end;
theorem
K < K + N iff 1 <= N & K <> +infty
proof
hereby
assume A1: K < K + N;
assume N < 1 or K = +infty;
then N = 0 or K = +infty by Th4c;
hence contradiction by A1, XXREAL_3:4, XXREAL_3:def 2;
end;
assume A2: 1 <= N & K <> +infty;
then reconsider k = K as Nat by Th3;
per cases by Th3;
suppose N is Nat;
then reconsider n = N as Nat;
k < (k qua ExtNat) + n by A2, NAT_1:19;
hence thesis;
end;
suppose N = +infty;
then A3: K + N = +infty by XXREAL_3:def 2;
k in REAL by XREAL_0:def 1;
hence thesis by A3, XXREAL_0:9;
end;
end;
theorem
K <> 0 & N = M * K implies M <= N
proof
assume A1: K <> 0 & N = M * K;
per cases by Th3;
suppose N is Nat;
then reconsider n = N as Nat;
per cases;
suppose M = 0;
hence thesis;
end;
suppose A2: M <> 0;
M <> +infty & K <> +infty
proof
assume M = +infty or K = +infty;
then n = +infty by A1, A2, XXREAL_3:def 5;
hence contradiction;
end;
then reconsider m = M, k = K as Nat by Th3;
n = m * (k qua ExtNat) by A1;
hence thesis by A1, NAT_1:24;
end;
end;
suppose N = +infty;
hence thesis by XXREAL_0:3;
end;
end;
theorem
M <= N implies M * K <= N * K by XXREAL_3:71;
theorem
K + M + N = K + (M + N) by XXREAL_3:29;
theorem
K * (N+M) = K*N + K*M
proof
per cases by Th3;
suppose K is Nat;
hence thesis by XXREAL_3:95;
end;
suppose A1: K = +infty;
per cases;
suppose A2: N is positive & M is positive;
hence K * (N+M) = +infty by A1, XXREAL_3:def 5
.= K*N + +infty by XXREAL_3:def 2
.= K*N + K*M by A1, A2, XXREAL_3:def 5;
end;
suppose M is non positive;
then A3: M = 0;
hence K * (N+M) = K*N by XXREAL_3:4
.= K*N + 0 by XXREAL_3:4
.= K*N + K*M by A3;
end;
suppose N is non positive;
then A3: N = 0;
hence K * (N+M) = K*M by XXREAL_3:4
.= 0 + K*M by XXREAL_3:4
.= K*N + K*M by A3;
end;
end;
end;
begin :: Sets of extended natural numbers
definition
let X be set;
attr X is ext-natural-membered means
:Def2:
for x being object holds x in X implies x is ext-natural;
end;
registration
cluster empty -> ext-natural-membered for set;
coherence;
cluster natural-membered -> ext-natural-membered for set;
coherence;
cluster ext-natural-membered -> ext-real-membered for set;
coherence
proof
let X be set;
assume A1: X is ext-natural-membered;
now
let x be object;
assume x in X;
then x is ext-natural by A1;
hence x is ext-real;
end;
hence thesis by MEMBERED:def 2;
end;
cluster ExtNAT -> ext-natural-membered;
coherence;
end;
registration
cluster non empty ext-natural-membered for set;
existence
proof
take ExtNAT;
thus thesis;
end;
end;
theorem ThSubset:
for X being set holds X is ext-natural-membered iff X c= ExtNAT
proof
let X be set;
hereby
assume A1: X is ext-natural-membered;
now
let x be object;
assume x in X;
then x is ext-natural by A1;
hence x in ExtNAT;
end;
hence X c= ExtNAT by TARSKI:def 3;
end;
assume X c= ExtNAT;
hence thesis;
end;
reserve X for ext-natural-membered set;
registration
let X;
cluster -> ext-natural for Element of X;
coherence
proof
let x be Element of X;
per cases;
suppose X is non empty;
hence thesis by Def2;
end;
suppose X is empty;
hence thesis;
end;
end;
end;
theorem ThEx:
for X being non empty ext-natural-membered set ex N st N in X
proof
let X be non empty ext-natural-membered set;
consider x being ExtReal such that
A1: x in X by MEMBERED:8;
reconsider N = x as ext-natural ExtReal by A1;
take N;
thus thesis by A1;
end;
theorem
(for N holds N in X) implies X = ExtNAT
proof
assume A1: for N holds N in X;
A2: X c= ExtNAT by ThSubset;
for x being object st x in ExtNAT holds x in X by A1;
then ExtNAT c= X by TARSKI:def 3;
hence thesis by A2, XBOOLE_0:def 10;
end;
theorem
for Y being set st Y c= X holds Y is ext-natural-membered;
registration
let X;
cluster -> ext-natural-membered for Subset of X;
coherence;
end;
registration
let N;
cluster {N} -> ext-natural-membered;
coherence by TARSKI:def 1;
let M;
cluster {N,M} -> ext-natural-membered;
coherence
proof
now
let x be object;
assume x in {N,M};
then per cases by TARSKI:def 2;
suppose x = N;
hence x is ext-natural;
end;
suppose x = M;
hence x is ext-natural;
end;
end;
hence thesis;
end;
let K;
cluster {N,M,K} -> ext-natural-membered;
coherence
proof
now
let x be object;
assume x in {N,M,K};
then per cases by ENUMSET1:def 1;
suppose x = N;
hence x is ext-natural;
end;
suppose x = M;
hence x is ext-natural;
end;
suppose x = K;
hence x is ext-natural;
end;
end;
hence thesis;
end;
end;
registration
let X; let Y be ext-natural-membered set;
cluster X \/ Y -> ext-natural-membered;
coherence
proof
now
let x be object;
assume x in X \/ Y;
then per cases by XBOOLE_0:def 3;
suppose x in X;
hence x is ext-natural;
end;
suppose x in Y;
hence x is ext-natural;
end;
end;
hence thesis;
end;
end;
registration
let X; let Y be set;
cluster X /\ Y -> ext-natural-membered;
coherence
proof
now
let x be object;
assume x in X /\ Y;
then x in X by XBOOLE_0:def 4;
hence x is ext-natural;
end;
hence thesis;
end;
cluster X \ Y -> ext-natural-membered;
coherence;
end;
registration
let X; let Y be ext-natural-membered set;
cluster X \+\ Y -> ext-natural-membered;
coherence
proof
X \+\ Y = (X \ Y) \/ (Y \ X) by XBOOLE_0:def 6;
hence thesis;
end;
end;
definition
let X; let Y be set;
redefine pred X c= Y means
N in X implies N in Y;
compatibility
proof
thus (X c= Y) implies for N st N in X holds N in Y;
assume A1: for N st N in X holds N in Y;
for x being object st x in X holds x in Y by A1;
hence thesis by TARSKI:def 3;
end;
end;
definition
let X; let Y be ext-natural-membered set;
redefine pred X = Y means
N in X iff N in Y;
compatibility
proof
thus (X = Y) implies for N holds N in X iff N in Y;
assume for N holds N in X iff N in Y;
then X c= Y & Y c= X;
hence thesis by XBOOLE_0:def 10;
end;
end;
definition
let X; let Y be ext-natural-membered set;
redefine pred X misses Y means
not ex N st N in X & N in Y;
compatibility
proof
thus X misses Y implies not ex N st N in X & N in Y by XBOOLE_0:3;
assume not ex N st N in X & N in Y;
then not ex x being object st x in X & x in Y;
hence thesis by XBOOLE_0:3;
end;
end;
theorem
for F being set st(for X being set st X in F holds X is ext-natural-membered)
holds union F is ext-natural-membered
proof
let F be set;
assume A1: for X being set st X in F holds X is ext-natural-membered;
let x be object;
assume x in union F;
