:: Simple-named complex-valued nominative data -- definition and basic
:: operations
:: by Ievgen Ivanov , Mykola Nikitchenko , Andrii Kryvolap and Artur Korni{\l}owicz
::
:: Received August 30, 2017
:: Copyright (c) 2017-2018 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 NOMIN_1, NUMBERS, SUBSET_1, XBOOLE_0, RELAT_1, FUNCT_1, FINSEQ_1,
XXREAL_0, NAT_1, ARYTM_3, CARD_1, PARTFUN1, TARSKI, CARD_3, FUNCOP_1,
ORDINAL4, ZFMISC_1, FINSET_1, ARYTM_1, FUNCT_4, ORDERS_5;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, ENUMSET1,
RELAT_1, RELSET_1, FUNCT_1, PARTFUN1, FUNCT_2, FINSEQ_1, FUNCOP_1,
TURING_1, XXREAL_0, XCMPLX_0, FUNCT_4, CARD_3, FINSEQ_4, FINSET_1;
constructors PROB_1, FINSEQ_4, RELSET_1, FUNCT_6, TURING_1;
registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1, FUNCOP_1,
FINSET_1, XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, CARD_1, RELSET_1, FUNCT_4,
INT_1;
requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM;
definitions TARSKI, XBOOLE_0, PARTFUN1, FUNCT_1, FUNCT_2, FINSEQ_1;
equalities FUNCOP_1, CARD_3, FINSEQ_1, FUNCT_4;
expansions FUNCT_1, RELAT_1, TARSKI;
theorems TARSKI, NAT_1, ZFMISC_1, RELAT_1, RELSET_1, FINSEQ_1, FUNCOP_1,
PARTFUN1, FUNCT_1, GRFUNC_1, FUNCT_4, FINSEQ_3, XBOOLE_0, XBOOLE_1,
XXREAL_0, XREAL_1, XTUPLE_0, INT_1, CARD_5, ENUMSET1, CARD_3, CGAMES_1;
schemes NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, FINSEQ_1;
begin :: Preliminaries
reserve a,a1,a2,v,v1,v2,x for object;
reserve V,A for set;
reserve m,n for Nat;
reserve S,S1,S2 for FinSequence;
theorem Th1:
for f being FinSequence st n in dom f holds (<*x*>^f).(n+1) = f.n
proof
let f be FinSequence;
len <*x*> = 1 by FINSEQ_1:39;
hence thesis by FINSEQ_1:def 7;
end;
theorem Th2:
for f being Function st dom f = NAT holds f|Seg(n) is FinSequence
proof
let f be Function;
assume dom f = NAT;
then dom(f|Seg(n)) = Seg n by RELAT_1:62;
hence thesis by FINSEQ_1:def 2;
end;
theorem Th3:
for f,g being FinSequence holds dom f c= dom g or dom g c= dom f
proof
let f,g be FinSequence;
consider n being Nat such that
A1: dom f = Seg n by FINSEQ_1:def 2;
consider m being Nat such that
A2: dom g = Seg m by FINSEQ_1:def 2;
m <= n or n <= m;
hence thesis by A1,A2,FINSEQ_1:5;
end;
registration
let f,g be FinSequence;
cluster f+*g -> FinSequence-like;
coherence
proof
consider n being Nat such that
A1: dom f = Seg n by FINSEQ_1:def 2;
consider m being Nat such that
A2: dom g = Seg m by FINSEQ_1:def 2;
A3: dom(f+*g) = dom f \/ dom g by FUNCT_4:def 1;
per cases;
suppose
A4: m <= n;
take n;
thus dom(f+*g) = Seg n by A1,A2,A3,A4,XBOOLE_1:12,FINSEQ_1:5;
end;
suppose
A5: m > n;
take m;
thus thesis by A1,A2,A3,A5,XBOOLE_1:12,FINSEQ_1:5;
end;
end;
end;
registration let f1, f2 be Function;
cluster f2 \/ (f1|(dom(f1)\dom(f2))) -> Function-like;
coherence
proof
dom(f1|(dom(f1)\dom(f2))) = dom(f1)\dom(f2) by RELAT_1:62;
hence thesis by GRFUNC_1:13,XBOOLE_1:79;
end;
end;
definition
let f,g be Function, x,y be object;
pred f,x =~ g,y means
(x in dom f iff y in dom g) & (x in dom f implies f.x = g.y);
end;
begin :: Definition of simple-named complex-valued nominative data
definition
let V,A;
mode NominativeSet of V,A is PartFunc of V,A;
end;
registration
let V,A;
cluster finite for NominativeSet of V,A;
existence
proof
take (the PartFunc of V,A) | {};
thus thesis;
end;
end;
definition
let V,A;
mode TypeSSNominativeData of V,A is finite NominativeSet of V,A;
end;
definition
let V,A;
func NDSS(V,A) -> set equals
the set of all d where d is TypeSSNominativeData of V,A;
coherence;
end;
registration
let V,A;
cluster NDSS(V,A) -> non empty;
coherence
proof
the TypeSSNominativeData of V,A in NDSS(V,A);
hence thesis;
end;
end;
theorem Th4:
x in NDSS(V,A) implies x is TypeSSNominativeData of V,A
proof
assume x in NDSS(V,A);
then ex w being TypeSSNominativeData of V,A st x = w;
hence thesis;
end;
theorem
NDSS(V,A) c= PFuncs(V,A)
proof
let x;
assume x in NDSS(V,A);
then ex w being TypeSSNominativeData of V,A st w = x;
hence thesis by PARTFUN1:45;
end;
theorem Th6:
{} in NDSS(V,A)
proof
{} is TypeSSNominativeData of V,A by RELSET_1:12;
hence thesis;
end;
theorem Th7:
for A, B being set st A c= B holds NDSS(V,A) c= NDSS(V,B)
proof
let A,B be set such that
A1: A c= B;
let d be object;
assume d in NDSS(V,A);
then ex w being TypeSSNominativeData of V,A st w = d;
then d is TypeSSNominativeData of V,B by A1,RELSET_1:7;
hence thesis;
end;
theorem Th8:
for f,g being finite Function holds
f tolerates g & f in NDSS(V,A) & g in NDSS(V,A) implies
f \/ g in NDSS(V,A)
proof
let f,g be finite Function;
assume
A1: f tolerates g;
assume f in NDSS(V,A) & g in NDSS(V,A);
then f is NominativeSet of V,A & g is NominativeSet of V,A by Th4;
then f \/ g is V-defined A-valued;
then f \/ g is PartFunc of V,A by A1,RELSET_1:4,PARTFUN1:51;
hence f \/ g in NDSS(V,A);
end;
definition
let V,A;
func FNDSC(V,A) -> Function means :Def3:
dom it = NAT & it.0 = A &
for n being Nat holds it.(n+1) = NDSS(V,A\/it.n);
existence
proof
deffunc G(set,set) = NDSS(V,A\/$2);
ex F being Function st dom F = NAT & F.0 = A &
for n being Nat holds F.(n+1) = G(n,F.n) from NAT_1:sch 11;
hence thesis;
end;
uniqueness
proof
let f,g being Function such that
A1: dom f = NAT and
A2: f.0 = A and
A3: for n being Nat holds f.(n+1) = NDSS(V,A\/f.n) and
A4: dom g = NAT and
A5: g.0 = A and
A6: for n being Nat holds g.(n+1) = NDSS(V,A\/g.n);
thus dom f = dom g by A1,A4;
let x be object;
assume x in dom f;
then reconsider n = x as Element of NAT by A1;
defpred P[Nat] means f.$1 = g.$1;
A7: P[0] by A2,A5;
A8: for n st P[n] holds P[n+1]
proof
let n;
assume P[n];
hence g.(n+1) = NDSS(V,A\/f.n) by A6
.= f.(n+1) by A3;
end;
for n holds P[n] from NAT_1:sch 2(A7,A8);
then f.n = g.n;
hence thesis;
end;
end;
theorem Th9:
FNDSC(V,A).1 = NDSS(V,A)
proof
FNDSC(V,A).(0+1) = NDSS(V,A \/ FNDSC(V,A).0) & FNDSC(V,A).0 = A by Def3;
hence thesis;
end;
theorem Th10:
FNDSC(V,A).2 = NDSS(V,A\/NDSS(V,A))
proof
FNDSC(V,A).(1+1) = NDSS(V,A\/FNDSC(V,A).1) by Def3;
