### The Mizar article:

### Minimization of Finite State Machines

**by****Miroslava Kaloper, and****Piotr Rudnicki**

- Received November 18, 1994
Copyright (c) 1994 Association of Mizar Users

- MML identifier: FSM_1
- [ MML identifier index ]

environ vocabulary FINSEQ_1, ARYTM_1, INT_1, FUNCT_1, RELAT_1, SGRAPH1, EQREL_1, TARSKI, FINSET_1, BOOLE, CARD_1, BORSUK_1, SETFAM_1, AMI_1, MATRIX_2, MCART_1, RELAT_2, FUNCOP_1, FUNCT_4, FSM_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XCMPLX_0, XREAL_0, NAT_1, INT_1, SETFAM_1, STRUCT_0, FINSET_1, GROUP_1, MCART_1, DOMAIN_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, FUNCT_4, FINSEQ_1, CARD_1, EQREL_1, BORSUK_1, MATRIX_2; constructors NAT_1, DOMAIN_1, FUNCT_4, BORSUK_1, MATRIX_2, PARTFUN1, GROUP_1, MEMBERED, XBOOLE_0; clusters SUBSET_1, FUNCT_1, INT_1, FINSET_1, CARD_1, XBOOLE_0, RELSET_1, FINSEQ_1, STRUCT_0, BORSUK_1, NAT_1, XREAL_0, MEMBERED, ZFMISC_1, FUNCT_2, PARTFUN1; requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM; definitions TARSKI, SETFAM_1, XBOOLE_0; theorems TARSKI, AXIOMS, SUBSET_1, SETFAM_1, MCART_1, FINSET_1, FUNCOP_1, RELSET_1, REAL_1, NAT_1, INT_1, CARD_1, CARD_2, FINSEQ_1, FINSEQ_2, FINSEQ_3, FINSEQ_4, FUNCT_1, FUNCT_2, FUNCT_4, RLVECT_1, EQREL_1, BORSUK_1, ZFMISC_1, MATRIX_2, GOBOARD1, STRUCT_0, GROUP_1, XBOOLE_0, XBOOLE_1, RELAT_1, XCMPLX_1; schemes NAT_1, FUNCT_1, FUNCT_2, FINSEQ_1, RECDEF_1, EQREL_1; begin :: Preliminaries reserve m, n, i, k for Nat; theorem Th1: for m, n being Nat holds m < n implies ex p being Nat st n = m+p & 1 <= p proof let m, n be Nat; assume A1: m < n; then consider p being Nat such that A2: n=m+p by NAT_1:28; take p; thus n = m+p by A2; assume p < 1; then p < 0+1; then p <= 0 & 0 <= p by NAT_1:18,38; then p = 0; hence contradiction by A1,A2; end; theorem Th2: i in Seg n implies i+m in Seg (n+m) proof assume i in Seg n; then 1 <= i & i <= n & i <= i+m by FINSEQ_1:3,NAT_1:29; then 1 <= i+m & i+m <= n+m by AXIOMS:22,REAL_1:55; hence thesis by FINSEQ_1:3; end; theorem Th3: i>0 & i+m in Seg (n+m) implies i in Seg n & i in Seg (n+m) proof assume A1: i > 0 & i+m in Seg (n+m); 1 = 0+1; then A2: 1 <= i by A1,NAT_1:38; A3: i+m <= n+m by A1,FINSEQ_1:3; then i <= n by REAL_1:53; hence i in Seg n by A2,FINSEQ_1:3; i <= i+m by NAT_1:29; then i <= n+m by A3,AXIOMS:22; hence thesis by A2,FINSEQ_1:3; end; theorem Th4: k < i implies ex j being Nat st j = i-k & 1 <= j proof assume k < i; then k - k < i - k by REAL_1:54; then A1: 0 < i - k by XCMPLX_1:14; then reconsider j = i - k as Nat by INT_1:16; take j; thus j = i - k; reconsider j' = j, 0' = 0 as Integer; 0' + 1 <= j' by A1,INT_1:20; hence 1 <= j; end; theorem Th5: for D being non empty set, df being FinSequence of D holds df is non empty implies ex d being Element of D, df1 being FinSequence of D st d = df.1 & df = <*d*>^df1 proof let D be non empty set, df be FinSequence of D; assume df is non empty; then len df <> 0 by FINSEQ_1:25; then A1: 1 <= len df by RLVECT_1:99; then 1 in dom df by FINSEQ_3:27; then reconsider d = df.1 as Element of D by FINSEQ_2:13; take d; reconsider lend1 = len df - 1 as Nat by A1,INT_1:18; deffunc F(Nat) = df.($1+1); consider dta being FinSequence such that A2: len dta = lend1 and A3: for j being Nat st j in Seg lend1 holds dta.j=F(j) from SeqLambda; now let j be Nat; assume j in dom dta; then A4: j in Seg len dta by FINSEQ_1:def 3; then j + 1 in Seg (len dta + 1) by Th2; then j+1 in Seg len df by A2,XCMPLX_1:27; then j+1 in dom df by FINSEQ_1:def 3; then df.(j+1) in D by FINSEQ_2:13; hence dta.j in D by A2,A3,A4; end; then reconsider dta as FinSequence of D by FINSEQ_2:14; take dta; thus d = df.1; now thus len df = len df; thus A5: len (<*d*>^dta) = len <*d*> + len dta by FINSEQ_1:35 .= 1 + (len df - 1) by A2,FINSEQ_1:57 .= len df by XCMPLX_1:27; let i be Nat; assume i in Seg len df; then A6: 1 <= i & i <= len df by FINSEQ_1:3; per cases by A6,REAL_1:def 5; suppose i = 1; hence df.i = (<*d*>^dta).i by FINSEQ_1:58; suppose A7: i > 1; then consider j being Nat such that A8: j = i-1 & 1 <= j by Th4; i - 1 <= lend1 by A6,REAL_1:49; then A9: j in Seg lend1 by A8,FINSEQ_1:3; len <*d*> = 1 by FINSEQ_1:57; then (<*d*>^dta).i = dta.j by A5,A6,A7,A8,FINSEQ_1:37 .= df.(j+1) by A3,A9 .= df.i by A8,XCMPLX_1:27; hence df.i = (<*d*>^dta).i; end; hence thesis by FINSEQ_2:10; end; theorem Th6: for df being FinSequence, d being set holds i in dom df implies (<*d*>^df).(i+1) = df.i proof let df be FinSequence, d be set; assume A1: i in dom df; then A2: i in Seg len df by FINSEQ_1:def 3; len (<*d*>^df) = len <*d*> + len df by FINSEQ_1:35 .= 1 + len df by FINSEQ_1:57; then i+1 in Seg len (<*d*>^df) by A2,Th2; then A3: i+1 in dom (<*d*>^df) by FINSEQ_1:def 3; A4: len <*d*> = 1 by FINSEQ_1:57; 1 <= i by A1,FINSEQ_3:27; then A5: 1 < i+1 by NAT_1:38; i+1 <= len (<*d*>^df) by A3,FINSEQ_3:27; hence (<*d*>^df).(i+1) = df.(i + 1 - len <*d*>) by A4,A5,FINSEQ_1:37 .= df.i by A4,XCMPLX_1:26; end; theorem Th7: for S being set, D1, D2 being non empty set, f1 being Function of S, D1, f2 being Function of D1, D2 holds f1 is bijective & f2 is bijective implies f2*f1 is bijective proof let S be set, D1, D2 be non empty set, f1 be Function of S, D1, f2 be Function of D1, D2; assume f1 is bijective & f2 is bijective; then A1: f1 is one-to-one onto & f2 is one-to-one onto by FUNCT_2:def 4; set f3 = f2*f1; rng f2 = D2 & rng f1 = D1 & dom f2 = D1 by A1,FUNCT_2:def 1,def 3; then rng f3 = D2 by RELAT_1:47; then f3 is one-to-one onto by A1,FUNCT_1:46,FUNCT_2:def 3; hence thesis by FUNCT_2:def 4; end; :: Partitions & Equivalence Relations theorem Th8: for Y being set, E1, E2 being Equivalence_Relation of Y holds Class E1 = Class E2 implies E1 = E2 proof let Y be set, E1, E2 be Equivalence_Relation of Y such that A1: Class E1 = Class E2; now let x be set; hereby assume A2: x in E1; then consider a, b being set such that A3: x = [a, b] & a in Y & b in Y by RELSET_1:6; reconsider a,b as Element of Y by A3; A4: b in Class (E1, b) by A3,EQREL_1:28; A5: a in Class (E1, b) by A2,A3,EQREL_1:27; Class (E1, b) in Class E2 by A1,A3,EQREL_1:def 5; then consider c being set such that A6: c in Y & Class (E1, b) = Class (E2, c) by EQREL_1:def 5; [a, c] in E2 & [b, c] in E2 by A4,A5,A6,EQREL_1:27; then [a, c] in E2 & [c, b] in E2 by EQREL_1:12; hence x in E2 by A3,EQREL_1:13; end; assume A7: x in E2; then consider a, b being set such that A8: x = [a, b] & a in Y & b in Y by RELSET_1:6; reconsider a, b as Element of Y by A8; A9: b in Class (E2, b) by A8,EQREL_1:28; A10: a in Class (E2, b) by A7,A8,EQREL_1:27; Class (E2, b) in Class E1 by A1,A8,EQREL_1:def 5; then consider c being set such that A11: c in Y & Class (E2, b) = Class (E1, c) by EQREL_1:def 5; [a, c] in E1 & [b, c] in E1 by A9,A10,A11,EQREL_1:27; then [a, c] in E1 & [c, b] in E1 by EQREL_1:12; hence x in E1 by A8,EQREL_1:13; end; hence thesis by TARSKI:2; end; theorem Th9: for W being non empty set, PW being a_partition of W holds PW is non empty proof let W be non empty set, PW be a_partition of W; consider x being Element of W; union PW = W by EQREL_1:def 6; then consider Y being set such that A1: x in Y & Y in PW by TARSKI:def 4; thus thesis by A1; end; theorem Th10: for Z being finite set, PZ being a_partition of Z holds PZ is finite proof let Z be finite set, PZ be a_partition of Z; bool Z is finite & PZ c= bool Z by FINSET_1:24; hence PZ is finite by FINSET_1:13; end; definition let W be non empty set; cluster -> non empty a_partition of W; coherence by Th9; end; definition let Z be finite set; cluster -> finite a_partition of Z; coherence by Th10; end; definition let X be non empty finite set; cluster non empty finite a_partition of X; existence proof consider p being a_partition of X; take p; thus p is non empty & p is finite; end; end; reserve X, A for non empty finite set, PX for a_partition of X, PA1, PA2 for a_partition of A; theorem Th11: for X being non empty set, PX being a_partition of X for Pi being set st Pi in PX ex x being Element of X st x in Pi proof let X be non empty set, PX be a_partition of X; let Pi be set; assume Pi in PX; then reconsider Pi as Element of PX; A1: Pi <> {} by EQREL_1:def 6; consider q being Element of Pi; q in Pi by A1; then reconsider q as Element of X; take q; thus thesis by A1; end; theorem Th12: card PX <= card X proof assume card PX > card X; then Card card X <` Card card PX by CARD_1:73; then Card X<`Card card PX by CARD_1:def 11; then Card X<`Card PX by CARD_1: def 11; then consider P_i being set such that A1: P_i in PX and A2: for x being set st x in X holds (proj PX).x <> P_i by FINSEQ_4:81; reconsider P_i as Element of PX by A1; consider q being Element of X such that A3: q in P_i by Th11; reconsider P_q = (proj PX).q as Element of PX; A4: q in P_q by BORSUK_1:def 1; P_q = P_i or P_q misses P_i by EQREL_1:def 6; hence contradiction by A2,A3,A4,XBOOLE_0:3; end; theorem Th13: PA1 is_finer_than PA2 implies card PA2 <= card PA1 proof assume A1: PA1 is_finer_than PA2; assume card PA1 < card PA2; then Card card PA1 <` Card card PA2 by CARD_1:73; then Card PA1 <` Card card PA2 by CARD_1:def 11; then A2: Card PA1 <` Card PA2 by CARD_1:def 11; defpred P[set,set] means $1 c= $2; A3: for e being set st e in PA1 ex u being set st u in PA2 & P[e,u] by A1,SETFAM_1:def 2; consider f being Function of PA1, PA2 such that A4: for e being set st e in PA1 holds P[e,f.e] from FuncEx1 (A3); consider p2_i being set such that A5: p2_i in PA2 and A6: for x being set st x in PA1 holds f.x <> p2_i by A2,FINSEQ_4:81; reconsider p2_i as Element of PA2 by A5; consider q being Element of A such that A7: q in p2_i by Th11; reconsider p2_q = f.((proj PA1).q) as Element of PA2; A8: q in (proj PA1).q by BORSUK_1:def 1; A9:(proj PA1).q c= p2_q by A4; p2_q=p2_i or p2_q misses p2_i by EQREL_1:def 6; hence contradiction by A6,A7,A8,A9,XBOOLE_0:3; end; theorem Th14: PA1 is_finer_than PA2 implies for p2 being Element of PA2 ex p1 being Element of PA1 st p1 c=p2 proof assume A1: PA1 is_finer_than PA2; consider E1 being Equivalence_Relation of A such that A2: PA1 = Class E1 by EQREL_1:43; consider E2 being Equivalence_Relation of A such that A3: PA2 = Class E2 by EQREL_1:43; let p2 be Element of PA2; reconsider p2' = p2 as Subset of A; consider a being set such that A4: a in A & p2' = Class (E2, a) by A3,EQREL_1:def 5; reconsider E1a = Class (E1, a) as Element of PA1 by A2,A4,EQREL_1:def 5; take E1a; consider p22 being set such that A5: p22 in PA2 & E1a c= p22 by A1,SETFAM_1:def 2; reconsider p22' = p22 as Subset of A by A5; consider b being set such that A6: b in A & p22' = Class (E2, b) by A3,A5,EQREL_1:def 5; a in Class (E1, a) by A4,EQREL_1:28; hence E1a c= p2 by A4,A5,A6,EQREL_1:31 ; end; theorem Th15: PA1 is_finer_than PA2 & card PA1 = card PA2 implies PA1 = PA2 proof defpred P[set,set] means $1 c= $2; assume A1: PA1 is_finer_than PA2 & card PA1 = card PA2; then A2: for e being set st e in PA1 ex u being set st u in PA2 & P[e,u] by SETFAM_1:def 2; consider f being Function of PA1, PA2 such that A3: for e being set st e in PA1 holds P[e,f.e] from FuncEx1 (A2); A4: dom f = PA1 & rng f c= PA2 by FUNCT_2:def 1,RELSET_1:12; A5: PA2 c= rng f proof let p2 be set; assume p2 in PA2; then reconsider p2' = p2 as Element of PA2; consider p1 being Element of PA1 such that A6: p1 c= p2' by A1,Th14; A7: p1 c= f.p1 by A3; A8: p2' meets f.p1 proof consider x being Element of A such that A9: x in p1 by Th11; thus thesis by A6,A7,A9,XBOOLE_0:3; end; reconsider fp1 = f.p1 as Subset of A by TARSKI:def 3; p2' = fp1 by A8,EQREL_1:def 6; hence p2 in rng f by A4,FUNCT_1:def 5; end; then rng f = PA2 by A4,XBOOLE_0:def 10; then A10: f is one-to-one by A1,FINSEQ_4:79; A11: for x being Element of PA1 holds x = f.x proof let x be Element of PA1; A12: x c= f.x by A3; reconsider fx = f.x as Subset of A by TARSKI:def 3; A13: fx c= A; assume x <> f.x; then consider a being set such that A14: a in x & not a in f.x or a in f.x & not a in x by TARSKI:2; consider E1 being Equivalence_Relation of A such that A15: PA1 = Class E1 by EQREL_1:43; A16: a in Class (E1, a) by A13,A14,EQREL_1:28; reconsider CE1a = Class (E1, a) as Element of PA1 by A13,A14,A15,EQREL_1:def 5; A17: CE1a c= f.CE1a by A3; reconsider fCE1a = f.CE1a as Subset of A by TARSKI:def 3; fx meets fCE1a by A12,A14,A16,A17,XBOOLE_0:3; then fx = fCE1a by EQREL_1:def 6; hence contradiction by A4,A10,A12,A14,A16,FUNCT_1:def 8; end; now let x be set; hereby assume x in PA1; then reconsider x' = x as Element of PA1; set fx = f.x'; fx in PA2; hence x in PA2 by A11; end; assume x in PA2; then consider y being set such that A18: y in dom f & x = f.y by A5,FUNCT_1:def 5; reconsider y' = y as Element of PA1 by A18,FUNCT_2:def 1; y' = f.y' by A11 ; hence x in PA1 by A18; end; hence thesis by TARSKI:2; end; begin :: Definitions and terminology for FSM definition let IAlph be set; struct (1-sorted) FSM over IAlph (# carrier -> set, Tran -> Function of [: the carrier, IAlph :], the carrier, InitS -> Element of the carrier #); end; definition let IAlph be set, fsm be FSM over IAlph; mode State of fsm is Element of fsm; end; definition let X be set; cluster non empty finite FSM over X; existence proof consider A being non empty finite set, T being Function of [: A, X :], A, I being Element of A; take S = FSM (# A, T, I #); thus S is non empty by STRUCT_0:def 1; thus S is finite by GROUP_1:def 13; end; end; definition :: WAYBEL11 cluster finite non empty 1-sorted; existence proof take 1-sorted(#{{}}#); thus thesis by GROUP_1:def 13,STRUCT_0:def 1; end; end; definition let S be finite 1-sorted; :: YELLOW13 cluster the carrier of S -> finite; coherence by GROUP_1:def 13; end; reserve IAlph, OAlph for non empty set, fsm for non empty FSM over IAlph, s for Element of IAlph, w, w1, w2 for FinSequence of IAlph, q, q', q1, q2 for State of fsm; definition let IAlph be non empty set; let fsm be non empty FSM over IAlph; let s be Element of IAlph, q be State of fsm; func s -succ_of q -> State of fsm equals :Def1: (the Tran of fsm).[q, s]; correctness; end; definition let IAlph be non empty set; let fsm be non empty FSM over IAlph; let q be State of fsm; let w be FinSequence of IAlph; func (q, w)-admissible -> FinSequence of the carrier of fsm means :Def2: it.1 = q & len it = len w + 1 & for i st 1 <= i & i <= len w ex wi being Element of IAlph, qi, qi1 being State of fsm st wi = w.i & qi = it.i & qi1 = it.(i+1) & wi-succ_of qi = qi1; existence proof set D = the carrier of fsm; set N = len w + 1; defpred P[Nat, set, set] means ex wn being Element of IAlph, q', q'' being Element of D st w.$1 = wn & q' = $2 & q'' = $3 & wn-succ_of q' = q''; A1: for n being Nat st 1 <= n & n < N holds for x being Element of D ex y being Element of D st P[n, x, y] proof let n be Nat; assume A2: 1 <= n; assume n < N; then n <= len w by NAT_1:38; then n in dom w by A2,FINSEQ_3:27; then reconsider wn = w.n as Element of IAlph by FINSEQ_2:13; let x be Element of D; wn-succ_of x = (the Tran of fsm).[x, wn] by Def1; hence thesis; end; consider p being FinSequence of D such that A3: len p = N & (p.1 = q or N = 0) & for n being Nat st 1 <= n & n < N holds P[n, p.n, p.(n+1)] from FinRecExD (A1); take p; thus p.1 = q by A3; thus len p = len w + 1 by A3; let i be Nat; assume A4: 1 <= i; assume i <= len w; then i < N by NAT_1:38; hence thesis by A3,A4; end; uniqueness proof let qs1, qs2 be FinSequence of the carrier of fsm such that A5: qs1.1 = q & len qs1 = len w + 1 & for i being Nat st 1 <= i & i <= len w ex wi being Element of IAlph, qs1i, qs1i1 being State of fsm st wi=w.i & qs1i=qs1.i & qs1i1 = qs1.(i+1) & wi-succ_of qs1i = qs1i1 and A6: qs2.1 = q & len qs2 = len w + 1 & for i being Nat st 1 <= i & i <= len w ex wi being Element of IAlph, qs2i, qs2i1 being State of fsm st wi = w.i & qs2i = qs2.i & qs2i1 = qs2.(i+1) & wi-succ_of qs2i = qs2i1; defpred P[Nat] means 1<=$1 & $1<=len qs1 implies qs1.$1 = qs2.$1; A7: P[0]; A8: now let k be Nat; assume A9: P[k]; thus P[k+1] proof assume A10: 1 <= k+1 & k+1 <= len qs1; A11: 0 = k or 0 < k & 0+1 = 1 by NAT_1:19; per cases by A11,NAT_1:38; suppose 0 = k; hence qs1.(k+1) = qs2.(k+1) by A5,A6; suppose A12: 1 <= k; A13: k <= len w by A5,A10,REAL_1:53; then consider i1 being Element of IAlph,qs1i,qs1i1 being State of fsm such that A14: i1 = w.k & qs1i = qs1.k & qs1i1 = qs1.(k+1) & i1-succ_of qs1i = qs1i1 by A5,A12; consider i2 being Element of IAlph,qs2i,qs2i1 being State of fsm such that A15: i2 = w.k & qs2i = qs2.k & qs2i1 = qs2.(k+1) & i2-succ_of qs2i = qs2i1 by A6,A12,A13; thus qs1.(k+1) = qs2.(k+1) by A5,A9,A12,A13,A14,A15,NAT_1:37; end; end; for k being Nat holds P[k] from Ind (A7,A8); hence qs1 = qs2 by A5,A6,FINSEQ_1:18; end; end; theorem Th16: (q, <*>IAlph)-admissible = <*q*> proof set eis = <*>IAlph; A1: len eis = 0 by FINSEQ_1:25; then A2: len <*q*> = len eis + 1 by FINSEQ_1:57; A3: <*q*>.1 = q by FINSEQ_1:57; for i being Nat st 1 <= i & i <= len eis ex wi being Element of IAlph, qi, qi1 being State of fsm st wi=eis.i & qi=<*q*>.i & qi1=<*q*>.(i+1)& wi-succ_of qi=qi1 by A1,AXIOMS:22; hence thesis by A2,A3,Def2; end; definition let IAlph be non empty set; let fsm be non empty FSM over IAlph; let w be FinSequence of IAlph; let q1, q2 be State of fsm; pred q1, w-leads_to q2 means :Def3: (q1, w)-admissible.(len w + 1) = q2; end; theorem Th17: q, <*>IAlph-leads_to q proof set eis = <*>IAlph; A1: (q, eis)-admissible = <*q*> by Th16; <*q*>.(len eis + 1) = <*q*>.(0+1) by FINSEQ_1:25 .= q by FINSEQ_1:57; hence thesis by A1,Def3; end; definition let IAlph be non empty set; let fsm be non empty FSM over IAlph; let w be FinSequence of IAlph; let q_seq be FinSequence of the carrier of fsm; pred q_seq is_admissible_for w means :Def4: ex q1 being State of fsm st q1 = q_seq.1 & (q1, w)-admissible = q_seq; end; theorem <*q*> is_admissible_for <*>IAlph proof (q, <*>IAlph)-admissible = <*q*> & q = <*q*>.1 by Th16,FINSEQ_1:57; hence thesis by Def4; end; definition let IAlph, fsm, q, w; func q leads_to_under w -> State of fsm means :Def5: q, w-leads_to it; existence proof len (q, w)-admissible=len w +1 & len w +1 in Seg (len w +1) by Def2,FINSEQ_1:6; then len w +1 in dom (q, w)-admissible by FINSEQ_1:def 3; then reconsider IT =(q,w)-admissible.(len w +1) as Element of fsm by FINSEQ_2:13; take IT; thus thesis by Def3; end; uniqueness proof let it1, it2 be Element of fsm; assume q, w-leads_to it1 & q, w-leads_to it2; then (q, w)-admissible.(len w + 1) = it1 & (q, w)-admissible.(len w + 1) = it2 by Def3; hence it1 = it2; end; end; theorem Th19: ((q, w)-admissible).(len (q, w)-admissible) = q' iff q, w-leads_to q' proof hereby set qs = (q, w)-admissible; assume A1: qs.(len qs) = q'; qs.1 = q & len qs = len w + 1 & for i being Nat st 1 <= i & i <= len w ex wi being Element of IAlph, qi, qi1 being State of fsm st wi=w.i & qi = qs.i & qi1 = qs.(i+1) & wi-succ_of qi = qi1 by Def2; hence q, w-leads_to q' by A1,Def3; end; assume q, w-leads_to q'; then (q, w)-admissible.(len w +1) = q' by Def3; hence thesis by Def2; end; theorem Th20: for k st 1 <= k & k <= len w1 holds (q1,w1^w2)-admissible.k = (q1,w1)-admissible.k proof set q1w = (q1,w1^w2)-admissible; set q1w1 = (q1,w1)-admissible; defpred P[Nat] means 1 <= $1 & $1 <= len w1 implies q1w.$1 = q1w1.$1; A1: P[0]; A2:now let k be Nat; assume A3: P[k]; thus P[k+1] proof assume A4: 1 <= k+1 & k+1 <= len w1; A5: 0 = k or 0 < k & 0 + 1= 1 by NAT_1:19; per cases by A4,A5,NAT_1:38; suppose A6: k = 0; hence q1w.(k+1) = q1 by Def2 .= q1w1.(k+1) by A6,Def2; suppose A7: 1 <= k & k <= len w1; len w1 <= len w1 + len w2 by NAT_1:29; then k <= len w1 + len w2 by A7,AXIOMS:22; then 1 <= k & k <= len (w1^w2) by A7,FINSEQ_1:35; then consider wk being Element of IAlph,qwk,qwk1 being State of fsm such that A8: wk = (w1^w2).k & qwk = q1w.k & qwk1 = q1w.(k+1) & wk-succ_of qwk = qwk1 by Def2; consider w1k being Element of IAlph, qw1k, qw1k1 being State of fsm such that A9: w1k = w1.k & qw1k = q1w1.k & qw1k1 = q1w1.(k+1) & w1k-succ_of qw1k = qw1k1 by A7,Def2; k in dom w1 by A7,FINSEQ_3:27; hence q1w.(k+1) = q1w1.(k+1) by A3,A7,A8,A9,FINSEQ_1:def 7; end; end; thus for k being Nat holds P[k] from Ind (A1,A2); end; theorem Th21: q1,w1-leads_to q2 implies (q1,w1^w2)-admissible.(len w1 + 1) = q2 proof assume A1: q1,w1-leads_to q2; set q1w = (q1,w1^w2)-admissible; set q1w1 = (q1,w1)-admissible; per cases by NAT_1:19; suppose A2: len w1 = 0; q1w1.1=q1 & q1w1.