:: Labelled State Transition Systems
:: by Micha{\l} Trybulec
::
:: Received May 5, 2009
:: Copyright (c) 2009 Association of Mizar Users
environ
vocabularies BOOLE, FUNCT_1, FINSEQ_1, RELAT_1, ALGSEQ_1, AFINSQ_1, FSM_1,
FINSET_1, PRELAMB, REWRITE1, REWRITE2, ARYTM, MCART_1, ARYTM_1, REWRITE3;
notations REWRITE2, CARD_1, TARSKI, XBOOLE_0, ZFMISC_1, XCMPLX_0, NAT_1,
DOMAIN_1, FUNCT_1, RELSET_1, FUNCT_2, XXREAL_0, FINSET_1, RELAT_1,
CARD_FIN, AFINSQ_1, SUBSET_1, REWRITE1, CATALAN2, FLANG_1, STRUCT_0,
NUMBERS, ORDINAL1, XREAL_0, REAL_1, MCART_1, FINSEQ_1;
constructors CARD_1, XXREAL_0, NAT_1, FSM_1, MEMBERED, CARD_FIN, SUBSET_1,
REWRITE2, REWRITE1, RELAT_1, FUNCT_1, FLANG_1, NUMBERS, XCMPLX_0,
XREAL_0, WELLORD2, MCART_1, FINSEQ_1, RELSET_1;
registrations CARD_1, RELSET_1, NAT_1, XREAL_0, XBOOLE_0, MEMBERED, XXREAL_0,
REWRITE2, STRUCT_0, SUBSET_1, REWRITE1, AFINSQ_1, RELAT_1, FUNCT_1,
ORDINAL1, NUMBERS, REAL_1, FINSET_1, FINSEQ_1;
requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM;
definitions TARSKI, STRUCT_0;
theorems AFINSQ_1, CALCUL_1, FLANG_1, FLANG_2, FUNCT_1, NAT_1, RELAT_1,
RELSET_1, REWRITE1, STRUCT_0, TARSKI, XREAL_1, XXREAL_0, ZFMISC_1,
FINSEQ_1, FINSEQ_2, FINSEQ_3, MCART_1, REWRITE2;
schemes FINSEQ_1, NAT_1, RELSET_1;
begin :: Preliminaries - FinSequences
reserve x, x1, x2, y, y1, y2, z, z1, z2, X, X1, X2, Y, Z for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u', u1, u2, v, v1, v2, w, w', w1, w2 for Element of E^omega;
reserve F, F1, F2 for Subset of E^omega;
reserve i, j, k, l, n for Nat;
theorem LmSeq10:
for p being FinSequence st k in dom p holds (<*x*>^p).(k + 1) = p.k
proof
let p be FinSequence such that
A: k in dom p;
len <*x*> + k in dom(<*x*>^p) by A, FINSEQ_1:41;
then k + 1 in dom(<*x*>^p) by FINSEQ_1:56;
then
B: k + 1 <= len(<*x*>^p) by FINSEQ_3:27;
k >= 1 by A, FINSEQ_3:27;
then k >= len <*x*> by FINSEQ_1:56;
then
C: len <*x*> + 0 < k + 1 by XREAL_1:10;
(<*x*>^p).(k + 1) = p.(k + 1 - len <*x*>) by B, C, FINSEQ_1:37
.= p.(k + 1 - 1) by FINSEQ_1:56
.= p.(k + (1 - 1));
hence thesis;
end;
theorem LmSeq20:
for p being FinSequence holds
p <> {} implies ex q being FinSequence, x st
p = q^<*x*> & len p = len q + 1
proof
let p be FinSequence such that
A: p <> {};
consider q being FinSequence, x such that
B: p = q^<*x*> by A, FINSEQ_1:63;
take q, x;
len p = len q + len <*x*> by B, FINSEQ_1:35
.= len q + 1 by FINSEQ_1:57;
hence thesis by B;
end;
theorem LmSeq40:
for p being FinSequence st k in dom p & not k + 1 in dom p holds len p = k
proof
let p be FinSequence such that
A: k in dom p & not k + 1 in dom p;
B: 1 <= k & k <= len p by A, FINSEQ_3:27;
C: 1 > k + 1 or k + 1 > len p by A, FINSEQ_3:27;
1 + 0 <= k + 1 by XREAL_1:9;
hence thesis by B, C, NAT_1:22;
end;
begin :: Preliminaries - RedSequences
theorem LmRed5:
for R being Relation, P being RedSequence of R, q1, q2 being FinSequence st
P = q1^q2 & len q1 > 0 & len q2 > 0 holds
q1 is RedSequence of R & q2 is RedSequence of R
proof
let R be Relation, P be RedSequence of R, q1, q2 be FinSequence such that
A1: P = q1^q2 and
A2: len q1 > 0 & len q2 > 0;
now
let i be Element of NAT;
assume i in dom q1 & i + 1 in dom q1; then
B: 1 <= i & i <= len q1 & 1 <= i + 1 & i + 1 <= len q1 by FINSEQ_3:27;
then
C: q1.i = (q1^q2).i & q1.(i + 1) = (q1^q2).(i + 1) by FINSEQ_1:85;
len q1 <= len P by A1, CALCUL_1:6;
then i <= len P & i + 1 <= len P by B, XXREAL_0:2;
then i in dom P & i + 1 in dom P by B, FINSEQ_3:27;
hence [q1.i, q1.(i + 1)] in R by A1, C, REWRITE1:def 2;
end;
hence q1 is RedSequence of R by A2, REWRITE1:def 2;
now
let i be Element of NAT;
assume i in dom q2 & i + 1 in dom q2;
then
B: 1 <= i & i <= len q2 & 1 <= i + 1 & i + 1 <= len q2 by FINSEQ_3:27;
then
C: q2.i = (q1^q2).(len q1 + i) &
q2.(i + 1) = (q1^q2).(len q1 + (i + 1)) by FINSEQ_1:86;
len q1 + len q2 = len P by A1, FINSEQ_1:35;
then
D: len q1 + i <= len P - len q2 + len q2 &
len q1 + (i + 1) <= len P - len q2 + len q2 by B, XREAL_1:9;
1 <= i + len q1 & 1 <= (i + 1) + len q1 by B, NAT_1:12;
then (len q1 + i) in dom P & (len q1 + i + 1) in dom P by D, FINSEQ_3:27;
hence [q2.i, q2.(i + 1)] in R by A1, C, REWRITE1:def 2;
end;
hence q2 is RedSequence of R by A2, REWRITE1:def 2;
end;
theorem LmRed10:
for R being Relation, P being RedSequence of R st len P > 1 holds
ex Q being RedSequence of R st <*P.1*>^Q = P & len Q + 1 = len P
proof
let R be Relation, P be RedSequence of R such that
A: len P > 1;
consider x being set, Q being FinSequence such that
B: P = <*x*>^Q & len P = len Q + 1 by REWRITE1:5;
C: P.1 = x by B, FINSEQ_1:58;
D: len <*x*> = 1 by FINSEQ_1:56;
1 + len Q > 1 + 0 by A, B;
then len Q > 0;
then Q is RedSequence of R by B, D, LmRed5;
hence thesis by B, C;
end;
theorem
for R being Relation, P being RedSequence of R st len P > 1 holds
ex Q being RedSequence of R st Q^<*P.len P*> = P & len Q + 1 = len P
proof
let R be Relation, P be RedSequence of R such that
A: len P > 1;
consider Q being FinSequence, x such that
B: P = Q^<*x*> & len Q + 1 = len P by LmSeq20;
C: P.len P = x by B, FINSEQ_1:59;
D: len <*x*> = 1 by FINSEQ_1:56;
1 + len Q > 1 + 0 by A, B;
then len Q > 0;
then Q is RedSequence of R by B, D, LmRed5;
hence thesis by B, C;
end;
theorem
for R being Relation, P being RedSequence of R st len P > 1 holds
ex Q being RedSequence of R st
len Q + 1 = len P &
for k st k in dom Q holds Q.k = P.(k + 1)
proof
let R be Relation, P be RedSequence of R such that
A: len P > 1;
consider Q being RedSequence of R such that
B: P = <*P.1*>^Q & len Q + 1 = len P by A, LmRed10;
take Q;
thus thesis by B, LmSeq10;
end;
theorem LmRed40:
for R being Relation holds
<*x, y*> is RedSequence of R implies [x, y] in R
proof
let R be Relation;
assume
A: <*x, y*> is RedSequence of R;
set P = <*x, y*>;
B: len P = 2 & P.1 = x & P.2 = y by FINSEQ_1:61;
then 1 in dom P & 1 + 1 in dom P by FINSEQ_3:27;
hence thesis by A, B, REWRITE1:def 2;
end;
begin :: Preliminaries - XFinSequences
theorem LmXSeq50:
w = u^v implies len u <= len w & len v <= len w
proof
assume w = u^v;
then len w = len u + len v by AFINSQ_1:20;
then len w + len u >= len u + len v + 0 &
len w + len v >= len u + len v + 0 by XREAL_1:9;
hence thesis by XREAL_1:8;
end;
theorem LmXSeq60:
w = u^v & u <> <%>E & v <> <%>E implies len u < len w & len v < len w
proof
assume that
A1: w = u^v and
A2: u <> <%>E & v <> <%>E;
len w = len u + len v by A1, AFINSQ_1:20;
then len w + len u > len u + len v + 0 &
len w + len v > len u + len v + 0 by A2, XREAL_1:10;
hence thesis by XREAL_1:8;
end;
theorem
w1^v1 = w2^v2 & ((len w1 = len w2) or (len v1 = len v2)) implies
w1 = w2 & v1 = v2
proof
assume that
A1: w1^v1 = w2^v2 and
A2: (len w1 = len w2) or (len v1 = len v2);
B: len w1 + len v1 = len (w2^v2) by A1, AFINSQ_1:20
.= len w2 + len v2 by AFINSQ_1:20;
now
let k be Element of NAT;
assume
C: k in dom w1;
hence w1.k = (w2^v2).k by A1, AFINSQ_1:def 4
.= w2.k by A2, B, C, AFINSQ_1:def 4;
end;
hence w1 = w2 by A2, B, AFINSQ_1:10;
hence thesis by A1, AFINSQ_1:31;
end;
theorem LmXSeq66:
w1^v1 = w2^v2 & ((len w1 <= len w2) or (len v1 >= len v2)) implies
ex u st w1^u = w2 & v1 = u^v2
proof
assume that
A1: w1^v1 = w2^v2 and
A2: (len w1 <= len w2) or (len v1 >= len v2);
A3: len w1 + len v1 = len (w2^v2) by A1, AFINSQ_1:20
.= len w2 + len v2 by AFINSQ_1:20;
len v1 >= len v2 implies
len w1 + len v1 - len v1 <= len w2 + len v2 - len v2 by A3, XREAL_1:15;
then consider u' being XFinSequence such that
B: w1^u' = w2 by A1, A2, AFINSQ_1:45;
reconsider u = u' as Element of E^omega by B, FLANG_1:5;
take u;
thus w1^u = w2 by B;
w2^v2 = w1^(u^v2) by B, AFINSQ_1:30;
hence thesis by A1, AFINSQ_1:31;