hence thesis by Th9;
end;
theorem Th11:
A c= Union FNDSC(V,A)
proof
set F = FNDSC(V,A);
A1: A = F.0 by Def3;
dom F = NAT by Def3;
then F.0 in rng F by FUNCT_1:def 3;
hence thesis by A1,ZFMISC_1:74;
end;
theorem
1 <= n implies {} in FNDSC(V,A).n
proof
set F = FNDSC(V,A);
assume 1 <= n;
then reconsider m = n-1 as Element of NAT by INT_1:5;
F.(m+1) = NDSS(V,A\/F.m) by Def3;
hence thesis by Th6;
end;
registration
let V,A,n;
cluster FNDSC(V,A) | Seg(n) -> FinSequence-like;
coherence
proof
dom FNDSC(V,A) = NAT by Def3;
hence thesis by Th2;
end;
end;
theorem Th13:
m <> 0 & m <= n implies FNDSC(V,A).m c= FNDSC(V,A).n
proof
assume
A1: m <> 0;
set S = FNDSC(V,A);
defpred P[Nat] means m <= $1 implies S.m c= S.$1;
A2: P[0];
A3: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat;
assume
A4: P[k];
assume that
A5: m <= k+1;
per cases by A5,NAT_1:8;
suppose m = k+1;
hence thesis;
end;
suppose
A6: m <= k;
per cases;
suppose k = 0;
hence S.m c= S.(k+1) by A1,A6;
end;
suppose
A7: k <> 0;
defpred R[Nat] means $1 <> 0 implies S.$1 c= S.($1+1);
A8: R[0];
A9: for z being Nat st R[z] holds R[z+1]
proof
let z be Nat such that
A10: R[z];
per cases;
suppose
A11: z = 0;
A12: S.1 = NDSS(V,A) by Th9;
S.(1+1) = NDSS(V,A\/NDSS(V,A)) by Th10;
hence thesis by A11,A12,Th7,XBOOLE_1:7;
end;
suppose z <> 0;
then
A13: A\/S.z c= A\/S.(z+1) by A10,XBOOLE_1:9;
A14: S.(z+1) = NDSS(V,A\/S.z) by Def3;
S.(z+1+1) = NDSS(V,A\/S.(z+1)) by Def3;
hence thesis by A13,A14,Th7;
end;
end;
for z being Nat holds R[z] from NAT_1:sch 2(A8,A9);
then S.k c= S.(k+1) by A7;
hence thesis by A4,A6;
end;
end;
end;
for k being Nat holds P[k] from NAT_1:sch 2(A2,A3);
hence thesis;
end;
definition
let V,A;
let S be FinSequence;
pred S IsNDRankSeq V,A means
S.1 = NDSS(V,A) &
for n being Nat st n in dom S & n+1 in dom S holds S.(n+1) = NDSS(V,A\/S.n);
end;
theorem
S IsNDRankSeq V,A implies S <> {};
theorem Th15:
S IsNDRankSeq V,A & S1 c= S & S1 <> {} implies S1 IsNDRankSeq V,A
proof
assume that
A1: S IsNDRankSeq V,A and
A2: S1 c= S and
A3: S1 <> {};
A4: dom S1 c= dom S by A2,XTUPLE_0:8;
rng S1 <> {} by A3;
then 1 in dom S1 by FINSEQ_3:32;
hence S1.1 = NDSS(V,A) by A1,A2,GRFUNC_1:2;
let n be Nat such that
A5: n in dom S1 and
A6: n+1 in dom S1;
S1.n = S.n by A2,A5,GRFUNC_1:2;
hence NDSS(V,A\/S1.n) = S.(n+1) by A1,A4,A5,A6
.= S1.(n+1) by A2,A6,GRFUNC_1:2;
end;
theorem Th16:
S IsNDRankSeq V,A & n in dom S implies S|n IsNDRankSeq V,A
proof
assume
A1: S IsNDRankSeq V,A;
assume
A2: n in dom S;
then n <= len S by FINSEQ_3:25;
then len(S|n) = n by FINSEQ_1:17;
then S|n <> {} by A2,FINSEQ_3:24;
hence thesis by A1,RELAT_1:59,Th15;
end;
theorem Th17:
S IsNDRankSeq V,A implies S ^ <* NDSS(V,A\/S.len S) *> IsNDRankSeq V,A
proof
assume
A1: S IsNDRankSeq V,A;
set S1 = S^<*NDSS(V,A\/S.len S)*>;
S <> {} by A1;
then rng S <> {};
then 1 in dom S by FINSEQ_3:32;
hence S1.1 = NDSS(V,A) by A1,FINSEQ_1:def 7;
let n be Nat such that
A2: n in dom S1 and
A3: n+1 in dom S1;
len <*NDSS(V,A\/S.len S)*> = 1 by FINSEQ_1:39;
then
A4: len S1 = 1 + len S by FINSEQ_1:22;
A5: 1 <= n by A2,FINSEQ_3:25;
A6: 1 <= n+1 by NAT_1:14;
A7: n+1 <= len S1 by A3,FINSEQ_3:25;
then n+1-1 <= 1 + len S - 1 by A4,XREAL_1:9;
then
A8: n in dom S by A5,FINSEQ_3:25;
then
A9: S1.n = S.n by FINSEQ_1:def 7;
per cases by A7,XXREAL_0:1;
suppose n+1 < len S1;
then n+1 <= len S by A4,NAT_1:13;
then
A10: n+1 in dom S by A6,FINSEQ_3:25;
then S1.(n+1) = S.(n+1) by FINSEQ_1:def 7;
hence thesis by A1,A8,A9,A10;
end;
suppose n+1 = len S1;
hence thesis by A4,A9,FINSEQ_1:42;
end;
end;
theorem Th18:
1 <= n implies FNDSC(V,A) | Seg(n) IsNDRankSeq V,A
proof
set F = FNDSC(V,A);
set S = F|Seg(n);
dom F = NAT by Def3;
then
A1: dom S = Seg(n) by RELAT_1:62;
assume 1 <= n;
then 1 in Seg(n);
hence S.1 = F.1 by FUNCT_1:49
.= NDSS(V,A) by Th9;
let n be Nat such that
A2: n in dom S;
assume n+1 in dom S;
hence S.(n+1) = F.(n+1) by A1,FUNCT_1:49
.= NDSS(V,A\/F.n) by Def3
.= NDSS(V,A\/S.n) by A1,A2,FUNCT_1:49;
end;
theorem Th19:
S IsNDRankSeq V,A & n in dom S implies S.n = FNDSC(V,A).n
proof
assume
A1: S IsNDRankSeq V,A;
set F = FNDSC(V,A);
defpred P[Nat] means $1 in dom S implies S.$1 = F.$1;
A2: P[0] by FINSEQ_3:24;
A3: for n st P[n] holds P[n+1]
proof
let n such that
A4: P[n] and
A5: n+1 in dom S;
per cases;
suppose n = 0;
hence thesis by A1,Th9;
end;
suppose n <> 0;
then
A6: 1 <= n by NAT_1:14;
A7: n+1 <= len S by A5,FINSEQ_3:25;
n <= n+1 by NAT_1:11;
then
A8: n <= len S by A7,XXREAL_0:2;
then n in dom S by A6,FINSEQ_3:25;
hence S.(n+1) = NDSS(V,A\/S.n) by A1,A5
.= F.(n+1) by A4,A6,A8,Def3,FINSEQ_3:25;
end;
end;
for n holds P[n] from NAT_1:sch 2(A2,A3);
hence thesis;
end;
theorem Th20:
S IsNDRankSeq V,A implies S = FNDSC(V,A) | dom S
proof
assume
A1: S IsNDRankSeq V,A;
set F = FNDSC(V,A);
set G = F|dom S;
dom F = NAT by Def3;
hence dom S = dom G by RELAT_1:62;
let x such that
A2: x in dom S;
thus S.x = F.x by A1,A2,Th19
.= G.x by A2,FUNCT_1:49;
end;
theorem Th21:
S1 IsNDRankSeq V,A & S2 IsNDRankSeq V,A implies S1 tolerates S2
proof
assume that
A1: S1 IsNDRankSeq V,A and
A2: S2 IsNDRankSeq V,A;
let x such that
A3: x in dom S1 /\ dom S2;
defpred P[Nat] means $1 in dom S1 /\ dom S2 implies S1.$1 = S2.$1;
A4: P[0]
proof
assume 0 in dom S1 /\ dom S2;
then 0 in dom S1 by XBOOLE_0:def 4;
hence thesis by FINSEQ_3:24;
end;
A5: for n st P[n] holds P[n+1]
proof
let n such that
A6: P[n] and
A7: n+1 in dom S1 /\ dom S2;
A8: n+1 in dom S1 by A7,XBOOLE_0:def 4;
A9: n+1 in dom S2 by A7,XBOOLE_0:def 4;
per cases;
suppose n = 0;
hence S1.(n+1) = S2.(n+1) by A1,A2;
end;
suppose n <> 0;
then
A10: 1 <= n by NAT_1:14;
A11: n <= n+1 by NAT_1:11;
n+1 <= len S1 by A8,FINSEQ_3:25;
then n <= len S1 by A11,XXREAL_0:2;
then
A12: n in dom S1 by A10,FINSEQ_3:25;
n+1 <= len S2 by A9,FINSEQ_3:25;
then n <= len S2 by A11,XXREAL_0:2;
then n in dom S2 by A10,FINSEQ_3:25;
hence S2.(n+1) = NDSS(V,A\/S1.n) by A2,A6,A9,A12,XBOOLE_0:def 4
.= S1.(n+1) by A1,A8,A12;
end;
end;
for n holds P[n] from NAT_1:sch 2(A4,A5);