(len w1 + 1)=q2 by A1,Def2,Def3; hence q1w.(len w1 + 1) = q2 by A2,Def2; suppose A3: len w1 > 0; 0+1 = 1; then A4: 1 <= len w1 by A3,NAT_1:38; len (w1^w2) = len w1 + len w2 by FINSEQ_1:35; then len w1 <= len (w1^w2) by NAT_1:37; then consider wk being Element of IAlph,qwk,qwk1 being State of fsm such that A5: wk = (w1^w2).(len w1) & qwk = q1w.(len w1) & qwk1 = q1w.((len w1)+1) & wk-succ_of qwk = qwk1 by A4,Def2; consider w1k being Element of IAlph, qw1k, qw1k1 being State of fsm such that A6: w1k = w1.(len w1) & qw1k = q1w1.(len w1) & qw1k1 = q1w1.((len w1)+1) & w1k-succ_of qw1k = qw1k1 by A4,Def2; len w1 in Seg len w1 by A3,FINSEQ_1:5; then len w1 in dom w1 by FINSEQ_1:def 3; then A7: wk = w1k by A5,A6,FINSEQ_1:def 7; A8: len q1w1 = len w1 + 1 by Def2; thus (q1,w1^w2)-admissible.(len w1 + 1) = (the Tran of fsm).[qwk, wk] by A5,Def1 .= (the Tran of fsm).[qw1k, w1k] by A4,A5,A6,A7,Th20 .= qw1k1 by A6,Def1 .= q2 by A1,A6,A8,Th19; end; theorem Th22: q1,w1-leads_to q2 implies for k st 1 <= k & k <= len w2 + 1 holds (q1,w1^w2)-admissible.(len w1 + k) = (q2,w2)-admissible.k proof assume A1: q1,w1-leads_to q2; set q1w = (q1,w1^w2)-admissible; set q2w2 = (q2,w2)-admissible; defpred P[Nat] means 1 <= $1 & $1 <= len w2 + 1 implies q1w.(len w1 + $1) = q2w2.$1; A2: P[0]; A3: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A4: 1 <= k & k <= len w2 + 1 implies q1w.(len w1 + k) = q2w2.k; assume A5: 1 <= k+1 & k+1 <= len w2 + 1; per cases by NAT_1:19; suppose A6: k = 0; hence q1w.(len w1 + (k+1)) = q2 by A1,Th21 .= q2w2.(k+1) by A6,Def2; suppose A7: 0 < k; A8: 0+1 = 1; then A9: 1 <= k by A7,NAT_1:38; A10: k <= len w2 by A5,REAL_1:53; A11: 1 <= len w1 + k by A9,NAT_1:37; len (w1^w2) = len w1 + len w2 by FINSEQ_1:35; then len w1 + k <= len (w1^w2) by A10,REAL_1:55; then consider wi being Element of IAlph, qi,qi1 being State of fsm such that A12: wi = (w1^w2).(len w1 + k) & qi = q1w.(len w1 + k) & qi1 = q1w.((len w1 + k)+1) & wi-succ_of qi = qi1 by A11,Def2; consider w2i being Element of IAlph,q2i,q2i1 being State of fsm such that A13: w2i = w2.k & q2i = q2w2.k & q2i1 = q2w2.(k+1) & w2i-succ_of q2i = q2i1 by A9,A10,Def2; k in dom w2 by A9,A10,FINSEQ_3:27; then wi = w2i by A12,A13,FINSEQ_1:def 7; hence q1w.(len w1 + (k+1)) = q2w2.(k+1) by A4,A5,A7,A8,A12,A13,NAT_1:38, XCMPLX_1:1; end; thus for n being Nat holds P[n] from Ind(A2,A3); end; theorem Th23: q1,w1-leads_to q2 implies (q1,w1^w2)-admissible = Del((q1,w1)-admissible,(len w1 + 1))^(q2,w2)-admissible proof assume A1: q1,w1-leads_to q2; set q1w = (q1,w1^w2)-admissible; set q1w1 = (q1,w1)-admissible; set q2w2 = (q2,w2)-admissible; set Dw1 = Del ((q1,w1)-admissible,(len w1+1)); A2: len q1w1 = len w1 + 1 by Def2; dom q1w1 = Seg (len q1w1) by FINSEQ_1:def 3; then len w1+1 in dom q1w1 by A2,FINSEQ_1:5; then consider m be Nat such that A3: len q1w1 = m + 1 & len Dw1 = m by MATRIX_2:8; A4: len q1w1 = len w1 + 1 by Def2; then A5: len Dw1 = len w1 by A3,XCMPLX_1:2; A6: len q1w = len (w1^w2) +1 by Def2 .= len w1 + len w2 +1 by FINSEQ_1:35 .= len Dw1 +(len w2 +1) by A5,XCMPLX_1:1 .=len Dw1 + len q2w2 by Def2 .= len (Dw1^q2w2) by FINSEQ_1:35; now let k be Nat; assume A7: 1 <= k & k <= len q1w; per cases by A7,NAT_1:38; suppose A8: 1 <= k & k <= len w1; then A9: k in dom Dw1 by A5,FINSEQ_3:27; A10: 1 <= k & k < len w1 + 1 by A8,NAT_1:38; thus q1w.k = q1w1.k by A8,Th20 .= Dw1.k by A4,A10,GOBOARD1:8 .= (Dw1^q2w2).k by A9,FINSEQ_1:def 7; suppose A11: len w1+1 <= k & k <= len q1w; then len w1 + 1 - len w1 <= k - len w1 by REAL_1:49; then 1 <= k - len w1 by XCMPLX_1:26; then 0 <= k - len w1 by AXIOMS:22 ; then reconsider i = k - len w1 as Nat by INT_1:16; A12: k = len w1 + i by XCMPLX_1:27; len q1w = len (w1^w2) + 1 by Def2; then len w1+1 <= k & k <= len w1 + len w2 +1 by A11,FINSEQ_1:35; then len w1+1 <= k & k <= len w1 + (len w2 +1) by XCMPLX_1:1; then A13: 1 <= i & i <= len w2 +1 by A12,REAL_1:53; len Dw1+1 <= k & k <= len Dw1 + len q2w2 by A3,A6,A11,Def2,FINSEQ_1:35 ; then (Dw1^q2w2).k = q2w2.(k - len w1) by A5,FINSEQ_1:36; hence q1w.k = (Dw1^q2w2).k by A1,A12,A13,Th22; end; hence thesis by A6,FINSEQ_1:18; end; begin :: Mealy and Moore machines and their responses definition let IAlph be set, OAlph be non empty set; struct (FSM over IAlph) Mealy-FSM over IAlph, OAlph (# carrier -> set, Tran -> Function of [: the carrier, IAlph :], the carrier, OFun -> Function of [: the carrier, IAlph :], OAlph, InitS -> Element of the carrier #); struct (FSM over IAlph) Moore-FSM over IAlph, OAlph (# carrier -> set, Tran -> Function of [: the carrier, IAlph :], the carrier, OFun -> Function of the carrier, OAlph, InitS -> Element of the carrier #); end; definition let IAlph be set, X be finite non empty set, T be Function of [: X, IAlph :], X, I be Element of X; cluster FSM (# X, T, I #) -> finite non empty; coherence by GROUP_1:def 13,STRUCT_0:def 1; end; definition let IAlph be set, OAlph be non empty set, X be finite non empty set, T be Function of [: X, IAlph :], X, O be Function of [: X, IAlph :], OAlph, I be Element of X; cluster Mealy-FSM (# X, T, O, I #) -> finite non empty; coherence by GROUP_1:def 13,STRUCT_0:def 1; end; definition let IAlph be set, OAlph be non empty set, X be finite non empty set, T be Function of [: X, IAlph :], X, O be Function of X, OAlph, I be Element of X; cluster Moore-FSM (# X, T, O, I #) -> finite non empty; coherence by GROUP_1:def 13,STRUCT_0:def 1; end; definition let IAlph be set, OAlph be non empty set; cluster finite non empty Mealy-FSM over IAlph, OAlph; existence proof consider X be finite non empty set, T be Function of [: X, IAlph :], X, O be Function of [: X, IAlph :], OAlph, I be Element of X; take Mealy-FSM (# X, T, O, I #); thus thesis; end; cluster finite non empty Moore-FSM over IAlph, OAlph; existence proof consider X be finite non empty set, T be Function of [: X, IAlph :], X, O be Function of X, OAlph, I be Element of X; take Moore-FSM (# X, T, O, I #); thus thesis; end; end; reserve tfsm, tfsm1, tfsm2, tfsm3 for non empty Mealy-FSM over IAlph, OAlph, sfsm for non empty Moore-FSM over IAlph, OAlph, qs for State of sfsm, q, q1, q2, q3, qa, qb, qc, qa', qt, q1t, q2t for State of tfsm, q11, q12 for State of tfsm1, q21, q22 for State of tfsm2; definition let IAlph, OAlph, tfsm, qt, w; func (qt, w)-response -> FinSequence of OAlph means :Def6: len it = len w & for i st i in dom w holds it.i = (the OFun of tfsm).[(qt, w)-admissible.i, w.i]; existence proof set qs = (qt, w)-admissible; A1: Seg len w = dom w by FINSEQ_1:def 3; deffunc F(Nat) = (the OFun of tfsm).[qs.$1, w.$1]; consider p being FinSequence such that A2: len p = len w & for i being Nat st i in Seg len w holds p.i = F(i) from SeqLambda; rng p c= OAlph proof let y be set; assume y in rng p; then consider x being set such that A3: x in dom p & y = p.x by FUNCT_1:def 5; reconsider x as Nat by A3; A4: dom p = Seg len p by FINSEQ_1:def 3; A5: 1 <= x & x <= len p by A3,FINSEQ_3:27; A6: len qs = len p + 1 by A2,Def2; x <= len p + 1 by A5,NAT_1:37; then x in dom qs by A5,A6,FINSEQ_3:27; then reconsider qsx = qs.x as Element of tfsm by FINSEQ_2:13 ; reconsider wx = w.x as Element of IAlph by A1,A2,A3,A4,FINSEQ_2:13; p.x = (the OFun of tfsm).[qsx, wx] by A2,A3,A4; hence y in OAlph by A3; end; then reconsider p' = p as FinSequence of OAlph by FINSEQ_1:def 4; len p' = len w & for i being Nat st i in dom w holds p'.i = (the OFun of tfsm).[qs.i, w.i] by A1,A2; hence thesis; end; uniqueness proof set qs = (qt, w)-admissible; let it1, it2 be FinSequence of OAlph; assume A7: len it1 = len w & for i being Nat st i in dom w holds it1.i = (the OFun of tfsm).[qs.i, w.i]; assume A8: len it2 = len w & for i being Nat st i in dom w holds it2.i = (the OFun of tfsm).[qs.i, w.i]; then A9: dom w = dom it1 & dom it1 = dom it2 by A7,FINSEQ_3:31; now thus len it1 = len it2 by A7,A8; let i be Nat; assume 1 <= i & i <= len it1; then i in dom it1 by FINSEQ_3:27; then it1.i = (the OFun of tfsm).[qs.i, w.i] & it2.i = (the OFun of tfsm).[qs.i, w.i] by A7,A8,A9; hence it1.i = it2.i; end; hence it1 = it2 by FINSEQ_1:18; end; end; theorem Th24: (qt, <*>IAlph)-response = <*>OAlph proof len (qt, <*>IAlph)-response = len <*>IAlph by Def6 .= 0 by FINSEQ_1:32; hence thesis by FINSEQ_1:32; end; definition let IAlph, OAlph, sfsm, qs, w; func (qs, w)-response -> FinSequence of OAlph means :Def7: len it = len w + 1 & for i st i in Seg (len w + 1) holds it.i = (the OFun of sfsm).((qs, w)-admissible.i); existence proof set qs' = (qs, w)-admissible; A1: Seg len qs' = dom qs' by FINSEQ_1:def 3; deffunc F(Nat) = (the OFun of sfsm).(qs'.$1); consider p being FinSequence such that A2: len p = len qs' & for i being Nat st i in Seg len qs' holds p.i = F(i) from SeqLambda; rng p c= OAlph proof let y be set; assume y in rng p; then consider x being set such that A3: x in dom p & y = p.x by FUNCT_1:def 5; reconsider x as Nat by A3; A4: dom p = Seg len p by FINSEQ_1:def 3; then reconsider qsx = qs'.x as State of sfsm by A1,A2,A3,FINSEQ_2:13; p.x = (the OFun of sfsm).qsx by A2,A3,A4; hence y in OAlph by A3; end; then reconsider p' = p as FinSequence of OAlph by FINSEQ_1:def 4; A5: len qs' = len w + 1 by Def2; len p' = len qs' & for i being Nat st i in dom qs' holds p'.i = (the OFun of sfsm).(qs'.i) by A1,A2; hence thesis by A1,A5; end; uniqueness proof set qs' = (qs, w)-admissible; let it1, it2 be FinSequence of OAlph; assume A6: len it1 = len w + 1 & for i being Nat st i in Seg (len w + 1) holds it1.i = (the OFun of sfsm).(qs'.i); assume A7: len it2 = len w + 1 & for i being Nat st i in Seg (len w + 1) holds it2.i = (the OFun of sfsm).(qs'.i); now thus len it1 = len it2 by A6,A7; let i be Nat; assume 1 <= i & i <= len it1; then i in Seg len it1 by FINSEQ_1:3; then it1.i = (the OFun of sfsm).(qs'.i) & it2.i = (the OFun of sfsm).(qs'.i) by A6,A7; hence it1.i = it2.i; end; hence it1 = it2 by FINSEQ_1:18; end; end; theorem Th25: ((qs, w)-response).1 = (the OFun of sfsm).qs proof 1 <= 1 & 1 <= len w + 1 by NAT_1:37; then 1 in Seg (len w + 1) by FINSEQ_1:3; hence ((qs, w)-response).1 = (the OFun of sfsm).((qs, w)-admissible.1) by Def7 .= (the OFun of sfsm).qs by Def2; end; theorem Th26: q1t,w1-leads_to q2t implies (q1t,w1^w2)-response = (q1t,w1)-response ^ (q2t,w2)-response proof assume A1: q1t,w1-leads_to q2t; set q1w1 = (q1t,w1)-response, q2w2 = (q2t,w2)-response; set q1w1w2 = (q1t, w1^w2)-response; set Dq1w1w2a = Del((q1t,w1)-admissible,(len w1 + 1)); set OF = the OFun of tfsm; A2: len q1w1w2 = len (w1^w2) by Def6 .= len w1 + len w2 by FINSEQ_1:35 .= len q1w1 + len w2 by Def6 .= len q1w1 + len q2w2 by Def6 .= len (q1w1 ^ q2w2) by FINSEQ_1:35; now let k be Nat; assume A3: 1 <= k & k <= len q1w1w2; A4: len (q1t,w1)-admissible = len w1 + 1 by Def2; dom (q1t,w1)-admissible =Seg (len (q1t,w1)-admissible)by FINSEQ_1:def 3 ; then dom (q1t,w1)-admissible = Seg (len w1 + 1) by Def2; then len w1 + 1 in dom (q1t,w1)-admissible by FINSEQ_1:5; then consider m being Nat such that A5: len (q1t,w1)-admissible = m+1 & (len Dq1w1w2a = m) by MATRIX_2:8; A6: m+1 = len w1 + 1 by A5,Def2; then A7: len Dq1w1w2a = len w1 by A5,XCMPLX_1:2; A8: len q1w1 = len w1 by Def6; per cases by A3,NAT_1:38; suppose A9: 1 <= k & k <= len q1w1; then A10: 1 <= k & k <= len w1 by Def6; then A11: k in dom w1 by FINSEQ_3:27; 1 <= k & k <= len (w1^w2) by A3,Def6; then A12: k in dom (w1^w2) by FINSEQ_3:27; A13: k in dom q1w1 by A9,FINSEQ_3:27; A14: 1 <= k & k < len w1 + 1 by A10,NAT_1:38; A15: k in dom Dq1w1w2a by A7,A10,FINSEQ_3:27; thus q1w1w2.k = OF.[(q1t,w1^w2)-admissible.k,(w1^w2).k] by A12,Def6 .= OF.[(Dq1w1w2a ^ (q2t,w2)-admissible).k,(w1^w2).k] by A1,Th23 .= OF.[Dq1w1w2a.k,(w1^w2).k] by A15,FINSEQ_1:def 7 .= OF.[Dq1w1w2a.k,w1.k] by A11,FINSEQ_1:def 7 .= OF.[(q1t,w1)-admissible.k, w1.k] by A4,A14,GOBOARD1:8 .= (q1t,w1)-response.k by A11,Def6 .= ((q1t,w1)-response ^ (q2t,w2)-response).k by A13,FINSEQ_1:def 7; suppose A16: len q1w1 + 1 <= k & k <= len q1w1w2; 1 <= k & k <= len (w1^w2) by A3,Def6; then A17: k in dom (w1^w2) by FINSEQ_3:27; A18: len q1w1w2 = len (w1^w2) by Def6 .= len w1 + len w2 by FINSEQ_1:35 .= len q1w1 + len w2 by Def6 .= len q1w1 + len q2w2 by Def6; then A19: k <= len Dq1w1w2a + len w2 by A7,A8,A16,Def6; len w2 <= len w2 + 1 by NAT_1:29; then len Dq1w1w2a + len w2 <= len Dq1w1w2a + (len w2 +1) by AXIOMS:24; then len Dq1w1w2a + len w2 <= (len Dq1w1w2a + len w2) +1 by XCMPLX_1:1; then k <= (len Dq1w1w2a + len w2) + 1 by A19,AXIOMS:22; then k <= len Dq1w1w2a + (len w2 + 1) by XCMPLX_1:1; then A20: len Dq1w1w2a + 1 <= k & k <= len Dq1w1w2a + len (q2t,w2)-admissible by A5,A6,A16,Def2,Def6; A21: len q1w1w2 = len (w1^w2) by Def6 .= len w1 + len w2 by FINSEQ_1:35; then A22: len w1 + 1 <= k & k <= len w1 + len w2 by A16,Def6; A23: q1w1w2.k = OF.[(q1t,w1^w2)-admissible.k,(w1^w2).k] by A17,Def6 .= OF.[( Del((q1t,w1)-admissible,(len w1 + 1)) ^ (q2t,w2)-admissible).k, (w1^w2).k] by A1,Th23 .= OF.[(q2t,w2)-admissible.(k - len Dq1w1w2a),(w1^w2).k] by A20,FINSEQ_1:36 .= OF.[(q2t,w2)-admissible.(k - len w1 ),w2.(k-len w1)] by A7,A22,FINSEQ_1: 36; len q1w1 + 1 - len q1w1 <= k - len q1w1 by A16,REAL_1:49; then A24: 1 <= k - len q1w1 by XCMPLX_1:26; then 0 <= k - len q1w1 by AXIOMS:22; then reconsider p = k - len q1w1 as Nat by INT_1:16; k <= len q1w1 + len w2 by A16,A21,Def6; then k - len q1w1 <= len q1w1 + len w2 - len q1w1 by REAL_1:49; then k - len q1w1 <= len w2 by XCMPLX_1:26; then A25: p in dom w2 by A24,FINSEQ_3:27; (q1w1^q2w2).k = (q2t,w2)-response.p by A16,A18,FINSEQ_1:36 .= OF.[(q2t,w2)-admissible.p,w2.p] by A25,Def6 .= OF.[(q2t,w2)-admissible.(k-len w1),w2.(k-len q1w1)] by Def6 .= OF.[(q2t,w2)-admissible.(k-len w1),w2.(k-len w1)] by Def6; hence q1w1w2.k = (q1w1 ^ q2w2).k by A23; end; hence thesis by A2,FINSEQ_1:18; end; theorem Th27: q11, w1 -leads_to q12 & q21, w1 -leads_to q22 & (q12,w2)-response <> (q22,w2)-response implies (q11,w1^w2)-response <> (q21,w1^w2)-response proof assume A1: q11, w1 -leads_to q12 & q21, w1 -leads_to q22 & (q12,w2)-response <> (q22,w2)-response; assume A2: (q11,w1^w2)-response = (q21,w1^w2)-response; set r12 = (q12,w2)-response, r22 = (q22,w2)-response; set r1w1 = (q11,w1)-response, r2w1 = (q21,w1)-response; set w = w1 ^ w2; set r11 = (q11,w)-response; set r21 = (q21,w)-response; A3: len r2w1 = len w1 & len r1w1 = len w1 by Def6; A4: len r12 = len w2 & len r22 = len w2 by Def6; then consider j being Nat such that A5: j in Seg len w2 & r12.j <> r22.j by A1,FINSEQ_2:10; A6: j in dom r12 & j in dom r22 by A4,A5,FINSEQ_1:def 3; r11 = r1w1 ^ r12 & r21 = r2w1 ^ r22 by A1,Th26; then r11.(len w1 + j)=r12.j & r21.(len w1 + j)=r22.j by A3,A6,FINSEQ_1:def 7 ; hence contradiction by A2,A5; end; reserve OAlph_f for finite non empty set, tfsm_f for finite non empty Mealy-FSM over IAlph, OAlph_f, sfsm_f for finite non empty Moore-FSM over IAlph, OAlph_f; definition let IAlph, OAlph; let tfsm be non empty Mealy-FSM over IAlph, OAlph; let sfsm be non empty Moore-FSM over IAlph, OAlph; pred tfsm is_similar_to sfsm means for tw being FinSequence of IAlph holds <*(the OFun of sfsm).(the InitS of sfsm)*>^((the InitS of tfsm, tw)-response) = (the InitS of sfsm, tw)-response; end; theorem for sfsm being non empty finite Moore-FSM over IAlph, OAlph ex tfsm being non empty finite Mealy-FSM over IAlph, OAlph st tfsm is_similar_to sfsm proof let sfsm be non empty finite Moore-FSM over IAlph, OAlph; set S = the carrier of sfsm, T = the Tran of sfsm; set sOF = the OFun of sfsm, IS = the InitS of sfsm; deffunc F(Element of S,Element of IAlph) = sOF.(T.[$1, $2]); consider tOF being Function of [: S, IAlph:], OAlph such that A1: for q being Element of S, s be Element of IAlph holds tOF.[q, s] = F(q,s) from Lambda2D; take tfsm = Mealy-FSM (# S, T, tOF, IS #); let tw be FinSequence of IAlph; set tIS = the InitS of tfsm; set twa = (tIS, tw)-admissible; set swa = (IS, tw)-admissible; defpred P[Nat] means $1 in Seg (len tw + 1) implies twa.$1 = swa.$1; A2: P[0] by FINSEQ_1:3; A3: for i being Nat st P[i] holds P[i+1] proof let i be Nat; assume A4: i in Seg (len tw + 1) implies twa.i = swa.i; assume (i+1) in Seg (len tw + 1); then A5: i+1 <= len tw + 1 by FINSEQ_1:3; per cases by A5,NAT_1:19,REAL_1:53; suppose A6: i = 0; twa.1=the InitS of tfsm & swa.1 = the InitS of sfsm by Def2; hence twa.(i+1) = swa.(i+1) by A6; suppose i > 0 & i <= len tw; then A7: 0+1=1 implies 1 <= i & i <= len tw by NAT_1:38; then A8: 1<=i & i<=len tw + 1 by NAT_1:37; consider twi being Element of IAlph, tqi, tqi1 being Element of tfsm such that A9: twi = tw.i & tqi = twa.i & tqi1 = twa.(i+1) & twi-succ_of tqi = tqi1 by A7,Def2; consider swi being Element of IAlph, sqi, sqi1 being Element of sfsm such that A10: swi = tw.i & sqi = swa.i & sqi1 = swa.(i+1) & swi-succ_of sqi = sqi1 by A7,Def2; thus twa.(i+1) = (the Tran of sfsm).[sqi, swi] by A4,A8,A9,A10,Def1, FINSEQ_1:3 .= swa.(i+1) by A10,Def1; end; A11: for i being Nat holds P[i] from Ind( A2, A3 ); now thus len (<*sOF.IS*>^((tIS, tw)-response)) = len <*sOF.IS*> + len ((tIS, tw)-response) by FINSEQ_1:35 .= 1+len ((tIS, tw)-response) by FINSEQ_1:57 .=len tw +1 by Def6; thus len ((IS, tw)-response) = len tw + 1 by Def7; let i be Nat; assume A12: i in Seg (len tw + 1); then A13: 1 <= i & i <= len tw + 1 by FINSEQ_1:3; per cases by A13,REAL_1:def 5; suppose A14: i = 1; then i in Seg 1 by FINSEQ_1:4,TARSKI:def 1; then i in dom <*sOF.IS*> by FINSEQ_1:55; hence (<*sOF.IS*>^((tIS, tw)-response)).i = <*sOF.IS*>.i by FINSEQ_1:def 7 .= sOF.IS by A14,FINSEQ_1:57 .= ((IS, tw)-response).i by A14,Th25; suppose 1 < i; then consider j being Nat such that A15: j = i - 1 & 1 <= j by Th4; A16: i = j + 1 by A15,XCMPLX_1:27; A17: len <*sOF.IS*> = 1 by FINSEQ_1:57; A18: j <= len tw by A13,A16,REAL_1:53; then A19: j in Seg len tw by A15,FINSEQ_1:3; 1 <= j & j <= len tw + 1 by A15,A18,NAT_1:37; then A20:j in Seg (len tw + 1) by FINSEQ_1:3; len ((tIS, tw)-response) = len tw by Def6; then A21:j in dom ((tIS, tw)-response) by A19,FINSEQ_1:def 3; consider swj being Element of IAlph, swaj, swai being Element of sfsm such that A22:swj = tw.j & swaj = swa.j & swai = swa.(j+1) & swj-succ_of swaj = swai by A15,A18,Def2; A23: T.[swaj, swj] = swai by A22,Def1; A24: j in dom tw by A19,FINSEQ_1:def 3; (<*sOF.IS*>^((tIS, tw)-response)).i = (tIS, tw)-response.j by A16,A17,A21,FINSEQ_1:def 7 .= (the OFun of tfsm).[(tIS, tw)-admissible.j, tw.j] by A24,Def6 .= tOF.[(IS, tw)-admissible.j, tw.j] by A11,A20 .= sOF.(T.[swaj, swj]) by A1,A22 .= (IS, tw)-response.i by A12,A16,A22,A23,Def7; hence (<*sOF.IS*>^((tIS, tw)-response)).i = (IS, tw)-response.i; end; hence thesis by FINSEQ_2:10; end; theorem ex sfsm_f st tfsm_f is_similar_to sfsm_f proof set S = the carrier of tfsm_f, T = the Tran of tfsm_f; set tOF = the OFun of tfsm_f, IS = the InitS of tfsm_f; set sS = [: S, OAlph_f :]; deffunc F(Element of sS,Element of IAlph) = [ T.[$1`1, $2], tOF.[$1`1, $2]]; consider sT being Function of [:sS, IAlph:], sS such that A1: for q being Element of sS, s being Element of IAlph holds sT.[q, s] = F(q,s) from Lambda2D; deffunc F(Element of S,Element of OAlph_f) = $2; consider sOF being Function of sS, OAlph_f such that A2: for q being Element of S, r being Element of OAlph_f holds sOF.[q, r] = F(q,r) from Lambda2D; consider r0 being Element of OAlph_f; set sSs = sS, sTs = sT; reconsider sOFs = sOF as Function of sSs, OAlph_f; set sI = [IS, r0]; reconsider sfsm_f = Moore-FSM (# sSs, sTs, sOFs, sI #) as finite non empty Moore-FSM over IAlph, OAlph_f; take sfsm_f; reconsider SI = sI as Element of sfsm_f; thus tfsm_f is_similar_to sfsm_f proof let tw be FinSequence of IAlph; set twa = (the InitS of tfsm_f, tw)-admissible; set swa = (the InitS of sfsm_f, tw)-admissible; defpred P[Nat] means $1 in Seg (len tw + 1) implies twa.$1 = (swa.$1)`1; A3: P[0] by FINSEQ_1:3; A4: for i being Nat st P[i] holds P[i+1] proof let i be Nat; assume A5: P[i]; assume i+1 in Seg (len tw +1); then A6: i+1 <= len tw +1 by FINSEQ_1:3; per cases by A6,NAT_1:19,REAL_1:53; suppose A7: i = 0; twa.1 = IS by Def2 .= [IS, r0]`1 by MCART_1:def 1 .= (swa.1)`1 by Def2; hence twa.(i+1) = (swa.(i+1))`1 by A7; suppose i > 0 & i <= len tw; then A8: 0+1=1 implies 1 <= i & i <= len tw by NAT_1:38; then A9: 1 <= i & i <= len tw + 1 by NAT_1:37; consider twi being Element of IAlph, tqi, tqi1 being Element of tfsm_f such that A10: twi = tw.i & tqi = twa.i & tqi1 = twa.