From: Wilmer Ricciotti <ricciott@cs.unibo.it>
Date: Tue, 15 Jan 2013 12:29:35 +0000 (+0000)
Subject: unistep_aux
X-Git-Tag: make_still_working~1346
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=789726e7f992ff6a37b91799fb081f8013703b49;p=helm.git

unistep_aux
---

diff --git a/matita/matita/lib/turing/multi_universal/moves_2.ma b/matita/matita/lib/turing/multi_universal/moves_2.ma
index 14f072b4e..d86b89a6a 100644
--- a/matita/matita/lib/turing/multi_universal/moves_2.ma
+++ b/matita/matita/lib/turing/multi_universal/moves_2.ma
@@ -12,6 +12,7 @@
 include "turing/turing.ma".
 include "turing/inject.ma".
 include "turing/while_multi.ma".
+include "turing/while_machine.ma".
 
 definition parmove_states ≝ initN 3.
 
@@ -252,4 +253,91 @@ lemma sem_parmoveL : ∀src,dst,sig,n.
   parmove src dst sig n L ⊨ R_parmoveL src dst sig n.
 #src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize 
 [/2/ | @wsem_parmoveL //]
+qed.
+
+(* while {
+     if current != null 
+        then move_r
+        else nop
+     }
+ *)
+ 
+definition mte_states ≝ initN 3.
+definition mte0 : mte_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition mte1 : mte_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition mte2 : mte_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
+definition mte_step ≝ 
+  λalpha:FinSet.λD.mk_TM alpha mte_states
+  (λp.let 〈q,a〉 ≝ p in
+    match a with
+    [ None ⇒ 〈mte1,None ?,N〉
+    | Some a' ⇒ match (pi1 … q) with
+      [ O ⇒ 〈mte2,Some ? a',D〉
+      | S q ⇒ 〈mte2,None ?,N〉 ] ])
+  mte0 (λq.q == mte1 ∨ q == mte2).
+  
+definition R_mte_step_true ≝ λalpha,D,t1,t2.
+  ∃ls,c,rs.
+    t1 = midtape alpha ls c rs ∧ t2 = tape_move ? t1 D.
+
+definition R_mte_step_false ≝ λalpha.λt1,t2:tape alpha.
+  current ? t1 = None ? ∧ t1 = t2.
+
+lemma sem_mte_step :
+  ∀alpha,D.mte_step alpha D ⊨ [ mte2 : R_mte_step_true alpha D, R_mte_step_false alpha ] .
+#alpha #D #intape @(ex_intro ?? 2) cases intape
+[ @ex_intro
+  [| % [ % [ % | normalize #H destruct ] | #_ % // ] ]
+|#a #al @ex_intro
+  [| % [ % [ % | normalize #H destruct ] | #_ % // ] ]
+|#a #al @ex_intro
+  [| % [ % [ % | normalize #H destruct ] | #_ % // ] ]
+| #ls #c #rs
+  @ex_intro [| % [ % [ % | #_ %{ls} %{c} %{rs} % // ]
+                     | normalize in ⊢ (?(??%?)→?); * #H @False_ind /2/ ] ] ]
+qed.
+
+definition move_to_end ≝ λsig,D.whileTM sig (mte_step sig D) mte2.
+
+definition R_move_to_end_r ≝ 
+  λsig,int,outt.
+  (current ? int = None ? → outt = int) ∧
+  ∀ls,c,rs.int = midtape sig ls c rs → outt = mk_tape ? (reverse ? rs@c::ls) (None ?) [ ].
+  
+lemma wsem_move_to_end_r : ∀sig. move_to_end sig R ⊫ R_move_to_end_r sig.
+#sig #ta #k #outc #Hloop
+lapply (sem_while … (sem_mte_step sig R) … Hloop) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ * #Hcurtb #Houtc % /2/ #ls #c #rs #Htb >Htb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H)
+| #tc #td * #ls * #c * #rs * #Htc >Htc cases rs
+  [ normalize in ⊢ (%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
+    lapply (IH Hfalse) -IH * #Htd1 #_ %
+    [ normalize in ⊢ (%→?); #H destruct (H)
+    | #ls0 #c0 #rs0 #H destruct (H) >Htd1 // ]
+  | #r0 #rs0 whd in ⊢ (???%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
+    lapply (IH Hfalse) -IH * #_ #IH %
+    [ normalize in ⊢ (%→?); #H destruct (H)
+    | #ls1 #c1 #rs1 #H destruct (H) >reverse_cons >associative_append @IH % ] ] ]
+qed.