hence thesis by A3;
end;
theorem Th22:
S1 IsNDRankSeq V,A & S2 IsNDRankSeq V,A implies S1 c= S2 or S2 c= S1
proof
assume S1 IsNDRankSeq V,A & S2 IsNDRankSeq V,A;
then
A1: S1 = FNDSC(V,A) | dom S1 & S2 = FNDSC(V,A) | dom S2 by Th20;
dom S1 c= dom S2 or dom S2 c= dom S1 by Th3;
hence thesis by A1,RELAT_1:75;
end;
theorem Th23:
S1 IsNDRankSeq V,A & S2 IsNDRankSeq V,A implies S1+*S2 = S1 or S1+*S2 = S2
proof
assume S1 IsNDRankSeq V,A & S2 IsNDRankSeq V,A;
then
A1: S1+*S2 = S2+*S1 by Th21,FUNCT_4:34;
dom S1 c= dom S2 or dom S2 c= dom S1 by Th3;
hence thesis by A1,FUNCT_4:19;
end;
theorem
S1 IsNDRankSeq V,A & S2 IsNDRankSeq V,A implies S1+*S2 IsNDRankSeq V,A
by Th23;
theorem Th25:
S IsNDRankSeq V,A & m <= n & n in dom S implies S.m c= S.n
proof
assume that
A1: S IsNDRankSeq V,A and
A2: m <= n and
A3: n in dom S;
per cases;
suppose
A4: m <> 0;
then
A5: 0+1 <= m by NAT_1:13;
n <= len S by A3,FINSEQ_3:25;
then m <= len S by A2,XXREAL_0:2;
then S.m = FNDSC(V,A).m & S.n = FNDSC(V,A).n
by A1,A3,A5,Th19,FINSEQ_3:25;
hence thesis by A2,A4,Th13;
end;
suppose
A6: m = 0;
not 0 in dom S by FINSEQ_3:24;
hence thesis by A6,FUNCT_1:def 2;
end;
end;
theorem Th26:
for F being FinSequence st F IsNDRankSeq V,A
ex S being FinSequence st len S = 1 + len F & S IsNDRankSeq V,A &
for n being Nat st n in dom S holds S.n = NDSS(V,A\/(<*A*>^F).n)
proof
let F be FinSequence such that
A1: F IsNDRankSeq V,A;
set G = <*A*>^F;
deffunc F(object) = NDSS(V,A\/G.$1);
consider S being FinSequence such that
A2: len S = len G and
A3: for n being Nat st n in dom S holds S.n = F(n) from FINSEQ_1:sch 2;
take S;
len <*A*> = 1 by FINSEQ_1:39;
hence
A4: len S = 1 + len F by A2,FINSEQ_1:22;
A5: G.1 = A by FINSEQ_1:41;
A6: for n being Nat st n in dom F holds G.(n+1) = F.n by Th1;
A7: 1 <= len S by A4,NAT_1:11;
thus S IsNDRankSeq V,A
proof
thus
A8: S.1 = NDSS(V,A\/G.1) by A3,A7,FINSEQ_3:25
.= NDSS(V,A) by A5;
let n be Nat such that
A9: n in dom S and
A10: n+1 in dom S;
A11: 1 <= n by A9,FINSEQ_3:25;
A12: n <= len F by A4,A10,FINSEQ_3:25,XREAL_1:6;
then
A13: n in dom F by A11,FINSEQ_3:25;
per cases by A11,XXREAL_0:1;
suppose n = 1;
then G.(n+1) = S.n by A8,A12,A1,Th1,FINSEQ_3:25;
hence NDSS(V,A\/S.n) = S.(n+1) by A3,A10;
end;
suppose
A14: n > 1;
then reconsider m = n-1 as Element of NAT by INT_1:5;
S.n = NDSS(V,A\/G.(m+1)) by A3,A9
.= NDSS(V,A\/F.m) by A13,A14,Th1,CGAMES_1:20
.= F.(m+1) by A1,A13,A14,CGAMES_1:20;
then G.(n+1) = S.n by A6,A12,A11,FINSEQ_3:25;
hence NDSS(V,A\/S.n) = S.(n+1) by A3,A10;
end;
end;
thus thesis by A3;
end;
Lm1:
for f being Function st
ex S being FinSequence st S IsNDRankSeq V,A & f in Union S
holds f in Union FNDSC(V,A)
proof
set F = FNDSC(V,A);
A1: dom F = NAT by Def3;
A2: F.0 = A\/A by Def3;
let f be Function;
given S being FinSequence such that
A3: S IsNDRankSeq V,A and
A4: f in Union S;
consider z being object such that
A5: z in dom S and
A6: f in S.z by A4,CARD_5:2;
reconsider z as Element of NAT by A5;
defpred P[Nat] means $1+1 in dom S implies S.($1+1) = F.($1+1);
A7: P[0] by A2,A3,Def3;
A8: for n st P[n] holds P[n+1]
proof
let n such that
A9: P[n] and
A10: n+1+1 in dom S;
A11: n+1+1 <= len S by A10,FINSEQ_3:25;
n+1 <= n+1+1 by NAT_1:11;
then
A12: n+1 <= len S by A11,XXREAL_0:2;
then n+1 in dom S by NAT_1:11,FINSEQ_3:25;
hence S.(n+1+1) = NDSS(V,A\/S.(n+1)) by A3,A10
.= F.(n+1+1) by A9,A12,Def3,NAT_1:11,FINSEQ_3:25;
end;
A13: for n holds P[n] from NAT_1:sch 2(A7,A8);
1 <= z by A5,FINSEQ_3:25;
then per cases by XXREAL_0:1;
suppose
A14: z = 1;
A15: F.(0+1) = NDSS(V,A\/F.0) by Def3;
F.1 in rng F by A1,FUNCT_1:def 3;
hence thesis by A2,A3,A6,A14,A15,TARSKI:def 4;
end;
suppose
A16: 1 < z;
then reconsider lZ = z-1 as Element of NAT by INT_1:5;
1-1 < lZ by A16,XREAL_1:14;
then 0+1 <= lZ by NAT_1:13;
then reconsider r = lZ-1 as Element of NAT by INT_1:5;
A17: S.(lZ+1) = NDSS(V,A\/S.lZ) by A3,A5,A16,CGAMES_1:20;
A18: F.(r+1) = S.(r+1) by A5,A13,A16,CGAMES_1:20;
F.(lZ+1) = NDSS(V,A\/F.lZ) by Def3;
then NDSS(V,A\/S.lZ) in rng F by A1,A18,FUNCT_1:def 3;
hence thesis by A6,A17,TARSKI:def 4;
end;
end;
theorem Th27:
<*NDSS(V,A)*> IsNDRankSeq V,A
proof
set S = <*NDSS(V,A)*>;
thus S.1 = NDSS(V,A) by FINSEQ_1:def 8;
let n be Nat such that
A1: n in dom S & n+1 in dom S;
dom S = {1} by FINSEQ_1:2,def 8;
then n = 1 & n+1 = 1 by A1,TARSKI:def 1;
hence thesis;
end;
theorem Th28:
<*NDSS(V,A),NDSS(V,A\/NDSS(V,A))*> IsNDRankSeq V,A
proof
A1: <*NDSS(V,A)*> IsNDRankSeq V,A by Th27;
len <*NDSS(V,A)*> = 1 by FINSEQ_1:40;
hence thesis by A1,Th17;
end;
theorem
<*NDSS(V,A),NDSS(V,A\/NDSS(V,A)),NDSS(V,A\/NDSS(V,A\/NDSS(V,A)))*>
IsNDRankSeq V,A
proof
set S = <*NDSS(V,A),NDSS(V,A\/NDSS(V,A))*>;
len S = 2 & S.2 = NDSS(V,A\/NDSS(V,A)) by FINSEQ_1:44;
hence thesis by Th17,Th28;
end;
definition
let V,A;
mode NonatomicND of V,A -> Function means :Def5:
ex S being FinSequence st S IsNDRankSeq V,A & it in Union S;
existence
proof
Union <*NDSS(V,A)*> = NDSS(V,A) by FINSEQ_3:135;
then (the PartFunc of V,A) | {} in Union <*NDSS(V,A)*>;
hence thesis by Th27;
end;
sethood
proof
take Union FNDSC(V,A);
thus thesis by Lm1;
end;
end;
reserve D,D1,D2 for NonatomicND of V,A;
theorem Th30:
{} is NonatomicND of V,A
proof
take S = <*NDSS(V,A)*>;
Union S = NDSS(V,A) by FINSEQ_3:135;
then (the PartFunc of V,A) | {} in Union S;
hence thesis by Th27;
end;
theorem Th31:
D in Union FNDSC(V,A)
proof
ex S being FinSequence st S IsNDRankSeq V,A & D in Union S by Def5;
hence thesis by Lm1;
end;
theorem Th32:
for d being set st d c= D holds d is NonatomicND of V,A
proof
let d be set;
assume
A1: d c= D;
then reconsider d as Function;
consider S being FinSequence such that
A2: S IsNDRankSeq V,A and
A3: D in Union S by Def5;
consider x such that
A4: x in dom S and
A5: D in S.x by A3,CARD_5:2;
reconsider x as Element of NAT by A4;
now
1 <= x by A4,FINSEQ_3:25;
then per cases by XXREAL_0:1;
suppose
A6: x = 1;
then D is TypeSSNominativeData of V,A by A2,A5,Th4;
then d is TypeSSNominativeData of V,A by A1,RELSET_1:1;
then d in NDSS(V,A);
