From e678e3b1a88e657401902bbddad02d3a4d70205a Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Tue, 20 Jul 2010 12:43:19 +0000
Subject: [PATCH] completed lemma 17

---
 helm/software/matita/nlibrary/re/re.ma | 159 ++++++++++++++-----------
 1 file changed, 90 insertions(+), 69 deletions(-)

diff --git a/helm/software/matita/nlibrary/re/re.ma b/helm/software/matita/nlibrary/re/re.ma
index 36abab4f6..2899fbc18 100644
--- a/helm/software/matita/nlibrary/re/re.ma
+++ b/helm/software/matita/nlibrary/re/re.ma
@@ -352,6 +352,34 @@ nlemma odot_dot_aux : ∀S.∀e1,e2: pre S.
     nrewrite > (cupA…); //;##]
 nqed.
 
+nlemma sub_dot_star : 
+  ∀S.∀X:word S → Prop.∀b. (X - ϵ b) · X^* ∪ {[]} = X^*.
+#S X b; napply extP; #w; @;
+##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
+    *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
+    @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
+    @; //; napply (subW … sube);
+##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl; ##[ #_; @2; //]
+    #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *;
+    ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2;
+        @; ncases b in H1; #H1; 
+        ##[##2: nrewrite > (sub0…); @w'; @(w1@w2);
+                nrewrite > (associative_append ? w' w1 w2);
+                nrewrite > defwl'; @; ##[@;//] @(wl'); @; //;
+           ##| ncases w' in Pw';
+               ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //;
+               ##| #x xs Px; @(x::xs); @(w1@w2); 
+                   nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct]
+                   @wl'; @; //; ##] ##]
+        ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl');
+            nrewrite < (wlnil); nrewrite > (append_nil…); ncases b;
+            ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]);
+                nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct]
+                @[]; @; //;
+            ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //] 
+                @; //; @; //; @; *;##]##]##] 
+nqed.
+
 (* theorem 16: 1 *)
 alias symbol "pc" (instance 13) = "cat lang".
 alias symbol "in_pl" (instance 23) = "in_pl".
@@ -387,32 +415,8 @@ ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S.  𝐋\p (•e) =  𝐋\p e ∪ 
         ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##]
       nrewrite > (cup_dotD…); nrewrite > (cupA…); 
       nrewrite > (?: ((?·?)∪{[]} = 𝐋 .|e^*|)); //;
-      nchange in ⊢ (???%) with ((𝐋. |e|)^* ); napply extP; #w; @;
-      ##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
-          *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
-          @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
-          @; //; napply (subW … sube);
-      ##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl;
-         ##[ #_; @2; //;
-         ##| #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *;
-            ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2;
-                @; ncases b' in H1; #H1; 
-                ##[##2: nrewrite > (sub0…); @w'; @(w1@w2);
-                        nrewrite > (associative_append ? w' w1 w2);
-                        nrewrite > defwl'; @; ##[@;//] @(wl'); @; //;
-                ##| ncases w' in Pw';
-                    ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //;
-                    ##| #x xs Px; @(x::xs); @(w1@w2); 
-                        nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct]
-                        @wl'; @; //; ##] ##]
-            ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl');
-                nrewrite < (wlnil); nrewrite > (append_nil…); ncases b';
-                ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]);
-                    nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct]
-                    @[]; @; //;
-                ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //] 
-                    @; //; @; //; @; *;##]##]##]##]##]
-nqed.
+      nchange in match (𝐋 .|e^*|) with ((𝐋. |e|)^* ); napply sub_dot_star;##]
+ nqed.
 
 (* theorem 16: 3 *)      
 nlemma odot_dot: 
@@ -438,58 +442,75 @@ nlemma cupID : ∀S.∀l:word S → Prop.l ∪ l = l.
 nlemma cup_star_nil : ∀S.∀l:word S → Prop. l^* ∪ {[]} = l^*.
 #S a; napply extP; #w; @; ##[*; //; #H; nrewrite < H; @[]; @; //] #;@; //;nqed.
 
-naxiom Admit : False.
-
-nlemma key_id : ∀S.∀e:pitem S. 𝐋\p e · 𝐋 .|e|^* ∪ {[]} = 𝐋\p e · 𝐋 .|e|^* ∪ 𝐋 .|e|^*.
-#S e; napply extP; #w; @;##[##2:*]
-##[*; #w1; *; #w2; *; *; #defw Hw1 Hw2;@; @w1; @w2; /3/;
-##|*; #wl; *; #H; nrewrite < H;
-(*
-   ngeneralize in match e;
-   nelim wl;##[#e;#_;@2;//] #x xs IH e;*; #Hx Hxs; ncases (IH Hxs); 
-   ##[##2: #H; nnormalize; nrewrite < H; nrewrite > (append_nil…);
-
-   ncases wl; ##[#_;@2; //] #x xs; *; #Hx Hxs; @; @x; @(flatten ? xs); @;
-   ##[@;//;##|@xs; @; //]
-   ngeneralize in match Hx; ngeneralize in match x; nelim e; nnormalize; //;
-   ##[#e1 e2 IH1 IH2 x; *; #w1; *; #w2; *; *; #defx Hw1 Hw2;
-      @; @w1; @w2; /4/ by conj;
-   ##|#e1 e2 IH1 IH2 y;*; #; ##[@|@2] /2/;
-   ##|#e IH y; *; #wl; *; #delwl Hw2; nrewrite < delwl;
-      nelim wl in Hw2; ##[#_;@[];@[];@;//; 
-*)
-   ncases Admit;
-##|*;##[##2: #H; nrewrite < H; @2; //] *; #w1; *; #w2; *; *; #defw Hw1 Hw2;
-   @; @w1; @w2; /3/;##]
+nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S .
+  ¬ (A b) → A ∪ { (b) } = C → A = C - { (b) }.
+#S A C b nbA defC; nrewrite < defC; napply extP; #w; @;
+##[ #Aw; /3/| *; *; //; #H nH; ncases nH; #abs; nlapply (abs H); *]
 nqed.
 