then consider Y being set such that
A2: x in Y & Y in F by TARSKI:def 4;
Y is ext-natural-membered by A1, A2;
hence x is ext-natural by A2;
end;
theorem
for F, X being set st X in F & X is ext-natural-membered
holds meet F is ext-natural-membered
proof
let F, X be set;
assume A1: X in F & X is ext-natural-membered;
let x be object;
assume x in meet F;
then x in X by A1, SETFAM_1:def 1;
hence thesis by A1;
end;
scheme
ENMSeparation {P[object]}:
ex X being ext-natural-membered set st for N holds N in X iff P[N]
proof
consider X being Subset of ExtNAT such that
A1: for N being set holds N in X iff N in ExtNAT & P[N]
from SUBSET_1:sch 1;
reconsider X as ext-natural-membered set;
take X;
let N be ExtNat;
thus N in X implies P[N] by A1;
assume A2: P[N];
N in ExtNAT by Def1;
hence N in X by A1, A2;
end;
definition
let X be ext-natural-membered set;
redefine mode UpperBound of X means
:DefU:
for N holds N in X implies N <= it;
compatibility
proof
let U be ExtReal;
thus U is UpperBound of X implies for N st N in X holds N <= U
by XXREAL_2:def 1;
assume for N holds N in X implies N <= U;
then for x being ExtReal st x in X holds x <= U;
hence U is UpperBound of X by XXREAL_2:def 1;
end;
redefine mode LowerBound of X means
:DefL:
for N holds N in X implies it <= N;
compatibility
proof
let L be ExtReal;
thus L is LowerBound of X implies for N st N in X holds L <= N
by XXREAL_2:def 2;
assume for N holds N in X implies L <= N;
then for x being ExtReal st x in X holds L <= x;
hence L is LowerBound of X by XXREAL_2:def 2;
end;
end;
registration
cluster -> bounded_below for ext-natural-membered set;
coherence
proof
let X be ext-natural-membered set;
for x being ExtReal st x in X holds 0 <= x;
then 0 is LowerBound of X by XXREAL_2:def 2;
hence thesis by XXREAL_2:def 9;
end;
cluster non empty -> left_end for ext-natural-membered set;
coherence
proof
let X be ext-natural-membered set;
assume X is non empty;
then consider N such that
A2: N in X by ThEx;
per cases;
suppose X is trivial;
hence thesis by A2;
end;
suppose A3: X is non trivial;
:: get smallest n in the set
defpred P[Nat] means $1 in X;
A4: ex n being Nat st P[n]
proof
per cases by Th3;
suppose N is Nat;
then reconsider n = N as Nat;
take n;
thus thesis by A2;
end;
suppose A5: N = +infty;
set n = the Element of X \ {N};
X \ {N} is non empty by A3, ZFMISC_1:139;
then A6: n in X & n <> +infty by A5, ZFMISC_1:56;
then reconsider n as Nat by Th3;
take n;
thus thesis by A6;
end;
end;
consider n being Nat such that
A7: P[n] & for m being Nat st P[m] holds n <= m from NAT_1:sch 5(A4);
:: n is lower bound
now
let x be ExtNat;
assume A7a: x in X;
per cases by Th3;
suppose x is Nat;
hence n <= x by A7, A7a;
end;
suppose x = +infty;
hence n <= x by XXREAL_0:3;
end;
end;
then A8: n is LowerBound of X by DefL;
:: n is biggest lower bound
for x being LowerBound of X holds x <= n by A7, XXREAL_2:def 2;
then inf X = n by A8, XXREAL_2:def 4;
hence thesis by A7, XXREAL_2:def 5;
end;
end;
end;
registration
let X;
cluster ext-natural for UpperBound of X;
existence
proof
for x being ExtReal st x in X holds x <= +infty by XXREAL_0:3;
then reconsider L = +infty as UpperBound of X by XXREAL_2:def 1;
take L;
thus thesis;
end;
cluster ext-natural for LowerBound of X;
existence
proof
for x being ExtReal st x in X holds 0 <= x;
then reconsider L = 0 as LowerBound of X by XXREAL_2:def 2;
take L;
thus thesis;
end;
cluster inf X -> ext-natural;
coherence
proof
per cases;
suppose X is non empty;
then inf X in X by XXREAL_2:def 5;
hence thesis;
end;
suppose X is empty;
hence thesis by XXREAL_2:38;
end;
end;
end;
registration
let X be non empty ext-natural-membered set;
cluster sup X -> ext-natural;
coherence
proof
consider N such that
A1: N in X by ThEx;
per cases;
suppose sup X = +infty;
hence thesis;
end;
suppose A2: sup X <> +infty;
A3: 0 <= sup X by A1, XXREAL_2:4;
then sup X in REAL by A2, XREAL_0:def 1, XXREAL_0:10;
then reconsider S = sup X as Real;
set s = [\ S /];
now
let x be ExtNat;
assume x in X;
then A4: x <= S by XXREAL_2:4;
then x <> +infty by XXREAL_0:4;
then reconsider n = x as Nat by Th3;
n < s + 1 by A4, INT_1:29, XXREAL_0:2;
hence x <= s by INT_1:7;
end;
then s is UpperBound of X by DefU;
then B1: S <= s by XXREAL_2:def 3;
s <= S by INT_1:def 6;
hence thesis by A3, B1, XXREAL_0:1;
end;
end;
end;
registration
cluster non empty bounded_above -> right_end for ext-natural-membered set;
coherence
proof
let X be ext-natural-membered set;
assume A1: X is non empty bounded_above;
then consider N such that
A2: N in X by ThEx;
per cases;
suppose X is trivial;
hence thesis by A2;
end;
suppose A3: X is non trivial;
:: get biggest n in the set
defpred P[Nat] means $1 in X;
A4: ex n being Nat st P[n]
proof
per cases by Th3;
suppose N is Nat;
then reconsider n = N as Nat;
take n;
thus thesis by A2;
end;
suppose A5: N = +infty;
set n = the Element of X \ {N};
X \ {N} is non empty by A3, ZFMISC_1:139;
then A6: n in X & n <> +infty by A5, ZFMISC_1:56;
then reconsider n as Nat by Th3;
take n;
thus thesis by A6;
end;
end;
consider R being Real such that
B1: R is UpperBound of X by A1, XXREAL_2:def 10;
set r = [\ R /];
C2: 0 <= R by A2, B1, XXREAL_2:def 1;
C3: R < r + 1 by INT_1:29;
then 0 <= r by C2, INT_1:7;
then r in NAT by ORDINAL1:def 12;
then reconsider r as Nat;
B2: for n being Nat st P[n] holds n <= r
proof
let n be Nat;
assume P[n];
then n <= R by B1, XXREAL_2:def 1;
then n < r + 1 by C3, XXREAL_0:2;
hence n <= r by NAT_1:13;
end;
consider n being Nat such that
A7: P[n] & for m being Nat st P[m] holds m <= n
from NAT_1:sch 6(B2,A4);
:: n is upper bound
now
let x be ExtNat;
assume A7a: x in X;
then x <= R by B1, XXREAL_2:def 1;
then x in REAL by XREAL_0:def 1, XXREAL_0:13;
then x is Nat by Th3;
hence x <= n by A7, A7a;
end;
then A8: n is UpperBound of X by DefU;
:: n is smallest upper bound
for x being UpperBound of X holds n <= x by A7, XXREAL_2:def 1;
then sup X = n by A8, XXREAL_2:def 3;
hence thesis by A7, XXREAL_2:def 6;
end;
end;
end;
definition
let X be left_end ext-natural-membered set;
redefine func min X -> ExtNat means
it in X & for N st N in X holds it <= N;
coherence;
compatibility
proof
let M be ExtNat;
thus M = min X implies M in X & for N st N in X holds M <= N