hence d in Union S by A2,A4,A6,CARD_5:2;
end;
suppose
A7: x > 1;
then reconsider n = x-1 as Element of NAT by INT_1:5;
A8: S.(n+1) = NDSS(V,A\/S.n) by A2,A4,A7,CGAMES_1:20;
then D is TypeSSNominativeData of V,A\/S.n by A5,Th4;
then d is TypeSSNominativeData of V,A\/S.n by A1,RELSET_1:1;
then d in NDSS(V,A\/S.n);
hence d in Union S by A4,A8,CARD_5:2;
end;
end;
hence thesis by A2,Def5;
end;
theorem Th33:
ex n being Nat st D is TypeSSNominativeData of V,A\/FNDSC(V,A).n
proof
set F = FNDSC(V,A);
consider S being FinSequence such that
A1: S IsNDRankSeq V,A and
A2: D in Union S by Def5;
consider x being object such that
A3: x in dom S and
A4: D in S.x by A2,CARD_5:2;
reconsider x as Element of NAT by A3;
reconsider n = x-1 as Element of NAT by A3,FINSEQ_3:25,INT_1:5;
take n;
A5: F.(n+1) = S.(n+1) by A1,A3,Th19;
F.(n+1) = NDSS(V,A\/F.n) by Def3;
hence thesis by A4,A5,Th4;
end;
registration
let V,A;
cluster -> finite for NonatomicND of V,A;
coherence
proof
let D;
ex n being Nat st D is TypeSSNominativeData of V,A\/FNDSC(V,A).n by Th33;
hence thesis;
end;
end;
theorem Th34:
D1 tolerates D2 & S IsNDRankSeq V,A & D1 in S.m & D2 in S.m
implies D1 \/ D2 in S.m
proof
set D = D1\/D2;
assume that
A1: D1 tolerates D2 and
A2: S IsNDRankSeq V,A and
A3: D1 in S.m & D2 in S.m;
A4: m in dom S by A3,FUNCT_1:def 2;
not 0 in dom S by FINSEQ_3:24;
then m <> 0 by A3,FUNCT_1:def 2;
then
A5: 0+1 <= m by NAT_1:13;
then reconsider z = m-1 as Element of NAT by INT_1:5;
per cases by A5,XXREAL_0:1;
suppose 1 < m;
then
A6: S.(z+1) = NDSS(V,A\/S.z) by A2,A4,CGAMES_1:20;
then D1 is PartFunc of V,A\/S.z & D2 is PartFunc of V,A\/S.z by A3,Th4;
then D is V-defined (A\/S.z)-valued;
then D is PartFunc of V,A\/S.z by A1,RELSET_1:4,PARTFUN1:51;
hence thesis by A6;
end;
suppose m = 1;
hence thesis by A1,A2,A3,Th8;
end;
end;
theorem Th35:
D1 tolerates D2 & S2 IsNDRankSeq V,A & S1 c= S2 &
D1 in Union S1 & D2 in Union S2
implies D1 \/ D2 in Union S2
proof
set D = D1 \/ D2;
set S = S1+*S2;
A1: dom S = dom S1 \/ dom S2 by FUNCT_4:def 1;
assume that
A2: D1 tolerates D2 and
A3: S2 IsNDRankSeq V,A and
A4: S1 c= S2 and
A5: D1 in Union S1 and
A6: D2 in Union S2;
Union S1 c= Union S2 by A4,CARD_3:24;
then consider i being object such that
A7: i in dom S2 and
A8: D1 in S2.i by A5,CARD_5:2;
consider j being object such that
A9: j in dom S2 and
A10: D2 in S2.j by A6,CARD_5:2;
reconsider i as Element of NAT by A7;
reconsider j as Element of NAT by A9;
set k = max(i,j);
dom S1 c= dom S2 by A4,XTUPLE_0:8;
then
A11: S = S2 by FUNCT_4:19;
k in dom S1 or k in dom S2 by A7,A9,XXREAL_0:16;
then
A12: k in dom S by A1,XBOOLE_0:def 3;
then
A13: S2.i c= S2.k by A3,A11,Th25,XXREAL_0:25;
S2.j c= S2.k by A3,A11,A12,Th25,XXREAL_0:25;
hence thesis by A11,A12,A2,A3,A10,A8,A13,Th34,CARD_5:2;
end;
theorem Th36:
D1 tolerates D2 implies D1 \/ D2 is NonatomicND of V,A
proof
set D = D1 \/ D2;
assume
A1: D1 tolerates D2;
then
A2: D is Function by PARTFUN1:51;
consider S1 being FinSequence such that
A3: S1 IsNDRankSeq V,A and
A4: D1 in Union S1 by Def5;
consider S2 being FinSequence such that
A5: S2 IsNDRankSeq V,A and
A6: D2 in Union S2 by Def5;
S1 c= S2 or S2 c= S1 by A3,A5,Th22;
hence thesis by A1,A2,A3,A4,A5,A6,Th35,Def5;
end;
theorem
D1 tolerates D2 implies D1 +* D2 is NonatomicND of V,A
proof
assume
A1: D1 tolerates D2;
then D1 \/ D2 = D1 +* D2 by FUNCT_4:30;
hence thesis by A1,Th36;
end;
definition
let V,A;
mode TypeSCNominativeData of V,A -> set means :Def6:
it in A or it is NonatomicND of V,A;
existence
proof
take the NonatomicND of V,A;
thus thesis;
end;
sethood
proof
set X = Union FNDSC(V,A);
take A \/ X;
let x be set;
assume x in A or x is NonatomicND of V,A;
then x in A or
x is Function & ex S being FinSequence st S IsNDRankSeq V,A & x in Union S
by Def5;
then x in A or x in X by Lm1;
hence thesis by XBOOLE_0:def 3;
end;
end;
definition
let V,A;
func ND(V,A) -> set equals
the set of all D where D is TypeSCNominativeData of V,A;
coherence;
end;
registration
let V,A;
cluster ND(V,A) -> non empty;
coherence
proof
the TypeSCNominativeData of V,A in ND(V,A);
hence thesis;
end;
end;
theorem
{} in ND(V,A)
proof
{} is NonatomicND of V,A by Th30;
then {} is TypeSCNominativeData of V,A by Def6;
hence thesis;
end;
theorem Th39:
x in ND(V,A) implies x is TypeSCNominativeData of V,A
proof
assume x in ND(V,A);
then ex D being TypeSCNominativeData of V,A st x = D;
hence thesis;
end;
theorem
ND(V,A) = Union FNDSC(V,A)
proof
set F = FNDSC(V,A);
thus ND(V,A) c= Union F
proof
let x;
assume x in ND(V,A);
then x is TypeSCNominativeData of V,A by Th39;
then
A1: x in A or x is NonatomicND of V,A by Def6;
A c= Union F by Th11;
hence thesis by A1,Th31;
end;
let x;
assume x in Union F;
then consider z being object such that
A2: z in dom F and
A3: x in F.z by CARD_5:2;
reconsider z as Element of NAT by A2,Def3;
per cases;
suppose z = 0;
then x in A by A3,Def3;
then x is TypeSCNominativeData of V,A by Def6;
hence thesis;
end;
suppose
A4: z <> 0;
then
A5: 1 <= z by NAT_1:14;
reconsider n = z-1 as Element of NAT by A4,INT_1:5,NAT_1:14;
A6: dom F = NAT by Def3;
set S = F|Seg(z);
F.(n+1) = NDSS(V,A\/F.n) by Def3;
then
A7: x is TypeSSNominativeData of V,A\/F.n by A3,Th4;
A8: dom S = Seg(z) by A6,RELAT_1:62;
A9: z in Seg(z) by A5;
then S.z = F.z by FUNCT_1:49;
then F.z in rng S by A8,A9,FUNCT_1:def 3;
then x in Union S by A3,TARSKI:def 4;
then x is NonatomicND of V,A by A7,A5,Th18,Def5;
then x is TypeSCNominativeData of V,A by Def6;
hence thesis;
end;
end;
theorem Th41:
D in ND(V,A)
proof
D is TypeSCNominativeData of V,A by Def6;
hence thesis;
end;
theorem Th42:
not D in A implies D in ND(V,A) \ A
proof
D in ND(V,A) by Th41;
hence thesis by XBOOLE_0:def 5;
end;
theorem Th43:
x in ND(V,A) \ A implies x is NonatomicND of V,A
proof
assume
A1: x in ND(V,A) \ A;
then x in ND(V,A);
then
A2: ex w being TypeSCNominativeData of V,A st x = w;
not x in A by A1,XBOOLE_0:def 5;
hence thesis by A2,Def6;
end;
theorem