(i+1) & twi-succ_of tqi = tqi1 by A8,Def2; consider swi being Element of IAlph, sqi, sqi1 being Element of sfsm_f such that A11: swi = tw.i & sqi = swa.i & sqi1 = swa.(i+1) & swi-succ_of sqi = sqi1 by A8,Def2; A12: sqi1 = sT.[sqi, swi] by A11,Def1 .= [T.[sqi`1, swi], tOF.[sqi`1, swi]] by A1; thus twa.(i+1) = T.[tqi, twi] by A10,Def1 .= (swa.(i+1))`1 by A5,A9,A10,A11,A12,FINSEQ_1:3,MCART_1:def 1; end; A13: for i being Nat holds P[i] from Ind ( A3, A4 ); now thus len (<*sOF.sI*>^((IS, tw)-response)) = len <*sOF.sI*> + len ((IS, tw)-response) by FINSEQ_1:35 .= 1 + len ((IS, tw)-response) by FINSEQ_1:57 .= len tw + 1 by Def6; thus len ((SI, tw)-response) = len tw + 1 by Def7; let i be Nat; assume A14: i in Seg (len tw + 1); then A15: 1 <= i & i <= len tw + 1 by FINSEQ_1:3; per cases by A15,REAL_1:def 5; suppose A16: i = 1; then i in Seg 1 by FINSEQ_1:4,TARSKI:def 1; then i in dom <*sOF.sI*> by FINSEQ_1:55; hence (<*sOF.sI*>^((IS, tw)-response)).i = <*sOF.sI*>.i by FINSEQ_1:def 7 .= sOF.sI by A16,FINSEQ_1:57 .= ((SI, tw)-response).i by A16,Th25; suppose 1 < i; then consider j being Nat such that A17:j = i-1 & 1 <= j by Th4; A18: i = j + 1 by A17,XCMPLX_1:27; A19:len <*sOF.sI*> = 1 by FINSEQ_1:57; A20: j <= len tw by A15,A18,REAL_1:53; then A21:j in Seg len tw by A17,FINSEQ_1:3; j <= len tw + 1 by A20,NAT_1:37; then A22:j in Seg (len tw + 1) by A17,FINSEQ_1:3; len ((IS, tw)-response) = len tw by Def6; then A23:j in dom ((IS, tw)-response) by A21,FINSEQ_1:def 3; consider swj being Element of IAlph, swaj, swai being Element of sfsm_f such that A24: swj = tw.j & swaj = swa.j & swai = swa.(j+1) & swj-succ_of swaj = swai by A17,A20,Def2; reconsider swaj as Element of sS; A25: swai = sT.[swaj, swj] by A24,Def1 .= [T.[swaj`1, swj], tOF.[swaj`1, swj]] by A1; A26: j in dom tw by A21,FINSEQ_1:def 3; (<*sOF.sI*>^((IS, tw)-response)).i = ((IS, tw)-response).j by A18,A19,A23,FINSEQ_1:def 7 .= (the OFun of tfsm_f).[(IS, tw)-admissible.j, tw.j] by A26,Def6 .= tOF.[swaj`1, swj] by A13,A22,A24 .= (the OFun of sfsm_f).(swa.i) by A2,A18,A24,A25 .= (SI, tw)-response.i by A14,Def7; hence (<*sOF.sI*>^((IS, tw)-response)).i = (SI, tw)-response.i; end; hence thesis by FINSEQ_2:10; end; end; begin :: Equivalence of states and machines (for Mealy machines) definition let IAlph, OAlph be non empty set, tfsm1, tfsm2 be non empty Mealy-FSM over IAlph, OAlph; pred tfsm1, tfsm2-are_equivalent means :Def9: for w being FinSequence of IAlph holds (the InitS of tfsm1,w)-response = (the InitS of tfsm2,w)-response; reflexivity; symmetry; end; theorem Th30: tfsm1, tfsm2-are_equivalent & tfsm2, tfsm3-are_equivalent implies tfsm1, tfsm3-are_equivalent proof assume A1: tfsm1, tfsm2-are_equivalent & tfsm2, tfsm3-are_equivalent; set IS1 = the InitS of tfsm1, IS2 = the InitS of tfsm2; set IS3 = the InitS of tfsm3; let w1 be FinSequence of IAlph; thus (IS1, w1)-response = (IS2, w1)-response by A1,Def9 .= (IS3, w1)-response by A1,Def9; end; definition let IAlph, OAlph, tfsm, qa, qb; pred qa, qb-are_equivalent means :Def10: for w holds (qa, w)-response = (qb, w)-response; reflexivity; symmetry; end; canceled 2; theorem q1, q2-are_equivalent & q2, q3-are_equivalent implies q1, q3-are_equivalent proof assume A1: q1, q2-are_equivalent & q2, q3-are_equivalent; thus q1, q3-are_equivalent proof let w be FinSequence of IAlph; (q1, w)-response = (q2, w)-response & (q2, w)-response = (q3, w)-response by A1,Def10; hence (q1, w)-response = (q3, w)-response; end; end; theorem Th34: qa' = (the Tran of tfsm).[qa, s] implies for i st i in Seg (len w + 1) holds (qa, <*s*>^w)-admissible.(i+1) = (qa', w)-admissible.i proof assume A1: qa' = (the Tran of tfsm).[qa, s]; set sw = (<*s*>^w); A2: len sw = len <*s*> + len w by FINSEQ_1:35 .= len w + 1 by FINSEQ_1:57; A3: sw.1 = s by FINSEQ_1:58; defpred P[Nat] means $1 in Seg (len w + 1) implies (qa, <*s*>^w)-admissible.($1+1) = (qa', w)-admissible.$1; A4: P[0] by FINSEQ_1:3; A5: for j being Nat st P[j] holds P[j+1] proof let j be Nat; assume A6: j in Seg (len w + 1) implies (qa, <*s*>^w)-admissible.(j+1) = (qa', w)-admissible.j; assume A7: j+1 in Seg (len w + 1); per cases; suppose A8: j = 0; A9: (qa', w)-admissible.1 = qa' by Def2; A10: 1 <= 1 & 1 <= len sw by A2,A7,A8,FINSEQ_1:3; set a_adm = (qa, sw)-admissible; consider swi1 being Element of IAlph, a1, a11 being Element of tfsm such that A11: swi1 = sw.1 & a1 = a_adm.1 & a11 = a_adm.(1+1) & swi1-succ_of a1 = a11 by A10,Def2; (the Tran of tfsm).[a1, swi1] = a11 by A11,Def1; hence (qa, <*s*>^w)-admissible.(j+1+1) = (qa', w)-admissible.(j+1) by A1,A3,A8,A9,A11,Def2; suppose j <> 0; then A12: j > 0 by NAT_1:19; then A13: j in Seg len w by A7,Th3; then A14: 1 <= j & j <= len w by FINSEQ_1:3; then A15: 1 <= j+1 & j+1 <= len sw by A2,NAT_1:37,REAL_1:55; set a_adm = (qa, sw)-admissible, a_adm' = (qa', w)-admissible; consider swj1 being Element of IAlph, aj1, aj11 being Element of tfsm such that A16: swj1 = sw.(j+1) & aj1 = a_adm.(j+1) & aj11 = a_adm.(j+1+1) & swj1-succ_of aj1 = aj11 by A15,Def2; consider wj being Element of IAlph, aj', aj1' being Element of tfsm such that A17: wj = w.j & aj' = a_adm'.j & aj1' = a_adm'.(j+1) & wj-succ_of aj' = aj1' by A14,Def2; j in dom w by A13,FINSEQ_1:def 3; hence (qa,<*s*>^w)-admissible.(j+1+1) = (qa',w)-admissible.(j+1) by A6,A7,A12,A16,A17,Th3,Th6; end; thus for i being Nat holds P[i] from Ind(A4,A5); end; theorem Th35: qa' = (the Tran of tfsm).[qa, s] implies (qa, <*s*>^w)-response = <*(the OFun of tfsm).[qa, s]*>^(qa', w)-response proof assume A1: qa' = (the Tran of tfsm).[qa, s]; set OF=the OFun of tfsm; set a_sresp = OF.[qa, s]; A2:len (<*s*>^w) = len <*s*> + len w by FINSEQ_1:35 .= 1+len w by FINSEQ_1:57; now thus len ((qa, <*s*>^w)-response) = 1 + len w by A2,Def6; thus A3: len (<*a_sresp*>^(qa', w)-response) = len <*a_sresp*> + len ((qa', w)-response) by FINSEQ_1:35 .= 1 + len ((qa', w)-response) by FINSEQ_1:57 .= 1 + len w by Def6; let j be Nat; assume A4: j in Seg (1 + len w); then A5: 1 <= j & j <= 1 + len w by FINSEQ_1:3; A6: j in dom(<*s*>^w) by A2,A4,FINSEQ_1:def 3; per cases by A5,REAL_1:def 5; suppose A7:j = 1; thus (qa, <*s*>^w)-response.j = OF.[(qa, <*s*>^w)-admissible.j, (<*s*>^w).j] by A6,Def6 .= OF.[qa,(<*s*>^w).j] by A7,Def2 .= a_sresp by A7,FINSEQ_1:58 .= (<*a_sresp*>^(qa', w)-response).j by A7,FINSEQ_1:58; suppose A8: j > 1; then consider i being Nat such that A9: i = j - 1 & 1 <= i by Th4; A10: len <*a_sresp*> = 1 by FINSEQ_1:57; j <= len w + 1 by A4,FINSEQ_1:3; then j - 1 <= len w + 1-1 by REAL_1:49; then i <= len w by A9,XCMPLX_1:26; then A11: i in Seg len w by A9,FINSEQ_1:3; then A12: i+1 in Seg (1 + len w) by Th2; A13: i in Seg (1 + len w) by A11,FINSEQ_2:9; A14: i in dom w by A11,FINSEQ_1:def 3; A15: i+1 in dom(<*s*>^w) by A2,A12,FINSEQ_1:def 3; (<*a_sresp*>^((qa', w)-response)).j = (qa', w)-response.i by A3,A5,A8,A9,A10,FINSEQ_1:37 .= OF.[(qa', w)-admissible.i, w.i] by A14,Def6 .= OF.[(qa', w)-admissible.i, (<*s*>^w).(i+1)] by A14,Th6 .= OF.[(qa, <*s*>^w)-admissible.(i+1), (<*s*>^w).(i+1)]by A1,A13,Th34 .= (qa, <*s*>^w)-response.(i+1) by A15,Def6 .= ((qa, <*s*>^w)-response).j by A9,XCMPLX_1:27; hence (qa, <*s*>^w)-response.j = (<*a_sresp*>^(qa', w)-response).j; end; hence thesis by FINSEQ_2:10; end; theorem Th36: qa, qb-are_equivalent iff for s holds (the OFun of tfsm).[qa, s] = (the OFun of tfsm).[qb, s] & (the Tran of tfsm).[qa, s], (the Tran of tfsm).[qb, s]-are_equivalent proof set OF= the OFun of tfsm; hereby assume A1:qa, qb-are_equivalent; let s be Element of IAlph; len <*s*> = 1 by FINSEQ_1:57; then A2: 1 in dom <*s*> by FINSEQ_3:27; thus A3: OF.[qa, s] = OF.[(qa, <*s*>)-admissible.1, s] by Def2 .= OF.[(qa, <*s*>)-admissible.1, <*s*>.1] by FINSEQ_1:57 .= (qa, <*s*>)-response.1 by A2,Def6 .= (qb, <*s*>)-response.1 by A1,Def10 .= OF.[(qb, <*s*>)-admissible.1, <*s*>.1] by A2,Def6 .= OF.[(qb,<*s*>)-admissible.1,s] by FINSEQ_1:57 .= OF.[qb,s] by Def2; set qa' = (the Tran of tfsm).[qa, s]; set qb' = (the Tran of tfsm).[qb, s]; now let w be FinSequence of IAlph; (qa, <*s*>^w)-response = <*OF.[qa,s]*>^(qa', w)-response & (qb, <*s*>^w)-response = <*OF.[qb,s]*>^(qb', w)-response & (qa, <*s*>^w)-response = (qb, <*s*>^w)-response by A1,Def10,Th35; hence (qa', w)-response = (qb', w)-response by A3,FINSEQ_1:46; end; hence (the Tran of tfsm).[qa, s], (the Tran of tfsm).[qb, s]-are_equivalent by Def10; end; assume A4: for s being Element of IAlph holds (the OFun of tfsm).[qa, s] = (the OFun of tfsm).[qb, s] & (the Tran of tfsm).[qa, s], (the Tran of tfsm).[qb, s]-are_equivalent; let w be FinSequence of IAlph; per cases; suppose A5: w = <*>IAlph; hence (qa, w)-response = <*>OAlph by Th24 .= (qb, w)-response by A5,Th24; suppose w <> {}; then consider s being Element of IAlph, wt being FinSequence of IAlph such that A6: s = w.1 & w = <*s*>^wt by Th5; set qa' = (the Tran of tfsm).[qa, s]; set qb' = (the Tran of tfsm).[qb, s]; qa', qb'-are_equivalent by A4; then A7: (qa', wt)-response = (qb', wt)-response by Def10; set a_sresp =(the OFun of tfsm).[qa,s]; set b_sresp =(the OFun of tfsm).[qb,s]; thus (qa, w)-response = <*a_sresp*>^(qa', wt)-response by A6,Th35 .= <*b_sresp*>^(qb',wt)-response by A4,A7.= (qb,w)-response by A6,Th35; end; theorem qa, qb-are_equivalent implies for w, i st i in dom w ex qai, qbi being State of tfsm st qai = (qa, w)-admissible.i & qbi = ((qb, w)-admissible.i) & qai, qbi-are_equivalent proof assume A1: qa, qb-are_equivalent; let w be FinSequence of IAlph; defpred P[Nat] means $1 in Seg len w implies ex qai, qbi being Element of tfsm st qai = (qa, w)-admissible.$1 & qbi = ((qb, w)-admissible.$1) & qai, qbi-are_equivalent; A2: P[0] by FINSEQ_1:3; A3: for i being Nat st P[i] holds P[i+1] proof let i be Nat; assume A4: i in Seg len w implies ex qai, qbi being Element of tfsm st qai = (qa, w)-admissible.i & qbi = (qb, w)-admissible.i & qai, qbi-are_equivalent; assume (i+1) in Seg len w; then A5: 1<=i+1 & i+1<=len w by FINSEQ_1:3; i=0 or 0<i & 0+1=1 by NAT_1:19; then A6: 0 = i or 1 <= i & i <= len w by A5,NAT_1:38; per cases by A6,FINSEQ_1:3; suppose A7: i = 0; reconsider qai = (qa,w)-admissible.1, qbi = (qb,w)-admissible.1 as Element of tfsm by Def2; take qai, qbi; A8: qai = qa by Def2; thus qai = (qa, w)-admissible.(i+1) by A7; thus qbi = (qb, w)-admissible.(i+1) by A7; thus qai, qbi-are_equivalent by A1,A8,Def2; suppose A9: i in Seg len w; then consider qai, qbi being Element of tfsm such that A10: qai = (qa, w)-admissible.i & qbi = (qb, w)-admissible.i & qai, qbi-are_equivalent by A4; A11: 1 <= i & i <= len w by A9,FINSEQ_1:3; then consider wi being Element of IAlph, qai', qai1' being Element of tfsm such that A12: wi = w.i & qai' = (qa, w)-admissible.i & qai1' = (qa, w)-admissible.(i+1) & wi-succ_of qai' = qai1' by Def2; consider wi' being Element of IAlph, qbi', qbi1' being Element of tfsm such that A13: wi' = w.i & qbi' = (qb, w)-admissible.i & qbi1' = (qb, w)-admissible.(i+1) & wi'-succ_of qbi' = qbi1' by A11,Def2; take qai1'; take qbi1'; thus qai1' = (qa, w)-admissible.(i+1) by A12; thus qbi1' = (qb, w)-admissible.(i+1) by A13; A14: (the Tran of tfsm).[qai',wi] = qai1' by A12,Def1; (the Tran of tfsm).[qbi',wi'] = qbi1' by A13,Def1; hence qai1', qbi1'-are_equivalent by A10,A12,A13,A14,Th36; end; A15: Seg len w = dom w by FINSEQ_1:def 3; for i being Nat holds P[i] from Ind (A2, A3); hence thesis by A15; end; definition let IAlph, OAlph,tfsm, qa, qb, k; pred k-equivalent qa, qb means :Def11: for w st len w<=k holds (qa,w)-response = (qb,w)-response; end; theorem Th38: k-equivalent qa, qa proof let w be FinSequence of IAlph; assume len w <= k; thus thesis; end; theorem Th39: k-equivalent qa, qb implies k-equivalent qb, qa proof assume A1: k-equivalent qa, qb; thus k-equivalent qb, qa proof let w be FinSequence of IAlph; assume len w <= k; hence thesis by A1,Def11; end; end; theorem Th40: k-equivalent qa, qb & k-equivalent qb, qc implies k-equivalent qa, qc proof assume A1: k-equivalent qa, qb & k-equivalent qb, qc; thus k-equivalent qa, qc proof let w be FinSequence of IAlph; assume len w <= k; then (qa, w)-response = (qb, w)-response & (qb, w)-response = (qc, w)-response by A1,Def11; hence thesis; end; end; theorem Th41: qa,qb-are_equivalent implies k-equivalent qa,qb proof assume A1: qa,qb-are_equivalent; thus k-equivalent qa,qb proof let w be FinSequence of IAlph; assume len w <= k; thus thesis by A1,Def10; end; end; theorem Th42: 0-equivalent qa, qb proof let w be FinSequence of IAlph; assume len w <= 0; then len w = 0 by NAT_1:18; then A1: w = <*>IAlph by FINSEQ_1:32; hence (qa,w)-response=<*>OAlph by Th24.=(qb,w)-response by A1,Th24; end; theorem Th43: (k+m)-equivalent qa, qb implies k-equivalent qa, qb proof assume A1: (k+m)-equivalent qa, qb; A2: k <= k+m by NAT_1:29;let w be FinSequence of IAlph; assume len w <= k; then len w <= k+m by A2,AXIOMS:22; hence thesis by A1,Def11; end; theorem Th44: 1 <= k implies (k-equivalent qa, qb iff 1-equivalent qa, qb & for s being Element of IAlph, k1 being Nat st k1 = k - 1 holds k1-equivalent (the Tran of tfsm).[qa, s], (the Tran of tfsm).[qb, s]) proof set OF = the OFun of tfsm; assume A1: 1 <= k; hereby assume A2: k-equivalent qa, qb; thus A3: (1-equivalent qa, qb) proof let w be FinSequence of IAlph; assume len w <= 1; then len w <= k by A1,AXIOMS:22; hence (qa, w)-response = (qb, w)-response by A2,Def11; end; let s be Element of IAlph, k1 be Nat such that A4: k1 = k - 1; set qa' = (the Tran of tfsm).[qa, s]; set qb' = (the Tran of tfsm).[qb, s]; thus k1-equivalent qa', qb' proof let w be FinSequence of IAlph; assume A5: len w <= k1;set sw = <*s*>^w; A6: len (<*s*>^w) = len <*s*> + len w by FINSEQ_1:35 .=1+ len w by FINSEQ_1:57; len w + 1 <= k1 + 1 by A5,REAL_1:55; then A7: len sw <= k by A4,A6,XCMPLX_1:27; A8: (qa, sw)-response = <*OF.[qa, s]*>^(qa', w)-response & (qb, sw)-response = <*OF.[qb, s]*>^(qb', w)-response by Th35; A9: 0+1 in Seg (0+1) & len <*s*> = 1 by FINSEQ_1:6,57; then A10: 1 in dom <*s*> by FINSEQ_1:def 3; A11: OF.[qa, s] = OF.[(qa, <*s*>)-admissible.1, s] by Def2 .= OF.[(qa, <*s*>)-admissible.1, <*s*>.1] by FINSEQ_1:def 8 .= (qa, <*s*>)-response.1 by A10,Def6 .= (qb, <*s*>)-response.1 by A3,A9,Def11 .= OF.[(qb, <*s*>)-admissible.1, <*s*>.1] by A10,Def6 .= OF.[(qb, <*s*>)-admissible.1, s] by FINSEQ_1:def 8 .= OF.[qb, s] by Def2; (qa, sw)-response = (qb, sw)-response by A2,A7,Def11; hence (qa', w)-response = (qb', w)-response by A8,A11,FINSEQ_1:46; end; end; assume that A12: (1-equivalent qa, qb) and A13: for s being Element of IAlph, k1 being Nat st k1 = k - 1 holds k1-equivalent (the Tran of tfsm).[qa, s],(the Tran of tfsm).[qb, s]; thus k-equivalent qa, qb proof let w be FinSequence of IAlph such that A14: len w <= k; per cases; suppose A15: w = <*>IAlph; hence (qa,w)-response=<*>OAlph by Th24.=(qb,w)-response by A15,Th24; suppose w is non empty; then consider s being Element of IAlph, w' being FinSequence of IAlph such that A16: s = w.1 & w = <*s*>^w' by Th5; set qa' = (the Tran of tfsm).[qa, s], qb' = (the Tran of tfsm).[qb, s]; A17: (qa, w)-response = <*OF.[qa, s]*>^(qa', w')-response & (qb, w)-response = <*OF.[qb, s]*>^(qb', w')-response by A16,Th35; A18: 0+1 in Seg (0+1) & len <*s*> = 1 by FINSEQ_1:6,57; then A19: 1 in dom <*s*> by FINSEQ_1:def 3; A20: OF.[qa, s] = OF.[(qa, <*s*>)-admissible.1, s] by Def2 .= OF.[(qa, <*s*>)-admissible.1, <*s*>.1] by FINSEQ_1:def 8 .= (qa, <*s*>)-response.1 by A19,Def6 .= (qb, <*s*>)-response.1 by A12,A18,Def11 .= OF.[(qb, <*s*>)-admissible.1, <*s*>.1] by A19,Def6 .= OF.[(qb, <*s*>)-admissible.1, s] by FINSEQ_1:def 8 .= OF.[qb, s] by Def2; reconsider k1 = k-1 as Nat by A1,INT_1:18; A21: k1-equivalent qa', qb' by A13; len w = len <*s*> + len w' by A16,FINSEQ_1:35 .= len w' + 1 by FINSEQ_1:57; then len w' + 1 - 1 <= k1 by A14,REAL_1:49; then len w' <= k1 by XCMPLX_1:26; hence (qa, w)-response = (qb, w)-response by A17,A20,A21,Def11; end; end; definition let IAlph, OAlph, tfsm, i; func i-eq_states_EqR tfsm -> Equivalence_Relation of the carrier of tfsm means :Def12: for qa, qb holds [qa, qb] in it iff i-equivalent qa, qb; existence proof set S = the carrier of tfsm; defpred P[set, set] means ex qa, qb being Element of S st qa = $1 & qb = $2 & i-equivalent qa, qb; A1: now let x be set; assume x in S; then reconsider q = x as Element of S; i-equivalent q, q by Th38; hence P[x, x]; end; A2: now let x, y be set; assume P[x,y]; then consider qa, qb being Element of S such that A3: qa=x & qb=y & i-equivalent qa, qb; i-equivalent qb, qa by A3,Th39; hence P[y, x] by A3; end; A4: now let x, y, z be set; assume P[x,y]; then consider qa, qb being Element of S such that A5: qa = x & qb = y & i-equivalent qa, qb; assume P[y,z]; then consider qb', qc being Element of S such that A6:qb'=y & qc=z & i-equivalent qb',qc; i-equivalent qa,qc by A5,A6,Th40; hence P[x, z] by A5,A6; end; consider EqR being Equivalence_Relation of S such that A7: for x, y being set holds [x, y] in EqR iff x in S & y in S & P[x,y] from Ex_Eq_Rel (A1, A2, A4); take EqR; let qa, qb be Element of S; hereby assume [qa, qb] in EqR; then consider qa', qb' being Element of S such that A8: qa' = qa & qb' = qb & i-equivalent qa', qb' by A7; thus i-equivalent qa, qb by A8; end; assume i-equivalent qa, qb; hence [qa, qb] in EqR by A7; end; uniqueness proof set S = the carrier of tfsm; let IT1, IT2 be Equivalence_Relation of S; assume A9: for qa, qb being Element of S holds [qa,qb] in IT1 iff i-equivalent qa, qb; assume A10: for qa, qb being Element of S holds [qa,qb] in IT2 iff i-equivalent qa, qb; assume IT1 <> IT2; then consider x being set such that A11: x in IT1 & not x in IT2 or x in IT2 & not x in IT1 by TARSKI:2; per cases by A11; suppose A12: x in IT1 & not x in IT2; then consider qa, qb being set such that A13: x = [qa, qb] & qa in S & qb in S by RELSET_1:6; reconsider qa, qb as Element of S by A13; i-equivalent qa, qb by A9,A12,A13; hence contradiction by A10,A12,A13; suppose A14: x in IT2 & not x in IT1; then consider qa, qb being set such that A15: x = [qa, qb] & qa in S & qb in S by RELSET_1:6; reconsider qa, qb as Element of S by A15; i-equivalent qa, qb by A10,A14,A15; hence contradiction by A9,A14,A15; end; end; definition let IAlph, OAlph; let tfsm be non empty Mealy-FSM over IAlph, OAlph; let i; func i-eq_states_partition tfsm -> non empty a_partition of the carrier of tfsm equals :Def13: Class (i-eq_states_EqR tfsm); coherence proof thus Class (i-eq_states_EqR tfsm) is a_partition of the carrier of tfsm by EQREL_1:42; end; end; theorem Th45: (k+1)-eq_states_partition tfsm is_finer_than k-eq_states_partition tfsm proof set K = k-eq_states_partition tfsm; set K1 = (k+1)-eq_states_partition tfsm; set S = the carrier of tfsm; let X be set; assume A1: X in K1; then reconsider X' = X as Subset of S; consider q being Element of S such that A2: q in X by A1,Th11; reconsider Y = (proj K).q as Element of K; A3: K = Class (k-eq_states_EqR tfsm ) by Def13; reconsider Y' = Y as Element of Class (k-eq_states_EqR tfsm) by Def13; take Y; thus Y in K; let x be set; assume A4: x in X; then x in X'; then reconsider x'=x as Element of S; A5: K1 = Class ((k+1)-eq_states_EqR tfsm) by Def13; reconsider X' as Element of Class ((k+1)-eq_states_EqR tfsm) by A1,Def13; consider Q being set such that A6: Q in S & X' = Class ((k+1)-eq_states_EqR tfsm, Q) by A5,EQREL_1:def 5; A7: [q, Q] in (k+1)-eq_states_EqR tfsm by A2,A6,EQREL_1:27; [x', Q] in (k+1)-eq_states_EqR tfsm by A4,A6,EQREL_1:27; then [Q, x'] in (k+1)-eq_states_EqR tfsm by EQREL_1:12; then [q, x'] in (k+1)-eq_states_EqR tfsm by A7,EQREL_1:13; then (k+1)-equivalent q, x' by Def12; then k-equivalent q, x' by Th43; then [q, x'] in k-eq_states_EqR tfsm by Def12; then A8: [x', q] in k-eq_states_EqR tfsm by EQREL_1:12; consider Q being set such that A9: Q in S & Y' = Class (k-eq_states_EqR tfsm, Q) by A3,EQREL_1:def 5; reconsider Q as Element of S by A9; q in Y by BORSUK_1:def 1; then Class(k-eq_states_EqR tfsm,Q)=Class(k-eq_states_EqR tfsm,q)by A9, EQREL_1:31 ; hence thesis by A8,A9,EQREL_1:27; end; theorem Th46: Class (k-eq_states_EqR tfsm) = Class ((k+1)-eq_states_EqR tfsm)implies for m holds Class ((k+m)-eq_states_EqR tfsm) = Class (k-eq_states_EqR tfsm) proof set S = the carrier of tfsm; set CEk = Class (k-eq_states_EqR tfsm); set Ek = (k-eq_states_EqR tfsm); set CEk1 = Class ((k+1)-eq_states_EqR tfsm); set Ek1 = ((k+1)-eq_states_EqR tfsm); assume CEk = CEk1; then A1: Ek = Ek1 by Th8; defpred P[Nat] means Class ((k+$1)-eq_states_EqR tfsm) = CEk; A2: P[0]; A3: for m being Nat st P[m] holds P[m+1] proof let m be Nat; set CEkm=Class ((k+m)-eq_states_EqR tfsm); set Ekm=((k+m)-eq_states_EqR tfsm); set CEkm1 = Class ((k+(m+1))-eq_states_EqR tfsm); set Ekm1 = ((k+(m+1))-eq_states_EqR tfsm); assume CEkm = CEk; then A4: Ekm = Ek by Th8; now let x be set; hereby assume A5: x in CEkm1; then reconsider x' = x as Subset of S; A6: x' c= S; consider qa being set such that A7: qa in S & x' = Class (Ekm1, qa) by A5,EQREL_1:def 5; reconsider qa as Element of S by A7; now let y be set; hereby assume A8: y in x; then reconsider y' = y as Element of S by A6; [y, qa] in Ekm1 by A7,A8,EQREL_1:27; then (k+(m+1))-equivalent y', qa by Def12; then k-equivalent y', qa by Th43; then [y', qa] in Ek by Def12; hence y in Class (Ek, qa) by EQREL_1:27; end; assume A9: y in Class (Ek, qa); then reconsider y' = y as Element of S; A10: [y', qa] in Ek by A9,EQREL_1:27; A11: 1 <= k+m+1 by NAT_1:29; A12: (k+1)-equivalent y', qa by A1,A10,Def12; A13: 1 <= k+1 by NAT_1:29; then A14: 1-equivalent y', qa by A12,Th44; now let s be Element of IAlph, k1 be Nat; assume k1 = k+m+1-1; then A15: k1 = k+m by XCMPLX_1:26; set Sy' = (the Tran of tfsm).