by XXREAL_2:def 7;
assume A1: M in X & for N st N in X holds M <= N;
then for x being ExtReal st x in X holds M <= x;
hence thesis by A1, XXREAL_2:def 7;
end;
end;
definition
let X be right_end ext-natural-membered set;
redefine func max X -> ExtNat means
it in X & for N st N in X holds N <= it;
coherence;
compatibility
proof
let M be ExtNat;
thus M = max X implies M in X & for N st N in X holds N <= M
by XXREAL_2:def 8;
assume A1: M in X & for N st N in X holds N <= M;
then for x being ExtReal st x in X holds x <= M;
hence thesis by A1, XXREAL_2:def 8;
end;
end;
begin :: Relations with extended natural numbers in range
definition
let R be Relation;
attr R is ext-natural-valued means
rng R c= ExtNAT;
end;
registration
cluster empty -> ext-natural-valued for Relation;
coherence
proof
let R be Relation;
assume R is empty;
then rng R = {};
hence thesis by XBOOLE_1:2;
end;
cluster natural-valued -> ext-natural-valued for Relation;
coherence
proof
let R be Relation;
assume R is natural-valued;
then rng R c< ExtNAT by Th1, VALUED_0:def 6, XBOOLE_1:59;
hence thesis by XBOOLE_0:def 8;
end;
cluster ext-natural-valued -> ExtNAT-valued ext-real-valued for Relation;
coherence by RELAT_1:def 19, XBOOLE_1:1, VALUED_0:def 2;
cluster ExtNAT-valued -> ext-natural-valued for Relation;
coherence by RELAT_1:def 19;
end;
registration
cluster ext-natural-valued for Function;
existence
proof
take the natural-valued Function;
thus thesis;
end;
end;
registration
let R be ext-natural-valued Relation;
cluster rng R -> ext-natural-membered;
coherence;
end;
theorem
for R being Relation, S being ext-natural-valued Relation st R c= S
holds R is ext-natural-valued;
registration
let R be ext-natural-valued Relation;
cluster -> ext-natural-valued for Subset of R;
coherence;
end;
registration
let R, S be ext-natural-valued Relation;
cluster R \/ S -> ext-natural-valued;
coherence;
end;
registration
let R be ext-natural-valued Relation, S be Relation;
cluster R /\ S -> ext-natural-valued;
coherence;
cluster R \ S -> ext-natural-valued;
coherence;
cluster S * R -> ext-natural-valued;
coherence;
end;
registration
let R, S be ext-natural-valued Relation;
cluster R \+\ S -> ext-natural-valued;
coherence
proof
R \+\ S = (R \ S) \/ (S \ R) by XBOOLE_0:def 6;
hence thesis;
end;
end;
registration
let R be ext-natural-valued Relation, X be set;
cluster R.:X -> ext-natural-membered;
coherence
proof
R.:X c= rng R & rng R c= ExtNAT by RELAT_1:111;
hence thesis;
end;
cluster R|X -> ext-natural-valued;
coherence;
cluster X|`R -> ext-natural-valued;
coherence;
end;
registration
let R be ext-natural-valued Relation, x be object;
cluster Im(R,x) -> ext-natural-membered;
coherence
proof
Im(R,x) = R.:{x} by RELAT_1:def 16;
hence thesis;
end;
end;
registration
let X;
cluster id X -> ext-natural-valued;
coherence by ThSubset;
end;
definition
let f be Function;
redefine attr f is ext-natural-valued means
for x being object st x in dom f holds f.x is ext-natural;
compatibility
proof
hereby
assume A1: f is ext-natural-valued;
let x be object;
assume x in dom f;
then f.x in rng f by FUNCT_1:3;
hence f.x is ext-natural by A1;
end;
assume A2: for x being object st x in dom f holds f.x is ext-natural;
now
let y be object;
assume y in rng f;
then consider x being object such that
A3: x in dom f & f.x = y by FUNCT_1:def 3;
y is ext-natural by A2, A3;
hence y in ExtNAT;
end;
hence thesis by TARSKI:def 3;
end;
end;
theorem ThFunc1:
for f being Function holds
f is ext-natural-valued iff for x being object holds f.x is ext-natural
proof
let f be Function;
hereby
assume A1: f is ext-natural-valued;
let x be object;
per cases;
suppose x in dom f;
hence f.x is ext-natural by A1;
end;
suppose not x in dom f;
hence f.x is ext-natural by FUNCT_1:def 2;
end;
end;
thus (for x being object holds f.x is ext-natural)
implies f is ext-natural-valued;
end;
registration
let f be ext-natural-valued Function;
let x be object;
cluster f.x -> ext-natural;
coherence by ThFunc1;
end;
registration
let X be set; let N;
cluster X --> N -> ext-natural-valued;
coherence
proof
rng(X --> N) c= {N} & {N} c= ExtNAT by ThSubset;
hence thesis;
end;
end;
registration
let f,g be ext-natural-valued Function;
cluster f +* g -> ext-natural-valued;
coherence
proof
rng(f +* g) c= rng f \/ rng g by FUNCT_4:17;
hence thesis by XBOOLE_1:1;
end;
end;
registration
let x be object; let N;
cluster x .--> N -> ext-natural-valued;
coherence
proof
x .--> N = {x} --> N by FUNCOP_1:def 9;
hence thesis;
end;
end;
registration
let Z be set; let X;
cluster -> ext-natural-valued for Relation of Z,X;
coherence
proof
let R be Relation of Z,X;
rng R c= X & X c= ExtNAT by ThSubset;
hence thesis;
end;
end;
registration
let Z be set; let X;
cluster [: Z, X :] -> ext-natural-valued for Relation of Z,X;
coherence;
end;
registration
cluster non empty constant ext-natural-valued for Function;
existence
proof
take the non empty set --> the ExtNat;
thus thesis;
end;
end;
theorem
for f being non empty constant ext-natural-valued Function
ex N st for x being object st x in dom f holds f.x = N
proof
let f be non empty constant ext-natural-valued Function;
consider R being ExtReal such that
A1: for x being object st x in dom f holds f.x = R by VALUED_0:14;
set x = the Element of dom f;
f.x = R & f.x is ext-natural by A1;
then reconsider N = R as ExtNat;
take N;
thus thesis by A1;
end;
begin :: Ordinal preliminaries
:: into ORDINAL1 ?
theorem Th2:
for f being Function holds f is Ordinal-yielding iff
for x being object st x in dom f holds f.x is Ordinal
proof
let f be Function;
hereby
assume f is Ordinal-yielding;
then consider A being Ordinal such that
A1: rng f c= A by ORDINAL2:def 4;
let x be object;
assume x in dom f;
then f.x in rng f by FUNCT_1:3;
hence f.x is Ordinal by A1;
end;
assume A2: for x being object st x in dom f holds f.x is Ordinal;
now
reconsider A = sup rng f as Ordinal;
take A;
now
let y be object;
assume A3: y in rng f;
then consider x being object such that
A4: x in dom f & y = f.x by FUNCT_1:def 3;
y is Ordinal by A2, A4;
then A5: y in On rng f by A3, ORDINAL1:def 9;
On rng f c= A by ORDINAL2:def 3;
hence y in A by A5;
end;
hence rng f c= A by TARSKI:def 3;
end;
hence thesis by ORDINAL2:def 4;
end;
:: into ORDINAL1 ?