for D being TypeSCNominativeData of V,A holds D in Union FNDSC(V,A)
proof
let D be TypeSCNominativeData of V,A;
A1: D in A or D is NonatomicND of V,A by Def6;
A c= Union FNDSC(V,A) by Th11;
hence thesis by A1,Th31;
end;
begin :: Examples of simple-named complex-valued nominative data
::$N [v->a]
definition
let v,a;
func ND_ex_1(v,a) -> set equals
v.-->a;
coherence;
end;
registration
let v,a;
cluster ND_ex_1(v,a) -> Function-like Relation-like;
coherence;
end;
theorem Th45:
v in V & a in A implies ND_ex_1(v,a) in NDSS(V,A)
proof
assume that
A1: v in V and
A2: a in A;
reconsider V1 = V, A1 = A as non empty set by A1,A2;
reconsider v as Element of V1 by A1;
reconsider a as Element of A1 by A2;
v.-->a in NDSS(V,A);
hence thesis;
end;
theorem Th46:
v in V & a in A implies ND_ex_1(v,a) is NonatomicND of V,A
proof
assume
A1: v in V & a in A;
take S = <*NDSS(V,A)*>;
thus S IsNDRankSeq V,A by Th27;
ND_ex_1(v,a) in NDSS(V,A) by A1,Th45;
hence thesis by FINSEQ_3:135;
end;
definition
let V,A be non empty set;
let v be Element of V;
let a be Element of A;
redefine func ND_ex_1(v,a) -> NonatomicND of V,A;
coherence by Th46;
end;
theorem Th47:
v in V & a in A implies ND_ex_1(v,a) is TypeSCNominativeData of V,A
proof
assume v in V & a in A;
then ND_ex_1(v,a) is NonatomicND of V,A by Th46;
hence thesis by Def6;
end;
::$N [v->[v1->a]]
definition
let v,v1,a1;
func ND_ex_2(v,v1,a1) -> set equals
v.-->(v1.-->a1);
coherence;
end;
registration
let v,v1,a1;
cluster ND_ex_2(v,v1,a1) -> Function-like Relation-like;
coherence;
end;
theorem Th48:
{v,v1} c= V & a1 in A implies ND_ex_2(v,v1,a1) in NDSS(V,A\/NDSS(V,A))
proof
assume that
A1: {v,v1} c= V and
A2: a1 in A;
reconsider V1 = V, A1 = A as non empty set by A1,A2;
reconsider v,v1 as Element of V1 by A1,ZFMISC_1:32;
reconsider a1 as Element of A1 by A2;
set d = ND_ex_2(v,v1,a1);
A3: dom(v.-->(v1.-->a1)) c= V;
A4: rng d = {v1.-->a1} by FUNCOP_1:88;
v1.-->a1 in NDSS(V,A);
then v1.-->a1 in A\/NDSS(V,A) by XBOOLE_0:def 3;
then d is NominativeSet of V,A\/NDSS(V,A) by A3,A4,ZFMISC_1:31,RELSET_1:4;
hence thesis;
end;
theorem Th49:
{v,v1} c= V & a1 in A implies ND_ex_2(v,v1,a1) is NonatomicND of V,A
proof
assume
A1: {v,v1} c= V & a1 in A;
take S = <*NDSS(V,A),NDSS(V,A\/NDSS(V,A))*>;
thus S IsNDRankSeq V,A by Th28;
A2: Union S = NDSS(V,A) \/ NDSS(V,A\/NDSS(V,A)) by FINSEQ_3:136;
ND_ex_2(v,v1,a1) in NDSS(V,A\/NDSS(V,A)) by A1,Th48;
hence thesis by A2,XBOOLE_0:def 3;
end;
definition
let V,A be non empty set;
let v,v1 be Element of V;
let a be Element of A;
redefine func ND_ex_2(v,v1,a) -> NonatomicND of V,A;
coherence
proof
{v,v1} c= V by ZFMISC_1:32;
hence thesis by Th49;
end;
end;
theorem
{v,v1} c= V & a1 in A implies
ND_ex_2(v,v1,a1) is TypeSCNominativeData of V,A
proof
assume {v,v1} c= V & a1 in A;
then ND_ex_2(v,v1,a1) is NonatomicND of V,A by Th49;
hence thesis by Def6;
end;
::$N [v->a,v1->a1]
definition
let v,v1,a,a1;
func ND_ex_3(v,v1,a,a1) -> set equals
(v,v1)-->(a,a1);
coherence;
end;
registration
let v,v1,a,a1;
cluster ND_ex_3(v,v1,a,a1) -> Function-like Relation-like;
coherence;
end;
theorem Th51:
{v,v1} c= V & {a,a1} c= A implies ND_ex_3(v,v1,a,a1) in NDSS(V,A)
proof
assume that
A1: {v,v1} c= V and
A2: {a,a1} c= A;
per cases;
suppose v = v1;
then
A3: ND_ex_3(v,v1,a,a1) = ND_ex_1(v1,a1) by FUNCT_4:81;
v1 in V & a1 in A by A1,A2,ZFMISC_1:32;
hence thesis by A3,Th45;
end;
suppose v <> v1;
then {v} misses {v1} by ZFMISC_1:11;
then
A4: ND_ex_1(v,a) tolerates ND_ex_1(v1,a1) by FUNCOP_1:87;
v in V & v1 in V & a in A & a1 in A by A1,A2,ZFMISC_1:32;
then
A5: ND_ex_1(v,a) in NDSS(V,A) & ND_ex_1(v1,a1) in NDSS(V,A) by Th45;
ND_ex_3(v,v1,a,a1) = ND_ex_1(v,a) \/ ND_ex_1(v1,a1) by A4,FUNCT_4:30;
hence thesis by A4,A5,Th8;
end;
end;
theorem Th52:
{v,v1} c= V & {a,a1} c= A implies
ND_ex_3(v,v1,a,a1) is NonatomicND of V,A
proof
assume
A1: {v,v1} c= V & {a,a1} c= A;
take <*NDSS(V,A)*>;
thus <*NDSS(V,A)*> IsNDRankSeq V,A by Th27;
ND_ex_3(v,v1,a,a1) in NDSS(V,A) by A1,Th51;
hence thesis by FINSEQ_3:135;
end;
definition
let V,A be non empty set;
let v,v1 be Element of V;
let a,a1 be Element of A;
redefine func ND_ex_3(v,v1,a,a1) -> NonatomicND of V,A;
coherence
proof
{v,v1} c= V & {a,a1} c= A by ZFMISC_1:32;
hence thesis by Th52;
end;
end;
theorem
{v,v1} c= V & {a,a1} c= A implies
ND_ex_3(v,v1,a,a1) is TypeSCNominativeData of V,A
proof
assume {v,v1} c= V & {a,a1} c= A;
then ND_ex_3(v,v1,a,a1) is NonatomicND of V,A by Th52;
hence thesis by Def6;
end;
::$N [v->a,v1->[v2->a1]]
definition
let v,v1,v2,a,a1;
func ND_ex_4(v,v1,v2,a,a1) -> set equals
(v,v1)-->(a,v2.-->a1);
coherence;
end;
registration
let v,v1,v2,a,a1;
cluster ND_ex_4(v,v1,v2,a,a1) -> Function-like Relation-like;
coherence;
end;
theorem Th54:
{v,v1,v2} c= V & {a,a1} c= A implies
ND_ex_4(v,v1,v2,a,a1) in NDSS(V,A\/NDSS(V,A))
proof
assume that
A1: {v,v1,v2} c= V and
A2: {a,a1} c= A;
v in {v,v1,v2} by ENUMSET1:def 1;
then
A3: v in V by A1;
v1 in {v,v1,v2} by ENUMSET1:def 1;
then
A4: v1 in V by A1;
A5: v2 in {v,v1,v2} by ENUMSET1:def 1;
A6: a in A by A2,ZFMISC_1:32;
A7: a1 in A by A2,ZFMISC_1:32;
set f = v2.-->a1;
set g = ND_ex_4(v,v1,v2,a,a1);
dom g = {v,v1} by FUNCT_4:62;
then
A8: dom g c= V by A3,A4,TARSKI:def 2;
A9: rng g c= {a,f} by FUNCT_4:62;
rng g c= A\/NDSS(V,A)
proof
let x;
assume x in rng g;
then per cases by A9,TARSKI:def 2;
suppose x = a;
hence thesis by A6,XBOOLE_0:def 3;
end;
suppose
A10: x = f;
dom f = {v2} by FUNCOP_1:13;
then
A11: dom f c= V by A1,A5,ZFMISC_1:31;
rng f = {a1} by FUNCOP_1:8;
then f is PartFunc of V,A by A7,A11,RELSET_1:4,ZFMISC_1:31;
then f in NDSS(V,A);
hence thesis by A10,XBOOLE_0:def 3;
end;
end;
then g is PartFunc of V,A\/NDSS(V,A) by A8,RELSET_1:4;
hence g in NDSS(V,A\/NDSS(V,A));
end;
theorem Th55:
{v,v1,v2} c= V & {a,a1} c= A implies
ND_ex_4(v,v1,v2,a,a1) is NonatomicND of V,A
proof
assume
A1: {v,v1,v2} c= V & {a,a1} c= A;
set R = ND_ex_4(v,v1,v2,a,a1);
set S1 = NDSS(V,A);
set S2 = NDSS(V,A\/NDSS(V,A));
R in S2 by A1,Th54;
then
A2: R in S1\/S2 by XBOOLE_0:def 3;