+
+definition R_move_to_end_l ≝ 
+  λsig,int,outt.
+  (current ? int = None ? → outt = int) ∧
+  ∀ls,c,rs.int = midtape sig ls c rs → outt = mk_tape ? [ ] (None ?) (reverse ? ls@c::rs).
+  
+lemma wsem_move_to_end_l : ∀sig. move_to_end sig L ⊫ R_move_to_end_l sig.
+#sig #ta #k #outc #Hloop
+lapply (sem_while … (sem_mte_step sig L) … Hloop) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ * #Hcurtb #Houtc % /2/ #ls #c #rs #Htb >Htb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H)
+| #tc #td * #ls * #c * #rs * #Htc >Htc cases ls
+  [ normalize in ⊢ (%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
+    lapply (IH Hfalse) -IH * #Htd1 #_ %
+    [ normalize in ⊢ (%→?); #H destruct (H)
+    | #ls0 #c0 #rs0 #H destruct (H) >Htd1 // ]
+  | #l0 #ls0 whd in ⊢ (???%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
+    lapply (IH Hfalse) -IH * #_ #IH %
+    [ normalize in ⊢ (%→?); #H destruct (H)
+    | #ls1 #c1 #rs1 #H destruct (H) >reverse_cons >associative_append @IH % ] ] ]
 qed.
\ No newline at end of file
diff --git a/matita/matita/lib/turing/multi_universal/unistep_aux.ma b/matita/matita/lib/turing/multi_universal/unistep_aux.ma
new file mode 100644
index 000000000..f240cea97
--- /dev/null
+++ b/matita/matita/lib/turing/multi_universal/unistep_aux.ma
@@ -0,0 +1,287 @@
+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic   
+    ||A||  Library of Mathematics, developed at the Computer Science 
+    ||T||  Department of the University of Bologna, Italy.           
+    ||I||                                                            
+    ||T||  
+    ||A||  
+    \   /  This file is distributed under the terms of the       
+     \ /   GNU General Public License Version 2   
+      V_____________________________________________________________*)
+
+include "turing/multi_universal/moves_2.ma".
+
+(*
+
+  in.obj : ...  x ...
+                ^
+  in.cfg : ...  ? ? ...
+                    ^
+                
+  out.cfg : ... 1 x ...
+                  ^
+                  
+  ---------------------
+  current (in.obj) = None
+  
+  in.cfg : ...  ? ? ...
+                    ^
+
+  out.cfg : ... 0 0 ...
+                  ^
+                  
+  obj_to_cfg ≝ 
+    move_l(cfg);
+    move_l(cfg);
+    (if (current(in.obj)) == None
+       then write(0,cfg);
+            move_r(cfg);
+            write(0,cfg);
+       else write(1,cfg);
+            move_r(cfg);
+            copy_step(obj,cfg);
+            move_l(obj);)
+    move_to_end_l(cfg);
+    move_r(cfg);
+       
+  
+  cfg_to_obj
+*)
+
+definition obj_to_cfg ≝ 
+  mmove cfg unialpha 3 L ·
+  mmove cfg unialpha 3 L ·
+  if_TM ?? (inject_TM ? (test_null ?) 3 obj)
+    (
+
+
+
+
+definition o2c_states ≝ initN 3.
+
+definition copy0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition copy1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition copy2 : copy_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
+
+definition trans_copy_step ≝ 
+ λsrc,dst.λsig:FinSet.λn.
+ λp:copy_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ match nth src ? a (None ?) with
+   [ None ⇒ 〈copy2,null_action sig n〉
+   | Some ai ⇒ match nth dst ? a (None ?) with 
+     [ None ⇒ 〈copy2,null_action ? n〉
+     | Some aj ⇒ 
+         〈copy1,change_vec ? (S n) 
+           (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
+           (〈Some ? ai,R〉) dst〉
+     ]
+   ]
+ | S q ⇒ match q with 
+   [ O ⇒ (* 1 *) 〈copy1,null_action ? n〉
+   | S _ ⇒ (* 2 *) 〈copy2,null_action ? n〉 ] ].