registration
cluster ordinal -> c=-linear for set;
coherence
proof
let A be set;
assume A is ordinal;
then for x,y being set st x in A & y in A holds x,y are_c=-comparable
by ORDINAL1:15;
hence thesis by ORDINAL1:def 8;
end;
end;
:: into ORDINAL1 ? ::: generalize in ORDINAL2
registration
let f be Ordinal-yielding Function, x be object;
cluster f.x -> ordinal;
coherence
proof
per cases;
suppose x in dom f;
hence thesis by Th2;
end;
suppose not x in dom f;
hence thesis by FUNCT_1:def 2;
end;
end;
end;
:: into ORDINAL4 ?
registration
let A, B be non-empty Sequence;
cluster A^B -> non-empty;
coherence
proof
now
let x be object;
assume A1: x in dom (A^B);
then reconsider c = x as Ordinal;
per cases;
suppose A2: c in dom A;
then (A^B).c = A.c by ORDINAL4:def 1;
hence (A^B).x is non empty by A2, FUNCT_1:def 9;
end;
suppose not c in dom A;
then consider d being Ordinal such that
A3: c = dom A +^ d by ORDINAL1:16, ORDINAL3:27;
c in dom A +^ dom B by A1, ORDINAL4:def 1;
then A4: d in dom B by A3, ORDINAL3:22;
then (A^B).(dom A +^ d) = B.d by ORDINAL4:def 1;
hence (A^B).x is non empty by A3, A4, FUNCT_1:def 9;
end;
end;
hence thesis by FUNCT_1:def 9;
end;
end;
:: into CARD_1 ? ::: version of CARD_1:64
theorem Th3:
for X being set, x being object holds card(X --> x) = card X
proof
let X be set, x be object;
deffunc F(object) = [$1, x];
consider f being Function such that
A1: dom f = X & for z being object st z in X holds f.z = F(z)
from FUNCT_1:sch 3;
now
let y be object;
hereby
assume y in rng f;
then consider z being object such that
A2: z in dom f & f.z = y by FUNCT_1:def 3;
A3: y = [z,x] by A1, A2;
z in X & x in {x} by A1, A2, TARSKI:def 1;
then y in [: X, {x} :] by A3, ZFMISC_1:87;
hence y in X --> x by FUNCOP_1:def 2;
end;
assume y in X --> x;
then consider z,x0 being object such that
A4: z in X & x0 in {x} & y = [z,x0] by ZFMISC_1:def 2;
y = [z,x] by A4, TARSKI:def 1
.= f.z by A1, A4;
hence y in rng f by A1, A4, FUNCT_1:3;
end;
then A5: rng f = X --> x by TARSKI:2;
now
let x1,x2 be object;
assume A6: x1 in dom f & x2 in dom f & f.x1 = f.x2;
then f.x1 = [x1,x] & f.x2 = [x2,x] by A1;
hence x1 = x2 by A6, XTUPLE_0:1;
end;
hence thesis by A1, A5, FUNCT_1:def 4, CARD_1:70;
end;
:: generalization of CARD_1:64
:: into CARD_1 ?
theorem
for c being Cardinal, x being object holds card(c --> x) = c
proof
let c be Cardinal, x be object;
thus card(c --> x) = card c by Th3
.= c;
end;
:: into CARD_1 ?
registration
let X be trivial set;
cluster card X -> trivial;
coherence
proof
per cases;
suppose X is empty;
hence thesis;
end;
suppose X is non empty;
then consider x being object such that
A1: X = {x} by ZFMISC_1:131;
thus thesis by A1, CARD_1:30, CARD_1:49;
end;
end;
end;
:: into CARD_2 ?
registration
let c1 be Cardinal, c2 be non empty Cardinal;
cluster c1 +` c2 -> non empty;
coherence
proof
c2 c= c1 +` c2 by CARD_2:94;
hence thesis;
end;
end;
:: into CARD_2 ?
theorem
for A being Ordinal holds A <> 0 & A <> 1 iff A is non trivial
proof
let A be Ordinal;
hereby
assume A1: A <> 0 & A <> 1;
then A in 1 or 1 in A by ORDINAL1:14;
then A2: 1 in A by A1, CARD_1:49, TARSKI:def 1;
0 in 1 by CARD_1:49, TARSKI:def 1;
then 0 in A by A2, ORDINAL1:10;
hence A is non trivial by A2, ZFMISC_1:def 10;
end;
assume A is non trivial;
then consider x, y being object such that
A3: x in A & y in A & x <> y by ZFMISC_1:def 10;
A4: card {x,y} c= card A by A3, ZFMISC_1:32, CARD_1:11;
card A c= A by CARD_1:8;
then card {x,y} c= A by A4, XBOOLE_1:1;
then {0,1} c= A by A3, CARD_2:57, CARD_1:50;
then 0 in A & 1 in A by ZFMISC_1:32;
hence thesis;
end;
:: into CARD_2 ?
theorem ThAdd:
for A being Ordinal, B being infinite Cardinal st A in B holds A +^ B = B
proof
let A being Ordinal, B be infinite Cardinal;
assume A1: A in B;
deffunc F(Ordinal) = A+^$1;
consider fi being Ordinal-Sequence such that
A2: dom fi = B & for C being Ordinal st C in B holds fi.C = F(C)
from ORDINAL2:sch 3;
A3: A +^ B = sup fi by A2, ORDINAL2:29;
now
let D be Ordinal;
assume D in rng fi;
then consider x being object such that
B1: x in dom fi & fi.x = D by FUNCT_1:def 3;
reconsider C = x as Ordinal by B1;
card A in B & card C in B by A1, A2, B1, CARD_1:9;
then card A +` card C in B +` B by CARD_2:96;
then B2: card A +` card C in card B by CARD_2:75;
D = A +^ C by A2, B1;
then card D = card A +` card C by CARD_2:13;
hence D in B by B2, CARD_3:43;
end;
then sup rng fi c= B by ORDINAL2:20;
then A4: sup fi c= B by ORDINAL2:def 5;
B c= A +^ B by ORDINAL3:24;
hence thesis by A3, A4, XBOOLE_0:def 10;
end;
:: into CARD_3 ?
registration
let f be Cardinal-yielding Function, g be Function;
cluster f*g -> Cardinal-yielding;
coherence
proof
now
let x be object;
assume A1: x in dom(f*g);
then x in dom g & g.x in dom f by FUNCT_1:11;
then f.(g.x) is Cardinal by CARD_3:def 1;
hence (f*g).x is Cardinal by A1, FUNCT_1:12;
end;
hence thesis by CARD_3:def 1;
end;
end;
:: into CARD_3 ?
registration
cluster natural-valued -> Cardinal-yielding for Function;
coherence
proof
let f be Function;
assume f is natural-valued;
then for x being object st x in dom f holds f.x is Cardinal;
hence thesis by CARD_3:def 1;
end;
end;
:: into CARD_3 ?
registration
let f be empty Function;
cluster disjoin f -> empty;
coherence by CARD_3:3;
end;
:: into CARD_3 ?
registration
let f be empty-yielding Function;
cluster disjoin f -> empty-yielding;
coherence
proof
now
let x be object;
assume x in dom disjoin f;
then A1: x in dom f by CARD_3:def 3;
[: f.x, {x} :] is empty;
hence (disjoin f).x is empty by A1, CARD_3:def 3;
end;
hence thesis by FUNCT_1:def 8;
end;
end;
:: into CARD_3 ?
registration
let f be non empty-yielding Function;
cluster disjoin f -> non empty-yielding;
coherence
proof
consider x being object such that
A1: x in dom f & f.x is non empty by FUNCT_1:def 8;
[: f.x, {x} :] is non empty by A1;
then (disjoin f).x is non empty by A1, CARD_3:def 3;
hence thesis;
end;
end;
:: into CARD_3 ?