Union <*S1,S2*> = S1\/S2 by FINSEQ_3:136;
hence thesis by A2,Def5,Th28;
end;
definition
let V,A be non empty set;
let v,v1,v2 be Element of V;
let a,a1 be Element of A;
redefine func ND_ex_4(v,v1,v2,a,a1) -> NonatomicND of V,A;
coherence
proof
A1: {v,v1,v2} c= V
proof
let x;
assume x in {v,v1,v2};
then x = v or x = v1 or x = v2 by ENUMSET1:def 1;
hence thesis;
end;
{a,a1} c= A by ZFMISC_1:32;
hence thesis by A1,Th55;
end;
end;
theorem
{v,v1,v2} c= V & {a,a1} c= A implies
ND_ex_4(v,v1,v2,a,a1) is TypeSCNominativeData of V,A
proof
assume {v,v1,v2} c= V & {a,a1} c= A;
then ND_ex_4(v,v1,v2,a,a1) is NonatomicND of V,A by Th55;
hence thesis by Def6;
end;
theorem
<*x*> is NonatomicND of {1},{x}
proof
take <*NDSS({1},{x})*>;
thus <*NDSS({1},{x})*> IsNDRankSeq {1},{x} by Th27;
<*x*> in NDSS({1},{x})
proof
1 in {1} & x in {x} by TARSKI:def 1;
then <*x*> is Relation of {1},{x} by RELSET_1:3;
hence thesis;
end;
hence thesis by FINSEQ_3:135;
end;
begin :: Operations on simple-named complex-valued nominative data
definition
let V,A,v,D such that
A1: v in dom D;
func denaming(v,D) -> TypeSCNominativeData of V,A equals :Def12:
D.v;
coherence
proof
assume
A2: not D.v in A;
consider S being FinSequence such that
A3: S IsNDRankSeq V,A and
A4: D in Union S by Def5;
consider Z being set such that
A5: D in Z and
A6: Z in rng S by A4,TARSKI:def 4;
consider z being object such that
A7: z in dom S and
A8: S.z = Z by A6,FUNCT_1:def 3;
reconsider z as Element of NAT by A7;
1 <= z by A7,FINSEQ_3:25;
then per cases by XXREAL_0:1;
suppose z = 1;
then ex w being TypeSSNominativeData of V,A st w = D by A3,A5,A8;
hence thesis by A1,A2,PARTFUN1:4;
end;
suppose
A9: z > 1;
reconsider lS = z - 1 as Element of NAT by A7,FINSEQ_3:25,INT_1:5;
set S1 = S|lS;
A10: dom S1 c= dom S by RELAT_1:60;
A11: z <= len S by A7,FINSEQ_3:25;
1-1 < lS by A9,XREAL_1:14;
then
A12: 0+1 <= lS by NAT_1:13;
lS <= z-0 by XREAL_1:10;
then len S1 = lS by A11,XXREAL_0:2,FINSEQ_1:59;
then
A13: lS in dom S1 by A12,FINSEQ_3:25;
then S.(lS+1) = NDSS(V,A\/S.lS) by A7,A3,A10;
then consider d being TypeSSNominativeData of V,A\/S.lS such that
A14: D = d by A5,A8;
A15: D.v in A \/ S.lS by A1,A14,PARTFUN1:4;
A16: not D.v in A implies D.v is Function
proof
assume
A17: not D.v in A;
per cases by A12,XXREAL_0:1;
suppose lS = 1;
then D.v in S.1 by A15,A17,XBOOLE_0:def 3;
then ex w being TypeSSNominativeData of V,A st w = D.v by A3;
hence D.v is Function;
end;
suppose
A18: lS > 1;
then reconsider c = lS - 1 as Element of NAT by INT_1:5;
c > 1-1 by A18,XREAL_1:14;
then
A19: 1 <= c by NAT_1:14;
A20: lS <= len S by A10,A13,FINSEQ_3:25;
c <= lS-0 by XREAL_1:10;
then c <= len S by A20,XXREAL_0:2;
then c in dom S by A19,FINSEQ_3:25;
then S.(c+1) = NDSS(V,A\/S.c) & D.v in S.lS
by A3,A10,A13,A15,A17,XBOOLE_0:def 3;
then ex w being TypeSSNominativeData of V,A\/S.c st w = D.v;
hence D.v is Function;
end;
end;
S1.lS = S.lS by FINSEQ_3:112;
then S.lS in rng S1 & D.v in S.lS
by A2,A13,A15,FUNCT_1:def 3,XBOOLE_0:def 3;
then D.v in Union S1 by TARSKI:def 4;
hence thesis by A2,A3,A10,A13,A16,Th16,Def5;
end;
end;
end;
definition
let V,A;
let v,D be object;
assume
A1: D is TypeSCNominativeData of V,A;
assume
A2: v in V;
func naming(V,A,v,D) -> NonatomicND of V,A equals :Def13:
v .--> D;
coherence
proof
A3: ND_ex_1(v,D) = v .--> D;
per cases by A1,Def6;
suppose D in A;
hence thesis by A2,A3,Th46;
end;
suppose D is NonatomicND of V,A;
then consider F being FinSequence such that
A4: F IsNDRankSeq V,A and
A5: D in Union F by Def5;
consider Z being set such that
A6: D in Z and
A7: Z in rng F by A5,TARSKI:def 4;
reconsider Z as non empty set by A6;
reconsider D1 = D as Element of Z by A6;
reconsider V as non empty set by A2;
reconsider v as Element of V by A2;
set d = v .--> D;
v.-->D1 is NominativeSet of V,Z;
then
A8: d in NDSS(V,Z);
A9: NDSS(V,Z) c= NDSS(V,A\/Z) by Th7,XBOOLE_1:7;
consider x being object such that
A10: x in dom F and
A11: F.x = Z by A7,FUNCT_1:def 3;
reconsider x as Nat by A10;
A12: x <= len F by A10,FINSEQ_3:25;
consider S being FinSequence such that
A13: len S = 1 + len F and
A14: S IsNDRankSeq V,A and
A15: for n being Nat st n in dom S holds S.n = NDSS(V,A\/(<*A*>^F).n)
by A4,Th26;
1 <= x+1 by NAT_1:11;
then
A16: x+1 in dom S by A13,A12,XREAL_1:6,FINSEQ_3:25;
then S.(x+1) = NDSS(V,A\/(<*A*>^F).(x+1)) by A15
.= NDSS(V,A\/Z) by A10,A11,Th1;
then NDSS(V,A\/Z) in rng S by A16,FUNCT_1:def 3;
then d in Union S by A8,A9,TARSKI:def 4;
hence thesis by A14,Def5;
end;
end;
end;
definition
let V,A;
let a be object;
let f be V-valued FinSequence;
assume
A1: len f > 0;
func namingSeq(V,A,f,a) -> FinSequence means :Def14:
len it = len f & it.1 = naming(V,A,f.len f,a) &
for n being Nat st 1 <= n < len it holds
it.(n+1) = naming(V,A,f.(len f-n),it.n);
existence
proof
defpred P[Nat,object,object] means
$3 = naming(V,A,f.(len f-$1),$2);
A2: for n being Nat st 1 <= n & n < len f
for x being set ex y being set st P[n,x,y];
ex g being FinSequence st len g = len f &
(g.1 = naming(V,A,f.len f,a) or len f = 0) &
for n being Nat st 1 <= n < len f holds P[n,g.n,g.(n+1)]
from RECDEF_1:sch 3(A2);
hence thesis by A1;
end;
uniqueness
proof
let g1,g2 be FinSequence such that
A3: len g1 = len f and
A4: g1.1 = naming(V,A,f.len f,a) and
A5: for n being Nat st 1 <= n < len g1 holds
g1.(n+1) = naming(V,A,f.(len f-n),g1.n) and
A6: len g2 = len f and
A7: g2.1 = naming(V,A,f.len f,a) and
A8: for n being Nat st 1 <= n < len g2 holds
g2.(n+1) = naming(V,A,f.(len f-n),g2.n);
thus len g1 = len g2 by A3,A6;
defpred P[Nat] means 1 <= $1 <= len g1 implies g1.$1 = g2.$1;
A9: P[0];
A10: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat such that
A11: P[k] and
1 <= k+1 and
A12: k+1 <= len g1;
per cases;
suppose k <> 0;
then
A13: 1 <= k by NAT_1:14;
hence g1.(k+1) = naming(V,A,f.(len f-k),g2.k)
by A5,A11,A12,NAT_1:13
.= g2.(k+1) by A3,A6,A8,A13,A12,NAT_1:13;
end;
suppose k = 0;
hence thesis by A4,A7;
end;
end;
for k being Nat holds P[k] from NAT_1:sch 2(A9,A10);