[y', s]; set Sqa = (the Tran of tfsm).[qa, s]; k = k+1-1 by XCMPLX_1:26; then k-equivalent Sy', Sqa by A12,A13, Th44; then [Sy', Sqa] in Ek by Def12; hence k1-equivalent Sy', Sqa by A4,A15,Def12; end; then (k+m+1)-equivalent y', qa by A11,A14,Th44; then (k+(m+1))-equivalent y', qa by XCMPLX_1:1; then [y', qa] in Ekm1 by Def12; hence y in x by A7,EQREL_1:27; end; then x = Class (Ek, qa) by TARSKI:2; hence x in CEk by EQREL_1:def 5; end; assume A16: x in CEk; then reconsider x' = x as Subset of S; consider qa being set such that A17: qa in S & x' = Class (Ek, qa) by A16,EQREL_1:def 5; reconsider qa as Element of S by A17; now let y be set; hereby assume A18: y in x; then reconsider y' = y as Element of S by A17; A19: [y', qa] in Ek by A17,A18,EQREL_1:27; A20: 1 <= k+m+1 by NAT_1:29; A21: (k+1)-equivalent y', qa by A1,A19,Def12; A22: 1 <= k+1 by NAT_1:29; then A23: 1-equivalent y', qa by A21,Th44; now let s be Element of IAlph, k1 be Nat; assume k1 = k+m+1-1; then A24: k1 = k+m by XCMPLX_1:26; set Sy' = (the Tran of tfsm).[y', s]; set Sqa = (the Tran of tfsm).[qa, s]; k = k+1-1 by XCMPLX_1:26; then k-equivalent Sy', Sqa by A21,A22, Th44; then [Sy', Sqa] in Ek by Def12; hence k1-equivalent Sy', Sqa by A4,A24,Def12; end; then (k+m+1)-equivalent y', qa by A20,A23,Th44; then (k+(m+1))-equivalent y', qa by XCMPLX_1:1; then [y', qa] in Ekm1 by Def12; hence y in Class (Ekm1, qa) by EQREL_1:27; end; assume A25: y in Class (Ekm1, qa); then reconsider y' = y as Element of S; [y, qa] in Ekm1 by A25,EQREL_1:27; then (k+(m+1))-equivalent y', qa by Def12; then k-equivalent y', qa by Th43; then [y', qa] in Ek by Def12; hence y in x by A17,EQREL_1:27; end; then x = Class (Ekm1, qa) by TARSKI:2; hence x in CEkm1 by EQREL_1:def 5; end; hence CEkm1 = CEk by TARSKI:2; end; thus for m being Nat holds P[m] from Ind(A2,A3); end; theorem Th47: k-eq_states_partition tfsm = (k+1)-eq_states_partition tfsm implies for m holds (k+m)-eq_states_partition tfsm = k-eq_states_partition tfsm proof assume A1: k-eq_states_partition tfsm = (k+1)-eq_states_partition tfsm; A2: k-eq_states_partition tfsm = Class (k-eq_states_EqR tfsm) & (k+1)-eq_states_partition tfsm = Class ((k+1)-eq_states_EqR tfsm) by Def13; let m be Nat; (k+m)-eq_states_partition tfsm = Class ((k+m)-eq_states_EqR tfsm) by Def13; hence thesis by A1,A2,Th46; end; theorem Th48: (k+1)-eq_states_partition tfsm <> k-eq_states_partition tfsm implies for i st i <= k holds (i+1)-eq_states_partition tfsm <> i-eq_states_partition tfsm proof assume A1: (k+1)-eq_states_partition tfsm <> k-eq_states_partition tfsm; let i be Nat such that A2: i <= k; assume A3: (i+1)-eq_states_partition tfsm = i-eq_states_partition tfsm; consider d being Nat such that A4: k = i+d by A2,NAT_1:28; A5: k-eq_states_partition tfsm = i-eq_states_partition tfsm by A3,A4,Th47; i <= k+1 by A2,NAT_1:37; then consider e being Nat such that A6: k+1 = i+e by NAT_1:28; thus contradiction by A1,A3,A5,A6,Th47; end; theorem Th49: for tfsm being finite non empty Mealy-FSM over IAlph, OAlph holds k-eq_states_partition tfsm = (k+1)-eq_states_partition tfsm or card (k-eq_states_partition tfsm) < card ((k+1)-eq_states_partition tfsm) proof let tfsm be finite non empty Mealy-FSM over IAlph, OAlph; set kp = k-eq_states_partition tfsm; set k1p = (k+1)-eq_states_partition tfsm; assume A1: kp <> k1p; A2: k1p is_finer_than kp by Th45; then card kp <= card k1p by Th13; then card kp = card k1p or card kp < card k1p by REAL_1:def 5; hence card kp < card k1p by A1,A2,Th15; end; theorem Th50: Class (0-eq_states_EqR tfsm, q) = the carrier of tfsm proof set 0e = 0-eq_states_EqR tfsm; set S = the carrier of tfsm; now let z be set; thus z in Class (0e, q) implies z in S; assume z in S; then reconsider z' = z as Element of S; 0-equivalent z', q by Th42; then [z,q] in 0e by Def12; hence z in Class (0e, q) by EQREL_1:27; end; hence thesis by TARSKI:2; end; theorem Th51: 0-eq_states_partition tfsm = { the carrier of tfsm } proof set S = the carrier of tfsm; set SS = { the carrier of tfsm }; set 0p = 0-eq_states_partition tfsm; set 0e = 0-eq_states_EqR tfsm; A1: 0p = Class 0e by Def13; now let x be set; hereby assume A2: x in 0p; then reconsider x' = x as Subset of S; consider y being set such that A3: y in S & x' = Class (0e, y) by A1,A2,EQREL_1:def 5; reconsider y as Element of S by A3; Class (0e, y) = S by Th50; hence x in SS by A3,TARSKI:def 1; end; assume x in SS; then A4: x = S by TARSKI:def 1; consider y being Element of S; Class (0e, y) in Class 0e by EQREL_1:def 5; hence x in 0p by A1,A4,Th50; end; hence thesis by TARSKI:2; end; theorem Th52: for tfsm being finite non empty Mealy-FSM over IAlph, OAlph st n+1 = card the carrier of tfsm holds (n+1)-eq_states_partition tfsm = n-eq_states_partition tfsm proof let tfsm be finite non empty Mealy-FSM over IAlph, OAlph; assume A1: n+1 = card the carrier of tfsm; assume A2: (n+1)-eq_states_partition tfsm <> n-eq_states_partition tfsm; defpred P[Nat] means $1 <= n+1 implies card ($1-eq_states_partition tfsm) > $1; A3: P[0] proof assume 0 <= n+1; A4: 0-eq_states_partition tfsm = { the carrier of tfsm } by Th51; set x = the carrier of tfsm; card { x } = 1 by CARD_1:79; hence card (0-eq_states_partition tfsm) > 0 by A4; end; A5: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A6: k <= n+1 implies card (k-eq_states_partition tfsm) > k; assume A7: k+1 <= n+1; then A8: k <= n by REAL_1:53; A9: k+1 <= card (k-eq_states_partition tfsm) by A6,A7,NAT_1:38; (k+1)-eq_states_partition tfsm<>k-eq_states_partition tfsm by A2,A8,Th48; then card((k+1)-eq_states_partition tfsm)>card(k-eq_states_partition tfsm) by Th49; hence card ((k+1)-eq_states_partition tfsm) > k+1 by A9,AXIOMS:22; end; for k being Nat holds P[k] from Ind(A3, A5); then card ((n+1)-eq_states_partition tfsm) > n+1; hence contradiction by A1,Th12; end; definition let IAlph, OAlph; let tfsm be non empty Mealy-FSM over IAlph, OAlph; let IT be a_partition of the carrier of tfsm; attr IT is final means :Def14: for qa, qb being State of tfsm holds qa, qb-are_equivalent iff ex X being Element of IT st qa in X & qb in X; end; theorem Th53: k-eq_states_partition tfsm is final implies (k+1)-eq_states_EqR tfsm = k-eq_states_EqR tfsm proof assume A1: k-eq_states_partition tfsm is final; set S = the carrier of tfsm; set k_eq = k-eq_states_EqR tfsm; set k1_eq = (k+1)-eq_states_EqR tfsm; set k_part = k-eq_states_partition tfsm; now let x be set; hereby assume A2: x in k1_eq; then consider a, b be set such that A3: x=[a,b] & a in S & b in S by RELSET_1:6; reconsider a as Element of S by A3; reconsider b as Element of S by A3; (k+1)-equivalent a,b by A2,A3,Def12; then k-equivalent a,b by Th43; hence x in k_eq by A3,Def12; end; assume A4: x in k_eq; then consider a, b be set such that A5: x=[a,b] & a in S & b in S by RELSET_1:6; reconsider a as Element of S by A5; reconsider b as Element of S by A5; Class(k_eq,b) in Class k_eq by EQREL_1:def 5; then reconsider cl = Class(k_eq,b) as Element of k_part by Def13; a in cl & b in cl by A4,A5,EQREL_1:27,28; then a,b-are_equivalent by A1,Def14; then (k+1)-equivalent a,b by Th41; hence x in k1_eq by A5,Def12; end; hence thesis by TARSKI:2; end; theorem Th54: k-eq_states_partition tfsm = (k+1)-eq_states_partition tfsm iff k-eq_states_partition tfsm is final proof set S = the carrier of tfsm; set k_part = k-eq_states_partition tfsm; set k1_part = (k+1)-eq_states_partition tfsm; set k_eq = k-eq_states_EqR tfsm; set k1_eq = (k+1)-eq_states_EqR tfsm; hereby assume A1: k_part = k1_part; now let qa, qb be Element of tfsm; hereby assume qa,qb-are_equivalent; then k-equivalent qa,qb by Th41; then A2: [qa,qb] in k_eq by Def12; Class(k_eq,qb) in Class k_eq by EQREL_1:def 5; then reconsider X= Class(k_eq,qb) as Element of k_part by Def13; take X; thus qa in X & qb in X by A2,EQREL_1:27,28; end; given X being Element of k_part such that A3: qa in X & qb in X; X in k_part; then X in Class(k_eq) by Def13; then consider qc being set such that A4: qc in S & X = Class(k_eq,qc) by EQREL_1:def 5; [qb,qc] in k_eq by A3,A4,EQREL_1:27; then A5: [qc,qb] in k_eq by EQREL_1:12; [qa,qc] in k_eq by A3,A4,EQREL_1:27; then A6: [qa,qb] in k_eq by A5,EQREL_1:13; then A7: k-equivalent qa,qb by Def12; thus qa,qb-are_equivalent proof let w be FinSequence of IAlph; consider n being Nat such that A8: dom w=Seg n by FINSEQ_1:def 2; A9: n = len w by A8,FINSEQ_1:def 3; per cases; suppose n <= k; hence (qa,w)-response = (qb,w)-response by A7,A9,Def11; suppose n > k; then consider m be Nat such that A10: n=k+m & 1 <= m by Th1; A11: n-eq_states_partition tfsm = k_part by A1,A10,Th47; n-eq_states_partition tfsm = Class(n-eq_states_EqR tfsm) by Def13; then Class(n-eq_states_EqR tfsm) = Class(k_eq) by A11,Def13; then [qa,qb] in n-eq_states_EqR tfsm by A6,Th8; then n-equivalent qa,qb by Def12; hence (qa,w)-response = (qb,w)-response by A9,Def11; end; end; hence k-eq_states_partition tfsm is final by Def14; end; assume k-eq_states_partition tfsm is final; then k1_eq = k_eq by Th53; hence k_part = Class(k1_eq) by Def13 .= k1_part by Def13; end; theorem for tfsm being finite non empty Mealy-FSM over IAlph, OAlph st n+1 = card the carrier of tfsm holds ex k being Nat st k <= n & k-eq_states_partition tfsm is final proof let tfsm be finite non empty Mealy-FSM over IAlph, OAlph; assume A1: n+1 = card the carrier of tfsm; take n; thus n <= n; n-eq_states_partition tfsm = (n+1)-eq_states_partition tfsm by A1,Th52; hence thesis by Th54; end; definition let IAlph, OAlph; let tfsm be finite non empty Mealy-FSM over IAlph, OAlph; func final_states_partition tfsm -> a_partition of the carrier of tfsm means :Def15: it is final; existence proof reconsider n = card the carrier of tfsm as Nat; card the carrier of tfsm <> 0 by CARD_2:59; then consider m being Nat such that A1: n=m+1 by NAT_1:22; A2: m-eq_states_partition tfsm=(m+1)-eq_states_partition tfsm by A1,Th52; take (m-eq_states_partition tfsm); thus thesis by A2,Th54; end; uniqueness proof let it1, it2 be a_partition of the carrier of tfsm; assume A3: it1 is final & it2 is final; now assume it1 <> it2; then not for X being set holds X in it1 iff X in it2 by TARSKI:2; then consider X being set such that A4: X in it1 & not X in it2 or not X in it1 & X in it2; per cases by A4; suppose A5: X in it1 & not X in it2; then reconsider X as Subset of tfsm; consider qx be Element of tfsm such that A6: qx in X by A5,Th11; union it2 = the carrier of tfsm by EQREL_1:def 6; then consider Z being set such that A7: qx in Z & Z in it2 by TARSKI:def 4; reconsider Z as Subset of tfsm by A7; set XZ= X\Z, ZX= Z\X, Y'= XZ \/ ZX; Y' <> {} proof assume Y' = {}; then X\Z = {} & Z\X = {} by XBOOLE_1:15; then X c= Z & Z c= X by XBOOLE_1:37 ; hence contradiction by A5,A7,XBOOLE_0:def 10 ; end; then consider qy be Element of tfsm such that A8: qy in Y' by SUBSET_1:10; reconsider X as Element of it1 by A5; A9: now assume A10: qy in X\Z; then A11: qy in X & not qy in Z by XBOOLE_0:def 4; not qx,qy-are_equivalent proof assume qx,qy-are_equivalent; then consider Z' be Element of it2 such that A12: qx in Z' & qy in Z' by A3,Def14; per cases; suppose Z = Z'; hence contradiction by A10,A12,XBOOLE_0:def 4; suppose Z <> Z'; then Z misses Z' by A7,EQREL_1:def 6; then Z /\ Z' = {} by XBOOLE_0:def 7; hence contradiction by A7,A12, XBOOLE_0:def 3; end; hence contradiction by A3,A6,A11,Def14; end; now assume A13: qy in Z\X; then A14: qy in Z & not qy in X by XBOOLE_0:def 4; not qx,qy-are_equivalent proof assume qx,qy-are_equivalent; then consider Z' be Element of it1 such that A15: qx in Z' & qy in Z' by A3,Def14; per cases; suppose X = Z'; hence contradiction by A13,A15,XBOOLE_0:def 4; suppose X <> Z'; then X misses Z' by EQREL_1:def 6; then X /\ Z' = {} by XBOOLE_0:def 7; hence contradiction by A6,A15,XBOOLE_0:def 3; end; hence contradiction by A3,A7,A14,Def14; end; hence contradiction by A8,A9,XBOOLE_0:def 2; suppose A16: not X in it1 & X in it2; then reconsider X as Subset of tfsm; consider qx be Element of tfsm such that A17: qx in X by A16,Th11; union it1 = the carrier of tfsm by EQREL_1:def 6; then consider Z being set such that A18: qx in Z & Z in it1 by TARSKI:def 4; reconsider Z as Subset of tfsm by A18; set XZ= X\Z; set ZX= Z\X; set Y'= XZ \/ ZX; Y' <> {} proof assume Y' = {}; then X\Z={} & Z\X={} by XBOOLE_1:15; then X c= Z & Z c= X by XBOOLE_1:37; hence contradiction by A16,A18,XBOOLE_0:def 10; end; then consider qy be Element of tfsm such that A19: qy in Y' by SUBSET_1:10; reconsider X as Element of it2 by A16; A20: now assume A21: qy in X\Z; then A22: qy in X & not qy in Z by XBOOLE_0:def 4; not qx,qy-are_equivalent proof assume qx,qy-are_equivalent; then consider Z' be Element of it1 such that A23: qx in Z' & qy in Z' by A3,Def14; per cases; suppose Z = Z'; hence contradiction by A21,A23,XBOOLE_0:def 4; suppose Z <> Z'; then Z misses Z' by A18,EQREL_1:def 6; then Z /\ Z' = {} by XBOOLE_0:def 7 ; hence contradiction by A18,A23,XBOOLE_0:def 3; end; hence contradiction by A3,A17,A22,Def14; end; now assume A24: qy in Z\X; then A25: qy in Z & not qy in X by XBOOLE_0:def 4; not qx,qy-are_equivalent proof assume qx,qy-are_equivalent; then consider Z' be Element of it2 such that A26: qx in Z' & qy in Z' by A3,Def14; per cases; suppose X = Z'; hence contradiction by A24,A26,XBOOLE_0:def 4; suppose X <> Z'; then X misses Z' by EQREL_1:def 6; then X /\ Z' = {} by XBOOLE_0:def 7 ; hence contradiction by A17,A26,XBOOLE_0:def 3; end; hence contradiction by A3,A18,A25,Def14; end; hence contradiction by A19,A20,XBOOLE_0:def 2; end; hence it1 = it2; end; end; theorem Th56: for tfsm being finite non empty Mealy-FSM over IAlph, OAlph holds n+1 = card the carrier of tfsm implies final_states_partition tfsm = n-eq_states_partition tfsm proof let tfsm be finite non empty Mealy-FSM over IAlph, OAlph; assume n+1 = card the carrier of tfsm; then (n+1)-eq_states_partition tfsm = n-eq_states_partition tfsm by Th52; then n-eq_states_partition tfsm is final by Th54; hence thesis by Def15; end; begin :: The reduction of a Mealy machine reserve tfsm, rtfsm for finite non empty Mealy-FSM over IAlph, OAlph, q for State of tfsm; definition let IAlph, OAlph be non empty set; let tfsm be finite non empty Mealy-FSM over IAlph, OAlph; let qf be Element of final_states_partition tfsm; let s be Element of IAlph; func (s,qf)-succ_class -> Element of final_states_partition tfsm means :Def16: ex q being State of tfsm, n being Nat st q in qf & (n+1) = card the carrier of tfsm & it = Class(n-eq_states_EqR tfsm, (the Tran of tfsm).[q,s]); existence proof set m = card the carrier of tfsm; card the carrier of tfsm <> 0 by CARD_2:59; then consider n being Nat such that A1: m=n+1 by NAT_1:22; consider q1 be Element of tfsm such that A2: q1 in qf by Th11; final_states_partition tfsm = n-eq_states_partition tfsm by A1,Th56; then A3: final_states_partition tfsm = Class(n-eq_states_EqR tfsm) by Def13; set q' = (the Tran of tfsm).[q1,s]; reconsider IT = Class(n-eq_states_EqR tfsm,q') as Element of final_states_partition tfsm by A3,EQREL_1:def 5; take IT; thus thesis by A1,A2; end; uniqueness proof let it1, it2 be Element of final_states_partition tfsm; given q1 be Element of tfsm, n1 being Nat such that A4: q1 in qf & (n1+1) = card the carrier of tfsm & it1 = Class(n1-eq_states_EqR tfsm, (the Tran of tfsm).[q1,s]); given q2 be Element of tfsm, n2 being Nat such that A5: q2 in qf & (n2+1) = card the carrier of tfsm & it2 = Class(n2-eq_states_EqR tfsm, (the Tran of tfsm).[q2,s]); set m = n1 + 1;A6: n1 = n2 by A4,A5,XCMPLX_1:2; A7: 1 <= m by INT_1:19; A8: n1 = m-1 by XCMPLX_1:26; set q1' = (the Tran of tfsm).[q1,s]; set q2' = (the Tran of tfsm).[q2,s]; final_states_partition tfsm is final by Def15; then q1,q2-are_equivalent by A4,A5,Def14; then m-equivalent q1,q2 by Th41; then n1-equivalent q1',q2' by A7,A8,Th44; then [q1',q2'] in n1-eq_states_EqR tfsm by Def12; hence it1 = it2 by A4,A5,A6,EQREL_1:44; end; end; definition let IAlph, OAlph, tfsm; let qf be Element of final_states_partition tfsm, s; func (qf,s)-class_response -> Element of OAlph means :Def17: ex q st q in qf & it = (the OFun of tfsm).[q,s]; existence proof consider q1 be Element of tfsm such that A1: q1 in qf by Th11; take (the OFun of tfsm).[q1,s]; thus thesis by A1; end; uniqueness proof let it1, it2 be Element of OAlph; given q1 be Element of tfsm such that A2: q1 in qf & it1 = (the OFun of tfsm).[q1,s]; given q2 be Element of tfsm such that A3: q2 in qf & it2 = (the OFun of tfsm).[q2,s]; final_states_partition tfsm is final by Def15; then q1,q2-are_equivalent by A2,A3,Def14; hence it1 = it2 by A2,A3,Th36; end; end; definition let IAlph, OAlph, tfsm; func the_reduction_of tfsm -> strict Mealy-FSM over IAlph, OAlph means :Def18: the carrier of it = final_states_partition tfsm & (for Q being State of it, s for q being State of tfsm st q in Q holds (the Tran of tfsm).[q, s] in (the Tran of it).[Q, s] & (the OFun of tfsm).[q, s] = (the OFun of it).[Q, s]) & the InitS of tfsm in the InitS of it; existence proof set TR = the Tran of tfsm; reconsider m = card the carrier of tfsm as Nat; card the carrier of tfsm <> 0 by CARD_2:59; then consider n be Nat such that A1: m = n+1 by NAT_1:22; set IS = Class(n-eq_states_EqR tfsm, the InitS of tfsm); set part=final_states_partition tfsm; deffunc F(Element of part,Element of IAlph) = ($2,$1)-succ_class; consider Trf being Function of [: part, IAlph:], part such that A2: for q being Element of part for i being Element of IAlph holds Trf.[q,i] = F(q,i) from Lambda2D; deffunc F(Element of part,Element of IAlph) = ($1,$2)-class_response; consider OF being Function of [:part,IAlph:],OAlph such that A3: for q being Element of part for i being Element of IAlph holds OF.[q,i] =F(q,i) from Lambda2D; A4: IS in Class (n-eq_states_EqR tfsm) by EQREL_1:def 5; part=n-eq_states_partition tfsm by A1,Th56; then reconsider IS as Element of part by A4,Def13; take IT = Mealy-FSM (# final_states_partition tfsm, Trf, OF, IS #); now let Q be Element of IT, s, q such that A5: q in Q; reconsider Q' = Q as Element of final_states_partition tfsm; reconsider s' = s as Element of IAlph; consider Q1 being Element of tfsm, n1 being Nat such that A6: Q1 in Q' & (n1+1) = card the carrier of tfsm & (s',Q')-succ_class = Class(n1-eq_states_EqR tfsm, TR.[Q1,s']) by Def16; A7: (the Tran of IT).[Q',s']=Class(n1-eq_states_EqR tfsm,TR.[Q1,s']) by A2,A6; final_states_partition tfsm is final by Def15; then q, Q1-are_equivalent by A5,A6,Def14; then A8: (n1+1)-equivalent q,Q1 by Th41; reconsider n1' = n1 as Integer; reconsider 1' = 1 as Integer; A9: 1 <= n1+1 by NAT_1:29; (n1' + 1') - 1' = n1' by XCMPLX_1:26; then n1-equivalent TR.[q,s'],TR.[Q1,s']by A8,A9,Th44; then [TR.[q,s'],TR.[Q1,s']] in n1-eq_states_EqR tfsm by Def12; hence TR.[q, s] in (the Tran of IT).[Q, s] by A7,EQREL_1:27; A10: (the OFun of IT).[Q',s'] = (Q',s')-class_response by A3; consider Q1 be Element of tfsm such that A11: Q1 in Q' & (Q',s')-class_response = (the OFun of tfsm).[Q1,s'] by Def17; final_states_partition tfsm is final by Def15; then q, Q1-are_equivalent by A5,A11,Def14; hence (the OFun of tfsm).[q, s] = (the OFun of IT).