end;
theorem LmXSeq70:
w1^v1 = w2^v2 implies
(ex u st w1^u = w2 & v1 = u^v2) or (ex u st w2^u = w1 & v2 = u^v1)
proof
assume
A: w1^v1 = w2^v2;
len w1 < len w2 or len w1 >= len w2;
hence thesis by A, LmXSeq66;
end;
begin :: Labelled State Transition Systems
definition
let X;
struct (1-sorted) transition-system over X
(#
carrier -> set,
Tran -> Relation of [: the carrier, X :], the carrier
#);
end;
:: Transition Systems over Subsets of E^omega
:: Deterministic Transition Systems
definition
let E, F;
let TS be transition-system over F;
attr TS is deterministic means
:DefDetTS:
(the Tran of TS) is Function &
not <%>E in rng dom (the Tran of TS) &
for s being Element of TS, u, v st
u <> v &
[s, u] in dom (the Tran of TS) &
[s, v] in dom (the Tran of TS) holds
not ex w st u^w = v or v^w = u;
end;
theorem ThDet10:
for TS being transition-system over F holds
dom (the Tran of TS) = {} implies TS is deterministic
proof
let TS be transition-system over F;
assume
A: dom (the Tran of TS) = {};
then
B: (the Tran of TS) = {};
for s being Element of TS, u, v st
u <> v &
[s, u] in dom (the Tran of TS) &
[s, v] in dom (the Tran of TS) holds
not ex w st u^w = v or v^w = u by A;
hence thesis by DefDetTS, B, RELAT_1:60;
end;
registration
let E, F;
cluster strict non empty finite deterministic transition-system over F;
existence
proof
set X = the non empty finite set;
reconsider T = {} as Relation of [: X, F :], X by RELSET_1:25;
take TS = transition-system (# X, T #);
thus TS is strict;
thus TS is non empty;
thus the carrier of TS is finite;
thus TS is deterministic by RELAT_1:60, ThDet10;
end;
end;
begin :: Productions
definition
let X;
let TS be transition-system over X;
let x, y, z;
pred x, y -->. z, TS means
:DefProd:
[[x, y], z] in the Tran of TS;
end;
theorem ThProd30:
for TS being transition-system over X holds
x, y -->. z, TS implies
x in TS & y in X & z in TS &
x in dom dom (the Tran of TS) & y in rng dom (the Tran of TS) &
z in rng (the Tran of TS)
proof
let TS be transition-system over X;
assume x, y -->. z, TS;
then
A: [[x, y], z] in the Tran of TS by DefProd;
then [x, y] in [: the carrier of TS, X :] by ZFMISC_1:106;
hence x in the carrier of TS & y in X & z in the carrier of TS
by A, ZFMISC_1:106;
[x, y] in dom (the Tran of TS) by A, RELAT_1:20;
hence x in dom dom (the Tran of TS) &
y in rng dom (the Tran of TS) by RELAT_1:20;
thus z in rng (the Tran of TS) by A, RELAT_1:20;
end;
theorem ThProd40:
for TS1 being transition-system over X1,
TS2 being transition-system over X2 st
the Tran of TS1 = the Tran of TS2 holds
x, y -->. z, TS1 implies x, y -->. z, TS2
proof
let TS1 be transition-system over X1,
TS2 be transition-system over X2 such that
A1: the Tran of TS1 = the Tran of TS2;
assume
A2: x, y -->. z, TS1;
[[x, y], z] in the Tran of TS1 by A2, DefProd;
hence thesis by A1, DefProd;
end;
theorem ThProd50:
for TS being transition-system over F st
the Tran of TS is Function holds
x, y -->. z1, TS & x, y -->. z2, TS implies z1 = z2
proof
let TS be transition-system over F such that
A: the Tran of TS is Function;
assume
B: x, y -->. z1, TS & x, y -->. z2, TS;
[[x, y], z1] in the Tran of TS & [[x, y], z2] in the Tran of TS
by B, DefProd;
hence thesis by A, FUNCT_1:def 1;
end;
theorem
for TS being transition-system over F st
not <%>E in rng dom (the Tran of TS) holds
not x, <%>E -->. y, TS
proof
let TS be transition-system over F such that
A: not <%>E in rng dom (the Tran of TS);
assume x, <%>E -->. y, TS;
then [[x, <%>E], y] in the Tran of TS by DefProd;
then [x, <%>E] in dom (the Tran of TS) by RELAT_1:20;
hence contradiction by A, RELAT_1:20;
end;
theorem ThProd70:
for TS being deterministic transition-system over F holds
u <> v & x, u -->. z1, TS & x, v -->. z2, TS implies
not ex w st u^w = v or v^w = u
proof
let TS be deterministic transition-system over F;
assume that
A1: u <> v and
A2: x, u -->. z1, TS & x, v -->. z2, TS;
x in TS by A2, ThProd30;
then reconsider x as Element of TS by STRUCT_0:def 5;
[[x, u], z1] in the Tran of TS & [[x, v], z2] in the Tran of TS
by A2, DefProd;
then [x, u] in dom the Tran of TS & [x, v] in dom the Tran of TS
by RELAT_1:20;
hence thesis by A1, DefDetTS;
end;
begin :: Direct Transitions
definition
let E, F;
let TS be transition-system over F;
let x1, x2, y1, y2;
pred x1, x2 ==>. y1, y2, TS means
:DefDir:
ex v, w st v = y2 & x1, w -->. y1, TS & x2 = w^v;
end;
theorem ThDir10:
for TS being transition-system over F holds
x1, x2 ==>. y1, y2, TS implies
x1 in TS & y1 in TS & x2 in E^omega & y2 in E^omega &
x1 in dom dom (the Tran of TS) & y1 in rng (the Tran of TS)
proof
let TS be transition-system over F;
assume x1, x2 ==>. y1, y2, TS;
then ex v, w st
v = y2 & x1, w -->. y1, TS & x2 = w^v by DefDir;
hence thesis by ThProd30;
end;
theorem ThDir15:
for TS1 being transition-system over F1,
TS2 being transition-system over F2 st
the Tran of TS1 = the Tran of TS2 &
x1, x2 ==>. y1, y2, TS1 holds x1, x2 ==>. y1, y2, TS2
proof
let TS1 be transition-system over F1,
TS2 be transition-system over F2 such that
A2: the Tran of TS1 = the Tran of TS2 and
A3: x1, x2 ==>. y1, y2, TS1;
consider v, w such that
B: v = y2 & x1, w -->. y1, TS1 & x2 = w^v by A3, DefDir;
take v, w;
thus thesis by A2, B, ThProd40;
end;
theorem ThDir25:
for TS being transition-system over F holds
x, u ==>. y, v, TS implies
ex w st x, w -->. y, TS & u = w^v
proof
let TS be transition-system over F;
assume x, u ==>. y, v, TS;
then consider v1, w such that
A: v1 = v & x, w -->. y, TS & u = w^v1 by DefDir;
take w;
thus thesis by A;
end;
theorem ThDir20:
for TS being transition-system over F holds
x, y -->. z, TS iff x, y ==>. z, <%>E, TS
proof
let TS be transition-system over F;
thus x, y -->. z, TS implies x, y ==>. z, <%>E, TS
proof
assume
A: x, y -->. z, TS;
y in F by A, ThProd30;
then reconsider w = y as Element of E^omega;
w = w^(<%>E) by AFINSQ_1:32;
hence thesis by A, DefDir;
end;
assume x, y ==>. z, <%>E, TS;
then ex v, w st v = <%>E & x, w -->. z, TS & y = w^v by DefDir;
hence thesis by AFINSQ_1:32;
end;
theorem ThDir26:
for TS being transition-system over F holds
x, v -->. y, TS iff x, v^w ==>. y, w, TS
proof
let TS be transition-system over F;
thus x, v -->. y, TS implies x, v^w ==>. y, w, TS by DefDir;
assume x, v^w ==>. y, w, TS;
then ex u st x, u -->. y, TS & v^w = u^w by ThDir25;
hence thesis by AFINSQ_1:31;
end;
theorem ThDir30:
for TS being transition-system over F holds
x, u ==>. y, v, TS implies x, u^w ==>. y, v^w, TS
proof
let TS be transition-system over F;
assume x, u ==>. y, v, TS;
then consider u1 such that
A: x, u1 -->. y, TS & u = u1^v by ThDir25;
u^w = u1^(v^w) by A, AFINSQ_1:30;
hence thesis by A, DefDir;
end;
theorem ThDir35:
for TS being transition-system over F holds
x, u ==>. y, v, TS implies len u >= len v
proof
let TS be transition-system over F;
assume
A: x, u ==>. y, v, TS;
ex w st x, w -->. y, TS & u = w^v by A, ThDir25;
hence thesis by LmXSeq50;
end;
theorem ThDir40:
for TS being transition-system over F st
the Tran of TS is Function holds
x1, x2 ==>. y1, z, TS & x1, x2 ==>. y2, z, TS implies y1 = y2
proof
let TS be transition-system over F such that
A: the Tran of TS is Function;
assume
B: x1, x2 ==>. y1, z, TS & x1, x2 ==>. y2, z, TS;
consider v1, w1 such that
C1: v1 = z & x1, w1 -->. y1, TS & x2 = w1^v1 by B, DefDir;
consider v2, w2 such that
C2: v2 = z & x1, w2 -->. y2, TS & x2 = w2^v2 by B, DefDir;
w1 = w2 by C1, C2, AFINSQ_1:31;
hence thesis by A, C1, C2, ThProd50;
end;
theorem ThDir60:
for TS being transition-system over F st
not <%>E in rng dom (the Tran of TS) holds not x, z ==>. y, z, TS
