Mercurial > hg > Members > kono > Proof > automaton

changeset 298:1b5c09f12373

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 31 Dec 2021 17:02:31 +0900
parents afc7db9b917d
children 841f4064e515
files automaton-in-agda/src/non-regular.agda
diffstat 1 files changed, 30 insertions(+), 14 deletions(-) [+]
line wrap: on
line diff
--- a/automaton-in-agda/src/non-regular.agda	Fri Dec 31 15:42:27 2021 +0900
+++ b/automaton-in-agda/src/non-regular.agda	Fri Dec 31 17:02:31 2021 +0900
@@ -105,9 +105,10 @@
 
 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 
 
-record TA { Q : Set } { Σ : Set  } (fa : Automaton Q  Σ ) (phase : ℕ)  ( q qd : Q )  : Set where
+record TA { Q : Set } { Σ : Set  } (fa : Automaton Q  Σ ) (phase : ℕ)  ( q qd : Q ) (is : List Σ)  : Set where
     field
        x y z : List Σ
+       xyz=is : x ++ y ++ z ≡ is 
        trace-z    : phase > 1 → Trace fa z  qd
        trace-yz   : phase > 0 → Trace fa (y ++ z)  qd
        trace-xyz  : phase ≡ 0 → Trace fa (x ++ y ++ z)  q
@@ -118,31 +119,46 @@
 make-TA : { Q : Set } { Σ : Set  } (fa : Automaton Q  Σ ) (finq : FiniteSet Q) (q qd : Q) (is : List Σ) 
      → (tr : Trace fa is q )
      → dup-in-list finq qd (tr→qs fa is q tr) ≡ true
-     → TA fa 0 q qd 
+     → TA fa 0 q qd is
 make-TA {Q} {Σ} fa finq q qd is tr dup = tra-phase1 q is tr dup where
    open TA
-   tra-phase2 : (q : Q)  → (is : List Σ)  → (tr : Trace fa is  q ) → phase2 finq qd (tr→qs fa is q tr) ≡ true  → TA fa 1 q qd
+   tra-phase2 : (q : Q)  → (is : List Σ)  → (tr : Trace fa is  q ) → phase2 finq qd (tr→qs fa is q tr) ≡ true  → TA fa 1 q qd is
    tra-phase2 q (i ∷ is) (tnext q tr) p with equal? finq qd q | inspect ( equal? finq qd) q
    ... | true | record { eq = eq } = {!!} 
    ... | false | record { eq = eq } = {!!} 
-   tra-phase1 : (q : Q)  → (is : List Σ)  → (tr : Trace fa is  q ) → phase1 finq qd (tr→qs fa is q tr) ≡ true  → TA fa 0 q qd 
+   tra-phase1 : (q : Q)  → (is : List Σ)  → (tr : Trace fa is  q ) → phase1 finq qd (tr→qs fa is q tr) ≡ true  → TA fa 0 q qd is
    tra-phase1 q (i ∷ is) (tnext q tr) p with equal? finq qd q | inspect (equal? finq qd) q
-   ... | true | record { eq = eq } = record { x = [] ; y = y TA0 ;  z = z TA0 ; trace-z = λ () ; trace-yz = λ _ → trace-yz TA0 a<sa 
+          | phase1 finq qd (tr→qs fa is (δ fa q i) tr) | inspect ( phase1 finq qd)  (tr→qs fa is (δ fa q i) tr) 
+   ... | true | record { eq = eq } | false | record { eq = np} = record { x = [] ; y = y TA0 ;  z = z TA0 ; xyz=is = {!!} -- cong (i ∷_ ) 
+           ; trace-z = λ () ; trace-yz = λ _ → trace-yz TA0 a<sa 
            ; trace-xyz  = λ _ → subst (λ k → Trace fa (y TA0 ++ z TA0) k ) (equal→refl finq eq) (trace-yz TA0 a<sa)
-           ; trace-xyyz = λ _ → subst (λ k → Trace fa (y TA0 ++ y TA0 ++ z TA0) k ) (equal→refl finq eq) (tra-01 (y TA0) (trace-yz TA0 a<sa)) } where