[Q, s] by A10,A11,Th36; end; hence thesis by EQREL_1:28; end; uniqueness proof set TR = the Tran of tfsm; let it1, it2 be strict Mealy-FSM over IAlph,OAlph; assume A12: the carrier of it1 = final_states_partition tfsm & (for Q being Element of it1, s, q st q in Q holds TR.[q, s] in (the Tran of it1).[Q, s] & (the OFun of tfsm).[q, s] = (the OFun of it1).[Q, s]) & the InitS of tfsm in the InitS of it1; assume A13: the carrier of it2 = final_states_partition tfsm & (for Q being Element of it2, s, q st q in Q holds TR.[q, s] in (the Tran of it2).[Q, s] & (the OFun of tfsm).[q, s] = (the OFun of it2).[Q, s]) & the InitS of tfsm in the InitS of it2; A14: the Tran of it1 = the Tran of it2 proof reconsider T1 = the Tran of it1 as Function of [: final_states_partition tfsm, IAlph:],final_states_partition tfsm by A12; reconsider T2 = the Tran of it2 as Function of [: final_states_partition tfsm, IAlph:],final_states_partition tfsm by A13; now let Q be Element of final_states_partition tfsm, s; consider q be Element of tfsm such that A15: q in Q by Th11; A16: TR.[q,s] in T1.[Q,s] by A12,A15; A17: TR.[q,s] in T2.[Q,s] by A13,A15; reconsider T1' = T1.[Q,s], T2' = T2.[Q,s] as Subset of tfsm by TARSKI:def 3; T1' = T2' or T1' misses T2' by EQREL_1:def 6; hence T1.[Q,s] = T2.[Q,s] by A16,A17,XBOOLE_0:3; end; hence thesis by FUNCT_2:120; end; A18: the OFun of it1 = the OFun of it2 proof reconsider OF1 = the OFun of it1 as Function of [: final_states_partition tfsm, IAlph :], OAlph by A12; reconsider OF2 = the OFun of it2 as Function of [: final_states_partition tfsm, IAlph :], OAlph by A13; now let Q be Element of final_states_partition tfsm, s; consider q be Element of tfsm such that A19: q in Q by Th11; thus OF1.[Q,s] = (the OFun of tfsm).[q,s] by A12,A19 .= OF2.[Q,s] by A13,A19; end; hence thesis by FUNCT_2:120; end; the InitS of it1 = the InitS of it2 proof the InitS of it1 in final_states_partition tfsm by A12; then reconsider IS1=the InitS of it1 as Subset of tfsm; the InitS of it2 in final_states_partition tfsm by A13; then reconsider IS2=the InitS of it2 as Subset of tfsm; IS1 = IS2 or IS1 misses IS2 by A12,A13,EQREL_1:def 6; hence thesis by A12,A13,XBOOLE_0:3; end; hence it1 = it2 by A12,A13,A14,A18; end; end; definition let IAlph, OAlph, tfsm; cluster the_reduction_of tfsm -> non empty finite; coherence proof the carrier of the_reduction_of tfsm = final_states_partition tfsm by Def18; hence thesis by GROUP_1:def 13,STRUCT_0:def 1; end; end; theorem Th57: for qr being State of rtfsm st rtfsm = the_reduction_of tfsm & q in qr holds for k being Nat st k in Seg (len w +1) holds (q,w)-admissible.k in (qr,w)-admissible.k proof let qr be State of rtfsm; set TR = the Tran of tfsm; assume A1: rtfsm = the_reduction_of tfsm & q in qr; defpred P[Nat] means $1 in Seg(len w +1) implies (q,w)-admissible.$1 in (qr,w)-admissible.$1; A2: P[0] by FINSEQ_1:3; A3: for k being Nat st P[k] holds P[k + 1] proof let k be Nat; assume A4: k in Seg (len w + 1) implies (q,w)-admissible.k in (qr,w)-admissible.k; assume A5: (k+1) in Seg (len w + 1); A6: k = 0 or 0 < k & 0+1 = 1 by NAT_1:19; per cases by A6,NAT_1:38; suppose A7: k = 0; then (q,w)-admissible.(k+1) = q by Def2; hence (q,w)-admissible.(k+1) in (qr,w)-admissible.(k+1) by A1,A7,Def2; suppose A8: 1 <= k; A9: 1 <= k+1 & k+1 <= len w + 1 by A5,FINSEQ_1:3; k <= k+1 by NAT_1:29; then A10: k <= len w + 1 by A9,AXIOMS:22; A11: 1 <= k & k <= len w by A8,A9,REAL_1:53; then consider w1i being Element of IAlph, q1i, q1i1 being Element of tfsm such that A12: w1i = w.k & q1i = (q,w)-admissible.k & q1i1 = (q,w)-admissible.(k+1) & w1i-succ_of q1i = q1i1 by Def2; consider w2i being Element of IAlph, q2i, q2i1 being Element of rtfsm such that A13: w2i = w.k & q2i = (qr,w)-admissible.k & q2i1 = (qr,w)-admissible.(k+1) & w2i-succ_of q2i = q2i1 by A11,Def2; A14: TR.[q1i,w1i] in (the Tran of rtfsm).[q2i,w2i] by A1,A4,A8,A10,A12,A13, Def18,FINSEQ_1:3; q1i1 = TR.[q1i,w1i] by A12,Def1; hence (q,w)-admissible.(k+1) in (qr,w)-admissible.(k+1) by A12,A13,A14, Def1; end; thus for k being Nat holds P[k] from Ind(A2,A3); end; theorem Th58: tfsm, the_reduction_of tfsm-are_equivalent proof now let w1 be FinSequence of IAlph; set rtfsm = the_reduction_of tfsm; set ad1 = (the InitS of tfsm,w1)-admissible; set ad2 = (the InitS of rtfsm, w1)-admissible; set r1=(the InitS of tfsm,w1)-response;set r2=(the InitS of rtfsm,w1)-response; A1: the InitS of tfsm in the InitS of rtfsm by Def18; A2: len r1 = len w1 by Def6 .= len r2 by Def6; now let k be Nat; assume 1 <= k & k <= len r1; then 1 <= k & k <= len w1 by Def6; then A3: k in Seg len w1 by FINSEQ_1:3; then A4: k in Seg (len w1 + 1) by FINSEQ_2:9; A5: k in dom w1 by A3,FINSEQ_1:def 3; k in Seg len ad1 by A4,Def2; then k in dom ad1 by FINSEQ_1:def 3; then A6: ad1.k is Element of tfsm & w1.k is Element of IAlph by A5,FINSEQ_2:13; k in Seg len ad2 by A4,Def2; then k in dom ad2 by FINSEQ_1:def 3; then A7: w1.k is Element of IAlph & ad2.k is Element of rtfsm by A5,FINSEQ_2:13; A8: ad1.k in ad2.k by A1,A4,Th57; thus r2.k = (the OFun of rtfsm).[ad2.k, w1.k] by A5,Def6 .= (the OFun of tfsm).[ad1.k, w1.k] by A6,A7,A8,Def18 .= r1.k by A5,Def6; end; hence r1 = r2 by A2,FINSEQ_1:18; end; hence thesis by Def9; end; begin :: Machine Isomorphism reserve qr1, qr2 for State of rtfsm, Tf for Function of the carrier of tfsm1, the carrier of tfsm2; definition let IAlph, OAlph, tfsm1, tfsm2; pred tfsm1, tfsm2-are_isomorphic means :Def19: ex Tf st Tf is bijective & Tf.the InitS of tfsm1 = the InitS of tfsm2 & for q11, s holds Tf.((the Tran of tfsm1).[q11,s])=(the Tran of tfsm2).[Tf.q11, s] & (the OFun of tfsm1).[q11,s] = (the OFun of tfsm2).[Tf.q11, s]; reflexivity proof let tfsm1 be non empty Mealy-FSM over IAlph, OAlph; take Tf = id the carrier of tfsm1; thus Tf is bijective; thus Tf.the InitS of tfsm1 = the InitS of tfsm1 by FUNCT_1:34; let q be Element of tfsm1, s; thus Tf.((the Tran of tfsm1).[q, s]) = (the Tran of tfsm1).[q, s] by FUNCT_1:34 .= (the Tran of tfsm1).[Tf.q, s] by FUNCT_1:34; thus thesis by FUNCT_1:34; end; symmetry proof let tfsm1, tfsm2 be non empty Mealy-FSM over IAlph, OAlph; given Tf being Function of the carrier of tfsm1, the carrier of tfsm2 such that A1: Tf is bijective & Tf.the InitS of tfsm1 = the InitS of tfsm2 & for q being Element of tfsm1, s being Element of IAlph holds Tf.((the Tran of tfsm1).[q, s]) = (the Tran of tfsm2).[Tf.q, s] & (the OFun of tfsm1).[q,s] = (the OFun of tfsm2).[Tf.q, s]; A2: Tf is one-to-one & Tf is onto by A1,FUNCT_2:def 4; then A3: rng Tf = the carrier of tfsm2 by FUNCT_2:def 3; then A4: the carrier of tfsm2 = dom(Tf") by A2,FUNCT_1:55; A5: dom Tf = the carrier of tfsm1 by FUNCT_2:def 1; then A6: rng(Tf") = the carrier of tfsm1 by A2,FUNCT_1:55; then reconsider Tf' = Tf" as Function of the carrier of tfsm2, the carrier of tfsm1 by A4,FUNCT_2:3; take Tf'; Tf' is one-to-one & Tf' is onto by A2,A6,FUNCT_1:62,FUNCT_2:def 3; hence Tf' is bijective by FUNCT_2:def 4; thus the InitS of tfsm1 = Tf'.the InitS of tfsm2 by A1,A2,A5,FUNCT_1:56; now let q be Element of tfsm2, s be Element of IAlph; A7: q = Tf.(Tf'.q) by A2,A3,FUNCT_1:57; thus (the Tran of tfsm1).[Tf'.q, s] = Tf'.(Tf.((the Tran of tfsm1).[Tf'.q, s])) by A2,A5,FUNCT_1:56 .= Tf'.((the Tran of tfsm2).[q,s]) by A1,A7; thus (the OFun of tfsm1).[Tf'.q,s]= (the OFun of tfsm2).[q,s] by A1,A7; end; hence thesis; end; end; theorem Th59: tfsm1,tfsm2-are_isomorphic & tfsm2,tfsm3-are_isomorphic implies tfsm1,tfsm3-are_isomorphic proof assume A1: tfsm1,tfsm2-are_isomorphic & tfsm2,tfsm3-are_isomorphic; then consider Tf1 being Function of the carrier of tfsm1, the carrier of tfsm2 such that A2: Tf1 is bijective & Tf1.the InitS of tfsm1 = the InitS of tfsm2 & for q being Element of tfsm1, s1 being Element of IAlph holds Tf1.((the Tran of tfsm1).[q, s1]) = (the Tran of tfsm2).[Tf1.q, s1] & (the OFun of tfsm1).[q,s1] = (the OFun of tfsm2).[Tf1.q, s1] by Def19; consider Tf2 being Function of the carrier of tfsm2, the carrier of tfsm3 such that A3: Tf2 is bijective & Tf2.the InitS of tfsm2 = the InitS of tfsm3 & for q being Element of tfsm2, s1 being Element of IAlph holds Tf2.((the Tran of tfsm2).[q, s1]) = (the Tran of tfsm3).[Tf2.q, s1] & (the OFun of tfsm2).[q,s1] = (the OFun of tfsm3).[Tf2.q, s1] by A1,Def19; take Tf = Tf2 * Tf1; thus Tf is bijective by A2,A3,Th7; A4: dom Tf1 = the carrier of tfsm1 by FUNCT_2:def 1; hence Tf.the InitS of tfsm1 = the InitS of tfsm3 by A2,A3,FUNCT_1:23; now let q be Element of tfsm1, s1 be Element of IAlph; thus (the Tran of tfsm3).[Tf.q, s1] = (the Tran of tfsm3).[Tf2.(Tf1.q), s1] by A4,FUNCT_1:23 .= Tf2.((the Tran of tfsm2).[Tf1.q, s1]) by A3 .= Tf2.(Tf1.((the Tran of tfsm1).[q,s1])) by A2 .= Tf.((the Tran of tfsm1).[q, s1]) by A4,FUNCT_1:23; thus (the OFun of tfsm3).[Tf.q, s1] = (the OFun of tfsm3).[Tf2.(Tf1.q), s1] by A4,FUNCT_1:23 .= (the OFun of tfsm2).[Tf1.q, s1] by A3 .= (the OFun of tfsm1).[q,s1] by A2; end; hence thesis; end; theorem Th60: (for q being State of tfsm1, s holds Tf.((the Tran of tfsm1).[q,s]) = (the Tran of tfsm2).[Tf.q,s]) implies for k st 1 <= k & k <= len w + 1 holds Tf.((q11,w)-admissible.k) = (Tf.q11,w)-admissible.k proof assume A1: for q being State of tfsm1, s holds Tf.((the Tran of tfsm1).[q,s]) = (the Tran of tfsm2).[Tf.q,s]; defpred P[Nat] means 1 <= $1 & $1 <= len w + 1 implies Tf.((q11,w)-admissible.$1) = (Tf.q11,w)-admissible.$1; A2: P[0]; A3: for k being Nat st P[k] holds P[k + 1] proof let k be Nat; assume A4: 1 <= k & k <= len w + 1 implies Tf.((q11,w)-admissible.k) = (Tf.q11,w)-admissible.k; assume A5: 1<=k+1 & k+1<=len w+1; A6: k=0 or 0<k & 0 + 1 = 1 by NAT_1:19; per cases by A6,NAT_1:38; suppose A7: k = 0; hence Tf.((q11,w)-admissible.(k+1)) = Tf.q11 by Def2 .= (Tf.q11,w)-admissible.(k+1) by A7,Def2; suppose A8: 1 <= k; A9: k <= len w by A5,REAL_1:53; A10: len w <= len w + 1 by NAT_1:29; consider wi being Element of IAlph,qi,qi1 being State of tfsm1 such that A11: wi = w.k & qi = (q11,w)-admissible.k & qi1 = (q11,w)-admissible.(k+1) & wi-succ_of qi =qi1 by A8,A9,Def2; consider wri being Element of IAlph, qri, qri1 being State of tfsm2 such that A12: wri = w.k & qri = (Tf.q11,w)-admissible.k & qri1 = (Tf.q11,w)-admissible.(k+1) & wri-succ_of qri = qri1 by A8,A9,Def2; thus Tf.((q11,w)-admissible.(k+1)) = Tf.((the Tran of tfsm1).[qi, wi]) by A11,Def1 .= (the Tran of tfsm2).[qri, wri] by A1,A4,A8,A9,A10,A11,A12,AXIOMS:22 .= (Tf.q11,w)-admissible.(k+1) by A12,Def1; end; thus for k being Nat holds P[k] from Ind(A2,A3); end; theorem Th61: (Tf.the InitS of tfsm1 = the InitS of tfsm2 & for q being State of tfsm1, s holds Tf.((the Tran of tfsm1).[q, s]) = (the Tran of tfsm2).[Tf.q, s] & (the OFun of tfsm1).[q,s] = (the OFun of tfsm2).[Tf.q, s]) implies (q11,q12-are_equivalent iff Tf.q11, Tf.q12 -are_equivalent) proof set S_tfsm1 = the carrier of tfsm1; set S_tfsm2 = the carrier of tfsm2; set OF1 = the OFun of tfsm1, OF2 = the OFun of tfsm2; assume A1: Tf.the InitS of tfsm1 = the InitS of tfsm2 & for q being Element of S_tfsm1, s holds Tf.((the Tran of tfsm1).[q, s]) = (the Tran of tfsm2).[Tf.q, s] & (the OFun of tfsm1).[q,s] = (the OFun of tfsm2).[Tf.q, s]; hereby assume A2: q11,q12-are_equivalent; reconsider Tq1 = Tf.q11 as Element of S_tfsm2; reconsider Tq2 = Tf.q12 as Element of S_tfsm2; now let w be FinSequence of IAlph; A3:len(Tq1,w)-response=len w by Def6.=len (Tq2,w)-response by Def6; now let k be Nat; assume A4: 1 <= k & k <= len (Tq1,w)-response; len (Tq1,w)-response = len w by Def6; then A5: k in Seg len w by A4,FINSEQ_1:3; then A6: k in Seg (len w + 1) by FINSEQ_2:9; then A7: 1 <= k & k <= len w + 1 by FINSEQ_1:3; A8: k in dom w by A5,FINSEQ_1:def 3; len (q11,w)-admissible = len w + 1 by Def2; then k in dom (q11,w)-admissible by A6,FINSEQ_1:def 3; then reconsider q1a = (q11,w)-admissible.k as Element of S_tfsm1 by FINSEQ_2:13; len (q12,w)-admissible = len w + 1 by Def2; then k in dom (q12,w)-admissible by A6,FINSEQ_1:def 3; then reconsider q2a = (q12,w)-admissible.k as Element of S_tfsm1 by FINSEQ_2:13; reconsider wk = w.k as Element of IAlph by A8,FINSEQ_2:13; thus (Tq1,w)-response.k=OF2.[(Tq1,w)-admissible.k, w.k] by A8,Def6 .= OF2.[Tf.q1a, wk] by A1,A7,Th60 .= OF1.[q1a, wk] by A1 .= (q11,w)-response.k by A8,Def6 .= (q12,w)-response.k by A2,Def10 .= OF1.[q2a, wk] by A8,Def6 .= OF2.[Tf.q2a, wk] by A1 .= OF2.[(Tq2,w)-admissible.k, w.k] by A1,A7,Th60 .= (Tq2,w)-response.k by A8,Def6; end; hence (Tq1,w)-response = (Tq2,w)-response by A3,FINSEQ_1:18; end; hence Tf.q11, Tf.q12-are_equivalent by Def10; end; assume A9: Tf.q11, Tf.q12-are_equivalent; now let w be FinSequence of IAlph; A10: len (q11,w)-response = len w by Def6 .= len (q12,w)-response by Def6; now let k be Nat; assume A11: 1 <= k & k <= len (q11,w)-response; len (q11,w)-response = len w by Def6; then A12: k in Seg len w by A11,FINSEQ_1:3; then A13: k in Seg (len w + 1) by FINSEQ_2:9; then A14: 1 <= k & k <= len w + 1 by FINSEQ_1:3; A15: k in dom w by A12,FINSEQ_1:def 3; len (q11,w)-admissible = len w + 1 by Def2; then k in dom (q11,w)-admissible by A13,FINSEQ_1:def 3; then reconsider q1a = (q11,w)-admissible.k as Element of S_tfsm1 by FINSEQ_2:13; len (q12,w)-admissible = len w + 1 by Def2; then k in dom (q12,w)-admissible by A13,FINSEQ_1:def 3; then reconsider q2a = (q12,w)-admissible.k as Element of S_tfsm1 by FINSEQ_2:13; reconsider wk = w.k as Element of IAlph by A15,FINSEQ_2:13; thus (q11,w)-response.k = OF1.[(q11,w)-admissible.k, w.k] by A15,Def6 .= OF2.[Tf.q1a, wk] by A1 .= OF2.[(Tf.q11,w)-admissible.k,wk] by A1,A14,Th60 .= (Tf.q11,w)-response.k by A15,Def6 .= (Tf.q12,w)-response.k by A9,Def10 .= OF2.[(Tf.q12,w)-admissible.k, w.k] by A15,Def6 .= OF2.[Tf.q2a, wk] by A1,A14,Th60 .= OF1.[q2a, w.k] by A1 .= (q12,w)-response.k by A15,Def6; end; hence (q11,w)-response = (q12,w)-response by A10,FINSEQ_1:18; end; hence thesis by Def10; end; theorem Th62: rtfsm = the_reduction_of tfsm & qr1<>qr2 implies not qr1,qr2 -are_equivalentproof assume A1: rtfsm = the_reduction_of tfsm & qr1 <> qr2; then A2: the carrier of rtfsm = final_states_partition tfsm by Def18; then reconsider q1' = qr1 as Subset of tfsm by TARSKI:def 3; consider x being Element of tfsm such that A3: x in q1' by A2,Th11; reconsider q2' = qr2 as Subset of tfsm by A2,TARSKI:def 3; consider y being Element of tfsm such that A4: y in q2' by A2,Th11; A5: final_states_partition tfsm is final by Def15; not x,y-are_equivalent proof assume x,y-are_equivalent; then consider X being Element of rtfsm such that A6: x in X & y in X by A2,A5,Def14; X is Subset of tfsm by A2,TARSKI:def 3; then A7: X = q1' or X misses q1' by A2,EQREL_1:def 6; q2' misses q1' by A1,A2,EQREL_1:def 6; hence contradiction by A3,A4,A6,A7,XBOOLE_0:3; end; then consider w being FinSequence of IAlph such that A8: (x,w)-response <> (y,w)-response by Def10; set xresp = (x,w)-response, yresp = (y,w)-response; set xadm = (x,w)-admissible, yadm = (y,w)-admissible; len xresp = len w by Def6 .= len yresp by Def6; then consider k be Nat such that A9: 1 <= k & k <= len xresp & xresp.k <> yresp.k by A8,FINSEQ_1:18; A10: k in Seg len xresp by A9,FINSEQ_1:3; set q1resp = (qr1,w)-response, q2resp = (qr2,w)-response; set q1adm = (qr1,w)-admissible, q2adm = (qr2,w)-admissible; len q1adm = len w + 1 by Def2 .= len xresp + 1 by Def6; then A11: k in Seg len q1adm by A10,FINSEQ_2:9; then k in dom q1adm by FINSEQ_1:def 3; then A12: q1adm.k is Element of rtfsm by FINSEQ_2:13; len xresp = len w by Def6; then A13: k in Seg len w by A9,FINSEQ_1:3; then k in dom w by FINSEQ_1:def 3; then A14: w.k is Element of IAlph by FINSEQ_2:13; k in Seg (len w + 1) by A13,FINSEQ_2:9; then A15: xadm.k in q1adm.k by A1,A3,Th57; A16: len q1adm = len w + 1 by Def2 .= len xadm by Def2; then k in dom xadm by A11,FINSEQ_1:def 3; then A17: xadm.k is Element of tfsm by FINSEQ_2:13; len q2adm = len w + 1 by Def2 .= len q1adm by Def2; then k in dom q2adm by A11,FINSEQ_1:def 3; then A18: q2adm.k is Element of rtfsm by FINSEQ_2:13; k in Seg (len w + 1) by A13,FINSEQ_2:9; then A19: yadm.k in q2adm.k by A1,A4,Th57; len yadm = len w + 1 by Def2 .= len xadm by Def2; then k in dom yadm by A11,A16,FINSEQ_1:def 3; then A20: yadm.k is Element of tfsm by FINSEQ_2:13; now assume A21: q1resp = q2resp; len w = len xresp by Def6; then A22: k in dom w by A9,FINSEQ_3:27; then A23: q1resp.k = (the OFun of rtfsm).[q1adm.k,w.k] by Def6 .= (the OFun of tfsm).[xadm.k,w.k] by A1,A12,A14,A15,A17,Def18; A24: q2resp.k = (the OFun of rtfsm).[q2adm.k,w.k] by A22,Def6 .= (the OFun of tfsm).[yadm.k,w.k] by A1,A14,A18,A19,A20,Def18; xresp.k = (the OFun of tfsm).[xadm.k,w.k] by A22,Def6; hence contradiction by A9,A21,A22,A23,A24,Def6; end; hence thesis by Def10; end; begin :: Reduced and Connected Machines definition let IAlph, OAlph be non empty set; let IT be non empty Mealy-FSM over IAlph,OAlph; attr IT is reduced means :Def20: for qa, qb being State of IT st qa <> qb holds not qa, qb-are_equivalent; end; theorem Th63: the_reduction_of tfsm is reduced proof let qa,qb be State of the_reduction_of tfsm; thus thesis by Th62; end; definition let IAlph, OAlph; cluster reduced finite (non empty Mealy-FSM over IAlph,OAlph); existence proof consider M being finite non empty Mealy-FSM over IAlph, OAlph; take the_reduction_of M; thus thesis by Th63; end; end; reserve Rtfsm for reduced (finite non empty Mealy-FSM over IAlph, OAlph); theorem Th64: Rtfsm, the_reduction_of Rtfsm-are_isomorphic proof set m = Rtfsm; set rm = the_reduction_of m; set fpm = final_states_partition m; A1: the carrier of rm = fpm by Def18; A2: the InitS of m in the InitS of rm by Def18; deffunc F(Element of m) = (proj fpm).$1; consider Tf being Function of the carrier of m, fpm such that A3:for q being Element of m holds Tf.q=F(q) from LambdaD; A4: now assume not Tf is one-to-one; then consider q1, q2 being set such that A5: q1 in the carrier of m & q2 in the carrier of m & Tf.q1 = Tf.q2 & q1 <> q2 by FUNCT_2:25; reconsider q1, q2 as State of m by A5; A6: not q1, q2-are_equivalent by A5,Def20; A7: fpm is final by Def15; Tf.q1 = (proj fpm).q1 & Tf.q2 = (proj fpm).q2 by A3; then q1 in Tf.q1 & q2 in Tf.q2 by BORSUK_1:def 1; hence contradiction by A5,A6,A7,Def14; end; rng Tf = fpm proof thus rng Tf c= fpm by RELSET_1:12; thus fpm c= rng Tf proof let x be set; assume A8: x in fpm; then reconsider pq=x as Subset of m; consider q being Element of m such that A9: q in pq by A8,Th11; pq = (proj fpm).q by A8,A9,BORSUK_1:29; then Tf.q = pq by A3; hence x in rng Tf by FUNCT_2:6; end; end; then Tf is onto by FUNCT_2:def 3; then A10: Tf is bijective by A4,FUNCT_2:def 4; set Im = the InitS of m; the InitS of rm = (proj fpm).Im by A1,A2,BORSUK_1:29; then A11: Tf.the InitS of m = the InitS of rm by A3; now let q be State of m, s be Element of IAlph; reconsider Tfq = Tf.q as State of rm by Def18; Tf.q = (proj fpm).q by A3; then A12: q in Tf.q by BORSUK_1:def 1; then A13: (the Tran of m).[q, s] in (the Tran of rm).[Tf.q, s] by A1,Def18; (the Tran of rm).[Tfq, s] is State of rm; then (the Tran of rm).[Tf.q, s] = (proj fpm).((the Tran of m).[q, s]) by A1,A13,BORSUK_1:29; hence Tf.((the Tran of m).[q, s]) = (the Tran of rm).[Tf.q, s] by A3; thus (the OFun of m).[q,s] = (the OFun of rm).[Tf.q, s] by A1,A12,Def18; end; hence thesis by A1,A10,A11,Def19; end; theorem Th65: tfsm is reduced iff ex M being finite non empty Mealy-FSM over IAlph,OAlph st tfsm, the_reduction_of M-are_isomorphic proof set M = tfsm; hereby assume M is reduced; then M, the_reduction_of M-are_isomorphic by Th64; hence ex M being finite non empty Mealy-FSM over IAlph,OAlph st tfsm, the_reduction_of M-are_isomorphic; end; given MM being finite non empty Mealy-FSM over IAlph,OAlph such that A1: M, the_reduction_of MM-are_isomorphic;let qa, qb be State of M such that A2: qa <> qb; set rMM = the_reduction_of MM; consider Tf being Function of the carrier of M, the carrier of rMM such that A3: Tf is bijective & Tf.the InitS of M = the InitS of rMM & for q being State of M, s being Element of IAlph holds Tf.((the Tran of M).[q, s]) = (the Tran of rMM).[Tf.q, s] & (the OFun of M).[q,s]=(the OFun of rMM).