proof
let TS be transition-system over F such that
A: not <%>E in rng dom (the Tran of TS);
assume x, z ==>. y, z, TS;
then consider v, w such that
C: v = z & x, w -->. y, TS & z = w^v by DefDir;
[[x, w], y] in the Tran of TS by C, DefProd;
then
D: [x, w] in dom (the Tran of TS) by RELAT_1:20;
w = <%>E by C, FLANG_2:4;
hence contradiction by A, D, RELAT_1:20;
end;
theorem ThDir50:
for TS being transition-system over F st
not <%>E in rng dom (the Tran of TS) holds
x, u ==>. y, v, TS implies len u > len v
proof
let TS be transition-system over F such that
A: not <%>E in rng dom (the Tran of TS);
assume
B: x, u ==>. y, v, TS;
consider w such that
C: x, w -->. y, TS & u = w^v by B, ThDir25;
D: w in rng dom (the Tran of TS) by C, ThProd30;
per cases;
suppose
E: v = <%>E;
then u <> <%>E by A, B, ThDir60;
then len u > 0;
hence thesis by E;
end;
suppose v <> <%>E;
hence thesis by A, C, D, LmXSeq60;
end;
end;
theorem ThDir70:
for TS being deterministic transition-system over F holds
x1, x2 ==>. y1, z1, TS & x1, x2 ==>. y2, z2, TS implies
y1 = y2 & z1 = z2
proof
let TS be deterministic transition-system over F;
assume
A: x1, x2 ==>. y1, z1, TS & x1, x2 ==>. y2, z2, TS;
consider v1, w1 such that
B1: v1 = z1 & x1, w1 -->. y1, TS & x2 = w1^v1 by A, DefDir;
consider v2, w2 such that
B2: v2 = z2 & x1, w2 -->. y2, TS & x2 = w2^v2 by A, DefDir;
(ex u' st w1^u' = w2 & v1 = u'^v2) or (ex u' st w2^u' = w1 & v2 = u'^v1)
by B1, B2, LmXSeq70;
then w1 = w2 by B1, B2, ThProd70;
then
C: v1 = v2 by B1, B2, AFINSQ_1:31;
the Tran of TS is Function by DefDetTS;
hence thesis by A, B1, B2, C, ThDir40;
end;
begin :: ==>.-relation is introduced to define transitions
:: using reduction relations from REWRITE1
reserve TS for non empty transition-system over F;
reserve s, s', s1, s2, t, t', t1, t2 for Element of TS;
reserve S, T for Subset of TS;
definition
let E, F, TS;
func ==>.-relation(TS) -> Relation of [: the carrier of TS, E^omega :] means
:DefRel:
[[x1, x2], [y1, y2]] in it iff x1, x2 ==>. y1, y2, TS;
existence
proof
defpred P[Element of [: the carrier of TS, E^omega :],
Element of [: the carrier of TS, E^omega :]] means
ex x1, x2, y1, y2 st
[x1, x2] = $1 & [y1, y2] = $2 & x1, x2 ==>. y1, y2, TS;
consider R being Relation of [: the carrier of TS, E^omega :] such that
A1: for s, t being Element of [: the carrier of TS, E^omega :] holds
[s, t] in R iff P[s, t] from RELSET_1:sch 2;
take R;
now
let x1, x2, y1, y2;
set s = [x1, x2];
set t = [y1, y2];
thus x1, x2 ==>. y1, y2, TS implies [[x1, x2], [y1, y2]] in R
proof
assume
A2: x1, x2 ==>. y1, y2, TS;
then
A3: x1 in TS & y1 in TS & x2 in E^omega & y2 in E^omega by ThDir10;
then x1 in the carrier of TS & y1 in the carrier of TS
by STRUCT_0:def 5;
then s in [: the carrier of TS, E^omega :] &
t in [: the carrier of TS, E^omega :] by A3, ZFMISC_1:def 2;
hence thesis by A1, A2;
end;
assume
A2: [[x1, x2], [y1, y2]] in R;
A3: [x1, x2] in dom R & [y1, y2] in rng R by A2, RELAT_1:20;
dom R c= [: the carrier of TS, E^omega :] &
rng R c= [: the carrier of TS, E^omega :] by RELAT_1:def 18,def 19;
then consider x1', x2', y1', y2' being set such that
A4: [x1', x2'] = s & [y1', y2'] = t & x1', x2' ==>. y1', y2', TS
by A1, A2, A3;
x1 = x1' & x2 = x2' & y1 = y1' & y2 = y2' by A4, ZFMISC_1:33;
hence x1, x2 ==>. y1, y2, TS by A4;
end;
hence thesis;
end;
uniqueness
proof
let R1, R2 be Relation of [: the carrier of TS, E^omega :] such that
A2: [[x1, x2], [y1, y2]] in R1 iff x1, x2 ==>. y1, y2, TS and
A3: [[x1, x2], [y1, y2]] in R2 iff x1, x2 ==>. y1, y2, TS;
now
let s, t be Element of [: the carrier of TS, E^omega :];
consider x, u being set such that
B2: x in the carrier of TS & u in E^omega & s = [x, u] by ZFMISC_1:def 2;
reconsider x as Element of the carrier of TS by B2;
reconsider u as Element of E^omega by B2;
consider y, v being set such that
B3: y in the carrier of TS & v in E^omega & t = [y, v] by ZFMISC_1:def 2;
reconsider y as Element of the carrier of TS by B3;
reconsider v as Element of E^omega by B3;
[s, t] in R1 iff x, u ==>. y, v, TS by A2, B2, B3;
hence [s, t] in R1 iff [s, t] in R2 by A3, B2, B3;
end;
hence thesis by RELSET_1:def 3;
end;
end;
theorem ThRel10:
[x, y] in ==>.-relation(TS) implies
ex s, v, t, w st x = [s, v] & y = [t, w]
proof
assume [x, y] in ==>.-relation(TS);
then
A: x in [: the carrier of TS, E^omega :] &
y in [: the carrier of TS, E^omega :] by ZFMISC_1:106;
consider x1, x2 such that
B1: x1 in the carrier of TS & x2 in E^omega & x = [x1, x2]
by A, ZFMISC_1:def 2;
consider y1, y2 such that
B2: y1 in the carrier of TS & y2 in E^omega & y = [y1, y2]
by A, ZFMISC_1:def 2;
thus thesis by B1, B2;
end;
theorem ThRel11:
[[x1, x2], [y1, y2]] in ==>.-relation(TS) implies
x1 in TS & y1 in TS & x2 in E^omega & y2 in E^omega &
x1 in dom dom (the Tran of TS) & y1 in rng (the Tran of TS)
proof
assume [[x1, x2], [y1, y2]] in ==>.-relation(TS);
then x1, x2 ==>. y1, y2, TS by DefRel;
hence thesis by ThDir10;
end;
theorem ThRel9:
x in ==>.-relation(TS) implies
ex s, t, v, w st x = [[s, v], [t, w]]
proof
assume
A: x in ==>.-relation(TS);
then consider y, z such that
B: x = [y, z] by RELAT_1:def 1;
consider s, v, t, w such that
C: y = [s, v] & z = [t, w] by A, B, ThRel10;
thus thesis by B, C;
end;
theorem ThRel20:
for TS1 being non empty transition-system over F1,
TS2 being non empty transition-system over F2 st
the carrier of TS1 = the carrier of TS2 &
the Tran of TS1 = the Tran of TS2 holds
==>.-relation(TS1) = ==>.-relation(TS2)
proof
let TS1 be non empty transition-system over F1,
TS2 be non empty transition-system over F2 such that
A1: the carrier of TS1 = the carrier of TS2 and
A2: the Tran of TS1 = the Tran of TS2;
B: now
let x;
assume
B0: x in ==>.-relation(TS1);
then consider s, t being Element of TS1, v, w such that
B1: x = [[s, v], [t, w]] by ThRel9;
s, v ==>. t, w, TS1 by B0, B1, DefRel;
then
B2: s, v ==>. t, w, TS2 by A2, ThDir15;
reconsider s as Element of TS2 by A1;
reconsider t as Element of TS2 by A1;
thus x in ==>.-relation(TS2) by B1, B2, DefRel;
end;
now
let x;
assume
B0: x in ==>.-relation(TS2);
then consider s, t being Element of TS2, v, w such that
B1: x = [[s, v], [t, w]] by ThRel9;
s, v ==>. t, w, TS2 by B0, B1, DefRel;
then
B2: s, v ==>. t, w, TS1 by A2, ThDir15;
reconsider s, t as Element of TS1 by A1;
thus x in ==>.-relation(TS1) by B1, B2, DefRel;
end;
hence thesis by B, TARSKI:2;
end;
theorem ThRel30:
[[x1, x2], [y1, y2]] in ==>.-relation(TS) implies
ex v, w st v = y2 & x1, w -->. y1, TS & x2 = w^v
proof
assume [[x1, x2], [y1, y2]] in ==>.-relation(TS);
then x1, x2 ==>. y1, y2, TS by DefRel;
hence thesis by DefDir;
end;
theorem ThRel40:
[[x, u], [y, v]] in ==>.-relation(TS) implies
ex w st x, w -->. y, TS & u = w^v
proof
assume [[x, u], [y, v]] in ==>.-relation(TS);
then x, u ==>. y, v, TS by DefRel;
hence thesis by ThDir25;
end;
theorem ThRel50:
x, y -->. z, TS iff [[x, y], [z, <%>E]] in ==>.-relation(TS)
proof
thus x, y -->. z, TS implies [[x, y], [z, <%>E]] in ==>.-relation(TS)
proof
assume x, y -->. z, TS;
then x, y ==>. z, <%>E, TS by ThDir20;
hence thesis by DefRel;
end;
assume [[x, y], [z, <%>E]] in ==>.-relation(TS);
then x, y ==>. z, <%>E, TS by DefRel;
hence thesis by ThDir20;
end;
theorem ThRel60:
x, v -->. y, TS iff [[x, v^w], [y, w]] in ==>.-relation(TS)
proof
thus x, v -->. y, TS implies [[x, v^w], [y, w]] in ==>.-relation(TS)
proof
assume x, v -->. y, TS;
then x, v^w ==>. y, w, TS by ThDir26;
hence thesis by DefRel;
end;
assume [[x, v^w], [y, w]] in ==>.-relation(TS);
then x, v^w ==>. y, w, TS by DefRel;
hence thesis by ThDir26;
end;
theorem