 (* theorem 16: 4 *)      
 nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 .|\fst e|)^*.
-#S e; ncases e; #e' b'; ncases b';
-##[ nchange in match (〈e',true〉^⊛) with 〈?,?〉;
-    nletin e'' ≝ (\fst (•e'));
+#S p; ncases p; #e b; ncases b;
+##[ nchange in match (〈e,true〉^⊛) with 〈?,?〉;
+    nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e));
     nchange in ⊢ (??%?) with (?∪?);
-    nchange in ⊢ (??(??%?)?) with (𝐋\p e'' · 𝐋 .|e''|^* );
-    nrewrite > (?:  𝐋\p e'' ·  𝐋.|e''|^* ∪ {[]} =  (𝐋\p e' ∪ 𝐋.|e'|) · 𝐋.|e''|^* ∪ {[]}); ##[##2:
-      nrewrite < (bull_cup…); nchange in ⊢ (???(??(??%?)?)) with (?∪?);
-      nchange in match e'' with e'';
-      ncases (\snd (•e')); ##[##2: nrewrite > (cup0…); //]
-      nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…);
-      nrewrite > (cupA…); nrewrite > (cup_star_nil…);
-      napply key_id;##]
-    nrewrite > (cup_dotD…); nrewrite > (cupA…);
-    nrewrite > (?: ?·? ∪ {[]} = 𝐋.|e'|^* ); ##[##2:
-      nrewrite > (erase_bull…); nrewrite > (dot_star_epsilon…); //]
-    nrewrite > (erase_bull…);  
+    nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 .|e'|^* );
+    nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 .|e| - ϵ b' )); ##[##2:
+      nlapply (bull_cup ? e); #bc;
+      nchange in match (𝐋\p (•e)) in bc with (?∪?);
+      nchange in match b' in bc with b';
+      ncases b' in bc; ##[##2: nrewrite > (cup0…); nrewrite > (sub0…); //]
+      nrewrite > (cup_sub…); ##[napply rcanc_sing] //;##]
+    nrewrite > (cup_dotD…); nrewrite > (cupA…);nrewrite > (erase_bull…);
+    nrewrite > (sub_dot_star…);
     nchange in match (𝐋\p 〈?,?〉) with (?∪?);
-    nrewrite > (cup_dotD…);nrewrite > (epsilon_dot…); //;
-##| nwhd in match (〈e',false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+    nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…); //;    
+##| nwhd in match (〈e,false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
     nrewrite > (cup0…);
-    nchange in ⊢ (??%?) with (𝐋\p e' · 𝐋 .|e'|^* );
-    nrewrite < (cup0 ? (𝐋\p e')); //;##]
+    nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 .|e|^* );
+    nrewrite < (cup0 ? (𝐋\p e)); //;##]
 nqed.
 
-(* corollary 17: non tipa *)
+nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝ 
+  match e with 
+  [ z ⇒ pz ?
+  | e ⇒ pe ?
+  | s x ⇒ ps ? x
+  | c e1 e2 ⇒ pc ? (pre_of_re ? e1) (pre_of_re ? e2)
+  | o e1 e2 ⇒ po ? (pre_of_re ? e1) (pre_of_re ? e2)
+  | k e1 ⇒ pk ? (pre_of_re ? e1)].
+
+nlemma notFalse : ¬False. @; //; nqed.
+
+nlemma dot0 : ∀S.∀A:word S → Prop. {} · A = {}.
+#S A; nnormalize; napply extP; #w; @; ##[##2: *]
+*; #w1; *; #w2; *; *; //; nqed.
+
+nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}.
+#S e; nelim e; ##[##1,2,3: //]
+##[ #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1 e2))) with (?∪?);
+    nrewrite > H1; nrewrite > H2; nrewrite > (dot0…); nrewrite > (cupID…);//
+##| #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1+e2))) with (?∪?);
+    nrewrite > H1; nrewrite > H2; nrewrite > (cupID…); //
+##| #e1 H1; nchange in match (𝐋\p (pre_of_re S (e1^* ))) with (𝐋\p (pre_of_re ??) · ?);
+    nrewrite > H1; napply dot0; ##]
+nqed.
+
+nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 .|pre_of_re S e| = 𝐋 e.
+#S A; nelim A; //; 
+##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?);
+    nrewrite < H1; nrewrite < H2; //
+##| #e1 e2 H1 H2; nchange in match (𝐋 (e1 + e2)) with (𝐋 e1 ∪ ?);
+    nrewrite < H1; nrewrite < H2; //
+##| #e1 H1; nchange in match (𝐋  (e1^* )) with ((𝐋 e1)^* );
+    nrewrite < H1; //]
+nqed.     
+
+(* corollary 17 *)
+nlemma L_Lp_bull : ∀S.∀e:re S.𝐋 e = 𝐋\p (•pre_of_re ? e).
+#S e; nrewrite > (bull_cup…); nrewrite > (Lp_pre_of_re…);
+nrewrite > (cupC…); nrewrite > (cup0…); nrewrite > (erase_pre_of_reK…); //;
+nqed.
 
 nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w.
 #S f g H; nrewrite > H; //; nqed.
-- 
2.39.2