registration
let f be empty-yielding Function;
cluster Union f -> empty;
coherence
proof
assume Union f is non empty;
then consider y being object such that
A1: y in Union f by XBOOLE_0:def 1;
y in union rng f by A1, CARD_3:def 4;
then consider Y being set such that
A2: y in Y & Y in rng f by TARSKI:def 4;
thus contradiction by A2;
end;
end;
:: into CARD_3 ?
registration
cluster Cardinal-yielding -> Ordinal-yielding for Function;
coherence
proof
let f be Function;
assume f is Cardinal-yielding;
then for x being object st x in dom f holds f.x is Ordinal by CARD_3:def 1;
hence thesis by Th2;
end;
end;
:: into CARD_3 ?
theorem
for f being Function, p being Permutation of dom f
holds Card(f*p) = (Card f)*p
proof
let f be Function, p be Permutation of dom f;
A1: rng p = dom f by FUNCT_2:def 3;
then A2: rng p = dom Card f by CARD_3:def 2;
A3: dom Card(f*p) = dom(f*p) by CARD_3:def 2
.= dom p by A1, RELAT_1:27
.= dom((Card f)*p) by A2, RELAT_1:27;
now
let x be object;
assume A4: x in dom Card(f*p);
then A5: x in dom(f*p) by CARD_3:def 2;
then A6: p.x in dom f by FUNCT_1:11;
thus Card(f*p).x = card((f*p).x) by A5, CARD_3:def 2
.= card(f.(p.x)) by A5, FUNCT_1:12
.= (Card f).(p.x) by A6, CARD_3:def 2
.= ((Card f)*p).x by A3, A4, FUNCT_1:12;
end;
hence thesis by A3, FUNCT_1:2;
end;
:: into CARD_3 ?
registration
let A be Sequence;
cluster Card A -> Sequence-like;
coherence
proof
dom Card A = dom A by CARD_3:def 2;
hence thesis by ORDINAL1:def 7;
end;
end;
:: into CARD_3 ?
theorem
for A, B being Sequence holds Card(A^B) = (Card A)^(Card B)
proof
let A, B be Sequence;
A1: dom Card(A^B) = dom(A^B) by CARD_3:def 2
.= (dom A)+^(dom B) by ORDINAL4:def 1
.= (dom Card A)+^(dom B) by CARD_3:def 2
.= (dom Card A)+^(dom Card B) by CARD_3:def 2
.= dom((Card A)^(Card B)) by ORDINAL4:def 1;
now
let x be object;
assume x in dom Card(A^B);
then A2: x in dom(A^B) by CARD_3:def 2;
then reconsider C = x as Ordinal;
x in (dom A)+^(dom B) by A2, ORDINAL4:def 1;
then per cases by ORDINAL3:38;
suppose A3: C in dom A;
then A4: C in dom Card A by CARD_3:def 2;
thus (Card(A^B)).x = card((A^B).x) by A2, CARD_3:def 2
.= card(A.x) by A3, ORDINAL4:def 1
.= (Card A).x by A3, CARD_3:def 2
.= ((Card A)^(Card B)).x by A4, ORDINAL4:def 1;
end;
suppose ex D being Ordinal st D in dom B & C = (dom A)+^D;
then consider D being Ordinal such that
A5: D in dom B & C = (dom A)+^D;
A6: D in dom Card B & C = (dom Card A)+^D by A5, CARD_3:def 2;
thus (Card(A^B)).x = card((A^B).x) by A2, CARD_3:def 2
.= card(B.D) by A5, ORDINAL4:def 1
.= (Card B).D by A5, CARD_3:def 2
.= ((Card A)^(Card B)).x by A6, ORDINAL4:def 1;
end;
end;
hence thesis by A1, FUNCT_1:2;
end;
:: into CARD_3 ?
registration
let f be trivial Function;
cluster Card f -> trivial;
coherence
proof
dom f is trivial;
then dom Card f is trivial by CARD_3:def 2;
hence thesis;
end;
end;
:: into CARD_3 ?
registration
let f be non trivial Function;
cluster Card f -> non trivial;
coherence
proof
dom f is non trivial;
then dom Card f is non trivial by CARD_3:def 2;
hence thesis;
end;
end;
:: into CARD_3 ?
registration
let A, B be Cardinal-yielding Sequence;
cluster A^B -> Cardinal-yielding;
coherence
proof
now
let x be object;
assume A1: x in dom (A^B);
then reconsider c = x as Ordinal;
per cases;
suppose A2: c in dom A;
then (A^B).c = A.c by ORDINAL4:def 1;
hence (A^B).x is Cardinal by A2, CARD_3:def 1;
end;
suppose not c in dom A;
then consider d being Ordinal such that
A3: c = dom A +^ d by ORDINAL1:16, ORDINAL3:27;
c in dom A +^ dom B by A1, ORDINAL4:def 1;
then A4: d in dom B by A3, ORDINAL3:22;
then (A^B).(dom A +^ d) = B.d by ORDINAL4:def 1;
hence (A^B).x is Cardinal by A3, A4, CARD_3:def 1;
end;
end;
hence thesis by CARD_3:def 1;
end;
end;
:: into CARD_3 or AFINSQ_1 ?
registration
let c1 be Cardinal;
cluster <% c1 %> -> Cardinal-yielding;
coherence
proof
now
let x be object;
assume x in dom <% c1 %>;
then x in {0} by AFINSQ_1:def 4, CARD_1:49;
then x = 0 by TARSKI:def 1;
hence <% c1 %>.x is Cardinal;
end;
hence thesis by CARD_3:def 1;
end;
let c2 be Cardinal;
cluster <% c1, c2 %> -> Cardinal-yielding;
coherence
proof
<% c1, c2 %> = <% c1 %> ^ <% c2 %> by AFINSQ_1:def 5;
hence thesis;
end;
let c3 be Cardinal;
cluster <% c1, c2, c3 %> -> Cardinal-yielding;
coherence
proof
<% c1, c2, c3 %> = <% c1 %> ^ <% c2 %> ^ <% c3 %> by AFINSQ_1:def 6
.= <% c1, c2 %> ^ <% c3 %> by AFINSQ_1:def 5;
hence thesis;
end;
end;
:: into CARD_3 or AFINSQ_1 ?
registration
let X1, X2, X3 be non empty set;
cluster <% X1, X2, X3 %> -> non-empty;
coherence
proof
<% X1, X2, X3 %> = <% X1 %> ^ <% X2 %> ^ <% X3 %> by AFINSQ_1:def 6
.= <% X1, X2 %> ^ <% X3 %> by AFINSQ_1:def 5;
hence thesis;
end;
end;
:: into AFINSQ_1 ?
registration
let A be infinite Ordinal;
cluster <% A %> -> non natural-valued;
coherence
proof
<% A %>.0 = A;
hence thesis;
end;
let x be object;
cluster <% A, x %> -> non natural-valued;
coherence
proof
<% A, x %>.0 = A;
hence thesis;
end;
cluster <% x, A %> -> non natural-valued;
coherence
proof
<% x, A %>.1 = A;
hence thesis;
end;
let y be object;
cluster <% A, x, y %> -> non natural-valued;
coherence
proof
<% A, x, y %>.0 = A;
hence thesis;
end;
cluster <% x, A, y %> -> non natural-valued;
coherence
proof
<% x, A, y %>.1 = A;
hence thesis;
end;
cluster <% x, y, A %> -> non natural-valued;
coherence
proof
<% x, y, A %>.2 = A;
hence thesis;
end;
end;
:: into AFINSQ_1 ?