[[x, u], [y, v]] in ==>.-relation(TS) implies
[[x, u^w], [y, v^w]] in ==>.-relation(TS)
proof
assume [[x, u], [y, v]] in ==>.-relation(TS);
then x, u ==>. y, v, TS by DefRel;
then x, u^w ==>. y, v^w, TS by ThDir30;
hence thesis by DefRel;
end;
theorem
[[x, u], [y, v]] in ==>.-relation(TS) implies len u >= len v
proof
assume [[x, u], [y, v]] in ==>.-relation(TS);
then x, u ==>. y, v, TS by DefRel;
hence thesis by ThDir35;
end;
theorem
the Tran of TS is Function implies
([x, [y1, z]] in ==>.-relation(TS) &
[x, [y2, z]] in ==>.-relation(TS) implies y1 = y2)
proof
assume
A: the Tran of TS is Function;
assume
B: [x, [y1, z]] in ==>.-relation(TS) & [x, [y2, z]] in ==>.-relation(TS);
then consider s, v, t, w such that
C: x = [s, v] & [y1, z] = [t, w] by ThRel10;
s, v ==>. y1, z, TS & s, v ==>. y2, z, TS by B, C, DefRel;
hence thesis by A, ThDir40;
end;
theorem
not <%>E in rng dom (the Tran of TS) implies
([[x, u], [y, v]] in ==>.-relation(TS) implies len u > len v)
proof
assume
A: not <%>E in rng dom (the Tran of TS);
assume [[x, u], [y, v]] in ==>.-relation(TS);
then x, u ==>. y, v, TS by DefRel;
hence thesis by A, ThDir50;
end;
theorem ThRel110:
not <%>E in rng dom (the Tran of TS) implies
not [[x, z], [y, z]] in ==>.-relation(TS)
proof
assume not <%>E in rng dom (the Tran of TS);
then not x, z ==>. y, z, TS by ThDir60;
hence thesis by DefRel;
end;
theorem ThRel120':
TS is deterministic implies
([x, y1] in ==>.-relation(TS) &
[x, y2] in ==>.-relation(TS) implies y1 = y2)
proof
assume
A: TS is deterministic;
assume
B: [x, y1] in ==>.-relation(TS) & [x, y2] in ==>.-relation(TS);
consider s1, v1, t1, w1 such that
C1: x = [s1, v1] & y1 = [t1, w1] by B, ThRel10;
consider s2, v2, t2, w2 such that
C2: x = [s2, v2] & y2 = [t2, w2] by B, ThRel10;
s1, v1 ==>. t1, w1, TS & s1, v1 ==>. t2, w2, TS by B, C1, C2, DefRel;
then t1 = t2 & w1 = w2 by A, ThDir70;
hence thesis by C1, C2;
end;
theorem
TS is deterministic implies
([x, [y1, z1]] in ==>.-relation(TS) &
[x, [y2, z2]] in ==>.-relation(TS) implies y1 = y2 & z1 = z2)
proof
assume
A: TS is deterministic;
assume
B: [x, [y1, z1]] in ==>.-relation(TS) & [x, [y2, z2]] in ==>.-relation(TS);
[y1, z1] = [y2, z2] by A, B, ThRel120';
hence thesis by ZFMISC_1:33;
end;
theorem
TS is deterministic implies ==>.-relation(TS) is Function-like
proof
assume TS is deterministic; then
for x, y, z st [x, y] in ==>.-relation(TS) & [x, z] in ==>.-relation(TS)
holds y = z by ThRel120';
hence thesis by FUNCT_1:def 1;
end;
begin :: Reduction Sequences of ==>.-relation
definition
let x, E;
func dim2(x, E) -> Element of E^omega equals
:DefDim2:
x`2 if ex y, u st x = [y, u] otherwise {};
coherence
proof
thus (ex y, u st x = [y, u]) implies
x`2 is Element of E^omega by MCART_1:7;
{} = <%>E;
hence thesis;
end;
consistency;
end;
theorem ThRedSeq4:
for P being RedSequence of ==>.-relation(TS),
k st k in dom P & k + 1 in dom P holds
ex s, v, t, w st P.k = [s, v] & P.(k + 1) = [t, w]
proof
let P be RedSequence of ==>.-relation(TS), k such that
A: k in dom P & k + 1 in dom P;
[P.k, P.(k + 1)] in ==>.-relation(TS) by A, REWRITE1:def 2;
hence thesis by ThRel10;
end;
theorem ThRedSeq5:
for P being RedSequence of ==>.-relation(TS),
k st k in dom P & k + 1 in dom P holds
P.k = [(P.k)`1, (P.k)`2] & P.(k + 1) = [(P.(k + 1))`1, (P.(k + 1))`2]
proof
let P be RedSequence of ==>.-relation(TS), k such that
A: k in dom P & k + 1 in dom P;
consider s, v, t, w such that
B: P.k = [s, v] & P.(k + 1) = [t, w] by A, ThRedSeq4;
thus thesis by B, MCART_1:8;
end;
theorem ThRedSeq10:
for P being RedSequence of ==>.-relation(TS),
k st k in dom P & k + 1 in dom P holds
(P.k)`1 in TS & (P.k)`2 in E^omega &
(P.(k + 1))`1 in TS & (P.(k + 1))`2 in E^omega &
(P.k)`1 in dom dom (the Tran of TS) &
(P.(k + 1))`1 in rng (the Tran of TS)
proof
let P be RedSequence of ==>.-relation(TS), k such that
A: k in dom P & k + 1 in dom P;
B: [P.k, P.(k + 1)] in ==>.-relation(TS) by A, REWRITE1:def 2;
then consider s, v, t, w such that
C: P.k = [s, v] & P.(k + 1) = [t, w] by ThRel10;
s in TS & v in E^omega & t in TS & w in E^omega &
s in dom dom (the Tran of TS) & t in rng (the Tran of TS)
by B, C, ThRel11;
hence thesis by C, MCART_1:7;
end;
theorem
for TS1 being non empty transition-system over F1,
TS2 being non empty transition-system over F2 st
the carrier of TS1 = the carrier of TS2 &
the Tran of TS1 = the Tran of TS2 holds
for P being RedSequence of ==>.-relation(TS1) holds
P is RedSequence of ==>.-relation(TS2)
proof
let TS1 be non empty transition-system over F1,
TS2 be non empty transition-system over F2 such that
A1: the carrier of TS1 = the carrier of TS2 and
A2: the Tran of TS1 = the Tran of TS2;
let P be RedSequence of ==>.-relation(TS1);
A3: len P > 0;
now
let i be Element of NAT such that
B: i in dom P & i + 1 in dom P;
[P.i, P.(i + 1)] in ==>.-relation(TS1) by B, REWRITE1:def 2;
hence [P.i, P.(i + 1)] in ==>.-relation(TS2) by A1, A2, ThRel20;
end;
hence thesis by A3, REWRITE1:def 2;
end;
theorem ThRedSeq30:
for P being RedSequence of ==>.-relation(TS) st ex x, u st P.1 = [x, u] holds
for k st k in dom P holds dim2(P.k, E) = (P.k)`2
proof
let P be RedSequence of ==>.-relation(TS);
given x, u such that
A: P.1 = [x, u];
let k such that
B: k in dom P;
per cases;
suppose
H: k > 1;
then consider l such that
E: k = l + 1 by NAT_1:6;
F: l >= 1 by E, H, NAT_1:19;
G: k <= len P by B, FINSEQ_3:27;
l <= k by E, NAT_1:11;
then l <= len P by G, XXREAL_0:2;
then l in dom P by F, FINSEQ_3:27;
then [P.l, P.k] in ==>.-relation(TS) by B, E, REWRITE1:def 2;
then
C: P.k in rng ==>.-relation(TS) by RELAT_1:20;
rng ==>.-relation(TS) c= [: the carrier of TS, E^omega :]
by RELAT_1:def 19;
then consider x1, y1 such that
D: x1 in the carrier of TS & y1 in E^omega & P.k = [x1, y1]
by C, ZFMISC_1:def 2;
thus thesis by D, DefDim2;
end;
suppose
E: k <= 1;
k >= 1 by B, FINSEQ_3:27;
then k = 1 by E, XXREAL_0:1;
hence thesis by A, DefDim2;
end;
end;
theorem ThRedSeq50:
for P being RedSequence of ==>.-relation(TS) st P.len P = [y, w] holds
for k st k in dom P ex u st (P.k)`2 = u^w
proof
let P be RedSequence of ==>.-relation(TS) such that
A: P.len P = [y, w];
defpred P[Nat] means
len P - $1 in dom P implies ex u st (P.(len P - $1))`2 = u^w;
B: P[0]
proof
(P.len P)`2 = w by A, MCART_1:7;
then (P.len P)`2 = <%>E^w by AFINSQ_1:32;
hence thesis;
end;
C: now
let k;
assume
C0: P[k];
now
assume
C1: len P - (k + 1) in dom P;
set len1 = len P - (k + 1);
set len2 = len P - k;
C2: len1 + 1 = len2;
thus ex u st (P.(len P - (k + 1)))`2 = u^w
proof
per cases;
suppose
D0: len P - k in dom P;
then consider u1 such that
D2: (P.len2)`2 = u1^w by C0;
D3: [P.len1, P.len2] in ==>.-relation(TS)
by C1, C2, D0, REWRITE1:def 2;
then consider x being Element of TS, v1 being Element of E^omega,
y being Element of TS, v2 such that
D4: P.len1 = [x, v1] & P.len2 = [y, v2] by ThRel10;
x, v1 ==>. y, v2, TS by D3, D4, DefRel;
then consider u2 such that
D5: x, u2 -->. y, TS & v1 = u2^v2 by ThDir25;
D6: (P.len1)`2 = u2^v2 by D4, D5, MCART_1:7
.= u2^(u1^w) by D2, D4, MCART_1:7
.= u2^u1^w by AFINSQ_1:30;
take u2^u1;
thus thesis by D6;
end;
suppose not len P - k in dom P;
then len1 = len P - 0 by C1, C2, LmSeq40;
hence thesis;
end;
end;
end;
hence P[k + 1];
end;
E: for k holds P[k] from NAT_1:sch 2(B, C);
hereby
let k such that
F0: k in dom P;
k <= len P by F0, FINSEQ_3:27;
then consider l such that
F1: k + l = len P by NAT_1:10;
k + l - l = len P - l by F1;
hence ex u st (P.k)`2 = u^w by F0, E;
end;
end;
theorem ThRedSeq60:
for P being RedSequence of ==>.-relation(TS) st
P.1 = [x, v] & P.len P = [y, w] holds ex u st v = u^w