hence thesis;
end;
end;
theorem Th58:
for f being V-valued FinSequence holds
1 <= n <= len f implies
namingSeq(V,A,f,a).n is NonatomicND of V,A
proof
let f be V-valued FinSequence;
assume that
A1: 1 <= n and
A2: n <= len f;
set g = namingSeq(V,A,f,a);
per cases by A1,XXREAL_0:1;
suppose n = 1;
then g.n = naming(V,A,f.len f,a) by A2,Def14;
hence thesis;
end;
suppose
A3: n > 1;
then reconsider k = n-1 as Element of NAT by INT_1:5;
1-1 < k by A3,XREAL_1:9;
then
A4: 0+1 <= k by INT_1:7;
A5: len f = len g by A1,A2,Def14;
k+0 < k+1 by XREAL_1:8;
then k < len g by A2,A5,XXREAL_0:2;
then g.(k+1) = naming(V,A,f.(len f-k),g.k) by A2,A4,Def14;
hence thesis;
end;
end;
definition
let V,A;
let f be V-valued FinSequence;
let a be object;
func naming(V,A,f,a) -> set equals
namingSeq(V,A,f,a).len(namingSeq(V,A,f,a));
coherence;
end;
theorem
for f being V-valued FinSequence st len f > 0
holds naming(V,A,f,a) is NonatomicND of V,A
proof
let f be V-valued FinSequence such that
A1: len f > 0;
A2: len(namingSeq(V,A,f,a)) = len f by A1,Def14;
then 0+1 <= len(namingSeq(V,A,f,a)) by A1,NAT_1:13;
hence thesis by A2,Th58;
end;
theorem
for V being non empty set
for v being Element of V holds
naming(V,A,<*v*>,a) = naming(V,A,v,a)
proof
let V be non empty set;
let v be Element of V;
set f = <*v*>;
A1: len f = 1 & f.1 = v by FINSEQ_1:40;
len(namingSeq(V,A,f,a)) = len f by Def14;
hence thesis by A1,Def14;
end;
theorem
for V being non empty set
for v1,v2 being Element of V st a is TypeSCNominativeData of V,A holds
naming(V,A,<*v1,v2*>,a) = v1.-->(v2.-->a)
proof
let V be non empty set;
let v1,v2 be Element of V such that
A1: a is TypeSCNominativeData of V,A;
set f = <*v1,v2*>;
set g = namingSeq(V,A,f,a);
A2: len f = 2 by FINSEQ_1:44;
A3: f.1 = v1 by FINSEQ_1:44;
A4: f.2 = v2 by FINSEQ_1:44;
A5: len g = len f by Def14;
g.1 is NonatomicND of V,A by A2,Th58;
then
A6: g.1 is TypeSCNominativeData of V,A by Def6;
A7: g.1 = naming(V,A,f.len f,a) by Def14
.= v2.-->a by A2,A4,A1,Def13;
thus naming(V,A,<*v1,v2*>,a) = naming(V,A,f.(len f-1),g.1) by A2,A5,Def14
.= v1.-->(v2.-->a) by A2,A3,A6,A7,Def13;
end;
theorem
for D being TypeSCNominativeData of V,A holds
v in V implies denaming(v,naming(V,A,v,D)) = D
proof
let D be TypeSCNominativeData of V,A;
assume
A1: v in V;
A2: naming(V,A,v,D) = v.-->D by A1,Def13;
dom(v.-->D) = {v} by FUNCOP_1:13;
then v in dom naming(V,A,v,D) by A2,TARSKI:def 1;
hence denaming(v,naming(V,A,v,D)) = (naming(V,A,v,D)).v by Def12
.= (v.-->D).v by A1,Def13
.= D by FUNCOP_1:72;
end;
theorem
v in dom D implies naming(V,A,v,denaming(v,D)) = v .--> D.v
proof
assume
A1: v in dom D;
ex n being Nat st D is TypeSSNominativeData of V,A\/FNDSC(V,A).n by Th33;
then dom D c= V by RELAT_1:def 18;
hence naming(V,A,v,denaming(v,D)) = v.-->denaming(v,D) by A1,Def13
.= v.-->D.v by A1,Def12;
end;
definition
let V,A;
let d1,d2 be object such that
A1: d1 is TypeSCNominativeData of V,A and
A2: d2 is TypeSCNominativeData of V,A;
func global_overlapping(V,A,d1,d2) -> TypeSCNominativeData of V,A means
:Def16:
ex f1,f2 being Function st f1 = d1 & f2 = d2 &
it = f2 \/ (f1|(dom(f1)\dom(f2))) if not d1 in A & not d2 in A
otherwise it = d2;
existence
proof
per cases;
suppose that
A3: not d1 in A and
A4: not d2 in A;
reconsider f1 = d1 as Function by A1,A3,Def6;
reconsider f2 = d2 as Function by A2,A4,Def6;
set X = dom(f1)\dom(f2);
set D = f2 \/ (f1|X);
D is NonatomicND of V,A
proof
d1 is NonatomicND of V,A by A1,A3,Def6;
then
A5: f1|X is NonatomicND of V,A by Th32,RELAT_1:59;
A6: d2 is NonatomicND of V,A by A2,A4,Def6;
A7: dom(f1|X) = dom(f1)\dom(f2) by RELAT_1:62;
dom(f1)\dom(f2) misses dom f2 by XBOOLE_1:79;
hence thesis by A5,A6,A7,Th36,PARTFUN1:56;
end;
then D is TypeSCNominativeData of V,A by Def6;
hence thesis by A2;
end;
suppose d1 in A or d2 in A;
hence thesis by A2;
end;
end;
uniqueness;
consistency;
end;
definition
let V,A;
let d1,d2,v be object;
func local_overlapping(V,A,d1,d2,v) -> TypeSCNominativeData of V,A equals
global_overlapping(V,A,d1,naming(V,A,v,d2));
coherence;
end;
theorem Th64:
not D1 in A & not D2 in A implies
global_overlapping(V,A,D1,D2) = D2 \/ (D1|(dom(D1)\dom(D2)))
proof
A1: D1 is TypeSCNominativeData of V,A by Def6;
A2: D2 is TypeSCNominativeData of V,A by Def6;
assume not D1 in A & not D2 in A;
then ex f1,f2 being Function st f1 = D1 & f2 = D2 &
global_overlapping(V,A,D1,D2) = f2 \/ (f1|(dom(f1)\dom(f2)))
by A1,A2,Def16;
hence thesis;
end;
theorem Th65:
not D1 in A & not D2 in A & dom D1 c= dom D2 implies
global_overlapping(V,A,D1,D2) = D2
proof
assume
A1: not D1 in A & not D2 in A;
assume
A2: dom D1 c= dom D2;
thus global_overlapping(V,A,D1,D2) = D2 \/ (D1|(dom(D1)\dom(D2)))
by A1,Th64
.= D2 \/ (D1|{}) by A2,XBOOLE_1:37
.= D2;
end;
theorem
not D in A implies global_overlapping(V,A,D,D) = D
proof
dom D c= dom D;
hence thesis by Th65;
end;
theorem
v in V & not v.-->a1 in A & not v.-->a2 in A &
a1 is TypeSCNominativeData of V,A & a2 is TypeSCNominativeData of V,A
implies
global_overlapping(V,A,v.-->a1,v.-->a2) = v.-->a2
proof
assume that
A1: v in V and
A2: not v.-->a1 in A & not v.-->a2 in A and
A3: a1 is TypeSCNominativeData of V,A &
a2 is TypeSCNominativeData of V,A;
A4: naming(V,A,v,a1) = v.-->a1 & naming(V,A,v,a2) = v.-->a2 by A1,A3,Def13;
dom(v.-->a1) = {v} & dom(v.-->a2) = {v} by FUNCOP_1:13;
hence thesis by A4,A2,Th65;
end;
theorem Th68:
{v1,v2} c= V & v1 <> v2 & not v1.-->a1 in A & not v2.-->a2 in A &
a1 is TypeSCNominativeData of V,A & a2 is TypeSCNominativeData of V,A
implies
global_overlapping(V,A,v1.-->a1,v2.-->a2) = (v2,v1)-->(a2,a1)
proof
set D1 = v1.-->a1;
set D2 = v2.-->a2;
assume that
A1: {v1,v2} c= V and
A2: v1 <> v2 and
A3: not D1 in A & not D2 in A and
A4: a1 is TypeSCNominativeData of V,A &
a2 is TypeSCNominativeData of V,A;
v1 in V & v2 in V by A1,ZFMISC_1:32;
then
A5: naming(V,A,v1,a1) = v1.-->a1 & naming(V,A,v2,a2) = v2.-->a2 by A4,Def13;
A6: {v1}\{v2} = {v1} by A2,ZFMISC_1:14;
A7: dom(D1) = {v1} & dom(D2) = {v2} by FUNCOP_1:13;