registration
cluster non empty non-empty natural-valued for XFinSequence;
existence
proof
take <% 1 %>;
thus thesis;
end;
let x be object;
cluster <% x %> -> one-to-one;
coherence
proof
<% x %> = 0 .--> x by AFINSQ_1:def 1;
hence thesis;
end;
end;
:: into AFINSQ_1 ?
theorem Th7:
for x, y being object holds dom <% x, y %> = {0,1} & rng <% x, y %> = {x,y}
proof
let x,y be object;
thus A1: dom <% x, y %> = {0,1} by AFINSQ_1:38, CARD_1:50;
now
let b be object;
hereby
assume b in rng <% x, y %>;
then consider a being object such that
A2: a in dom <% x, y %> & <% x, y %>.a = b by FUNCT_1:def 3;
per cases by A1, A2, TARSKI:def 2;
suppose a = 0;
hence b in {x,y} by A2, TARSKI:def 2;
end;
suppose a = 1;
hence b in {x,y} by A2, TARSKI:def 2;
end;
end;
assume b in {x, y};
then per cases by TARSKI:def 2;
suppose b = x;
then 0 in dom <% x, y %> & <% x, y %>.0 = b
by A1, TARSKI:def 2;
hence b in rng <% x, y %> by FUNCT_1:3;
end;
suppose b = y;
then 1 in dom <% x, y %> & <% x, y %>.1 = b
by A1, TARSKI:def 2;
hence b in rng <% x, y %> by FUNCT_1:3;
end;
end;
hence thesis by TARSKI:2;
end;
:: into AFINSQ_1 ?
theorem Th8:
for x,y,z being object holds dom<% x,y,z %>={0,1,2} & rng<% x,y,z %>={x,y,z}
proof
let x,y,z be object;
thus A1: dom <% x, y, z %> = {0,1,2} by CARD_1:51, AFINSQ_1:39;
now
let b be object;
hereby
assume b in rng <% x, y, z %>;
then consider a being object such that
A2: a in dom <% x, y, z %> & <% x, y, z %>.a = b by FUNCT_1:def 3;
per cases by A1, A2, ENUMSET1:def 1;
suppose a = 0;
hence b in {x,y,z} by A2, ENUMSET1:def 1;
end;
suppose a = 1;
hence b in {x,y,z} by A2, ENUMSET1:def 1;
end;
suppose a = 2;
hence b in {x,y,z} by A2, ENUMSET1:def 1;
end;
end;
assume b in {x, y, z};
then per cases by ENUMSET1:def 1;
suppose b = x;
then 0 in dom <% x, y, z %> & <% x, y, z %>.0 = b
by A1, ENUMSET1:def 1;
hence b in rng <% x, y, z %> by FUNCT_1:3;
end;
suppose b = y;
then 1 in dom <% x, y, z %> & <% x, y, z %>.1 = b
by A1, ENUMSET1:def 1;
hence b in rng <% x, y, z %> by FUNCT_1:3;
end;
suppose b = z;
then 2 in dom <% x, y, z %> & <% x, y, z %>.2 = b
by A1, ENUMSET1:def 1;
hence b in rng <% x, y, z %> by FUNCT_1:3;
end;
end;
hence thesis by TARSKI:2;
end;
:: into AFINSQ_1 ?
registration
let x be object;
cluster <% x %> -> trivial;
coherence;
let y be object;
cluster <% x, y %> -> non trivial;
coherence
proof
dom <% x, y %> = {0,1} by AFINSQ_1:38, CARD_1:50;
then 0 in dom <% x, y %> & 1 in dom <% x, y %> by TARSKI:def 2;
hence thesis by ZFMISC_1:def 10;
end;
let z be object;
cluster <% x, y, z %> -> non trivial;
coherence
proof
dom <% x, y, z %> = {0,1,2} by AFINSQ_1:39, CARD_1:51;
then 0 in dom <% x, y, z %> & 1 in dom <% x, y, z %> by ENUMSET1:def 1;
hence thesis by ZFMISC_1:def 10;
end;
end;
:: into AFINSQ_1 ?
registration
cluster non empty trivial for XFinSequence;
existence
proof
take <% the object %>;
thus thesis;
end;
let D be non empty set;
cluster non empty trivial for XFinSequence of D;
existence
proof
take <% the Element of D %>;
thus thesis;
end;
end;
:: into AFINSQ_1 ?
theorem Th9:
for p being non empty trivial Sequence
ex x being object st p = <% x %>
proof
let p be non empty trivial XFinSequence;
consider x being object such that
A1: rng p = {x} by ZFMISC_1:131;
take x;
consider z being object such that
A2: dom p = {z} by ZFMISC_1:131;
dom p = card {z} by A2;
then A3: dom p = 1 by CARD_1:30;
p = (dom p) --> x by A1, FUNCOP_1:9
.= 0 .--> x by A3, CARD_1:49, FUNCOP_1:def 9;
hence thesis by AFINSQ_1:def 1;
end;
:: into AFINSQ_1 ?
theorem
for D being non empty set, p being non empty trivial Sequence of D
ex x being Element of D st p = <% x %>
proof
let D be non empty set, p be non empty trivial Sequence of D;
consider x being object such that
A1: p = <% x %> by Th9;
rng p = {x} by A1, AFINSQ_1:33;
then x in rng p by TARSKI:def 1;
then reconsider x as Element of D;
take x;
thus thesis by A1;
end;
:: into AFINSQ_1 ?
theorem
<% 0 %> = id 1
proof
thus <% 0 %> = {[0,0]} by AFINSQ_1:32
.= id 1 by SYSREL:13, CARD_1:49;
end;
:: into AFINSQ_1 ?
theorem
<% 0, 1 %> = id 2
proof
A1: dom <% 0, 1 %> = dom id 2 by Th7, CARD_1:50;
now
let x be object;
assume x in dom <% 0, 1 %>;
then A2: x in 2 & x in {0,1} by A1, Th7;
then per cases by TARSKI:def 2;
suppose x = 0;
hence <% 0, 1 %>.x = (id 2).x by A2, FUNCT_1:18;
end;
suppose x = 1;
hence <% 0, 1 %>.x = (id 2).x by A2, FUNCT_1:18;
end;
end;
hence thesis by A1, FUNCT_1:2;
end;
:: into AFINSQ_1 ?
theorem
<% 0, 1, 2 %> = id 3
proof
A1: dom <% 0, 1, 2 %> = dom id 3 by Th8, CARD_1:51;
now
let x be object;
assume x in dom <% 0, 1, 2 %>;
then A2: x in 3 & x in {0,1,2} by A1, Th8;
then per cases by ENUMSET1:def 1;
suppose x = 0;
hence <% 0, 1, 2 %>.x = (id 3).x by A2, FUNCT_1:18;
end;
suppose x = 1;
hence <% 0, 1, 2 %>.x = (id 3).x by A2, FUNCT_1:18;
end;
suppose x = 2;
hence <% 0, 1, 2 %>.x = (id 3).x by A2, FUNCT_1:18;
end;
end;
hence thesis by A1, FUNCT_1:2;
end;
:: into AFINSQ_1 ?