proof
let P be RedSequence of ==>.-relation(TS) such that
A: P.1 = [x, v] & P.len P = [y, w];
0 + 1 <= len P by NAT_1:8;
then 1 in dom P by FINSEQ_3:27;
then consider u such that
B: (P.1)`2 = u^w by A, ThRedSeq50;
take u;
thus thesis by A, B, MCART_1:7;
end;
theorem ThRedSeq70:
for P being RedSequence of ==>.-relation(TS) st
P.1 = [x, u] & P.len P = [y, u] holds
(for k st k in dom P holds (P.k)`2 = u)
proof
defpred P[Nat] means
for P being RedSequence of ==>.-relation(TS), x, y st
P.1 = [x, u] & P.len P = [y, u] & len P = $1 holds
(for k st k in dom P holds (P.k)`2 = u);
C: P[0];
D: now
let k;
assume
D0: P[k];
now
let P be RedSequence of ==>.-relation(TS), x, y such that
D1: P.1 = [x, u] & P.len P = [y, u] and
D2: len P = k + 1;
per cases;
suppose
D3: k = 0;
hereby
let l;
assume l in dom P;
then l <= 1 & 1 <= l by D2, D3, FINSEQ_3:27;
then l = 1 by XXREAL_0:1;
hence (P.l)`2 = u by D1, MCART_1:7;
end;
end;
suppose k <> 0;
then
D2'A: k + 1 > 0 + 1 by XREAL_1:8;
then consider Q being RedSequence of ==>.-relation(TS) such that
D3: <*P.1*>^Q = P & len P = len Q + 1 by D2, LmRed10;
D2': len <*P.1*> = 1 by FINSEQ_1:57;
D6': len Q >= 0 + 1 by NAT_1:8;
then
D6: Q.len Q = [y, u] by D1, D2', D3, FINSEQ_1:86;
len P >= 1 + 1 by D2, D2'A, NAT_1:8;
then
D6'A: 1 in dom P & 1 + 1 in dom P by D2, D2'A, FINSEQ_3:27;
then
D6'C: P.(1 + 1) = [(P.(1 + 1))`1, (P.(1 + 1))`2] by ThRedSeq5;
then
D6'B: Q.1 = [(P.(1 + 1))`1, (P.(1 + 1))`2] by D2', D3, D6', FINSEQ_1:86;
reconsider u1 = (P.(1 + 1))`2 as Element of E^omega
by D6'A, ThRedSeq10;
consider v such that
E1: u1 = v^u by D6, D6'B, ThRedSeq60;
==>.-relation(TS) reduces P.1, P.(1 + 1) by D6'A, REWRITE1:18;
then consider P' being RedSequence of ==>.-relation(TS) such that
E1': P'.1 = P.1 & P'.len P' = P.(1 + 1) by REWRITE1:def 3;
consider w such that
E2: u = w^u1 by D1, D6'C, E1', ThRedSeq60;
u = w^v^u by E1, E2, AFINSQ_1:30;
then w^v = {} by FLANG_2:4;
then w = {} by AFINSQ_1:33;
then
D5: Q.1 = [(P.(1 + 1))`1, u] by D6'B, E2, AFINSQ_1:32;
hereby
let l;
assume l in dom P;
then
D7'A: 1 <= l & l <= len P by FINSEQ_3:27;
then 1 + 1 <= l + 1 & l <= len P by XREAL_1:8;
then (1 + 1 = l + 1 or 1 + 1 <= l) & l <= len P by NAT_1:8;
then
D7': 1 = l or (1 + 1 - 1 <= l - 1 & l - 1 <= len Q + 1 - 1)
by D3, XREAL_1:11;
reconsider l' = l - 1 as Element of NAT by D7'A, NAT_1:21;
D7: l = 1 or l' in dom Q by D7', FINSEQ_3:27;
per cases by D7;
suppose l = 1;
hence (P.l)`2 = u by D1, MCART_1:7;
end;
suppose
D8: (l - 1) in dom Q;
then
D9: (Q.(l - 1))`2 = u by D0, D2, D3, D5, D6;
1 <= l - 1 & l - 1 <= len Q by D8, FINSEQ_3:27;
then 1 + 0 < l - 1 + 1 & l - 1 + 1 <= len Q + 1 by XREAL_1:8;
hence (P.l)`2 = u by D2', D3, D9, FINSEQ_1:37;
end;
end;
end;
end;
hence P[k + 1];
end;
for k holds P[k] from NAT_1:sch 2(C, D);
hence thesis;
end;
theorem
for P being RedSequence of ==>.-relation(TS), k st
k in dom P & k + 1 in dom P holds
ex v, w st
v = (P.(k + 1))`2 & (P.k)`1, w -->. (P.(k + 1))`1, TS & (P.k)`2 = w^v
proof
let P be RedSequence of ==>.-relation(TS), k such that
A: k in dom P & k + 1 in dom P;
consider s, u, t, v such that
B: P.k = [s, u] & P.(k + 1) = [t, v] by A, ThRedSeq4;
[[s, u], [t, v]] in ==>.-relation(TS) by A, B, REWRITE1:def 2;
then consider v1, w1 such that
C: v1 = v & s, w1 -->. t, TS & u = w1^v1 by ThRel30;
take v1, w1;
thus v1 = (P.(k + 1))`2 by B, C, MCART_1:7;
(P.k)`1, w1 -->. t, TS by B, C, MCART_1:7;
hence thesis by B, C, MCART_1:7;
end;
theorem
for P being RedSequence of ==>.-relation(TS), k st
k in dom P & k + 1 in dom P & P.k = [x, u] & P.(k + 1) = [y, v] holds
ex w st x, w -->. y, TS & u = w^v
proof
let P be RedSequence of ==>.-relation(TS), k such that
A1: k in dom P & k + 1 in dom P and
A2: P.k = [x, u] & P.(k + 1) = [y, v];
[[x, u], [y, v]] in ==>.-relation(TS) by A1, A2, REWRITE1:def 2;
hence thesis by ThRel40;
end;
theorem ThRedSeq77:
x, y -->. z, TS iff
<*[x, y], [z, <%>E]*> is RedSequence of ==>.-relation(TS)
proof
thus x, y -->. z, TS implies
<*[x, y], [z, <%>E]*> is RedSequence of ==>.-relation(TS)
proof
assume x, y -->. z, TS;
then [[x, y], [z, <%>E]] in ==>.-relation(TS) by ThRel50;
hence thesis by REWRITE1:8;
end;
assume <*[x, y], [z, <%>E]*> is RedSequence of ==>.-relation(TS);
then [[x, y], [z, <%>E]] in ==>.-relation(TS) by LmRed40;
hence thesis by ThRel50;
end;
theorem ThRedSeq78:
x, v -->. y, TS iff
<*[x, v^w], [y, w]*> is RedSequence of ==>.-relation(TS)
proof
thus x, v -->. y, TS implies
<*[x, v^w], [y, w]*> is RedSequence of ==>.-relation(TS)
proof
assume x, v -->. y, TS;
then [[x, v^w], [y, w]] in ==>.-relation(TS) by ThRel60;
hence thesis by REWRITE1:8;
end;
assume <*[x, v^w], [y, w]*> is RedSequence of ==>.-relation(TS);
then [[x, v^w], [y, w]] in ==>.-relation(TS) by LmRed40;
hence thesis by ThRel60;
end;
theorem ThRedSeq79:
for P being RedSequence of ==>.-relation(TS) st
P.1 = [x, v] & P.len P = [y, w] holds len v >= len w
proof
let P be RedSequence of ==>.-relation(TS) such that
A: P.1 = [x, v] & P.len P = [y, w];
consider u such that
B: v = u^w by A, ThRedSeq60;
thus thesis by B, LmXSeq50;
end;
theorem ThRedSeq80:
not <%>E in rng dom (the Tran of TS) implies
for P being RedSequence of ==>.-relation(TS) st
P.1 = [x, u] & P.len P = [y, u] holds len P = 1 & x = y
proof
assume
A: not <%>E in rng dom (the Tran of TS);
defpred P[Nat] means
for P being RedSequence of ==>.-relation(TS), x, y st
P.1 = [x, u] & P.len P = [y, u] &
len P = $1 & $1 <> 1 holds contradiction;
C: P[0];
D: now
let k;
assume P[k];
now
let P be RedSequence of ==>.-relation(TS), x, y such that
D1: P.1 = [x, u] & P.len P = [y, u] and
D2: len P = k + 1 & k + 1 <> 1;
D2'A: len P > 1 by D2, NAT_1:26;
consider Q being RedSequence of ==>.-relation(TS) such that
D3: <*P.1*>^Q = P & len P = len Q + 1 by D2, LmRed10, NAT_1:26;
D4'A: len Q >= 0 + 1 by NAT_1:8;
len Q + 1 >= 1 + 1 by D4'A, XREAL_1:8;
then
D5: 1 in dom P & 1 + 1 in dom P by D2'A, D3, FINSEQ_3:27;
then
P.(1 + 1) = [(P.(1 + 1))`1, (P.(1 + 1))`2] by ThRedSeq5
.= [(P.(1 + 1))`1, u] by D1, D5, ThRedSeq70;
then [[x, u], [(P.(1 + 1))`1, u]] in ==>.-relation(TS)
by D1, D5, REWRITE1:def 2;
then x, u ==>. (P.(1 + 1))`1, u, TS by DefRel;
hence contradiction by A, ThDir60;
end;
hence P[k + 1];
end;
E: for k holds P[k] from NAT_1:sch 2(C, D);
let P be RedSequence of ==>.-relation(TS) such that
F: P.1 = [x, u] & P.len P = [y, u];
thus
G: len P = 1 by E, F;
thus thesis by F, G, ZFMISC_1:33;
end;
theorem ThRedSeq81:
not <%>E in rng dom (the Tran of TS) implies
for P being RedSequence of ==>.-relation(TS) st
(P.1)`2 = (P.len P)`2 holds len P = 1
proof
assume
A: not <%>E in rng dom (the Tran of TS);
let P be RedSequence of ==>.-relation(TS) such that
B: (P.1)`2 = (P.len P)`2;
per cases;
suppose
C: len P <= 1;
len P >= 0 + 1 by NAT_1:13;
hence len P = 1 by C, XXREAL_0:1;
end;
suppose
C: len P > 1;
then reconsider p1 = len P - 1 as Nat by NAT_1:21;
C1: p1 + 1 = len P;
1 <= len P & 1 + 1 <= len P by C, NAT_1:13;
then 1 in dom P & 1 + 1 in dom P by FINSEQ_3:27;
then consider s1, v1, t1, w1 such that
D1: P.1 = [s1, v1] & P.(1 + 1) = [t1, w1] by ThRedSeq4;
D1'a: p1 <= len P by C1, NAT_1:13;
D1'b: 0 + 1 <= p1 + 1 by XREAL_1:8;
1 <= p1 by C, C1, NAT_1:13;
then p1 in dom P & p1 + 1 in dom P by D1'a, D1'b, FINSEQ_3:27;
then consider s2, v2, t2, w2 such that
D2: P.p1 = [s2, v2] & P.(p1 + 1) = [t2, w2] by ThRedSeq4;
(P.len P)`2 = w2 by D2, MCART_1:7;
then v1 = w2 by B, D1, MCART_1:7;
hence len P = 1 by A, D1, D2, ThRedSeq80;
end;
end;
theorem ThRedSeq90:
not <%>E in rng dom (the Tran of TS) implies
for P being RedSequence of ==>.-relation(TS) st
P.1 = [x, u] & P.len P = [y, <%>E] holds len P <= len u + 1