{v1} misses {v2} by A2,ZFMISC_1:11;
hence (v2,v1)-->(a2,a1) = D2 \/ (D1|(dom(D1)\dom(D2)))
by A6,A7,FUNCT_4:30,PARTFUN1:56
.= global_overlapping(V,A,D1,D2) by A3,A5,Th64;
end;
theorem
{v,v1,v2} c= V & v <> v1 & a2 in A &
not v1.-->a1 in A & not v.-->(v2.-->a2) in A &
a1 is TypeSCNominativeData of V,A implies
local_overlapping(V,A,v1.-->a1,v2.-->a2,v) = (v,v1)-->(v2.-->a2,a1)
proof
set d1 = v1.-->a1;
set d2 = v2.-->a2;
assume that
A1: {v,v1,v2} c= V and
A2: v <> v1 and
A3: a2 in A and
A4: not d1 in A and
A5: not v.-->d2 in A and
A6: a1 is TypeSCNominativeData of V,A;
A7: v in {v,v1,v2} by ENUMSET1:def 1;
v2 in {v,v1,v2} by ENUMSET1:def 1;
then
A8: ND_ex_1(v2,a2) is TypeSCNominativeData of V,A by A1,A3,Th47;
then
A9: naming(V,A,v,d2) = v.-->d2 by A1,A7,Def13;
{v1,v} c= V
proof
let x;
assume x in {v1,v};
then x = v1 or x = v by TARSKI:def 2;
then x in {v,v1,v2} by ENUMSET1:def 1;
hence thesis by A1;
end;
hence thesis by A4,A5,A6,A2,A8,A9,Th68;
end;
definition
let V,A,v;
set Dom = {d where d is NonatomicND of V,A: v in dom d};
func denaming(V,A,v) -> PartFunc of ND(V,A),ND(V,A) means
dom it = {d where d is NonatomicND of V,A: v in dom d} &
for D being NonatomicND of V,A st D in dom it holds it.D = denaming(v,D);
existence
proof
defpred P[object,object] means
ex D being NonatomicND of V,A st D = $1 & $2 = denaming(v,D);
A1: for x being object st x in Dom
ex y being object st y in ND(V,A) & P[x,y]
proof
let x;
assume x in Dom;
then consider d being NonatomicND of V,A such that
A2: x = d and v in dom d;
take denaming(v,d);
thus thesis by A2;
end;
consider f being Function such that
A3: dom f = Dom and
A4: rng f c= ND(V,A) and
A5: for x being object st x in Dom holds P[x,f.x]
from FUNCT_1:sch 6(A1);
Dom c= ND(V,A)
proof
let x;
assume x in Dom;
then ex d being NonatomicND of V,A st x = d & v in dom d;
hence thesis by Th41;
end;
then reconsider f as PartFunc of ND(V,A),ND(V,A) by A3,A4,RELSET_1:4;
take f;
thus dom f = Dom by A3;
let D be NonatomicND of V,A;
assume D in dom f;
then P[D,f.D] by A3,A5;
hence thesis;
end;
uniqueness
proof
let f,g be PartFunc of ND(V,A),ND(V,A) such that
A6: dom f = Dom and
A7: for D being NonatomicND of V,A st D in dom f holds f.D = denaming(v,D) and
A8: dom g = Dom and
A9: for D being NonatomicND of V,A st D in dom g holds g.D = denaming(v,D);
thus dom f = dom g by A6,A8;
let D be object;
assume
A10: D in dom f;
then consider D1 being NonatomicND of V,A such that
A11: D = D1 & v in dom D1 by A6;
thus f.D = denaming(v,D1) by A7,A10,A11
.= g.D by A6,A8,A9,A10,A11;
end;
end;
definition
let V,A,v;
func naming(V,A,v) -> Function of ND(V,A),ND(V,A) means
for D being TypeSCNominativeData of V,A holds it.D = naming(V,A,v,D);
existence
proof
defpred P[object,object] means $2 = naming(V,A,v,$1);
A1: for x being Element of ND(V,A) ex y being Element of ND(V,A) st P[x,y]
proof
let x be Element of ND(V,A);
set y = naming(V,A,v,x);
y is TypeSCNominativeData of V,A by Def6;
then y in ND(V,A);
then reconsider y as Element of ND(V,A);
take y;
thus thesis;
end;
consider f being Function of ND(V,A),ND(V,A) such that
A2: for x being Element of ND(V,A) holds P[x,f.x] from FUNCT_2:sch 3(A1);
take f;
let D be TypeSCNominativeData of V,A;
D in ND(V,A);
hence thesis by A2;
end;
uniqueness
proof
let f,g be Function of ND(V,A),ND(V,A) such that
A3: for D being TypeSCNominativeData of V,A holds f.D = naming(V,A,v,D) and
A4: for D being TypeSCNominativeData of V,A holds g.D = naming(V,A,v,D);
let D be Element of ND(V,A);
A5: D is TypeSCNominativeData of V,A by Th39;
hence f.D = naming(V,A,v,D) by A3
.= g.D by A4,A5;
end;
end;
definition
let V,A,v;
func local_overlapping(V,A,v) -> PartFunc of [:ND(V,A),ND(V,A):],ND(V,A)
means
dom it = [: ND(V,A) \ A , ND(V,A) :] &
for d1 being NonatomicND of V,A
for d2 being object st not d1 in A & d2 in ND(V,A) holds
it. [d1,d2] = local_overlapping(V,A,d1,d2,v);
existence
proof
defpred P[object,object] means
ex d1 being NonatomicND of V,A, d2 being object st $1 = [d1,d2] &
$2 = local_overlapping(V,A,d1,d2,v);
A1: for x being object st x in [: ND(V,A) \ A , ND(V,A) :]
ex y being object st y in ND(V,A) & P[x,y]
proof
let x;
assume x in [: ND(V,A) \ A , ND(V,A) :];
then consider d1,d2 being object such that
A2: d1 in ND(V,A) \ A & d2 in ND(V,A) and
A3: x = [d1,d2] by ZFMISC_1:def 2;
reconsider d1 as NonatomicND of V,A by A2,Th43;
take local_overlapping(V,A,d1,d2,v);
thus thesis by A3;
end;
consider f being Function such that
A4: dom f = [: ND(V,A) \ A , ND(V,A) :] and
A5: rng f c= ND(V,A) and
A6: for x being object st x in [: ND(V,A) \ A , ND(V,A) :] holds P[x,f.x]
from FUNCT_1:sch 6(A1);
reconsider f as PartFunc of [:ND(V,A),ND(V,A):],ND(V,A)
by A4,A5,RELSET_1:4,ZFMISC_1:96;
take f;
thus dom f = [: ND(V,A) \ A , ND(V,A) :] by A4;
let D1 be NonatomicND of V,A;
let D2 be object;
set D = [D1,D2];
assume not D1 in A & D2 in ND(V,A);
then D1 in ND(V,A) \ A & D2 in ND(V,A) by Th42;
then D in dom f by A4,ZFMISC_1:def 2;
then consider d1 being NonatomicND of V,A, d2 being object such that
A7: D = [d1,d2] and
A8: f.D = local_overlapping(V,A,d1,d2,v) by A4,A6;
d1 = D1 & d2 = D2 by A7,XTUPLE_0:1;
hence thesis by A8;
end;
uniqueness
proof
let f,g be PartFunc of [:ND(V,A),ND(V,A):],ND(V,A) such that
A9: dom f = [: ND(V,A) \ A , ND(V,A) :] and
A10: for d1 being NonatomicND of V,A, d2 being object
st not d1 in A & d2 in ND(V,A) holds
f. [d1,d2] = local_overlapping(V,A,d1,d2,v) and
A11: dom g = [: ND(V,A) \ A , ND(V,A) :] and
A12: for d1 being NonatomicND of V,A, d2 being object
st not d1 in A & d2 in ND(V,A) holds
g. [d1,d2] = local_overlapping(V,A,d1,d2,v);
thus dom f = dom g by A9,A11;
let D be object such that
A13: D in dom f;
consider D1,D2 being object such that
A14: D1 in ND(V,A) \ A & D2 in ND(V,A) and
A15: D = [D1,D2] by A9,A13,ZFMISC_1:def 2;
reconsider D1 as NonatomicND of V,A by A14,Th43;
A16: not D1 in A by A14,XBOOLE_0:def 5;
hence f.D = local_overlapping(V,A,D1,D2,v) by A10,A14,A15
.= g.D by A12,A14,A15,A16;
end;
end;