theorem
for x, y being object holds <% x, y %> * <% 1, 0 %> = <% y, x %>
proof
let x,y be object;
rng <% 1, 0 %> = {1,0} by Th7
.= dom <% x, y %> by Th7;
then A1: dom(<% x, y %> * <% 1, 0 %>) = dom <% 1, 0 %> by RELAT_1:27
.= {0,1} by Th7;
then A2: dom(<% x, y %> * <% 1, 0 %>) = dom <% y, x %> by Th7;
now
let a be object;
assume A3: a in dom(<% x, y %> * <% 1, 0 %>);
then per cases by A1, TARSKI:def 2;
suppose A4: a = 0;
thus (<% x, y %> * <% 1, 0 %>).a
= <% x, y %>.(<% 1, 0 %>.a) by A3, FUNCT_1:12
.= <% y, x %>.a by A4;
end;
suppose A5: a = 1;
thus (<% x, y %> * <% 1, 0 %>).a
= <% x, y %>.(<% 1, 0 %>.a) by A3, FUNCT_1:12
.= <% y, x %>.a by A5;
end;
end;
hence thesis by A2, FUNCT_1:2;
end;
:: into AFINSQ_1 ?
theorem
for x, y, z being object holds <% x, y, z %> * <% 0, 2, 1 %> = <% x, z, y %>
proof
let x,y,z be object;
rng <% 0, 2, 1 %> = {0,2,1} by Th8
.= {0,1,2} by ENUMSET1:57
.= dom <% x, y, z %> by Th8;
then A1: dom(<% x, y, z %> * <% 0, 2, 1 %>) = dom <% 0, 2, 1 %> by RELAT_1:27
.= {0,1,2} by Th8;
then A2: dom(<% x, y, z %> * <% 0, 2, 1 %>) = dom <% x, z, y %> by Th8;
now
let a be object;
assume A3: a in dom(<% x, y, z %> * <% 0, 2, 1 %>);
then per cases by A1, ENUMSET1:def 1;
suppose A4: a = 0;
thus (<% x, y, z %> * <% 0, 2, 1 %>).a
= <% x, y, z %>.(<% 0, 2, 1 %>.a) by A3, FUNCT_1:12
.= <% x, z, y %>.a by A4;
end;
suppose A5: a = 1;
thus (<% x, y, z %> * <% 0, 2, 1 %>).a
= <% x, y, z %>.(<% 0, 2, 1 %>.a) by A3, FUNCT_1:12
.= <% x, z, y %>.a by A5;
end;
suppose A6: a = 2;
thus (<% x, y, z %> * <% 0, 2, 1 %>).a
= <% x, y, z %>.(<% 0, 2, 1 %>.a) by A3, FUNCT_1:12
.= <% x, z, y %>.a by A6;
end;
end;
hence thesis by A2, FUNCT_1:2;
end;
:: into AFINSQ_1 ?
theorem
for x, y, z being object holds <% x, y, z %> * <% 1, 0, 2 %> = <% y, x, z %>
proof
let x,y,z be object;
rng <% 1, 0, 2 %> = {1,0,2} by Th8
.= {0,1,2} by ENUMSET1:58
.= dom <% x, y, z %> by Th8;
then A1: dom(<% x, y, z %> * <% 1, 0, 2 %>) = dom <% 1, 0, 2 %> by RELAT_1:27
.= {0,1,2} by Th8;
then A2: dom(<% x, y, z %> * <% 1, 0, 2 %>) = dom <% y, x, z %> by Th8;
now
let a be object;
assume A3: a in dom(<% x, y, z %> * <% 1, 0, 2 %>);
then per cases by A1, ENUMSET1:def 1;
suppose A4: a = 0;
thus (<% x, y, z %> * <% 1, 0, 2 %>).a
= <% x, y, z %>.(<% 1, 0, 2 %>.a) by A3, FUNCT_1:12
.= <% y, x, z %>.a by A4;
end;
suppose A5: a = 1;
thus (<% x, y, z %> * <% 1, 0, 2 %>).a
= <% x, y, z %>.(<% 1, 0, 2 %>.a) by A3, FUNCT_1:12
.= <% y, x, z %>.a by A5;
end;
suppose A6: a = 2;
thus (<% x, y, z %> * <% 1, 0, 2 %>).a
= <% x, y, z %>.(<% 1, 0, 2 %>.a) by A3, FUNCT_1:12
.= <% y, x, z %>.a by A6;
end;
end;
hence thesis by A2, FUNCT_1:2;
end;
:: into AFINSQ_1 ?
theorem
for x, y, z being object holds <% x, y, z %> * <% 1, 2, 0 %> = <% y, z, x %>
proof
let x,y,z be object;
rng <% 1, 2, 0 %> = {1,2,0} by Th8
.= {0,1,2} by ENUMSET1:59
.= dom <% x, y, z %> by Th8;
then A1: dom(<% x, y, z %> * <% 1, 2, 0 %>) = dom <% 1, 2, 0 %> by RELAT_1:27
.= {0,1,2} by Th8;
then A2: dom(<% x, y, z %> * <% 1, 2, 0 %>) = dom <% y, z, x %> by Th8;
now
let a be object;
assume A3: a in dom(<% x, y, z %> * <% 1, 2, 0 %>);
then per cases by A1, ENUMSET1:def 1;
suppose A4: a = 0;
thus (<% x, y, z %> * <% 1, 2, 0 %>).a
= <% x, y, z %>.(<% 1, 2, 0 %>.a) by A3, FUNCT_1:12
.= <% y, z, x %>.a by A4;
end;
suppose A5: a = 1;
thus (<% x, y, z %> * <% 1, 2, 0 %>).a
= <% x, y, z %>.(<% 1, 2, 0 %>.a) by A3, FUNCT_1:12
.= <% y, z, x %>.a by A5;
end;
suppose A6: a = 2;
thus (<% x, y, z %> * <% 1, 2, 0 %>).a
= <% x, y, z %>.(<% 1, 2, 0 %>.a) by A3, FUNCT_1:12
.= <% y, z, x %>.a by A6;
end;
end;
hence thesis by A2, FUNCT_1:2;
end;
:: into AFINSQ_1 ?
theorem
for x, y, z being object holds <% x, y, z %> * <% 2, 0, 1 %> = <% z, x, y %>
proof
let x,y,z be object;
rng <% 2, 0, 1 %> = {2,0,1} by Th8
.= {2,1,0} by ENUMSET1:57
.= {0,1,2} by ENUMSET1:60
.= dom <% x, y, z %> by Th8;
then A1: dom(<% x, y, z %> * <% 2, 0, 1 %>) = dom <% 2, 0, 1 %> by RELAT_1:27
.= {0,1,2} by Th8;
then A2: dom(<% x, y, z %> * <% 2, 0, 1 %>) = dom <% z, x, y %> by Th8;
now
let a be object;
assume A3: a in dom(<% x, y, z %> * <% 2, 0, 1 %>);
then per cases by A1, ENUMSET1:def 1;
suppose A4: a = 0;
thus (<% x, y, z %> * <% 2, 0, 1 %>).a
= <% x, y, z %>.(<% 2, 0, 1 %>.a) by A3, FUNCT_1:12
.= <% z, x, y %>.a by A4;
end;
suppose A5: a = 1;
thus (<% x, y, z %> * <% 2, 0, 1 %>).a
= <% x, y, z %>.(<% 2, 0, 1 %>.a) by A3, FUNCT_1:12
.= <% z, x, y %>.a by A5;
end;
suppose A6: a = 2;
thus (<% x, y, z %> * <% 2, 0, 1 %>).a
= <% x, y, z %>.(<% 2, 0, 1 %>.a) by A3, FUNCT_1:12
.= <% z, x, y %>.a by A6;
end;
end;
hence thesis by A2, FUNCT_1:2;
end;
:: into AFINSQ_1 ?