proof
assume
A: not <%>E in rng dom (the Tran of TS);
defpred P[Nat] means
for P being RedSequence of ==>.-relation(TS), u, x st
len u = $1 & P.1 = [x, u] & P.len P = [y, <%>E] holds
len P <= len u + 1;
C: now
let k;
assume
C1: for n st n < k holds P[n];
now
let P be RedSequence of ==>.-relation(TS), u, x such that
C2: len u = k & P.1 = [x, u] & P.len P = [y, <%>E];
per cases;
suppose len u < 1;
then u = <%>E by NAT_1:26;
then len P = 1 by A, C2, ThRedSeq80;
hence len P <= len u + 1 by NAT_1:26;
end;
suppose
C3: len u >= 1;
C4': len P <> 1
proof
assume len P = 1;
then u = <%>E by C2, ZFMISC_1:33;
hence contradiction by C3;
end;
then
C5: len P > 1 by NAT_1:26;
consider P' being RedSequence of ==>.-relation(TS) such that
D: <*P.1*>^P' = P & len P' + 1 = len P by C4', LmRed10, NAT_1:26;
len P >= 1 & len P >= 1 + 1 by C5, NAT_1:13;
then
D2: 1 in dom P & 1 + 1 in dom P by FINSEQ_3:27;
then
D3: P.(1 + 1) = [(P.(1 + 1))`1, (P.(1 + 1))`2] by ThRedSeq5;
then
D4: [[x, u], [(P.(1 + 1))`1, (P.(1 + 1))`2]] in ==>.-relation(TS)
by C2, D2, REWRITE1:def 2;
then reconsider u1 = (P.(1 + 1))`2 as Element of E^omega by ThRel11;
x, u ==>. (P.(1 + 1))`1, u1, TS by D4, DefRel;
then consider v such that
D5: x, v -->. (P.(1 + 1))`1, TS & u = v^u1 by ThDir25;
D6: v <> <%>E by A, D5, ThProd30;
len u = len u1 + len v by D5, AFINSQ_1:20;
then
E0: len u - 0 > len u1 + len v - len v by D6, XREAL_1:17;
E2: len <*P.1*> = 1 by FINSEQ_1:56;
E3: len P' >= 1 by NAT_1:26;
E4: P'.1 = [(P.(1 + 1))`1, u1] by D, D3, E2, E3, FINSEQ_1:86;
P'.len P' = [y, <%>E] by C2, D, E2, E3, FINSEQ_1:86;
then
F: len P' <= len u1 + 1 by E0, E4, C1, C2;
len u1 + 1 <= len u by E0, NAT_1:13;
then len P' <= len u by F, XXREAL_0:2;
hence len P <= len u + 1 by D, XREAL_1:8;
end;
end;
hence P[k];
end;
for k holds P[k] from NAT_1:sch 4(C);
hence thesis;
end;
theorem ThRedSeq100:
not <%>E in rng dom (the Tran of TS) implies
for P being RedSequence of ==>.-relation(TS) st
P.1 = [x, <%e%>] & P.len P = [y, <%>E] holds len P = 2
proof
assume
A: not <%>E in rng dom (the Tran of TS);
let P be RedSequence of ==>.-relation(TS) such that
B: P.1 = [x, <%e%>] & P.len P = [y, <%>E];
C: len P <> 1 by B, ZFMISC_1:33;
len P <= len <%e%> + 1 by A, B, ThRedSeq90;
then len P <= 1 + 1 by AFINSQ_1:38;
hence thesis by C, NAT_1:27;
end;
theorem ThRedSeq110:
not <%>E in rng dom (the Tran of TS) implies
for P being RedSequence of ==>.-relation(TS) st
P.1 = [x, v] & P.len P = [y, w] holds
len v > len w or (len P = 1 & x = y & v = w)
proof
assume
A: not <%>E in rng dom (the Tran of TS);
let P be RedSequence of ==>.-relation(TS) such that
B: P.1 = [x, v] & P.len P = [y, w];
consider u such that
C: v = u^w by B, ThRedSeq60;
D: len v >= len w by B, ThRedSeq79;
per cases;
suppose len v > len w;
hence thesis;
end;
suppose len v <= len w;
then
D1: len v = len w by D, XXREAL_0:1;
len v = len u + len w by C, AFINSQ_1:20;
then u = <%>E by D1;
then v = w by C, AFINSQ_1:32;
hence thesis by A, B, ThRedSeq80;
end;
end;
theorem
not <%>E in rng dom (the Tran of TS) implies
for P being RedSequence of ==>.-relation(TS), k st
k in dom P & k + 1 in dom P holds (P.k)`2 <> (P.(k + 1))`2
proof
assume
A: not <%>E in rng dom (the Tran of TS);
let P be RedSequence of ==>.-relation(TS), k such that
B: k in dom P & k + 1 in dom P;
consider s, u, t, v such that
C: P.k = [s, u] & P.(k + 1) = [t, v] by B, ThRedSeq4;
[[s, u], [t, v]] in ==>.-relation(TS) by B, C, REWRITE1:def 2;
then u <> v by A, ThRel110;
then (P.k)`2 <> v by C, MCART_1:7;
hence thesis by C, MCART_1:7;
end;
theorem ThRedSeq125:
not <%>E in rng dom (the Tran of TS) implies
for P being RedSequence of ==>.-relation(TS), k, l st
k in dom P & l in dom P & k < l holds (P.k)`2 <> (P.l)`2
proof
assume
A: not <%>E in rng dom (the Tran of TS);
defpred P[Nat] means
for P being RedSequence of ==>.-relation(TS), k, l st
len P = $1 & k in dom P & l in dom P & k < l holds (P.k)`2 <> (P.l)`2;
B: P[0];
C: now
let i;
assume
D: P[i];
now
let P be RedSequence of ==>.-relation(TS), k, l such that
D1: len P = i + 1 & k in dom P & l in dom P & k < l;
D2: i < len P by D1, NAT_1:13;
D3: 1 <= k & k <= len P & 1 <= l & l <= len P by D1, FINSEQ_3:27;
per cases;
suppose
k = 1 & l = len P;
hence (P.k)`2 <> (P.l)`2 by A, D1, ThRedSeq81;
end;
suppose
E: k <> 1 & l = len P;
then
E0'a: k > 1 by D3, XXREAL_0:1;
E0'b: l > 1 by D3,D1,XXREAL_0:1;
then consider Q being RedSequence of ==>.-relation(TS) such that
E0: <*P.1*>^Q = P & len Q + 1 = len P by E, LmRed10;
E0': len <*P.1*> = 1 by FINSEQ_1:57;
reconsider k1 = k - 1 as Nat by D3, NAT_1:21;
reconsider l1 = l - 1 as Nat by D3, NAT_1:21;
k1 > 1 - 1 by E0'a, XREAL_1:11; then
E2'a: k1 >= 0 + 1 by NAT_1:13;
l1 > 1 - 1 by E0'b, XREAL_1:11; then
E2'b: l1 >= 0 + 1 by NAT_1:13;
E2'c: k1 <= len Q + 1 - 1 by D3, E0, XREAL_1:11;
E2'd: l1 <= len Q + 1 - 1 by D3, E0, XREAL_1:11;
E3: k1 in dom Q & l1 in dom Q by E2'a, E2'b, E2'c, E2'd, FINSEQ_3:27;
E4: k1 < l1 by D1, XREAL_1:11;
P.k = (Q.k1) & P.l = (Q.l1) by D3, E0, E0', E0'a, E0'b, FINSEQ_1:37;
hence (P.k)`2 <> (P.l)`2 by D, D1, E0, E3, E4;
end;
suppose l <> len P;
then l < i + 1 by D1, D3, XXREAL_0:1;
then k < i + 1 & l <= i by D1, NAT_1:13, XXREAL_0:2;
then
E0: k <= i & l <= i by NAT_1:13;
then reconsider Q = P | i as RedSequence of ==>.-relation(TS)
by D3, REWRITE2:3, XXREAL_0:2;
E0'a: k <= len Q & l <= len Q by D2, E0, FINSEQ_1:80;
E1: len Q = i by D2, FINSEQ_1:80;
E2: k in dom Q by D3, E0'a, FINSEQ_3:27;
E3: l in dom Q by D3, E0'a, FINSEQ_3:27;
P.k = Q.k & P.l = Q.l by E0, FINSEQ_3:121;
hence (P.k)`2 <> (P.l)`2 by D, D1, E1, E2, E3;
end;
end;
hence P[i + 1];
end;
D: for i holds P[i] from NAT_1:sch 2(B, C);
let P be RedSequence of ==>.-relation(TS), k, l such that
E: k in dom P & l in dom P & k < l;
len P = len P;
hence thesis by D, E;
end;
theorem ThRedSeq130:
TS is deterministic implies
for P, Q being RedSequence of ==>.-relation(TS) st P.1 = Q.1 holds
for k st k in dom P & k in dom Q holds P.k = Q.k
proof
assume
A: TS is deterministic;
let P, Q be RedSequence of ==>.-relation(TS) such that
B: P.1 = Q.1;
defpred P[Nat] means
$1 in dom P & $1 in dom Q implies P.$1 = Q.$1;
D: P[0] by FINSEQ_3:27;
E: now
let k;
assume
E1: P[k];
now
assume
F0: k + 1 in dom P & k + 1 in dom Q;
per cases;
suppose
F1: k in dom P & k in dom Q;
G1: [P.k, P.(k + 1)] in ==>.-relation(TS) by F0, F1, REWRITE1:def 2;
G2: [Q.k, Q.(k + 1)] in ==>.-relation(TS) by F0, F1, REWRITE1:def 2;
thus P.(k + 1) = Q.(k + 1) by A, E1, F1, G1, G2, ThRel120';
end;
suppose not k in dom P or not k in dom Q;
then k = 0 by F0, REWRITE2:1;
hence P.(k + 1) = Q.(k + 1) by B;
end;
end;
hence P[k + 1];
end;
for k holds P[k] from NAT_1:sch 2(D, E);
hence thesis;
end;
theorem ThRedSeq135:
TS is deterministic implies
for P, Q being RedSequence of ==>.-relation(TS) st
P.1 = Q.1 & len P = len Q holds P = Q
proof
assume
A: TS is deterministic;
let P, Q be RedSequence of ==>.-relation(TS) such that
B1: P.1 = Q.1 and
B2: len P = len Q;
now
let k;
assume
C: k in dom P;
then 1 <= k & k <= len P by FINSEQ_3:27;
then k in dom Q by B2, FINSEQ_3:27;
hence P.k = Q.k by A, B1, C, ThRedSeq130;
end;
hence thesis by B2, FINSEQ_2:10;
end;
theorem ThRedSeq140:
TS is deterministic implies
for P, Q being RedSequence of ==>.-relation(TS) st
P.1 = Q.1 & (P.len P)`2 = (Q.len Q)`2 holds P = Q
proof
assume
A: TS is deterministic;
then
A': not <%>E in rng dom (the Tran of TS) by DefDetTS;
let P, Q be RedSequence of ==>.-relation(TS) such that
B1: P.1 = Q.1 and
B2: (P.len P)`2 = (Q.len Q)`2;
per cases by XXREAL_0:1;
suppose len P = len Q;
hence thesis by A, B1, ThRedSeq135;