[Tf.q, s] by A1,Def19; Tf is one-to-one by A3,FUNCT_2:def 4; then Tf.qa <> Tf.qb by A2,FUNCT_2:25 ; then not Tf.qa, Tf.qb-are_equivalent by Th62; hence thesis by A3,Th61; end; definition let IAlph, OAlph; let tfsm be non empty Mealy-FSM over IAlph,OAlph; let IT be State of tfsm; attr IT is accessible means :Def21: ex w st the InitS of tfsm, w-leads_to IT; end; definition let IAlph, OAlph; let IT be non empty Mealy-FSM over IAlph, OAlph; attr IT is connected means :Def22: for q being State of IT holds q is accessible; end; definition let IAlph, OAlph; cluster connected (finite non empty Mealy-FSM over IAlph,OAlph); existence proof reconsider S = {1} as finite non empty set; reconsider IS = 1 as Element of S by TARSKI:def 1; set dF = [:S, IAlph:]; set Tr = dF --> 1; dom Tr = dF & rng Tr c= S by FUNCOP_1:19; then reconsider T = Tr as Function of dF, S by FUNCT_2:def 1,RELSET_1:11; consider out being Element of OAlph; reconsider OF = dF --> out as Function of [:S, IAlph:], OAlph; reconsider M = Mealy-FSM(#S, T, OF, IS#) as finite non empty Mealy-FSM over IAlph, OAlph; take M; let q be State of M; A1: q = 1 by TARSKI:def 1; the InitS of M, <*>IAlph-leads_to the InitS of M by Th17; hence q is accessible by A1,Def21; end; end; reserve Ctfsm, Ctfsm1, Ctfsm2 for connected (finite non empty Mealy-FSM over IAlph, OAlph); theorem Th66: the_reduction_of Ctfsm is connected proof set c = Ctfsm; set rtfsm = the_reduction_of c; assume not rtfsm is connected; then consider Q be State of rtfsm such that A1: not Q is accessible by Def22; A2: the carrier of rtfsm = final_states_partition c by Def18; A3: Q in the carrier of rtfsm; reconsider Q' = Q as Subset of c by A2,TARSKI:def 3; Q' in final_states_partition c by A3,Def18; then consider q be Element of c such that A4: q in Q by Th11; q is accessible by Def22; then consider w such that A5: the InitS of c,w-leads_to q by Def21; A6: (the InitS of c,w)-admissible.(len w + 1) = q by A5,Def3; A7: the InitS of c in the InitS of rtfsm by Def18; 1 <= len w + 1 & len w + 1 <= len w + 1 by NAT_1:29; then A8: len w + 1 in Seg (len w + 1) by FINSEQ_1:3; then A9: q in (the InitS of rtfsm,w)-admissible.(len w + 1) by A6,A7,Th57; len w + 1 in Seg (len (the InitS of rtfsm,w)-admissible) by A8,Def2; then len w + 1 in dom (the InitS of rtfsm,w)-admissible by FINSEQ_1:def 3; then A10: (the InitS of rtfsm,w)-admissible.(len w + 1) in the carrier of rtfsm by FINSEQ_2:13; then reconsider QQ = (the InitS of rtfsm,w)-admissible.(len w + 1) as Subset of c by A2; Q' = QQ or Q' misses QQ by A2,A10,EQREL_1:def 6; then the InitS of rtfsm,w-leads_to Q by A4,A9,Def3,XBOOLE_0:3; hence contradiction by A1,Def21; end; definition let IAlph, OAlph; cluster connected reduced finite (non empty Mealy-FSM over IAlph,OAlph); existence proof consider cm being connected finite (non empty Mealy-FSM over IAlph, OAlph); take the_reduction_of cm; thus thesis by Th63,Th66;end; end; definition let IAlph, OAlph; let tfsm be non empty Mealy-FSM over IAlph,OAlph; func accessibleStates tfsm equals :Def23: { q where q is State of tfsm : q is accessible }; coherence; end; definition let IAlph, OAlph, tfsm; cluster accessibleStates tfsm -> finite non empty; coherence proof set m = tfsm; set AS = { q where q is State of m : q is accessible }; AS c= the carrier of m proof let x be set; assume x in AS; then consider q being State of m such that A1: x = q & q is accessible; thus x in the carrier of m by A1; end; then A2: AS is finite by FINSET_1:13; the InitS of m,<*>IAlph-leads_to the InitS of m by Th17; then the InitS of m is accessible by Def21; then the InitS of m in AS; hence thesis by A2,Def23; end; end; theorem Th67: accessibleStates tfsm c= the carrier of tfsm & for q holds q in accessibleStates tfsm iff q is accessible proof set AS = { q where q is State of tfsm: q is accessible }; A1: AS = accessibleStates tfsm by Def23; AS c= the carrier of tfsm proof let x be set; assume x in AS; then consider q being State of tfsm such that A2: x = q & q is accessible; thus x in the carrier of tfsm by A2; end; hence accessibleStates tfsm c= the carrier of tfsm by Def23; let q be State of tfsm; hereby assume q in accessibleStates tfsm; then consider q' being State of tfsm such that A3: q' = q & q' is accessible by A1; thus q is accessible by A3; end; assume q is accessible; hence thesis by A1; end; theorem Th68: (the Tran of tfsm)|[:accessibleStates tfsm, IAlph:] is Function of [:accessibleStates tfsm, IAlph:], accessibleStates tfsm proof set cTran = (the Tran of tfsm) | [:accessibleStates tfsm, IAlph:]; A1:accessibleStates tfsm c= the carrier of tfsm & IAlph c= IAlph by Th67; then [:accessibleStates tfsm, IAlph:] c= [:the carrier of tfsm, IAlph:] by ZFMISC_1:119; then cTran is Function of [:accessibleStates tfsm, IAlph:], the carrier of tfsm by FUNCT_2:38; then A2: dom cTran = [:accessibleStates tfsm, IAlph:] by FUNCT_2:def 1; rng cTran c= accessibleStates tfsm proof let x be set; assume A3: x in rng cTran; rng cTran c= rng the Tran of tfsm & rng the Tran of tfsm c= the carrier of tfsm by FUNCT_1:76,RELSET_1:12; then rng cTran c= the carrier of tfsm by XBOOLE_1:1; then reconsider q1=x as State of tfsm by A3; consider d being set such that A4: d in dom cTran & x = cTran.d by A3,FUNCT_1:def 5; A5: d`1 in accessibleStates tfsm & d`2 in IAlph by A2,A4,MCART_1:10; then reconsider q = d`1 as State of tfsm by A1; reconsider s = d`2 as Element of IAlph by A2,A4,MCART_1:10; set I = the InitS of tfsm; q is accessible by A5,Th67; then consider w being FinSequence of IAlph such that A6: I, w-leads_to q by Def21; A7: len (w^<*s*>) = len w + len <*s*> by FINSEQ_1:35; set qsa = (q, <*s*>)-admissible; A8: qsa.1 = q by Def2; A9: <*s*>.1 = s by FINSEQ_1:57; 1 <= len <*s*> by FINSEQ_1:56; then consider wi being Element of IAlph, qi, qi1 being State of tfsm such that A10: wi=<*s*>.1 & qi=qsa.1 & qi1=qsa.(1+1) & wi-succ_of qi = qi1 by Def2; (the Tran of tfsm).d = q1 by A2,A4,FUNCT_1:72; then (the Tran of tfsm).[q, s] = q1 by A2,A4,MCART_1:23; then A11: qsa.(1+1) = q1 by A8,A9,A10,Def1; A12: len(w^<*s*>)+1 =len w +1+1 by A7,FINSEQ_1:56 .= len w +(1+1) by XCMPLX_1:1; 1+1 =2; then 1 <= 2 & 2 <= len <*s*> + 1 by FINSEQ_1:56; then (I, w^<*s*>)-admissible.(len (w^<*s*>) + 1) = q1 by A6,A11,A12,Th22; then I,w^<*s*>-leads_to q1 by Def3; then q1 is accessible by Def21; hence x in accessibleStates tfsm by Th67; end; hence thesis by A2,FUNCT_2:def 1,RELSET_1:11; end; theorem for cTran being Function of [:accessibleStates tfsm, IAlph:], accessibleStates tfsm, cOFun being Function of [:accessibleStates tfsm, IAlph:], OAlph, cInitS being Element of accessibleStates tfsm st cTran = (the Tran of tfsm) | [:accessibleStates tfsm, IAlph:] & cOFun = (the OFun of tfsm) | [:accessibleStates tfsm, IAlph:] & cInitS = the InitS of tfsm holds tfsm, Mealy-FSM(#accessibleStates tfsm, cTran, cOFun, cInitS#) -are_equivalent proof set M = tfsm; let cTran be Function of [:accessibleStates M, IAlph:], accessibleStates M, cOFun be Function of [:accessibleStates M, IAlph:], OAlph, cInitS be Element of accessibleStates M such that A1: cTran = (the Tran of M) | [:accessibleStates M, IAlph:] and A2: cOFun = (the OFun of M) | [:accessibleStates M, IAlph:] and A3: cInitS = the InitS of M; set cm = Mealy-FSM(#accessibleStates M, cTran, cOFun, cInitS#); let w be FinSequence of IAlph; set ma= (the InitS of M,w)-admissible;set cma= (the InitS of cm, w)-admissible; A4: now thus len ma = len w + 1 & len cma = len w + 1 by Def2; defpred P[Nat] means $1 in Seg (len w + 1) implies ma.$1 = cma.$1; A5: P[0] by FINSEQ_1:3; A6: for k being Nat st P[k] holds P[k + 1] proof let j be Nat such that A7: j in Seg (len w + 1) implies ma.j = cma.j; assume j+1 in Seg (len w + 1); then 1<=j+1 & j+1<=len w + 1 by FINSEQ_1:3; then A8: j<=len w&j<len w +1 by NAT_1:38,REAL_1:53; A9: 0=j or 0<j & 0+1=1 by NAT_1:19; per cases by A9,NAT_1:38; suppose A10:0 = j; hence ma.(j+1) = the InitS of M by Def2 .= cma.(j+1) by A3,A10,Def2; suppose A11: 1 <= j; then consider mwj being Element of IAlph, mqj,mqj1 being State of M such that A12: mwj = w.j & mqj = ma.j & mqj1 = ma.(j+1) & mwj-succ_of mqj = mqj1 by A8,Def2; consider cmwj being Element of IAlph,cmqj,cmqj1 being State of cm such that A13: cmwj = w.j & cmqj = cma.j & cmqj1 = cma.(j+1) & cmwj-succ_of cmqj=cmqj1 by A8,A11,Def2; mwj-succ_of mqj = (the Tran of M).[mqj, mwj] by Def1 .= cTran.[cmqj, cmwj] by A1,A7,A8,A11,A12,A13,FINSEQ_1:3, FUNCT_1:72 .= cmwj-succ_of cmqj by Def1; hence ma.(j+1) = cma.(j+1) by A12,A13; end; thus for j be Nat holds P[j] from Ind (A5, A6); end; then A14: ma = cma by FINSEQ_2:10; set mr = (the InitS of M, w)-response; set cmr = (the InitS of cm, w)-response; now thus len mr = len w & len cmr = len w by Def6; let j be Nat; assume A15: j in Seg len w; then A16: j in dom w by FINSEQ_1:def 3; A17: j in Seg (len w + 1) by A15,FINSEQ_2:9; then j in dom ma by A4,FINSEQ_1:def 3; then A18: ma.j in the carrier of M & w.j in IAlph by A16,FINSEQ_2:13; j in dom cma by A4,A17,FINSEQ_1:def 3; then cma.j in accessibleStates M by FINSEQ_2:13; then A19: [cma.j, w.j] in [:accessibleStates M, IAlph:] by A18,ZFMISC_1:106; thus mr.j = (the OFun of M).[ma.j, w.j] by A16,Def6 .= cOFun.[cma.j, w.j] by A2,A14,A19,FUNCT_1:72 .= cmr.j by A16,Def6; end; hence thesis by FINSEQ_2:10; end; theorem ex Ctfsm st the Tran of Ctfsm = (the Tran of tfsm)|[:accessibleStates tfsm, IAlph:] & the OFun of Ctfsm = (the OFun of tfsm)|[:accessibleStates tfsm, IAlph:] & the InitS of Ctfsm = the InitS of tfsm & tfsm, Ctfsm-are_equivalent proof set M = tfsm; reconsider cTran = (the Tran of M) | [:accessibleStates M, IAlph:] as Function of [:accessibleStates M,IAlph:],accessibleStates M by Th68; accessibleStates M c= the carrier of M & IAlph c= IAlph by Th67; then [:accessibleStates M, IAlph:] c= [:the carrier of M, IAlph:] by ZFMISC_1:119; then reconsider cOFun = (the OFun of M) | [:accessibleStates M, IAlph:] as Function of [:accessibleStates M, IAlph:], OAlph by FUNCT_2:38; set I = the InitS of M; I, <*>IAlph-leads_to I by Th17; then I is accessible by Def21; then reconsider cInitS = the InitS of M as Element of accessibleStates M by Th67; set cm = Mealy-FSM(#accessibleStates M, cTran, cOFun, cInitS#); A1: now let w be FinSequence of IAlph; set ma= (the InitS of M,w)-admissible; set cma= (the InitS of cm,w)-admissible; now thus len ma = len w + 1 & len cma = len w + 1 by Def2; defpred P[Nat] means $1 in Seg (len w + 1) implies ma.$1 = cma.$1; A2: P[0] by FINSEQ_1:3; A3: for k being Nat st P[k] holds P[k + 1] proof let j be Nat such that A4: j in Seg (len w + 1) implies ma.j = cma.j; assume j+1 in Seg(len w +1); then 1<=j+1&j+1<=len w +1 by FINSEQ_1:3; then A5: j<=len w&j<len w +1 by NAT_1:38,REAL_1:53; A6: 0=j or 0<j & 0+1 = 1 by NAT_1:19; per cases by A6,NAT_1:38; suppose A7:0 = j; hence ma.(j+1) = the InitS of M by Def2 .= cma.(j+1) by A7,Def2; suppose A8: 1 <= j; then consider mwj being Element of IAlph, mqj,mqj1 being State of M such that A9: mwj = w.j & mqj = ma.j & mqj1 = ma.(j+1) & mwj-succ_of mqj = mqj1 by A5,Def2; consider cmwj being Element of IAlph, cmqj, cmqj1 being State of cm such that A10: cmwj = w.j & cmqj = cma.j & cmqj1 = cma.(j+1) & cmwj-succ_of cmqj=cmqj1 by A5,A8,Def2; mwj-succ_of mqj = (the Tran of M).[mqj, mwj] by Def1 .= cTran.[cmqj, cmwj] by A4,A5,A8,A9,A10,FINSEQ_1:3, FUNCT_1:72 .= cmwj-succ_of cmqj by Def1; hence ma.(j+1) = cma.(j+1) by A9,A10; end; thus for j be Nat holds P[j] from Ind (A2, A3); end; hence ma = cma by FINSEQ_2:10; end; now let q be State of cm; q in accessibleStates M & accessibleStates M c= the carrier of M by Th67; then reconsider q' = q as State of M; q' is accessible by Th67; then consider w being FinSequence of IAlph such that A11: the InitS of M, w-leads_to q' by Def21; (the InitS of M, w)-admissible.(len w + 1) = q' by A11,Def3; then (the InitS of cm, w)-admissible.(len w + 1) = q by A1; then the InitS of cm, w-leads_to q by Def3; hence q is accessible by Def21; end; then reconsider cm as connected (finite non empty Mealy-FSM over IAlph, OAlph) by Def22; take cm; thus the Tran of cm = (the Tran of M) | [:accessibleStates M, IAlph:]; thus the OFun of cm = (the OFun of M) | [:accessibleStates M, IAlph:]; thus the InitS of cm = the InitS of M; let w be FinSequence of IAlph; set ma= (the InitS of M,w)-admissible; set cma= (the InitS of cm,w)-admissible; A12: len ma = len w + 1 & len cma = len w + 1 by Def2; set mr = (the InitS of M, w)-response; set cmr = (the InitS of cm, w)-response; now thus len mr = len w & len cmr = len w by Def6;let j be Nat; assume A13: j in Seg len w; then A14: j in Seg (len w + 1) by FINSEQ_2:9; A15: [ma.j, w.j] = [cma.j, w.j] by A1; A16: j in dom w by A13,FINSEQ_1:def 3; j in dom ma by A12,A14,FINSEQ_1:def 3; then A17: ma.j in the carrier of M & w.j in IAlph by A16,FINSEQ_2:13; j in dom cma by A12,A14,FINSEQ_1:def 3; then cma.j in accessibleStates M by FINSEQ_2:13; then A18: [cma.j, w.j] in [:accessibleStates M, IAlph:] by A17,ZFMISC_1:106; thus mr.j = (the OFun of M).[ma.j, w.j] by A16,Def6 .= cOFun.[cma.j, w.j] by A15,A18,FUNCT_1:72 .= cmr.j by A16,Def6; end; hence thesis by FINSEQ_2:10; end; begin :: Machine union definition let IAlph be set, OAlph be non empty set; let tfsm1, tfsm2 be non empty Mealy-FSM over IAlph, OAlph; func tfsm1-Mealy_union tfsm2 -> strict Mealy-FSM over IAlph, OAlph means :Def24: the carrier of it = (the carrier of tfsm1) \/ (the carrier of tfsm2) & the Tran of it = (the Tran of tfsm1) +* (the Tran of tfsm2) & the OFun of it = (the OFun of tfsm1) +* (the OFun of tfsm2) & the InitS of it = the InitS of tfsm1; existence proof set S_tfsm1 = the carrier of tfsm1, S_tfsm2 = the carrier of tfsm2; set Tr_tfsm1 = the Tran of tfsm1, Tr_tfsm2 = the Tran of tfsm2; set Of_tfsm1 = the OFun of tfsm1, Of_tfsm2 = the OFun of tfsm2; set Tr = Tr_tfsm1 +* Tr_tfsm2, Of = Of_tfsm1 +* Of_tfsm2; reconsider S = S_tfsm1 \/ S_tfsm2 as non empty set; A1: dom Tr_tfsm1 = [: S_tfsm1, IAlph :] by FUNCT_2:def 1; dom Tr_tfsm2 = [: S_tfsm2, IAlph :] by FUNCT_2:def 1; then A2: dom Tr = [: S_tfsm1, IAlph :] \/ [: S_tfsm2, IAlph :] by A1,FUNCT_4: def 1 .= [: S_tfsm1 \/ S_tfsm2, IAlph :] by ZFMISC_1:120; A3: rng Tr_tfsm1 c= S_tfsm1 & rng Tr_tfsm2 c= S_tfsm2 by RELSET_1:12; A4: rng Tr c= rng Tr_tfsm1 \/ rng Tr_tfsm2 by FUNCT_4:18; rng Tr_tfsm1 \/ rng Tr_tfsm2 c= S_tfsm1 \/ S_tfsm2 by A3,XBOOLE_1:13; then rng Tr c= S by A4,XBOOLE_1:1; then reconsider Tr as Function of [: S, IAlph :], S by A2,FUNCT_2:4; A5: dom Of_tfsm1 = [: S_tfsm1, IAlph :] by FUNCT_2:def 1; dom Of_tfsm2 = [: S_tfsm2, IAlph :] by FUNCT_2:def 1; then A6: dom Of = [: S_tfsm1, IAlph :] \/ [: S_tfsm2, IAlph :] by A5,FUNCT_4: def 1 .= [: S_tfsm1 \/ S_tfsm2, IAlph :] by ZFMISC_1:120; A7: rng Of_tfsm1 c= OAlph & rng Of_tfsm2 c= OAlph by RELSET_1:12; A8: rng Of c= rng Of_tfsm1 \/ rng Of_tfsm2 by FUNCT_4:18; rng Of_tfsm1 \/ rng Of_tfsm2 c= OAlph \/ OAlph by A7,XBOOLE_1:13; then rng Of c= OAlph \/ OAlph by A8,XBOOLE_1:1; then reconsider Of as Function of [: S, IAlph :], OAlph by A6,FUNCT_2:4; reconsider IS = the InitS of tfsm1 as Element of S by XBOOLE_0:def 2; take Mealy-FSM (# S, Tr, Of, IS #); thus thesis; end; uniqueness; end; definition let IAlph be set, OAlph be non empty set; let tfsm1, tfsm2 be non empty finite Mealy-FSM over IAlph, OAlph; cluster tfsm1-Mealy_union tfsm2 -> non empty finite; coherence proof the carrier of tfsm1-Mealy_union tfsm2 = (the carrier of tfsm1) \/ (the carrier of tfsm2) by Def24; hence thesis by GROUP_1:def 13,STRUCT_0:def 1; end; end; theorem Th71: tfsm = tfsm1-Mealy_union tfsm2 & the carrier of tfsm1 misses the carrier of tfsm2 & q11 = q implies (q11,w)-admissible = (q,w)-admissible proof assume A1: tfsm = tfsm1-Mealy_union tfsm2 & (the carrier of tfsm1) misses (the carrier of tfsm2) & q11 = q; set ad1 = (q11,w)-admissible, ad = (q,w)-admissible; A2: len ad1 = len w + 1 by Def2 .= len ad by Def2; defpred P[Nat] means 1 <= $1 & $1 <= len ad1 implies ad1.$1 = ad.$1; A3: P[0]; A4: for k being Nat st P[k] holds P[k + 1] proof let k be Nat; assume A5: 1 <= k & k <= len ad1 implies ad1.k = ad.k; assume A6: 1 <= (k+1) & (k+1) <= len ad1; A7: k = 0 or 0 < k & 0 + 1 = 1 by NAT_1:19; per cases by A7,NAT_1:38; suppose A8: k = 0; hence ad1.(k+1) = q11 by Def2 .= ad.(k+1) by A1,A8,Def2; suppose A9: 1 <= k; (k+1) <= len w + 1 by A6,Def2; then A10: 1 <= k & k <= len w by A9,REAL_1:53; then consider w1k being Element of IAlph, q1k, q1k1 being Element of tfsm1 such that A11: w1k = w.k & q1k = ad1.k & q1k1 = ad1.(k+1) & w1k-succ_of q1k = q1k1 by Def2; consider wk being Element of IAlph, qk, qk1 being Element of tfsm such that A12: wk = w.k & qk = ad.k & qk1 = ad.(k+1) & wk-succ_of qk = qk1 by A10,Def2; len w <= len w + 1 by NAT_1:29; then A13: k <= len w + 1 by A10,AXIOMS:22; A14: dom (the Tran of tfsm2) = [:the carrier of tfsm2,IAlph:]by FUNCT_2:def 1; [:the carrier of tfsm1, IAlph :] misses [:the carrier of tfsm2, IAlph :] by A1,ZFMISC_1:127; then A15: not [q1k,w1k] in [:the carrier of tfsm2,IAlph:] by XBOOLE_0:3; thus ad1.(k+1) = (the Tran of tfsm1).[q1k,w1k] by A11,Def1 .= ((the Tran of tfsm1) +* (the Tran of tfsm2)).[q1k,w1k] by A14,A15,FUNCT_4:12 .= (the Tran of tfsm).[qk,wk] by A1,A5,A9,A11,A12,A13,Def2,Def24 .= ad.(k+1) by A12,Def1; end; for k being Nat holds P[k] from Ind(A3, A4); hence thesis by A2,FINSEQ_1:18; end; theorem Th72: tfsm = tfsm1-Mealy_union tfsm2 & the carrier of tfsm1 misses the carrier of tfsm2 & q11 = q implies (q11,w)-response = (q,w)-response proof set q1 = q11; assume A1: tfsm = tfsm1-Mealy_union tfsm2 & (the carrier of tfsm1) misses (the carrier of tfsm2) & q1 = q; set res = (q,w)-response, res1 = (q1,w)-response; set ad1 = (q1,w)-admissible; A2: len res1 = len w by Def6 .= len res by Def6; A3: len res1 = len w by Def6; now let k be Nat; assume 1 <= k & k <= len res1; then A4: k in Seg len w by A3,FINSEQ_1:3; then A5: k in dom w by FINSEQ_1:def 3; k in Seg (len w + 1) by A4,FINSEQ_2:9; then k in Seg len ad1 by Def2; then A6: k in dom ad1 by FINSEQ_1:def 3; A7: dom (the OFun of tfsm2) = [:the carrier of tfsm2,IAlph :] by FUNCT_2:def 1; A8: ad1.k in the carrier of tfsm1 by A6,FINSEQ_2:13; w.k in IAlph by A5,FINSEQ_2:13; then A9: [ad1.k,w.k] in [: the carrier of tfsm1, IAlph :] by A8,ZFMISC_1:106; [:the carrier of tfsm1,IAlph :] misses [:the carrier of tfsm2,IAlph:] by A1,ZFMISC_1:127; then A10: not [ad1.k,w.k] in [: the carrier of tfsm2, IAlph :] by A9,XBOOLE_0:3; res1.k = (the OFun of tfsm1).[(q1,w)-admissible.k,w.k] by A5,Def6 .= ((the OFun of tfsm1) +* (the OFun of tfsm2)).[ad1.k,w.k] by A7,A10,FUNCT_4:12 .= ((the OFun of tfsm1) +* (the OFun of tfsm2)). [(q,w)-admissible.k,w.k] by A1,Th71 .= (the OFun of tfsm).[(q,w)-admissible.k, w.k] by A1,Def24 .= res.k by A5,Def6; hence res1.k = res.k; end; hence thesis by A2,FINSEQ_1:18; end; theorem Th73: tfsm = tfsm1-Mealy_union tfsm2 & the carrier of tfsm1 misses the carrier of tfsm2 & q21 = q implies (q21,w)-admissible = (q,w)-admissible proof set q' = q21; assume A1: tfsm = tfsm1-Mealy_union tfsm2 & (the carrier of tfsm1) /\ (the carrier of tfsm2) = {} & q' = q; set ad' = (q',w)-admissible, ad = (q,w)-admissible; A2: len ad' = len w + 1 by Def2 .= len ad by Def2; defpred P[Nat] means 1<=$1 & $1 <= len ad' implies ad'.$1= ad.$1; A3: P[0]; A4: for k being Nat st P[k] holds P[k + 1] proof let k be Nat; assume A5: 1 <= k & k <= len ad' implies ad'.k = ad.k; assume A6: 1 <= (k+1) & (k+1) <= len ad'; A7: k = 0 or 0 < k & 0 + 1 = 1 by NAT_1:19; per cases by A7,NAT_1:38; suppose A8: k = 0; hence ad'.(k+1) = q' by Def2 .= ad.(k+1) by A1,A8,Def2; suppose A9: 1 <= k; k+1 <= len w + 1 by A6,Def2; then A10: 1 <= k & k <= len w by A9,REAL_1:53; then consider w'k being Element of IAlph, q'k, q'k1 being Element of tfsm2 such that A11: w'k = w.k & q'k = ad'.k & q'k1 = ad'.(k+1) & w'k-succ_of q'k = q'k1 by Def2; consider wk being Element of IAlph, qk, qk1 being Element of tfsm such that A12: wk = w.k & qk = ad.k & qk1 = ad.(k+1) & wk-succ_of qk = qk1 by A10,Def2; len w <= len w + 1 by NAT_1:29; then A13: k <= len w + 1 by A10,AXIOMS:22; A14:dom (the Tran of tfsm2) = [:the carrier of tfsm2,IAlph:]by FUNCT_2:def 1; thus ad'.(k+1) = (the Tran of tfsm2).[q'k,w'k] by A11,Def1 .= ((the Tran of tfsm1) +* (the Tran of tfsm2)).[q'k,w'k]by A14,FUNCT_4:14 .= (the Tran of tfsm).[qk,wk] by A1,A5,A9,A11,A12,A13,Def2,Def24 .= ad.(k+1) by A12,Def1; end; for k being Nat holds P[k] from Ind(A3, A4); hence thesis by A2,FINSEQ_1:18; end; theorem Th74: tfsm = tfsm1-Mealy_union tfsm2 & the carrier of tfsm1 misses the carrier of tfsm2 & q21 = q implies (q21,w)-response = (q,w)-response proof set q' = q21; assume A1: tfsm = tfsm1-Mealy_union tfsm2 & (the carrier of tfsm1) misses (the carrier of tfsm2) & q' = q; set res = (q,w)-response, res' = (q',w)-response; set ad' = (q',w)-admissible; A2: len res' = len w by Def6 .