theorem
for x, y, z being object holds <% x, y, z %> * <% 2, 1, 0 %> = <% z, y, x %>
proof
let x,y,z be object;
rng <% 2, 1, 0 %> = {2,1,0} by Th8
.= {0,1,2} by ENUMSET1:60
.= dom <% x, y, z %> by Th8;
then A1: dom(<% x, y, z %> * <% 2, 1, 0 %>) = dom <% 2, 1, 0 %> by RELAT_1:27
.= {0,1,2} by Th8;
then A2: dom(<% x, y, z %> * <% 2, 1, 0 %>) = dom <% z, y, x %> by Th8;
now
let a be object;
assume A3: a in dom(<% x, y, z %> * <% 2, 1, 0 %>);
then per cases by A1, ENUMSET1:def 1;
suppose A4: a = 0;
thus (<% x, y, z %> * <% 2, 1, 0 %>).a
= <% x, y, z %>.(<% 2, 1, 0 %>.a) by A3, FUNCT_1:12
.= <% z, y, x %>.a by A4;
end;
suppose A5: a = 1;
thus (<% x, y, z %> * <% 2, 1, 0 %>).a
= <% x, y, z %>.(<% 2, 1, 0 %>.a) by A3, FUNCT_1:12
.= <% z, y, x %>.a by A5;
end;
suppose A6: a = 2;
thus (<% x, y, z %> * <% 2, 1, 0 %>).a
= <% x, y, z %>.(<% 2, 1, 0 %>.a) by A3, FUNCT_1:12
.= <% z, y, x %>.a by A6;
end;
end;
hence thesis by A2, FUNCT_1:2;
end;
:: into AFINSQ_1 ?
theorem
for x,y being object st x <> y holds <% x, y %> is one-to-one
proof
let x,y be object;
assume A1: x <> y;
now
let x1, x2 be object;
assume A2: x1 in dom <%x,y%> & x2 in dom <%x,y%> & <%x,y%>.x1=<%x,y%>.x2;
then x1 in {0,1} & x2 in {0,1} by Th7;
then per cases by TARSKI:def 2;
suppose x1 = 0 & x2 = 0;
hence x1 = x2;
end;
suppose x1 = 0 & x2 = 1;
hence x1 = x2 by A1, A2; :: by contradiction
end;
suppose x1 = 1 & x2 = 0;
hence x1 = x2 by A1, A2; :: by contradiction
end;
suppose x1 = 1 & x2 = 1;
hence x1 = x2;
end;
end;
hence thesis by FUNCT_1:def 4;
end;
:: into AFINSQ_1 ?
theorem
for x,y,z being object st x <> y & x <> z & y <> z
holds <% x, y, z %> is one-to-one
proof
let x,y,z be object;
assume A1: x <> y & x <> z & y <> z;
now
let x1, x2 be object;
assume A2: x1 in dom <%x,y,z%> & x2 in dom <%x,y,z%> &
<%x,y,z%>.x1 = <%x,y,z%>.x2;
then x1 in {0,1,2} & x2 in {0,1,2} by Th8;
then per cases by ENUMSET1:def 1;
suppose x1 = 0 & (x2 = 0 or x2 = 1 or x2 = 2);
hence x1 = x2 by A1, A2; :: by contradiction
end;
suppose x1 = 1 & (x2 = 0 or x2 = 1 or x2 = 2);
hence x1 = x2 by A1, A2; :: by contradiction
end;
suppose x1 = 2 & (x2 = 0 or x2 = 1 or x2 = 2);
hence x1 = x2 by A1, A2; :: by contradiction
end;
end;
hence thesis by FUNCT_1:def 4;
end;
begin :: Relations with cardinal domain
definition
let R be Relation;
attr R is with_cardinal_domain means
:Def1:
ex c being Cardinal st dom R = c;
end;
registration
cluster empty -> with_cardinal_domain for Relation;
coherence;
cluster finite Sequence-like -> with_cardinal_domain for Relation;
coherence
proof
let R be Relation;
assume A1: R is finite Sequence-like;
dom R is epsilon-transitive epsilon-connected by A1, ORDINAL1:def 7;
hence thesis by A1;
end;
cluster with_cardinal_domain -> Sequence-like for Relation;
coherence by ORDINAL1:def 7;
let c be Cardinal;
cluster -> with_cardinal_domain for ManySortedSet of c;
coherence by PARTFUN1:def 2;
let x be object;
cluster c --> x -> with_cardinal_domain;
coherence;
end;
registration
let X be set;
cluster -> with_cardinal_domain for Denumeration of X;
coherence
proof
let f be Denumeration of X;
per cases;
suppose X = {};
hence thesis;
end;
suppose X <> {};
hence thesis;
end;
end;
end;
registration
let c be Cardinal;
cluster -> with_cardinal_domain for Permutation of c;
coherence;
end;
registration
cluster non empty trivial non-empty with_cardinal_domain
Cardinal-yielding for Function;
existence
proof
take <% 1 %>;
thus thesis;
end;
cluster non empty non trivial non-empty finite with_cardinal_domain
Cardinal-yielding for Function;
existence
proof
take <% 1, 1 %>;
thus thesis;
end;
cluster non empty non-empty infinite with_cardinal_domain
natural-valued for Function;
existence
proof
take omega --> 1;
thus thesis;
end;
cluster non trivial non-empty with_cardinal_domain
Cardinal-yielding non natural-valued for Function;
existence
proof
take <% omega, 1 %>;
thus thesis;
end;
end;
registration
let R be with_cardinal_domain Relation;
cluster dom R -> cardinal;
coherence
proof
consider c being Cardinal such that
A1: dom R = c by Def1;
thus thesis by A1;
end;
end;
registration
let f be with_cardinal_domain Function;
identify dom f with card f;
correctness
proof
consider c being Cardinal such that
A1: dom f = c;
c = card dom f by A1;
hence thesis by CARD_1:62;
end;
end;
registration
let R be with_cardinal_domain Relation, P be total (rng R)-defined Relation;
cluster R * P -> with_cardinal_domain;
coherence
proof
dom P = rng R by PARTFUN1:def 2;
then dom(R*P) = dom R by RELAT_1:27;
hence thesis;
end;
end;
registration
let g be Function, f be Denumeration of dom g;
cluster g * f -> with_cardinal_domain;
coherence
proof
rng f = dom g by FUNCT_2:def 3;
then dom(g*f) = dom f by RELAT_1:27;
hence thesis;
end;
end;
registration
let f be with_cardinal_domain Function, p be Permutation of dom f;
cluster f * p -> with_cardinal_domain;
coherence
proof
rng p = dom f by FUNCT_2:def 3;
then dom(f*p) = dom p by RELAT_1:27;
hence thesis;
end;
end;
theorem Th56:
for A, B being with_cardinal_domain Sequence st dom A in dom B
holds A^B is with_cardinal_domain
proof
let A, B be with_cardinal_domain Sequence;
assume A1: dom A in dom B;
A2: dom(A^B) = (dom A)+^(dom B) by ORDINAL4:def 1;
per cases;
suppose dom B is infinite;
hence thesis by A1, A2, ThAdd;
end;
suppose A3: dom B is finite;
dom A c= dom B by A1, ORDINAL1:def 2;
hence thesis by A2, A3;
end;
end;
registration
let p be XFinSequence, B be with_cardinal_domain Sequence;
cluster p^B -> with_cardinal_domain;
coherence
proof
per cases;
suppose dom B is infinite;
then omega c= dom B & dom p in omega by ORDINAL1:16, CARD_1:61;
hence thesis by Th56;
end;
suppose dom B is finite;
then B is XFinSequence;
hence thesis;
end;
end;
end;
definition
mode Counters is non empty with_cardinal_domain Cardinal-yielding Function;
end;
definition
mode Counters+ is
non empty non-empty with_cardinal_domain Cardinal-yielding Function;
end;