end;
suppose
C: len P > len Q;
set k = len Q;
k >= 0 + 1 by NAT_1:13;
then
D: k in dom P & k in dom Q by C, FINSEQ_3:27;
then
E: (P.len Q)`2 = (P.len P)`2 by A, B1, B2, ThRedSeq130;
len P >= 0 + 1 by NAT_1:13;
then len P in dom P by FINSEQ_3:27;
hence thesis by A', C, D, E, ThRedSeq125;
end;
suppose
C: len P < len Q;
set k = len P;
k >= 0 + 1 by NAT_1:13; then
D: k in dom Q & k in dom P by C, FINSEQ_3:27; then
E: (Q.len P)`2 = (Q.len Q)`2 by A, B1, B2, ThRedSeq130;
len Q >= 0 + 1 by NAT_1:13;
then len Q in dom Q by FINSEQ_3:27;
hence thesis by A', C, D, E, ThRedSeq125;
end;
end;
begin :: Reductions
theorem ThRed60:
==>.-relation(TS) reduces [x, v], [y, w] implies ex u st v = u^w
proof
assume ==>.-relation(TS) reduces [x, v], [y, w];
then consider P being RedSequence of ==>.-relation(TS) such that
A: P.1 = [x, v] & P.len P = [y, w] by REWRITE1:def 3;
thus thesis by A, ThRedSeq60;
end;
theorem ThRed70:
==>.-relation(TS) reduces [x, u], [y, v] implies
==>.-relation(TS) reduces [x, u^w], [y, v^w]
proof
assume ==>.-relation(TS) reduces [x, u], [y, v];
then consider P being RedSequence of ==>.-relation(TS) such that
A: P.1 = [x, u] & P.len P = [y, v] by REWRITE1:def 3;
deffunc F(set) = [(P.$1)`1, dim2(P.$1, E)^w];
consider Q being FinSequence such that
B1: len Q = len P and
B2: for k st k in dom Q holds Q.k = F(k) from FINSEQ_1:sch 2;
for k being Element of NAT st k in dom Q & k + 1 in dom Q holds
[Q.k, Q.(k + 1)] in ==>.-relation(TS)
proof
let k be Element of NAT such that
C1: k in dom Q & k + 1 in dom Q;
1 <= k & k <= len Q & 1 <= k + 1 & k + 1 <= len Q by C1, FINSEQ_3:27;
then
C2: k in dom P & k + 1 in dom P by B1, FINSEQ_3:27;
[P.k, P.(k + 1)] in ==>.-relation(TS) by C2, REWRITE1:def 2;
then [[(P.k)`1, (P.k)`2], P.(k + 1)] in ==>.-relation(TS)
by C2, ThRedSeq5;
then [[(P.k)`1, (P.k)`2], [(P.(k + 1))`1, (P.(k + 1))`2]]
in ==>.-relation(TS) by C2, ThRedSeq5;
then [[(P.k)`1, dim2(P.k, E)], [(P.(k + 1))`1, (P.(k + 1))`2]]
in ==>.-relation(TS) by A, C2, ThRedSeq30;
then [[(P.k)`1, dim2(P.k, E)], [(P.(k + 1))`1, dim2(P.(k + 1), E)]]
in ==>.-relation(TS) by A, C2, ThRedSeq30;
then (P.k)`1, dim2(P.k, E) ==>. (P.(k + 1))`1, dim2(P.(k + 1), E), TS
by DefRel;
then (P.k)`1, dim2(P.k, E)^w ==>. (P.(k + 1))`1, dim2(P.(k + 1), E)^w, TS
by ThDir30;
then [[(P.k)`1, dim2(P.k, E)^w], [(P.(k + 1))`1, dim2(P.(k + 1), E)^w]]
in ==>.-relation(TS) by DefRel;
then [[(P.k)`1, dim2(P.k, E)^w], Q.(k + 1)] in ==>.-relation(TS)
by C1, B2;
hence thesis by C1, B2;
end;
then reconsider Q as RedSequence of ==>.-relation(TS)
by B1, REWRITE1:def 2;
D3: len P >= 0 + 1 by NAT_1:13;
then 1 in dom P by FINSEQ_3:27;
then
D1: dim2(P.1, E) = (P.1)`2 by A, ThRedSeq30
.= u by A, MCART_1:7;
D0: len Q >= 0 + 1 by NAT_1:13;
then 1 in dom Q by FINSEQ_3:27;
then
D: Q.1 = [(P.1)`1, dim2(P.1, E)^w] by B2
.= [x, u^w] by A, D1, MCART_1:7;
len P in dom P by D3, FINSEQ_3:27;
then
D2: dim2(P.len P, E) = (P.len P)`2 by A, ThRedSeq30
.= v by A, MCART_1:7;
len Q in dom Q by D0, FINSEQ_3:27;
then Q.len Q = [(P.len Q)`1, dim2(P.len Q, E)^w] by B2;
then Q.len Q = [y, v^w] by A, B1, D2, MCART_1:7;
hence thesis by D, REWRITE1:def 3;
end;
theorem ThRed81:
x, y -->. z, TS implies ==>.-relation(TS) reduces [x, y], [z, <%>E]
proof
assume x, y -->. z, TS;
then <*[x, y], [z, <%>E]*> is RedSequence of ==>.-relation(TS)
by ThRedSeq77;
then [[x, y], [z, <%>E]] in ==>.-relation(TS) by LmRed40;
hence thesis by REWRITE1:16;
end;
theorem ThRed82:
x, v -->. y, TS implies ==>.-relation(TS) reduces [x, v^w], [y, w]
proof
assume x, v -->. y, TS;
then <*[x, v^w], [y, w]*> is RedSequence of ==>.-relation(TS)
by ThRedSeq78;
then [[x, v^w], [y, w]] in ==>.-relation(TS) by LmRed40;
hence thesis by REWRITE1:16;
end;
theorem ThRed83:
x1, x2 ==>. y1, y2, TS implies ==>.-relation(TS) reduces [x1, x2], [y1, y2]
proof
assume x1, x2 ==>. y1, y2, TS;
then [[x1, x2], [y1, y2]] in ==>.-relation(TS) by DefRel;
hence thesis by REWRITE1:16;
end;
theorem ThRed84:
==>.-relation(TS) reduces [x, v], [y, w] implies len v >= len w
proof
assume ==>.-relation(TS) reduces [x, v], [y, w];
then consider P being RedSequence of ==>.-relation(TS) such that
A: P.1 = [x, v] & P.len P = [y, w] by REWRITE1:def 3;
thus thesis by A, ThRedSeq79;
end;
theorem
==>.-relation(TS) reduces [x, w], [y, v^w] implies v = <%>E
proof
assume ==>.-relation(TS) reduces [x, w], [y, v^w];
then len w >= len(v^w) by ThRed84;
then len v + len w <= 0 + len w by AFINSQ_1:20;
hence thesis by XREAL_1:8;
end;
theorem ThRed110:
not <%>E in rng dom (the Tran of TS) implies
(==>.-relation(TS) reduces [x, v], [y, w] implies
len v > len w or (x = y & v = w))
proof
assume
A: not <%>E in rng dom (the Tran of TS);
assume ==>.-relation(TS) reduces [x, v], [y, w];
then consider P being RedSequence of ==>.-relation(TS) such that
B: P.1 = [x, v] & P.len P = [y, w] by REWRITE1:def 3;
thus thesis by A, B, ThRedSeq110;
end;
theorem
not <%>E in rng dom (the Tran of TS) implies
(==>.-relation(TS) reduces [x, u], [y, u] implies x = y)
proof
assume
A: not <%>E in rng dom (the Tran of TS);
assume ==>.-relation(TS) reduces [x, u], [y, u];
then len u > len u or (x = y) by A, ThRed110;
hence thesis;
end;
theorem ThRed130:
not <%>E in rng dom (the Tran of TS) implies
(==>.-relation(TS) reduces [x, <%e%>], [y, <%>E] implies
[[x, <%e%>], [y, <%>E]] in ==>.-relation(TS))
proof
assume
A: not <%>E in rng dom (the Tran of TS);
assume ==>.-relation(TS) reduces [x, <%e%>], [y, <%>E];
then consider P being RedSequence of ==>.-relation(TS) such that
B: P.1 = [x, <%e%>] & P.len P = [y, <%>E] by REWRITE1:def 3;
C: len P = 1 + 1 by A, B, ThRedSeq100;
then 1 in dom P & 1 + 1 in dom P by FINSEQ_3:27;
hence thesis by B, C, REWRITE1:def 2;
end;
theorem ThRed140:
TS is deterministic implies
(==>.-relation(TS) reduces x, [y1, z] &
==>.-relation(TS) reduces x, [y2, z] implies y1 = y2)
proof
assume
A: TS is deterministic;
assume that
B1: ==>.-relation(TS) reduces x, [y1, z] and
B2: ==>.-relation(TS) reduces x, [y2, z];
consider P being RedSequence of ==>.-relation(TS) such that
C1: P.1 = x & P.len P = [y1, z] by B1, REWRITE1:def 3;
consider Q being RedSequence of ==>.-relation(TS) such that
C2: Q.1 = x & Q.len Q = [y2, z] by B2, REWRITE1:def 3;
(P.len P)`2 = z & (Q.len Q)`2 = z by C1, C2, MCART_1:7;
then P = Q by A, C1, C2, ThRedSeq140;
hence thesis by C1, C2, ZFMISC_1:33;
end;
begin :: Transitions
definition
let E, F, TS, x1, x2, y1, y2;
pred x1, x2 ==>* y1, y2, TS means :DefTran:
==>.-relation(TS) reduces [x1, x2], [y1, y2];
end;
theorem ThTran5:
for TS1 being non empty transition-system over F1,
TS2 being non empty transition-system over F2 st
the carrier of TS1 = the carrier of TS2 &
the Tran of TS1 = the Tran of TS2 holds
x1, x2 ==>* y1, y2, TS1 implies x1, x2 ==>* y1, y2, TS2
proof
let TS1 be non empty transition-system over F1,
TS2 be non empty transition-system over F2 such that
A1: the carrier of TS1 = the carrier of TS2 and
A2: the Tran of TS1 = the Tran of TS2;
assume x1, x2 ==>* y1, y2, TS1;
then ==>.-relation(TS1) reduces [x1, x2], [y1, y2] by DefTran;
then ==>.-relation(TS2) reduces [x1, x2], [y1, y2] by A1, A2, ThRel20;
hence thesis by DefTran;
end;
theorem ThTran10:
x, y ==>* x, y, TS
proof
==>.-relation(TS) reduces [x, y], [x, y] by REWRITE1:13;
hence thesis by DefTran;
end;
theorem ThTran20:
x1, x2 ==>* y1, y2, TS & y1, y2 ==>* z1, z2, TS implies
x1, x2 ==>* z1, z2, TS
proof
assume x1, x2 ==>* y1, y2, TS & y1, y2 ==>* z1, z2, TS;
then ==>.-relation(TS) reduces [x1, x2], [y1, y2] &
==>.-relation(TS) reduces [y1, y2], [z1, z2] by DefTran;
then ==>.-relation(TS) reduces [x1, x2], [z1, z2] by REWRITE1:17;