= len res by Def6; A3: len res' = len w by Def6; now let k be Nat; assume 1 <= k & k <= len res'; then A4: k in Seg len w by A3,FINSEQ_1:3; then A5: k in dom w by FINSEQ_1:def 3; k in Seg (len w + 1) by A4,FINSEQ_2:9; then k in Seg len ad' by Def2; then A6: k in dom ad' by FINSEQ_1:def 3; A7: dom (the OFun of tfsm2) = [:the carrier of tfsm2,IAlph:] by FUNCT_2:def 1; A8: w.k in IAlph by A5,FINSEQ_2:13; ad'.k in the carrier of tfsm2 by A6,FINSEQ_2:13; then A9: [ad'.k, w.k ] in dom (the OFun of tfsm2) by A7,A8,ZFMISC_1:106; res'.k = (the OFun of tfsm2).[(q',w)-admissible.k,w.k] by A5,Def6 .= ((the OFun of tfsm1) +* (the OFun of tfsm2)).[ad'.k,w.k] by A9,FUNCT_4:14 .= ((the OFun of tfsm1) +* (the OFun of tfsm2)). [(q,w)-admissible.k,w.k] by A1,Th73 .= (the OFun of tfsm).[(q,w)-admissible.k, w.k] by A1,Def24 .= res.k by A5,Def6; hence res'.k = res.k; end; hence thesis by A2,FINSEQ_1:18; end; reserve Rtfsm1, Rtfsm2 for reduced (non empty Mealy-FSM over IAlph, OAlph); theorem Th75: tfsm = Rtfsm1-Mealy_union Rtfsm2 & the carrier of Rtfsm1 misses the carrier of Rtfsm2 & Rtfsm1, Rtfsm2-are_equivalent implies ex Q being State of the_reduction_of tfsm st the InitS of Rtfsm1 in Q & the InitS of Rtfsm2 in Q & Q = the InitS of the_reduction_of tfsm proof set rtfsm1 = Rtfsm1; set rtfsm2 = Rtfsm2; assume A1: tfsm = rtfsm1-Mealy_union rtfsm2 & (the carrier of rtfsm1) misses (the carrier of rtfsm2) & rtfsm1, rtfsm2-are_equivalent; set RUN = the_reduction_of tfsm; set SRUN = the carrier of RUN; set S_tfsm = the carrier of tfsm, S_rtfsm1 = the carrier of rtfsm1; set S_rtfsm2 = the carrier of rtfsm2; A2: S_tfsm = S_rtfsm1 \/ S_rtfsm2 by A1,Def24; reconsider IS1 = the InitS of rtfsm1 as Element of S_tfsm by A1,Def24; reconsider IS2 = the InitS of rtfsm2 as Element of S_tfsm by A2,XBOOLE_0: def 2 ; A3: SRUN = final_states_partition tfsm by Def18; A4: final_states_partition tfsm is final by Def15; now let w be FinSequence of IAlph; (the InitS of rtfsm1,w)-response = (the InitS of rtfsm2,w)-response by A1,Def9; hence (IS1, w)-response = (the InitS of rtfsm2,w)-response by A1,Th72 .= (IS2, w)-response by A1,Th74; end; then IS1,IS2-are_equivalent by Def10; then consider X being Element of SRUN such that A5: IS1 in X & IS2 in X by A3,A4,Def14; take X; thus the InitS of rtfsm1 in X & the InitS of rtfsm2 in X by A5; the InitS of tfsm = the InitS of rtfsm1 by A1,Def24; then A6: the InitS of rtfsm1 in the InitS of RUN by Def18; A7: X is Subset of S_tfsm by A3,TARSKI:def 3; the InitS of RUN is Subset of S_tfsm by A3,TARSKI:def 3; then X = the InitS of RUN or X misses the InitS of RUN by A3,A7,EQREL_1:def 6; hence thesis by A5,A6,XBOOLE_0:3; end; reserve CRtfsm1, CRtfsm2 for connected reduced (non empty Mealy-FSM over IAlph, OAlph), q1u, q2u for State of tfsm; theorem Th76: for crq11, crq12 being State of CRtfsm1 holds crq11 = q1u & crq12 = q2u & the carrier of CRtfsm1 misses the carrier of CRtfsm2 & CRtfsm1, CRtfsm2-are_equivalent & tfsm = CRtfsm1-Mealy_union CRtfsm2 & not crq11, crq12-are_equivalent implies not q1u,q2u-are_equivalent proof let crq11, crq12 be State of CRtfsm1; set rtfsm1 = CRtfsm1, rtfsm2 = CRtfsm2, q1 = crq11, q2 = crq12; assume A1: q1 = q1u & q2 = q2u & (the carrier of rtfsm1) misses (the carrier of rtfsm2) & rtfsm1, rtfsm2-are_equivalent & tfsm = rtfsm1-Mealy_union rtfsm2; assume not q1,q2-are_equivalent; then consider w be FinSequence of IAlph such that A2: (q1,w)-response <> (q2,w)-response by Def10; (q1u,w)-response = (q1,w)-response by A1,Th72; then (q1u,w)-response <> (q2u,w)-response by A1,A2,Th72; hence thesis by Def10; end; theorem Th77: for crq21, crq22 being State of CRtfsm2 holds crq21 = q1u & crq22 = q2u & the carrier of CRtfsm1 misses the carrier of CRtfsm2 & CRtfsm1, CRtfsm2-are_equivalent & tfsm = CRtfsm1-Mealy_union CRtfsm2 & not crq21, crq22-are_equivalent implies not q1u, q2u-are_equivalent proof let crq21, crq22 be State of CRtfsm2; set rtfsm1 = CRtfsm1, rtfsm2 = CRtfsm2, q1 = crq21, q2 = crq22; assume A1: q1 = q1u & q2 = q2u & (the carrier of rtfsm1) misses (the carrier of rtfsm2) & rtfsm1, rtfsm2-are_equivalent & tfsm = rtfsm1-Mealy_union rtfsm2; assume not q1,q2-are_equivalent; then consider w be FinSequence of IAlph such that A2: (q1,w)-response <> (q2,w)-response by Def10; (q1u,w)-response = (q1,w)-response by A1,Th74; then (q1u,w)-response <> (q2u,w)-response by A1,A2,Th74; hence thesis by Def10; end; reserve CRtfsm1, CRtfsm2 for connected reduced (finite non empty Mealy-FSM over IAlph, OAlph); theorem Th78: the carrier of CRtfsm1 misses the carrier of CRtfsm2 & CRtfsm1, CRtfsm2-are_equivalent & tfsm = CRtfsm1-Mealy_union CRtfsm2 implies for Q being State of the_reduction_of tfsm holds not ex q1, q2 being Element of Q st q1 in the carrier of CRtfsm1 & q2 in the carrier of CRtfsm1 & q1 <> q2 proof set rtfsm1 = CRtfsm1; set rtfsm2 = CRtfsm2; assume A1: (the carrier of rtfsm1) misses (the carrier of rtfsm2) & rtfsm1, rtfsm2-are_equivalent & tfsm = rtfsm1-Mealy_union rtfsm2; consider tfsm1 be finite non empty Mealy-FSM over IAlph, OAlph such that A2: rtfsm1, the_reduction_of tfsm1-are_isomorphic by Th65; given Q be Element of the_reduction_of tfsm, q1, q2 being Element of Q such that A3: q1 in the carrier of rtfsm1 & q2 in the carrier of rtfsm1 & q1 <> q2; set tfsm1_r = the_reduction_of tfsm1; consider Tf being Function of the carrier of rtfsm1, the carrier of tfsm1_r such that A4: Tf is bijective & Tf.the InitS of rtfsm1 = the InitS of tfsm1_r & for q being State of rtfsm1, s being Element of IAlph holds Tf.((the Tran of rtfsm1).[q, s]) = (the Tran of tfsm1_r).[Tf.q, s] & (the OFun of rtfsm1).[q,s] = (the OFun of tfsm1_r).[Tf.q, s] by A2,Def19; Tf is one-to-one onto by A4,FUNCT_2:def 4; then A5: Tf is one-to-one & rng Tf=the carrier of tfsm1_r by FUNCT_2:def 3; A6: dom Tf = the carrier of rtfsm1 by FUNCT_2:def 1; then A7: Tf.q1 in the carrier of tfsm1_r & Tf.q2 in the carrier of tfsm1_r & Tf.q1 <> Tf.q2 by A3,A5,FUNCT_1:def 5,def 8; reconsider Tq1 = Tf.q1, Tq2 = Tf.q2 as Element of tfsm1_r by A3,A5,A6,FUNCT_1:def 5; A8: not Tq1, Tq2 -are_equivalent by A7,Th62; set rtfsm = the_reduction_of tfsm; the carrier of tfsm = (the carrier of rtfsm1) \/ (the carrier of rtfsm2) by A1,Def24; then reconsider q1, q2 as Element of tfsm by A3,XBOOLE_0:def 2; A9: final_states_partition tfsm is final by Def15; A10:the carrier of rtfsm = final_states_partition tfsm by Def18; then reconsider Q as Subset of tfsm by TARSKI:def 3; Q <> {} by A10,EQREL_1:def 6; then A11:q1,q2-are_equivalent by A9,A10,Def14; reconsider q1' = q1, q2' = q2 as Element of rtfsm1 by A3; not q1',q2'-are_equivalent by A4,A8,Th61; hence contradiction by A1,A11,Th76; end; theorem Th79: the carrier of CRtfsm1 misses the carrier of CRtfsm2 & CRtfsm1, CRtfsm2-are_equivalent & tfsm = CRtfsm1-Mealy_union CRtfsm2 implies for Q being State of the_reduction_of tfsm holds not ex q1, q2 being Element of Q st q1 in the carrier of CRtfsm2 & q2 in the carrier of CRtfsm2 & q1 <> q2 proof set rtfsm1 = CRtfsm1; set rtfsm2 = CRtfsm2; assume A1: (the carrier of rtfsm1) misses (the carrier of rtfsm2) & rtfsm1, rtfsm2-are_equivalent & tfsm = rtfsm1-Mealy_union rtfsm2; consider tfsm2 be finite non empty Mealy-FSM over IAlph, OAlph such that A2: rtfsm2, the_reduction_of tfsm2-are_isomorphic by Th65; given Q be Element of the_reduction_of tfsm, q1, q2 being Element of Q such that A3: q1 in the carrier of rtfsm2 & q2 in the carrier of rtfsm2 & q1 <> q2; set tfsm2_r = the_reduction_of tfsm2; consider Tf being Function of the carrier of rtfsm2, the carrier of tfsm2_r such that A4: Tf is bijective & Tf.the InitS of rtfsm2 = the InitS of tfsm2_r & for q being State of rtfsm2, s being Element of IAlph holds Tf.((the Tran of rtfsm2).[q, s]) = (the Tran of tfsm2_r).[Tf.q, s] & (the OFun of rtfsm2).[q,s] = (the OFun of tfsm2_r).[Tf.q, s] by A2,Def19; Tf is one-to-one onto by A4,FUNCT_2:def 4; then A5: Tf is one-to-one & rng Tf=the carrier of tfsm2_r by FUNCT_2:def 3; A6: dom Tf = the carrier of rtfsm2 by FUNCT_2:def 1; then A7: Tf.q1 in the carrier of tfsm2_r & Tf.q2 in the carrier of tfsm2_r & Tf.q1 <> Tf.q2 by A3,A5,FUNCT_1:def 5,def 8; reconsider Tq1 = Tf.q1, Tq2 = Tf.q2 as Element of tfsm2_r by A3,A5,A6,FUNCT_1:def 5; A8: not Tq1, Tq2 -are_equivalent by A7,Th62; set rtfsm = the_reduction_of tfsm; A9: the carrier of tfsm = (the carrier of rtfsm1) \/ (the carrier of rtfsm2) by A1,Def24; then reconsider q1 as Element of tfsm by A3,XBOOLE_0:def 2; reconsider q2 as Element of tfsm by A3,A9,XBOOLE_0:def 2; A10: final_states_partition tfsm is final by Def15; A11: the carrier of rtfsm = final_states_partition tfsm by Def18; then reconsider Q as Subset of tfsm by TARSKI:def 3; Q <> {} by A11,EQREL_1:def 6; then A12: q1,q2-are_equivalent by A10,A11,Def14; reconsider q1' = q1 as Element of rtfsm2 by A3; reconsider q2' = q2 as Element of rtfsm2 by A3; not q1',q2'-are_equivalent by A4,A8,Th61; hence contradiction by A1,A12,Th77; end; theorem Th80: the carrier of CRtfsm1 misses the carrier of CRtfsm2 & CRtfsm1, CRtfsm2-are_equivalent & tfsm = CRtfsm1-Mealy_union CRtfsm2 implies for Q being State of the_reduction_of tfsm ex q1, q2 being Element of Q st q1 in the carrier of CRtfsm1 & q2 in the carrier of CRtfsm2 & for q being Element of Q holds q = q1 or q = q2 proof set rtfsm1 = CRtfsm1; set rtfsm2 = CRtfsm2; assume A1: (the carrier of rtfsm1) misses (the carrier of rtfsm2) & rtfsm1, rtfsm2-are_equivalent & tfsm = rtfsm1-Mealy_union rtfsm2; assume not thesis; then consider Q be Element of the_reduction_of tfsm such that A2: for q1, q2 being Element of Q holds not q1 in the carrier of rtfsm1 or not q2 in the carrier of rtfsm2 or not for q being Element of Q holds q = q1 or q = q2; A3: (the carrier of rtfsm1) /\ (the carrier of rtfsm2) = {} by A1,XBOOLE_0:def 7; set rtfsm = the_reduction_of tfsm; set S_rtfsm1 = the carrier of rtfsm1; set IS_rtfsm1 = the InitS of rtfsm1; set S_rtfsm2 = the carrier of rtfsm2; set IS_rtfsm2 = the InitS of rtfsm2; set S_tfsm = the carrier of tfsm; set S_rtfsm = the carrier of rtfsm; A4: S_rtfsm = final_states_partition tfsm by Def18; then reconsider Q as Subset of S_tfsm by TARSKI:def 3; Q in S_rtfsm; then A5: Q in final_states_partition tfsm by Def18; then A6: Q <> {} by EQREL_1:def 6; then consider q being Element of S_tfsm such that A7: q in Q by SUBSET_1:10; union final_states_partition tfsm = S_tfsm by EQREL_1:def 6 .= S_rtfsm1 \/ S_rtfsm2 by A1,Def24; then A8: Q c= S_rtfsm1 \/ S_rtfsm2 by A4,ZFMISC_1:92; per cases by A7,A8,XBOOLE_0:def 2; suppose A9: q in S_rtfsm1; set Q' = Q \ {q}; A10: now assume A11: Q' = {}; reconsider q as Element of S_rtfsm1 by A9; q is accessible by Def22; then consider w be FinSequence of IAlph such that A12: IS_rtfsm1, w-leads_to q by Def21; set ad_l = IS_rtfsm2 leads_to_under w; A13: now assume A14: ad_l in Q; ad_l <> q by A3,XBOOLE_0:def 3; hence contradiction by A11,A14,ZFMISC_1:64; end; now assume A15: not ad_l in Q; A16: ad_l in S_rtfsm1 \/ S_rtfsm2 by XBOOLE_0:def 2; reconsider q as Element of S_tfsm; reconsider ad_l as Element of S_tfsm by A1,A16,Def24; A17: final_states_partition tfsm is final by Def15; not ex Y being Element of S_rtfsm st q in Y & ad_l in Y proof assume not thesis; then consider Y be Element of S_rtfsm such that A18: q in Y & ad_l in Y; reconsider Y as Subset of S_tfsm by A4,TARSKI:def 3; reconsider Q as Subset of S_tfsm; Y in S_rtfsm; then Y in final_states_partition tfsm by Def18; then Y misses Q by A5,A15,A18,EQREL_1:def 6; hence contradiction by A7,A18,XBOOLE_0:3; end; then not q, ad_l-are_equivalent by A4,A17,Def14; then consider dseq being FinSequence of IAlph such that A19: (q,dseq)-response <> (ad_l,dseq)-response by Def10; reconsider ad_l2 = ad_l as Element of S_rtfsm2; reconsider q_1 = q as Element of S_rtfsm1; A20: IS_rtfsm2, w -leads_to ad_l2 by Def5; (q,dseq)-response = (q_1,dseq)-response by A1,Th72; then (q_1,dseq)-response <> (ad_l2,dseq)-response by A1,A19,Th74; then (IS_rtfsm1,w^dseq)-response <> (IS_rtfsm2,w^dseq)-response by A12,A20,Th27; hence contradiction by A1,Def9; end; hence contradiction by A13; end; now assume A21: Q' <> {}; A22: Q' c= Q by XBOOLE_1:36; reconsider Q as Subset of S_tfsm; reconsider Q' as Subset of S_tfsm; consider x being Element of S_tfsm such that A23: x in Q' by A21,SUBSET_1:10; A24: Q' c= S_rtfsm1 \/ S_rtfsm2 by A8,A22,XBOOLE_1:1; per cases by A23,A24,XBOOLE_0:def 2; suppose A25: x in S_rtfsm1; reconsider q as Element of Q by A7; reconsider x as Element of Q by A23,XBOOLE_0:def 4; not q in S_rtfsm1 or not x in S_rtfsm1 or q = x by A1,Th78; hence contradiction by A9,A23,A25,ZFMISC_1:64; suppose A26: x in S_rtfsm2; set Q'' = Q' \ {x}; A27: now assume A28: Q'' = {}; A29: for z being Element of Q holds z = q or z = x proof given z being Element of Q such that A30: z <> q & z <> x; z in Q' by A6,A30,ZFMISC_1:64; hence contradiction by A28,A30,ZFMISC_1:64; end; reconsider q as Element of Q by A7; reconsider x as Element of Q by A23,ZFMISC_1:64; not q in S_rtfsm1 or not x in S_rtfsm2 or ex z being Element of Q st z <> q & z <> x by A2; hence contradiction by A9,A26,A29; end; now assume A31: Q'' <> {}; A32: Q'' c= Q' by XBOOLE_1:36; reconsider Q' as Subset of S_tfsm; reconsider Q'' as Subset of S_tfsm; consider y being Element of S_tfsm such that A33: y in Q'' by A31,SUBSET_1:10; A34: Q'' c= S_rtfsm1 \/ S_rtfsm2 by A24,A32,XBOOLE_1:1; per cases by A33,A34,XBOOLE_0:def 2; suppose A35: y in S_rtfsm1; A36: y in Q' by A33,ZFMISC_1:64; reconsider q as Element of Q by A7; reconsider y as Element of Q by A36,ZFMISC_1:64; not q in S_rtfsm1 or not y in S_rtfsm1 or q = y by A1,Th78; hence contradiction by A9,A35,A36,ZFMISC_1:64; suppose A37: y in S_rtfsm2; reconsider x as Element of Q by A23,ZFMISC_1:64; y in Q' by A33,ZFMISC_1:64; then reconsider y as Element of Q by ZFMISC_1:64; not x in S_rtfsm2 or not y in S_rtfsm2 or x = y by A1,Th79; hence contradiction by A26,A33,A37,ZFMISC_1:64; end; hence contradiction by A27; end; hence contradiction by A10; suppose A38: q in S_rtfsm2; set Q' = Q \ {q}; A39: now assume A40: Q' = {}; reconsider q as Element of S_rtfsm2 by A38; q is accessible by Def22; then consider w be FinSequence of IAlph such that A41: IS_rtfsm2, w-leads_to q by Def21; set ad_l = IS_rtfsm1 leads_to_under w; A42: now assume A43: ad_l in Q; ad_l <> q by A1,XBOOLE_0:3; hence contradiction by A40,A43,ZFMISC_1:64; end; now assume A44: not ad_l in Q; A45: ad_l in S_rtfsm1 \/ S_rtfsm2 by XBOOLE_0:def 2; reconsider q as Element of S_tfsm; reconsider ad_l as Element of S_tfsm by A1,A45,Def24; A46: final_states_partition tfsm is final by Def15; not ex Y being Element of S_rtfsm st q in Y & ad_l in Y proof assume not thesis; then consider Y be Element of S_rtfsm such that A47: q in Y & ad_l in Y; reconsider Y as Subset of S_tfsm by A4,TARSKI:def 3; Y in S_rtfsm; then Y in final_states_partition tfsm by Def18; then Y misses Q by A5,A44,A47,EQREL_1:def 6; hence contradiction by A7,A47,XBOOLE_0:3; end; then not q, ad_l-are_equivalent by A4,A46,Def14; then consider dseq being FinSequence of IAlph such that A48: (q,dseq)-response <> (ad_l,dseq)-response by Def10; reconsider ad_l2 = ad_l as Element of S_rtfsm1; reconsider q_1 = q as Element of S_rtfsm2; A49: IS_rtfsm1, w -leads_to ad_l2 by Def5; (q,dseq)-response = (q_1,dseq)-response by A1,Th74; then (q_1,dseq)-response <> (ad_l2,dseq)-response by A1,A48,Th72; then (IS_rtfsm2,w^dseq)-response <> (IS_rtfsm1,w^dseq)-response by A41,A49,Th27; hence contradiction by A1,Def9; end; hence contradiction by A42; end; now assume A50: Q' <> {}; A51: Q' c= Q by XBOOLE_1:36; reconsider Q as Subset of S_tfsm; reconsider Q' as Subset of S_tfsm; consider x being Element of S_tfsm such that A52: x in Q' by A50,SUBSET_1:10; A53: Q' c= S_rtfsm1 \/ S_rtfsm2 by A8,A51,XBOOLE_1:1; per cases by A52,A53,XBOOLE_0:def 2; suppose A54: x in S_rtfsm1; set Q'' = Q' \ {x}; A55: now assume A56: Q'' = {}; A57: for z being Element of Q holds z = q or z = x proof given z being Element of Q such that A58: z <> q & z <> x; z in Q' by A6,A58,ZFMISC_1:64; hence contradiction by A56,A58,ZFMISC_1:64; end; reconsider q as Element of Q by A7; reconsider x as Element of Q by A52,ZFMISC_1:64; not x in S_rtfsm1 or not q in S_rtfsm2 or ex z being Element of Q st z <> x & z <> q by A2; hence contradiction by A38,A54,A57; end; now assume A59: Q'' <> {}; A60: Q'' c= Q' by XBOOLE_1:36; reconsider Q' as Subset of S_tfsm; reconsider Q'' as Subset of S_tfsm; consider y being Element of S_tfsm such that A61: y in Q'' by A59,SUBSET_1:10; A62: Q'' c= S_rtfsm1 \/ S_rtfsm2 by A53,A60,XBOOLE_1:1; per cases by A61,A62,XBOOLE_0:def 2; suppose A63: y in S_rtfsm1; reconsider x as Element of Q by A52,ZFMISC_1:64; y in Q' by A61,ZFMISC_1:64; then reconsider y as Element of Q by ZFMISC_1:64; not x in S_rtfsm1 or not y in S_rtfsm1 or x = y by A1,Th78; hence contradiction by A54,A61,A63,ZFMISC_1:64; suppose A64: y in S_rtfsm2; A65: y in Q' by A61,ZFMISC_1:64; reconsider q as Element of Q by A7; reconsider y as Element of Q by A65,ZFMISC_1:64; not q in S_rtfsm2 or not y in S_rtfsm2 or q = y by A1,Th79; hence contradiction by A38,A64,A65,ZFMISC_1:64; end; hence contradiction by A55; suppose A66: x in S_rtfsm2; reconsider q as Element of Q by A7; reconsider x as Element of Q by A52,XBOOLE_0:def 4; not q in S_rtfsm2 or not x in S_rtfsm2 or q = x by A1,Th79; hence contradiction by A38,A52,A66,ZFMISC_1:64; end; hence contradiction by A39; end; begin :: The minimization theorem theorem Th81: for tfsm1, tfsm2 being finite non empty Mealy-FSM over IAlph, OAlph ex fsm1, fsm2 being finite non empty Mealy-FSM over IAlph, OAlph st the carrier of fsm1 misses the carrier of fsm2 & fsm1, tfsm1-are_isomorphic & fsm2,tfsm2-are_isomorphic proof let tfsm1, tfsm2 be finite non empty Mealy-FSM over IAlph, OAlph; set S_tfsm1 = the carrier of tfsm1, Tr1 = the Tran of tfsm1; set Of1 = the OFun of tfsm1, S_tfsm2 = the carrier of tfsm2; set Tr2 = the Tran of tfsm2, Of2 = the OFun of tfsm2; set S_fsm1 = [:{0}, S_tfsm1:], S_fsm2 = [:{1}, S_tfsm2:]; deffunc F(Element of S_fsm1, Element of IAlph) = [ $1`1, Tr1.[$1`2,$2] ]; consider T1 be Function of [: S_fsm1, IAlph :], S_fsm1 such that A1: for q being Element of S_fsm1, s being Element of IAlph holds T1.[q,s] = F(q,s) from Lambda2D; deffunc F(Element of S_fsm1, Element of IAlph) = Of1.[$1`2, $2]; consider O1 be Function of [: S_fsm1, IAlph :], OAlph such that A2: for q being Element of S_fsm1, s being Element of IAlph holds O1.[q,s] = F(q,s) from Lambda2D; deffunc F(Element of S_fsm2, Element of IAlph) = [ $1`1, Tr2.[$1`2,$2] ]; consider T2 be Function of [: S_fsm2, IAlph :], S_fsm2 such that A3: for q being Element of S_fsm2, s being Element of IAlph holds T2.[q,s] = F(q,s) from Lambda2D; deffunc F(Element of S_fsm2, Element of IAlph) = Of2.[$1`2,$2]; consider O2 be Function of [: S_fsm2, IAlph :], OAlph such that A4: for q being Element of S_fsm2, s being Element of IAlph holds O2.[q,s] = F(q,s) from Lambda2D; set z = 0, zz = 1, A = {z}, B = {zz}; set sS_fsm1 = [: A, S_tfsm1 :]; reconsider sT1 = T1 as Function of [: sS_fsm1, IAlph :], sS_fsm1; reconsider sO1 = O1 as Function of [: sS_fsm1, IAlph :], OAlph; reconsider I1 = [ 0, the InitS of tfsm1 ] as Element of S_fsm1 by ZFMISC_1 :128; set sI1 = I1; take fsm1 = Mealy-FSM (# sS_fsm1, sT1, sO1, sI1 #); reconsider sS_fsm2 = [: B, S_tfsm2 :] as finite non empty set; reconsider sT2 = T2 as Function of [: sS_fsm2, IAlph :], sS_fsm2; reconsider sO2 = O2 as Function of [: sS_fsm2, IAlph :], OAlph; reconsider I2 = [ 1, the InitS of tfsm2 ] as Element of S_fsm2 by ZFMISC_1 :128; reconsider sI2 = I2 as Element of sS_fsm2; take fsm2 = Mealy-FSM (# sS_fsm2, sT2, sO2, sI2 #); A5: A misses B by ZFMISC_1:17; S_fsm1 /\ S_fsm2 = [: A /\ B, S_tfsm1 /\ S_tfsm2 :] by ZFMISC_1:123 .= [: {}, S_tfsm1 /\ S_tfsm2 :] by A5,XBOOLE_0:def 7 .= {} by ZFMISC_1:113; hence (the carrier of fsm1) /\ (the carrier of fsm2) = {}; thus fsm1, tfsm1-are_isomorphic proof deffunc F(set) = $1`2; consider Tf1 being Function such that A6: dom Tf1 = the carrier of fsm1 & for q being set st q in the carrier of fsm1 holds Tf1.q = F(q) from Lambda; A7: rng Tf1 = S_tfsm1 proof assume A8: not rng Tf1 = S_tfsm1; per cases by A8,XBOOLE_0:def 10; suppose not rng Tf1 c= S_tfsm1; then consider q being set such that A9: q in rng Tf1 & not q in S_tfsm1 by TARSKI:def 3; consider x be set such that A10: x in dom Tf1 & q = Tf1.x by A9,FUNCT_1:def 5; reconsider x2 = x`2 as Element of S_tfsm1 by A6,A10,MCART_1:10; x2 in S_tfsm1; hence contradiction by A6,A9,A10; suppose not S_tfsm1 c= rng Tf1; then consider x being set such that A11: x in S_tfsm1 & not x in rng Tf1 by TARSKI:def 3; 0 in {0} by TARSKI:def 1; then A12: [0,x] in sS_fsm1 by A11,ZFMISC_1:106; then A13: Tf1.