-        TA0 : TA fa 1 (δ fa q i ) qd 
+           ; trace-xyyz = λ _ → subst (λ k → Trace fa (y TA0 ++ y TA0 ++ z TA0) k ) (equal→refl finq eq) (tra-02 (y TA0) qd (trace-yz TA0 a<sa) {!!} {!!}) } where
+--  : phase2 finq qd (tr→qs fa (y TA0 ++ z TA0) qd (trace-yz TA0 a<sa))
+--    ≡ true
+--  : phase1 finq qd (tr→qs fa (y TA0 ++ z TA0) qd (trace-yz TA0 a<sa))
+--    ≡ false
+        TA0 : TA fa 1 (δ fa q i ) qd  is
         TA0 = tra-phase2 (δ fa q i ) is tr p
         tra-02 : (y1 : List Σ) → (q : Q) → (tr : Trace fa (y1 ++ z TA0) q)
-            → phase2 finq qd (tr→qs fa (y1 ++ z TA0) q tr) ≡ true  → Trace fa (y1 ++ y TA0 ++ z TA0) q
-        tra-02 [] q tr p with equal? finq qd q | inspect ( equal? finq qd) q
-        ... | true | record { eq = eq } = subst (λ k →  Trace fa (y TA0 ++ z TA0) k ) (equal→refl finq eq) (trace-yz TA0 a<sa )
-        ... | false | record { eq = eq } = {!!}
-        tra-02 (y1 ∷ ys) q (tnext q tr) p = tnext q (tra-02 ys (δ fa q y1) tr {!!} )
+            → phase2 finq qd (tr→qs fa (y1 ++ z TA0) q tr) ≡ true
+            → phase1 finq qd (tr→qs fa (y1 ++ z TA0) q tr) ≡ false
+            → Trace fa (y1 ++ y TA0 ++ z TA0) q
+        tra-02 [] q tr p np with equal? finq qd q | inspect ( equal? finq qd) q
+        ... | true  | record { eq = eq } = subst (λ k →  Trace fa (y TA0 ++ z TA0) k ) (equal→refl finq eq) (trace-yz TA0 a<sa )
+        ... | false | record { eq = ne } = {!!}
+        tra-02 (y1 ∷ ys) q (tnext q tr) p np with equal? finq qd q | inspect ( equal? finq qd) q
+        ... | true  | record { eq = eq } = {!!}
+        ... | false | record { eq = ne } = tnext q (tra-02 ys (δ fa q y1) tr p np )
         tra-01 : (y1 : List Σ) → Trace fa (y1 ++ z TA0) qd → Trace fa (y1 ++ y TA0 ++ z TA0) qd
         tra-01 = {!!}
-   ... | false | _ = record { x = i ∷ x TA0 ; y = y TA0 ; z = z TA0 ; trace-z = λ () ; trace-yz = λ ()
+   ... | true | record { eq = eq } | true | record { eq = np} = record { x = i ∷ x TA0 ; y = y TA0 ; z = z TA0 ; xyz=is = cong (i ∷_ ) (xyz=is TA0)
+            ; trace-z = λ () ; trace-yz = λ ()
             ; trace-xyz = λ _ → tnext q (trace-xyz TA0 refl ) ; trace-xyyz = λ _ → tnext q (trace-xyyz TA0 refl )} where
-        TA0 : TA fa 0 (δ fa q i ) qd 
+        TA0 : TA fa 0 (δ fa q i ) qd  is
+        TA0 = tra-phase1 (δ fa q i ) is tr np
+   ... | false | _ | _ | _ = record { x = i ∷ x TA0 ; y = y TA0 ; z = z TA0 ; xyz=is = cong (i ∷_ ) (xyz=is TA0) ; trace-z = λ () ; trace-yz = λ ()
+            ; trace-xyz = λ _ → tnext q (trace-xyz TA0 refl ) ; trace-xyyz = λ _ → tnext q (trace-xyyz TA0 refl )} where
+        TA0 : TA fa 0 (δ fa q i ) qd  is
         TA0 = tra-phase1 (δ fa q i ) is tr p
 
 open RegularLanguage