hence thesis by DefTran;
end;
theorem ThTran21:
x, y -->. z, TS implies x, y ==>* z, <%>E, TS
proof
assume x, y -->. z, TS;
then ==>.-relation(TS) reduces [x, y], [z, <%>E] by ThRed81;
hence thesis by DefTran;
end;
theorem
x, v -->. y, TS implies x, v^w ==>* y, w, TS
proof
assume x, v -->. y, TS;
then ==>.-relation(TS) reduces [x, v^w], [y, w] by ThRed82;
hence thesis by DefTran;
end;
theorem ThTran30:
x, u ==>* y, v, TS implies x, u^w ==>* y, v^w, TS
proof
assume x, u ==>* y, v, TS;
then ==>.-relation(TS) reduces [x, u], [y, v] by DefTran;
then ==>.-relation(TS) reduces [x, u^w], [y, v^w] by ThRed70;
hence thesis by DefTran;
end;
theorem ThTran40:
x1, x2 ==>. y1, y2, TS implies x1, x2 ==>* y1, y2, TS
proof
assume x1, x2 ==>. y1, y2, TS;
then ==>.-relation(TS) reduces [x1, x2], [y1, y2] by ThRed83;
hence thesis by DefTran;
end;
theorem ThTran50:
x, v ==>* y, w, TS implies ex u st v = u^w
proof
assume x, v ==>* y, w, TS;
then ==>.-relation(TS) reduces [x, v], [y, w] by DefTran;
hence thesis by ThRed60;
end;
theorem ThTran60:
x, v ==>* y, w, TS implies len w <= len v
proof
assume x, v ==>* y, w, TS;
then consider u such that
A: v = u^w by ThTran50;
thus thesis by A, LmXSeq50;
end;
theorem
x, w ==>* y, v^w, TS implies v = <%>E
proof
assume x, w ==>* y, v^w, TS;
then len(v^w) <= len w by ThTran60;
then len v + len w <= 0 + len w by AFINSQ_1:20;
hence thesis by XREAL_1:8;
end;
theorem ThTran70:
not <%>E in rng dom (the Tran of TS) implies
(x, u ==>* y, u, TS iff x = y)
proof
assume
A: not <%>E in rng dom (the Tran of TS);
thus x, u ==>* y, u, TS implies x = y
proof
assume x, u ==>* y, u, TS;
then ==>.-relation(TS) reduces [x, u], [y, u] by DefTran;
then consider p being RedSequence of ==>.-relation(TS) such that
B: p.1 = [x, u] & p.len p = [y, u] by REWRITE1:def 3;
thus thesis by A, B, ThRedSeq80;
end;
assume x = y;
hence thesis by ThTran10;
end;
theorem ThTran80:
not <%>E in rng dom (the Tran of TS) implies
(x, <%e%> ==>* y, <%>E, TS implies x, <%e%> ==>. y, <%>E, TS)
proof
assume
A1: not <%>E in rng dom (the Tran of TS);
assume x, <%e%> ==>* y, <%>E, TS;
then ==>.-relation(TS) reduces [x, <%e%>], [y, <%>E] by DefTran;
then [[x, <%e%>], [y, <%>E]] in ==>.-relation(TS) by A1, ThRed130;
hence thesis by DefRel;
end;
theorem ThTran90:
TS is deterministic implies
((x1, x2 ==>* y1, z, TS & x1, x2 ==>* y2, z, TS) implies y1 = y2)
proof
assume
A: TS is deterministic;
assume that
B1: x1, x2 ==>* y1, z, TS and
B2: x1, x2 ==>* y2, z, TS;
C1: ==>.-relation(TS) reduces [x1, x2], [y1, z] by B1, DefTran;
C2: ==>.-relation(TS) reduces [x1, x2], [y2, z] by B2, DefTran;
thus thesis by A, C1, C2, ThRed140;
end;
begin :: Acceptance of Words
definition
let E, F, TS, x1, x2, y;
pred x1, x2 ==>* y, TS means :DefAcc:
x1, x2 ==>* y, <%>E, TS;
end;
theorem ThAcc5:
for TS1 being non empty transition-system over F1,
TS2 being non empty transition-system over F2 st
the carrier of TS1 = the carrier of TS2 &
the Tran of TS1 = the Tran of TS2 holds
x, y ==>* z, TS1 implies x, y ==>* z, TS2
proof
let TS1 be non empty transition-system over F1,
TS2 be non empty transition-system over F2 such that
A1: the carrier of TS1 = the carrier of TS2 and
A2: the Tran of TS1 = the Tran of TS2;
assume x, y ==>* z, TS1;
then x, y ==>* z, <%>E, TS1 by DefAcc;
then x, y ==>* z, <%>E, TS2 by A1, A2, ThTran5;
hence thesis by DefAcc;
end;
theorem ThAcc10:
x, <%>E ==>* x, TS
proof
x, <%>E ==>* x, <%>E, TS by ThTran10;
hence thesis by DefAcc;
end;
theorem ThAcc20:
x, u ==>* y, TS implies x, u^v ==>* y, v, TS
proof
assume x, u ==>* y, TS;
then x, u ==>* y, <%>E, TS by DefAcc;
then x, u^v ==>* y, <%>E^v, TS by ThTran30;
hence thesis by AFINSQ_1:32;
end;
theorem
x, y -->. z, TS implies x, y ==>* z, TS
proof
assume x, y -->. z, TS;
then x, y ==>* z, <%>E, TS by ThTran21;
hence thesis by DefAcc;
end;
theorem
x1, x2 ==>. y, <%>E, TS implies x1, x2 ==>* y, TS
proof
assume x1, x2 ==>. y, <%>E, TS;
then x1, x2 ==>* y, <%>E, TS by ThTran40;
hence thesis by DefAcc;
end;
theorem
x, u ==>* y, TS & y, v ==>* z, TS implies x, u^v ==>* z, TS
proof
assume x, u ==>* y, TS & y, v ==>* z, TS;
then x, u^v ==>* y, v, TS & y, v ==>* z, <%>E, TS
by ThAcc20, DefAcc;
then x, u^v^<%>E ==>* y, v, TS & y, v^<%>E ==>* z, <%>E, TS by AFINSQ_1:32;
then x, u^v^<%>E ==>* y, v^<%>E, TS & y, v^<%>E ==>* z, <%>E, TS
by AFINSQ_1:32;
then x, u^v^<%>E ==>* z, <%>E, TS by ThTran20;
then x, u^v ==>* z, <%>E, TS by AFINSQ_1:32;
hence thesis by DefAcc;
end;
theorem ThAcc50:
not <%>E in rng dom (the Tran of TS) implies
(x, <%>E ==>* y, TS iff x = y)
proof
assume
A: not <%>E in rng dom (the Tran of TS);
thus x, <%>E ==>* y, TS implies x = y
proof
assume x, <%>E ==>* y, TS;
then x, <%>E ==>* y, <%>E, TS by DefAcc;
hence thesis by A, ThTran70;
end;
assume x = y;
hence thesis by ThAcc10;
end;
theorem
not <%>E in rng dom (the Tran of TS) implies
(x, <%e%> ==>* y, TS implies x, <%e%> ==>. y, <%>E, TS)
proof
assume
A: not <%>E in rng dom (the Tran of TS);
assume x, <%e%> ==>* y, TS;
then x, <%e%> ==>* y, <%>E, TS by DefAcc;
hence thesis by A, ThTran80;
end;
theorem
TS is deterministic implies
((x1, x2 ==>* y1, TS & x1, x2 ==>* y2, TS) implies y1 = y2)
proof
assume
A: TS is deterministic;
assume x1, x2 ==>* y1, TS & x1, x2 ==>* y2, TS;
then x1, x2 ==>* y1, <%>E, TS & x1, x2 ==>* y2, <%>E, TS by DefAcc;
hence thesis by A, ThTran90;
end;
begin :: Reachable States
definition
let E, F, TS, x, X;
func x-succ_of (X, TS) -> Subset of TS equals
{ s : ex t st t in X & t, x ==>* s, TS };
coherence
proof
set Y = { s : ex t st t in X & t, x ==>* s, TS };
Y c= the carrier of TS
proof
let y;
assume y in Y;
then ex s st s = y & ex t st t in X & t, x ==>* s, TS;
hence thesis;
end;
hence thesis;
end;
end;
theorem ThSucc10:
s in x-succ_of (X, TS) iff ex t st t in X & t, x ==>* s, TS
proof
thus s in x-succ_of (X, TS) implies ex t st t in X & t, x ==>* s, TS
proof
assume s in x-succ_of (X, TS);
then ex s' st s' = s & (ex t st t in X & t, x ==>* s', TS);
hence thesis;
end;
thus thesis;
end;
theorem
not <%>E in rng dom (the Tran of TS) implies <%>E-succ_of (S, TS) = S
proof
assume
A: not <%>E in rng dom (the Tran of TS);
B: now
let x;
assume x in <%>E-succ_of (S, TS);
then consider s such that
C1: s in S and
C2: s, <%>E ==>* x, TS by ThSucc10;
thus x in S by A, C1, C2, ThAcc50;
end;
now
let x;
assume
C: x in S;
x, <%>E ==>* x, TS by ThAcc10;
hence x in <%>E-succ_of (S, TS) by C;
end;
hence thesis by B, TARSKI:2;
end;
theorem
for TS1 being non empty transition-system over F1,
TS2 being non empty transition-system over F2 st
the carrier of TS1 = the carrier of TS2 &
the Tran of TS1 = the Tran of TS2 holds
x-succ_of (X, TS1) = x-succ_of (X, TS2)
proof
let TS1 be non empty transition-system over F1,
TS2 be non empty transition-system over F2 such that
A1: the carrier of TS1 = the carrier of TS2 and
A2: the Tran of TS1 = the Tran of TS2;
B: now
let y;
assume
C0: y in x-succ_of (X, TS1);
then reconsider q = y as Element of TS1;
consider p being Element of TS1 such that
C: p in X & p, x ==>* q, TS1 by C0, ThSucc10;
reconsider p as Element of TS2 by A1;
reconsider q as Element of TS2 by A1;
p, x ==>* q, TS2 by C, A1, A2, ThAcc5;
hence y in x-succ_of (X, TS2) by C;
end;
now
let y;
assume
C0: y in x-succ_of (X, TS2);
then reconsider q = y as Element of TS2;
consider p being Element of TS2 such that
C: p in X & p, x ==>* q, TS2 by C0, ThSucc10;
reconsider p as Element of TS1 by A1;
reconsider q as Element of TS1 by A1;
p, x ==>* q, TS1 by C, A1, A2, ThAcc5;
hence y in x-succ_of (X, TS1) by C;
end;
hence thesis by B, TARSKI:2;
end;