[0,x] in rng Tf1 by A6,FUNCT_1:def 5; Tf1.[0,x]=[0,x]`2 by A6, A12; hence contradiction by A11,A13,MCART_1:7; end; then reconsider Tf1s = Tf1 as Function of the carrier of fsm1, the carrier of tfsm1 by A6,FUNCT_2:3; take Tf1s; now let x1, x2 be set; assume A14: x1 in dom Tf1 & x2 in dom Tf1 & Tf1.x1 = Tf1.x2; then A15: x1 = [x1`1, x1`2] & x2 = [x2`1,x2`2] by A6,MCART_1:23; A16: Tf1.x1 = x1`2 & Tf1.x2 = x2`2 by A6,A14; A17: x1`1 in A by A6,A14,MCART_1:10; x2`1 in A by A6,A14,MCART_1:10; then x1`1 = 0 & x2`1 = 0 by A17,TARSKI:def 1; hence x1 = x2 by A14,A15,A16; end; then Tf1s is one-to-one onto by A7,FUNCT_1:def 8,FUNCT_2:def 3; hence Tf1s is bijective by FUNCT_2:def 4; thus Tf1s.the InitS of fsm1 = sI1`2 by A6 .= the InitS of tfsm1 by MCART_1:7; now let q be State of fsm1, s be Element of IAlph; A18: q`1 in A & q`2 in S_tfsm1 by MCART_1:10; reconsider q1 = q`2 as Element of S_tfsm1 by MCART_1:10; reconsider Tq1 = Tr1.[q1, s] as Element of S_tfsm1; A19: [q`1,Tq1] in [: A, S_tfsm1 :] by A18,ZFMISC_1:106; thus Tf1s.((the Tran of fsm1).[q, s]) = Tf1s.[q`1, Tr1.[q`2, s]] by A1 .= [q`1, Tr1.[q`2, s]]`2 by A6,A19 .= Tr1.[q`2, s] by MCART_1:7 .= (the Tran of tfsm1).[Tf1s.q, s] by A6; thus (the OFun of fsm1).[q,s] = Of1.[q`2, s] by A2 .= (the OFun of tfsm1).[Tf1s.q, s] by A6; end; hence for q being State of fsm1, s being Element of IAlph holds Tf1s.((the Tran of fsm1).[q, s]) = (the Tran of tfsm1).[Tf1s.q, s] & (the OFun of fsm1).[q,s] = (the OFun of tfsm1).[Tf1s.q, s]; end; deffunc F(set) = $1`2; consider Tf1 being Function such that A20: dom Tf1 = the carrier of fsm2 & for q being set st q in the carrier of fsm2 holds Tf1.q = F(q) from Lambda; A21:rng Tf1 = S_tfsm2 proof assume A22:not rng Tf1 = S_tfsm2; per cases by A22,XBOOLE_0:def 10; suppose not rng Tf1 c= S_tfsm2; then consider q being set such that A23: q in rng Tf1 & not q in S_tfsm2 by TARSKI:def 3; consider x be set such that A24: x in dom Tf1 & q = Tf1.x by A23,FUNCT_1:def 5; reconsider x2 = x`2 as Element of S_tfsm2 by A20,A24,MCART_1:10; x2 in S_tfsm2; hence contradiction by A20,A23,A24; suppose not S_tfsm2 c= rng Tf1; then consider x being set such that A25: x in S_tfsm2 & not x in rng Tf1 by TARSKI:def 3; 1 in {1} by TARSKI:def 1; then A26: [1,x] in sS_fsm2 by A25,ZFMISC_1:106; then A27: Tf1.[1,x] in rng Tf1 by A20,FUNCT_1:def 5; Tf1.[1,x] = [1,x]`2 by A20,A26; hence contradiction by A25,A27,MCART_1:7; end; then reconsider Tf1s = Tf1 as Function of the carrier of fsm2, the carrier of tfsm2 by A20,FUNCT_2:3; take Tf1s; now let x1, x2 be set; assume A28: x1 in dom Tf1 & x2 in dom Tf1 & Tf1.x1 = Tf1.x2; then A29: x1 = [x1`1, x1`2] & x2 = [x2`1,x2`2] by A20,MCART_1:23; A30: Tf1.x1 = x1`2 & Tf1.x2 = x2`2 by A20,A28; A31: x1`1 in B by A20,A28,MCART_1:10; x2`1 in B by A20,A28,MCART_1:10; then x1`1 = 1 & x2`1 = 1 by A31,TARSKI:def 1; hence x1 = x2 by A28,A29,A30; end; then Tf1s is one-to-one onto by A21,FUNCT_1:def 8,FUNCT_2:def 3; hence Tf1s is bijective by FUNCT_2:def 4; thus Tf1s.the InitS of fsm2 = sI2`2 by A20 .= the InitS of tfsm2 by MCART_1:7; now let q be State of fsm2, s be Element of IAlph; A32: q`1 in B & q`2 in S_tfsm2 by MCART_1:10; reconsider q1 = q`2 as Element of S_tfsm2 by MCART_1:10; set Tq1 = Tr2.[q1, s]; A33: [q`1,Tq1] in [: B, S_tfsm2 :] by A32,ZFMISC_1:106; thus Tf1s.((the Tran of fsm2).[q, s]) = Tf1s.[q`1, Tr2.[q`2, s]] by A3 .= [q`1, Tr2.[q`2, s]]`2 by A20,A33.= Tr2.[q`2, s] by MCART_1:7 .= (the Tran of tfsm2).[Tf1s.q, s] by A20; thus (the OFun of fsm2).[q,s] = Of2.[q`2, s] by A4 .= (the OFun of tfsm2).[Tf1s.q, s] by A20; end; hence thesis; end; theorem Th82: tfsm1, tfsm2-are_isomorphic implies tfsm1, tfsm2-are_equivalent proof assume tfsm1, tfsm2-are_isomorphic; then consider Tf such that A1: Tf is bijective & Tf.the InitS of tfsm1 = the InitS of tfsm2 & for q being Element of tfsm1, s being Element of IAlph holds Tf.((the Tran of tfsm1).[q, s]) = (the Tran of tfsm2).[Tf.q, s] & (the OFun of tfsm1).[q,s] = (the OFun of tfsm2).[Tf.q, s] by Def19; now let w1 be FinSequence of IAlph; set rw1 = (the InitS of tfsm1,w1)-response; set rw2 = (the InitS of tfsm2, w1)-response; set aw1 = (the InitS of tfsm1,w1)-admissible; set aw2 = (the InitS of tfsm2,w1)-admissible; A2:for k being Nat st 1<=k & k<= len w1 +1 holds Tf.(aw1.k) = aw2.k proof defpred P[Nat] means 1 <= $1 & $1 <= len w1 + 1 implies Tf.(aw1.$1) = aw2.$1; A3: P[0]; A4: for k being Nat st P[k] holds P[k + 1] proof let k be Nat; assume A5: 1 <= k & k <= len w1 + 1 implies Tf.(aw1.k) = aw2.k; assume A6: 1 <= (k+1) & (k+1) <= len w1 + 1; A7: 0 = k or 0 < k & 0+1 = 1 by NAT_1:19; per cases by A7,NAT_1:38; suppose A8: 0 = k; hence Tf.(aw1.(k+1)) = the InitS of tfsm2 by A1,Def2 .= aw2.(k+1) by A8,Def2; suppose 1 <= k; then A9: 1 <= k & k <= len w1 by A6,REAL_1:53; then consider w1k be Element of IAlph, qk, qk1 be Element of tfsm1 such that A10: w1k= w1.k & qk = aw1.k & qk1 = aw1.(k+1) & w1k-succ_of qk = qk1 by Def2; consider w2k be Element of IAlph, q2k, q2k1 be Element of tfsm2 such that A11: w2k= w1.k & q2k = aw2.k & q2k1 = aw2.(k+1) & w2k-succ_of q2k = q2k1 by A9,Def2; A12: len w1 <= len w1 + 1 by NAT_1:29; thus Tf.(aw1.(k+1)) = Tf.((the Tran of tfsm1).[qk, w1k]) by A10,Def1 .= (the Tran of tfsm2).[q2k, w2k] by A1,A5,A9,A10,A11,A12, AXIOMS:22 .= aw2.(k+1) by A11,Def1; end; thus for i being Nat holds P[i] from Ind (A3, A4); end; A13: len rw1 = len w1 by Def6 .= len rw2 by Def6; now let k be Nat; assume 1 <= k & k <= len rw1; then 1 <= k & k <= len w1 by Def6; then A14: k in dom w1 by FINSEQ_3:27; then k in Seg len w1 by FINSEQ_1:def 3; then k in Seg (len w1 + 1) by FINSEQ_2:9; then k in Seg len aw1 by Def2; then A15: k in dom aw1 by FINSEQ_1:def 3; then 1 <= k & k <= len aw1 by FINSEQ_3:27; then A16: 1 <= k & k <= len w1 + 1 by Def2; A17: aw1.k is Element of tfsm1 & w1.k is Element of IAlph by A14,A15,FINSEQ_2:13; thus rw2.k = (the OFun of tfsm2).[aw2.k, w1.k] by A14,Def6 .= (the OFun of tfsm2).[Tf.(aw1.k), w1.k] by A2,A16 .= (the OFun of tfsm1).[aw1.k, w1.k] by A1,A17 .= rw1.k by A14,Def6; end; hence rw1 = rw2 by A13,FINSEQ_1:18; end; hence thesis by Def9; end; theorem Th83: the carrier of CRtfsm1 misses the carrier of CRtfsm2 & CRtfsm1, CRtfsm2-are_equivalent implies CRtfsm1, CRtfsm2-are_isomorphic proof set rtfsm1 = CRtfsm1; set rtfsm2 = CRtfsm2; assume A1: (the carrier of rtfsm1) misses (the carrier of rtfsm2) & rtfsm1, rtfsm2-are_equivalent; set UN = rtfsm1-Mealy_union rtfsm2; set RUN = the_reduction_of UN; set SRUN = the carrier of RUN; set S_rtfsm1 = the carrier of rtfsm1; set S_rtfsm2 = the carrier of rtfsm2; A2: SRUN = final_states_partition UN by Def18; defpred P[set, set] means ex part being Element of SRUN st $1 in part & $2 in part \ { $1 }; A3: for x,y1,y2 being set st x in S_rtfsm1 & P[x,y1] & P[x,y2] holds y1 = y2 proof let x, y1, y2 be set; assume A4: x in S_rtfsm1 & P[x,y1] & P[x,y2]; then reconsider x' = x as Element of S_rtfsm1; consider part1 being Element of SRUN such that A5: x' in part1 & y1 in part1 \ { x' } by A4; consider part2 being Element of SRUN such that A6: x' in part2 & y2 in part2 \ { x' } by A4; assume A7: y1 <> y2; A8: y1 in part1 by A5,ZFMISC_1:64; part1 is Subset of UN by A2,TARSKI:def 3; then part1 is Subset of S_rtfsm1 \/ S_rtfsm2 by Def24; then A9: y1 in S_rtfsm1 or y1 in S_rtfsm2 by A8,XBOOLE_0:def 2; A10: not y1 in S_rtfsm1 proof assume A11: y1 in S_rtfsm1; reconsider y1 as Element of part1 by A5,ZFMISC_1:64; reconsider x' as Element of part1 by A5; y1 <> x' by A5,ZFMISC_1:64; hence contradiction by A1,A11,Th78; end; A12: y2 in part2 by A6,ZFMISC_1:64; part2 is Subset of UN by A2,TARSKI:def 3; then part2 is Subset of S_rtfsm1 \/ S_rtfsm2 by Def24; then A13: y2 in S_rtfsm1 or y2 in S_rtfsm2 by A12,XBOOLE_0:def 2; A14: not y2 in S_rtfsm1 proof assume A15: y2 in S_rtfsm1; reconsider y2 as Element of part2 by A6,ZFMISC_1:64; reconsider x' as Element of part2 by A6; y2 <> x' by A6,ZFMISC_1:64; hence contradiction by A1,A15,Th78; end; A16: part1 is Subset of UN by A2,TARSKI:def 3; part2 is Subset of UN by A2,TARSKI:def 3; then part1 = part2 or part1 misses part2 by A2,A16,EQREL_1:def 6; hence contradiction by A1,A5,A6,A7,A8,A9,A10,A12,A13,A14,Th79,XBOOLE_0:3; end; A17: for x being set st x in S_rtfsm1 ex y being set st P[x,y] proof let x be set; assume A18: x in S_rtfsm1; then x in S_rtfsm1 \/ S_rtfsm2 by XBOOLE_0:def 2; then A19: x in the carrier of UN by Def24; union SRUN= the carrier of UN by A2,EQREL_1:def 6; then consider part being set such that A20: x in part & part in SRUN by A19,TARSKI:def 4; reconsider part as Element of SRUN by A20; set xs = { x }; consider p, y be Element of part such that A21: p in S_rtfsm1 & y in S_rtfsm2 & for z being Element of part holds z = p or z = y by A1,Th80; x <> y by A1,A18,A21,XBOOLE_0:3; then A22: y in part \ xs by A20,ZFMISC_1:64; set IT = y; take IT; thus thesis by A20,A22; end; consider f being Function such that A23: dom f = S_rtfsm1 & for x being set st x in S_rtfsm1 holds P[x,f.x] from FuncEx( A3,A17 ); A24: f is one-to-one proof now let x1, x2 be set; assume A25: x1 in dom f & x2 in dom f & f.x1 = f.x2; assume A26: x1 <> x2; consider part1 be Element of SRUN such that A27: x1 in part1 & f.x1 in part1 \ { x1 } by A23,A25; consider part2 be Element of SRUN such that A28: x2 in part2 & f.x2 in part2 \ { x2 } by A23,A25; A29: f.x1 in part1 by A27,ZFMISC_1:64; A30: f.x1 in part2 by A25,A28,ZFMISC_1:64; A31: part1 is Subset of UN by A2,TARSKI:def 3; part2 is Subset of UN by A2,TARSKI:def 3; then part1 = part2 or part1 misses part2 by A2,A31,EQREL_1:def 6; hence contradiction by A1,A23,A25,A26,A27,A28,A29,A30,Th78,XBOOLE_0:3; end; hence thesis by FUNCT_1:def 8; end; A32:rng f = S_rtfsm2 proof now assume A33: not rng f c= S_rtfsm2 or not S_rtfsm2 c= rng f; per cases by A33; suppose not rng f c= S_rtfsm2; then consider y1 being set such that A34: y1 in rng f & not y1 in S_rtfsm2 by TARSKI:def 3; consider x1 being set such that A35: x1 in S_rtfsm1 & y1 = f.x1 by A23,A34,FUNCT_1:def 5; consider part1 be Element of SRUN such that A36: x1 in part1 & f.x1 in part1 \ { x1 } by A23,A35; A37: y1 in part1 by A35,A36,ZFMISC_1:64; part1 is Subset of UN by A2,TARSKI:def 3; then A38: part1 is Subset of S_rtfsm1 \/ S_rtfsm2 by Def24; A39: y1 is Element of part1 by A35,A36,ZFMISC_1:64; now assume A40: y1 in S_rtfsm1; y1 <> x1 by A35,A36,ZFMISC_1:64; hence contradiction by A1,A35,A36,A39,A40,Th78; end; hence contradiction by A34,A37,A38,XBOOLE_0:def 2; suppose not S_rtfsm2 c= rng f; then consider y1 being set such that A41: y1 in S_rtfsm2 & not y1 in rng f by TARSKI:def 3; y1 in S_rtfsm1 \/ S_rtfsm2 by A41,XBOOLE_0:def 2; then A42: y1 in the carrier of UN by Def24; union SRUN = the carrier of UN by A2,EQREL_1:def 6; then consider Z being set such that A43: y1 in Z & Z in SRUN by A42,TARSKI:def 4; reconsider Z as Element of SRUN by A43; consider q1, q2 being Element of Z such that A44: q1 in S_rtfsm1 & q2 in S_rtfsm2 & for q being Element of Z holds q = q1 or q = q2 by A1,Th80; consider F being Element of SRUN such that A45: q1 in F & f.q1 in F \ {q1} by A23,A44; A46: Z is Subset of UN by A2,TARSKI:def 3; F is Subset of UN by A2,TARSKI:def 3; then A47: Z = F or Z misses F by A2,A46,EQREL_1:def 6; A48: f.q1 in F by A45,ZFMISC_1:64; now assume A49: y1 <> f.q1; Z is Subset of S_rtfsm1 \/ S_rtfsm2 by A46,Def24; then A50: f.q1 in S_rtfsm1 or f.q1 in S_rtfsm2 by A43,A45,A47,A48,XBOOLE_0: 3,def 2; now assume A51: f.q1 in S_rtfsm1; A52: f.q1 is Element of Z by A43,A45,A47,XBOOLE_0:3,ZFMISC_1:64; f.q1 <> q1 by A45,ZFMISC_1:64; hence contradiction by A1,A44,A51,A52,Th78; end; hence contradiction by A1,A41,A43,A45,A47,A48,A49,A50,Th79, XBOOLE_0:3; end; hence contradiction by A23,A41,A44,FUNCT_1:def 5; end; hence thesis by XBOOLE_0:def 10; end; then reconsider f as Function of S_rtfsm1, S_rtfsm2 by A23,FUNCT_2:3; f is one-to-one onto by A24,A32,FUNCT_2:def 3; then A53: f is bijective by FUNCT_2:def 4; A54: f.the InitS of rtfsm1 = the InitS of rtfsm2 proof assume A55: f.the InitS of rtfsm1 <> the InitS of rtfsm2; consider Q being Element of SRUN such that A56: the InitS of rtfsm1 in Q & the InitS of rtfsm2 in Q & Q = the InitS of RUN by A1,Th75; consider part being Element of SRUN such that A57: the InitS of rtfsm1 in part & f.the InitS of rtfsm1 in part \ { the InitS of rtfsm1 } by A23; A58: Q is Subset of UN by A2,TARSKI:def 3; part is Subset of UN by A2,TARSKI:def 3; then Q = part or Q misses part by A2,A58,EQREL_1:def 6; then f.the InitS of rtfsm1 is Element of Q by A56,A57,XBOOLE_0:3,ZFMISC_1:64; hence contradiction by A1,A55,A56,Th79; end; for q being State of rtfsm1, s being Element of IAlph holds f.((the Tran of rtfsm1).[q, s]) = (the Tran of rtfsm2).[f.q, s] & (the OFun of rtfsm1).[q,s] = (the OFun of rtfsm2).[f.q, s] proof let q be State of rtfsm1, s be Element of IAlph; A59: the carrier of UN = S_rtfsm1 \/ S_rtfsm2 by Def24; then reconsider q' = q as Element of UN by XBOOLE_0:def 2; f.q in rng f by A23,FUNCT_1:def 5; then reconsider fq' = f.q' as Element of UN by A59, XBOOLE_0:def 2; A60: final_states_partition UN is final by Def15; consider part be Element of SRUN such that A61: q in part & f.q in part \ { q } by A23; f.q in part by A61,ZFMISC_1:64; then A62: q', fq'-are_equivalent by A2,A60,A61,Def14; set qu = (the Tran of UN).[q',s], qfu = (the Tran of UN).[fq',s]; set q1 = (the Tran of rtfsm1).[q,s], fq = f.q; set q2 = (the Tran of rtfsm2).[fq,s]; qu,qfu-are_equivalent by A62,Th36; then consider X being Element of SRUN such that A63: qu in X & qfu in X by A2,A60,Def14; A64: dom (the Tran of rtfsm1) = [: S_rtfsm1,IAlph :] by FUNCT_2:def 1; A65: (the carrier of rtfsm1) /\ (the carrier of rtfsm2) = {} by A1,XBOOLE_0: def 7; dom (the Tran of rtfsm1) /\ dom (the Tran of rtfsm2) = [: S_rtfsm1,IAlph :] /\ [: S_rtfsm2,IAlph :] by A64,FUNCT_2:def 1 .= [: {} ,IAlph :] by A65,ZFMISC_1:122 .= {} by ZFMISC_1:113; then A66: dom (the Tran of rtfsm1) misses dom (the Tran of rtfsm2) by XBOOLE_0:def 7; A67: qu = ((the Tran of rtfsm1)+*(the Tran of rtfsm2)).[q',s] by Def24 .=q1 by A64,A66,FUNCT_4:17; A68: dom (the Tran of rtfsm2) = [: S_rtfsm2,IAlph :] by FUNCT_2:def 1; A69: qfu = ((the Tran of rtfsm1)+*(the Tran of rtfsm2)).[fq',s] by Def24 .= q2 by A68,FUNCT_4:14; consider XX be Element of SRUN such that A70: q1 in XX & f.q1 in XX \ { q1 } by A23; A71: f.q1 in XX by A70,ZFMISC_1:64; A72: X is Subset of UN by A2,TARSKI:def 3; now assume A73: f.q1 <> q2; XX is Subset of UN by A2,TARSKI:def 3; then X = XX or X misses XX by A2,A72,EQREL_1:def 6; hence contradiction by A1,A63,A67,A69,A70,A71,A73,Th79,XBOOLE_0:3; end; hence f.((the Tran of rtfsm1).[q, s]) = (the Tran of rtfsm2).[f.q, s]; A74: dom (the OFun of rtfsm1) = [: S_rtfsm1,IAlph :] by FUNCT_2:def 1; A75: dom (the OFun of rtfsm2) = [: S_rtfsm2,IAlph :] by FUNCT_2:def 1; then dom (the OFun of rtfsm1) /\ dom (the OFun of rtfsm2) = [: S_rtfsm1 /\ S_rtfsm2, IAlph :] by A74,ZFMISC_1:122 .= {} by A65,ZFMISC_1:113; then dom (the OFun of rtfsm1) misses dom (the OFun of rtfsm2) by XBOOLE_0 :def 7; hence (the OFun of rtfsm1).[q,s] = ((the OFun of rtfsm1) +* (the OFun of rtfsm2)).[q,s] by A74,FUNCT_4:17 .= (the OFun of UN).[q',s] by Def24 .= (the OFun of UN).[fq',s] by A62,Th36 .= ((the OFun of rtfsm1) +* (the OFun of rtfsm2)).[f.q,s] by Def24 .= (the OFun of rtfsm2).[f.q, s] by A75,FUNCT_4:14; end; hence thesis by A53,A54,Def19; end; theorem Th84: Ctfsm1, Ctfsm2-are_equivalent implies the_reduction_of Ctfsm1, the_reduction_of Ctfsm2-are_isomorphic proof assume A1: Ctfsm1, Ctfsm2-are_equivalent; set rtfsm1 = the_reduction_of Ctfsm1; set rtfsm2 = the_reduction_of Ctfsm2; consider fsm1, fsm2 being finite non empty Mealy-FSM over IAlph, OAlph such that A2: (the carrier of fsm1) misses (the carrier of fsm2) & fsm1,rtfsm1-are_isomorphic & fsm2,rtfsm2-are_isomorphic by Th81; set IS_fsm1= the InitS of fsm1, IS_fsm2= the InitS of fsm2; set IS_rtfsm1= the InitS of rtfsm1, IS_rtfsm2= the InitS of rtfsm2; set S_fsm1 = the carrier of fsm1, S_fsm2 = the carrier of fsm2; set S_rtfsm1 = the carrier of rtfsm1, S_rtfsm2 = the carrier of rtfsm2; A3: rtfsm1 is connected by Th66; A4: fsm1 is connected proof assume not fsm1 is connected; then consider q1 being Element of S_fsm1 such that A5: not q1 is accessible by Def22; consider Tf being Function of the carrier of fsm1, S_rtfsm1 such that A6: Tf is bijective & Tf.the InitS of fsm1 = IS_rtfsm1 & for q being State of fsm1, s being Element of IAlph holds Tf.((the Tran of fsm1).[q, s]) = (the Tran of rtfsm1).[Tf.q, s] & (the OFun of fsm1).[q,s] = (the OFun of rtfsm1).[Tf.q, s] by A2,Def19; A7: Tf is one-to-one by A6,FUNCT_2:def 4; set q = Tf.q1; q is accessible by A3,Def22; then consider w being FinSequence of IAlph such that A8: IS_rtfsm1,w-leads_to q by Def21; A9: 1 <= len w + 1 & len w + 1 <= len w + 1 by NAT_1:29; then A10: Tf.((IS_fsm1,w)-admissible.(len w + 1)) = (IS_rtfsm1,w)-admissible.(len w + 1) by A6,Th60; A11: dom Tf = S_fsm1 by FUNCT_2:def 1; len (IS_fsm1,w)-admissible = len w + 1 by Def2; then len w + 1 in Seg (len (IS_fsm1,w)-admissible) by A9,FINSEQ_1:3; then len w + 1 in dom (IS_fsm1,w)-admissible by FINSEQ_1:def 3; then A12: (the InitS of fsm1,w)-admissible.(len w + 1) in dom Tf by A11, FINSEQ_2:13; Tf".(Tf.((IS_fsm1,w)-admissible.(len w + 1))) = Tf".q by A8,A10,Def3; then (IS_fsm1,w)-admissible.(len w + 1) = Tf".(Tf.q1) by A7,A12,FUNCT_1:56 .= q1 by A7,A11,FUNCT_1:56; then IS_fsm1,w-leads_to q1 by Def3; hence contradiction by A5,Def21; end; A13: rtfsm2 is connected by Th66; A14: fsm2 is connected proof assume not fsm2 is connected; then consider q1 being Element of S_fsm2 such that A15: not q1 is accessible by Def22; consider Tf being Function of S_fsm2, S_rtfsm2 such that A16: Tf is bijective & Tf.IS_fsm2 = IS_rtfsm2 & for q being State of fsm2, s being Element of IAlph holds Tf.((the Tran of fsm2).[q, s]) = (the Tran of rtfsm2).[Tf.q, s] & (the OFun of fsm2).[q,s] = (the OFun of rtfsm2).[Tf.q, s] by A2,Def19; A17: Tf is one-to-one by A16,FUNCT_2:def 4; set q = Tf.q1; q is accessible by A13,Def22; then consider w being FinSequence of IAlph such that A18: IS_rtfsm2,w-leads_to q by Def21; A19: 1 <= len w + 1 & len w + 1 <= len w + 1 by NAT_1:29; then A20: Tf.((IS_fsm2,w)-admissible.(len w + 1)) = (IS_rtfsm2,w)-admissible.(len w + 1) by A16,Th60; A21: dom Tf = S_fsm2 by FUNCT_2:def 1; len (IS_fsm2,w)-admissible = len w + 1 by Def2; then len w + 1 in Seg (len (IS_fsm2,w)-admissible) by A19,FINSEQ_1:3; then len w + 1 in dom (IS_fsm2,w)-admissible by FINSEQ_1:def 3; then A22: (IS_fsm2,w)-admissible.(len w + 1) in dom Tf by A21,FINSEQ_2:13; Tf".(Tf.((IS_fsm2,w)-admissible.(len w + 1))) = Tf".q by A18,A20,Def3; then (IS_fsm2,w)-admissible.(len w + 1) = Tf".(Tf.q1) by A17,A22,FUNCT_1:56 .= q1 by A17,A21,FUNCT_1:56; then IS_fsm2,w-leads_to q1 by Def3; hence contradiction by A15,Def21; end; A23:rtfsm1,Ctfsm1-are_equivalent & rtfsm2,Ctfsm2-are_equivalent by Th58; fsm1,rtfsm1-are_equivalent & fsm2,rtfsm2-are_equivalent by A2,Th82; then A24: fsm1,Ctfsm1-are_equivalent & fsm2,Ctfsm2-are_equivalent by A23,Th30; then fsm1,Ctfsm2-are_equivalent by A1,Th30; then A25: fsm1,fsm2-are_equivalent by A24,Th30; reconsider fsm1, fsm2 as reduced (finite non empty Mealy-FSM over IAlph,OAlph) by A2,Th65; fsm1, fsm2-are_isomorphic by A2,A4,A14,A25,Th83; then fsm1,rtfsm2-are_isomorphic by A2,Th59; hence thesis by A2,Th59; end; theorem for M1, M2 being connected reduced (finite non empty Mealy-FSM over IAlph, OAlph) holds M1, M2-are_isomorphic iff M1, M2-are_equivalent proof let M1, M2 be connected reduced (finite non empty Mealy-FSM over IAlph, OAlph); thus M1, M2-are_isomorphic implies M1, M2-are_equivalent by Th82; assume M1, M2-are_equivalent; then A1: the_reduction_of M1, the_reduction_of M2-are_isomorphic by Th84; A2: M1, the_reduction_of M1-are_isomorphic & M2, the_reduction_of M2-are_isomorphic by Th64; then M1, the_reduction_of M2-are_isomorphic by A1,Th59; hence thesis by A2,Th59; end;

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