+
+definition copy_step ≝ 
+  λsrc,dst,sig,n.
+  mk_mTM sig n copy_states (trans_copy_step src dst sig n) 
+    copy0 (λq.q == copy1 ∨ q == copy2).
+
+definition R_comp_step_true ≝ 
+  λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+  ∃x,y.
+   current ? (nth src ? int (niltape ?)) = Some ? x ∧
+   current ? (nth dst ? int (niltape ?)) = Some ? y ∧
+   outt = change_vec ?? 
+            (change_vec ?? int
+              (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
+            (tape_move_mono ? (nth dst ? int (niltape ?)) 〈Some ? x, R〉) dst.
+
+definition R_comp_step_false ≝ 
+  λsrc,dst:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
+    (current ? (nth src ? int (niltape ?)) = None ? ∨
+     current ? (nth dst ? int (niltape ?)) = None ?) ∧ outt = int.
+
+lemma copy_q0_q2_null :
+  ∀src,dst,sig,n,v.src < S n → dst < S n → 
+  (nth src ? (current_chars ?? v) (None ?) = None ? ∨
+   nth dst ? (current_chars ?? v) (None ?) = None ?) → 
+  step sig n (copy_step src dst sig n) (mk_mconfig ??? copy0 v) 
+  = mk_mconfig ??? copy2 v.
+#src #dst #sig #n #v #Hi #Hj
+whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?);
+* #Hcurrent
+[ @eq_f2
+  [ whd in ⊢ (??(???%)?); >Hcurrent %
+  | whd in ⊢ (??(????(???%))?); >Hcurrent @tape_move_null_action ]
+| @eq_f2
+  [ whd in ⊢ (??(???%)?); >Hcurrent cases (nth src ?? (None sig)) //
+  | whd in ⊢ (??(????(???%))?); >Hcurrent
+    cases (nth src ?? (None sig)) [|#x] @tape_move_null_action ] ]
+qed.
+
+lemma copy_q0_q1 :
+  ∀src,dst,sig,n,v,a,b.src ≠ dst → src < S n → dst < S n → 
+  nth src ? (current_chars ?? v) (None ?) = Some ? a →
+  nth dst ? (current_chars ?? v) (None ?) = Some ? b → 
+  step sig n (copy_step src dst sig n) (mk_mconfig ??? copy0 v) =
+    mk_mconfig ??? copy1 
+     (change_vec ? (S n) 
+       (change_vec ?? v
+         (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
+            (tape_move_mono ? (nth dst ? v (niltape ?)) 〈Some ? a, R〉) dst).
+#src #dst #sig #n #v #a #b #Heq #Hsrc #Hdst #Ha1 #Ha2
+whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+[ whd in match (trans ????);
+  >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) //
+| whd in match (trans ????);
+  >Ha1 >Ha2 whd in ⊢ (??(????(???%))?); >(\b ?) //
+  change with (change_vec ?????) in ⊢ (??(????%)?);
+  <(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
+  <(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
+  >tape_move_multi_def 
+  >pmap_change >pmap_change <tape_move_multi_def
+  >tape_move_null_action
+  @eq_f2 // >nth_change_vec_neq //
+]
+qed.
+
+lemma sem_copy_step :
+  ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → 
+  copy_step src dst sig n ⊨ 
+    [ copy1: R_comp_step_true src dst sig n, 
+            R_comp_step_false src dst sig n ].
+#src #dst #sig #n #Hneq #Hsrc #Hdst #int
+lapply (refl ? (current ? (nth src ? int (niltape ?))))
+cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?);
+[ #Hcur_src %{2} %
+  [| % [ %
+    [ whd in ⊢ (??%?); >copy_q0_q2_null /2/ 
+    | normalize in ⊢ (%→?); #H destruct (H) ]
+  | #_ % // % // ] ]
+| #a #Ha lapply (refl ? (current ? (nth dst ? int (niltape ?))))
+  cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?);
+  [ #Hcur_dst %{2} %
+    [| % [ %
+       [ whd in ⊢ (??%?); >copy_q0_q2_null /2/ 
+       | normalize in ⊢ (%→?); #H destruct (H) ]
+       | #_ % // %2 >Hcur_dst % ] ]
+  | #b #Hb %{2} %
+    [| % [ % 
+      [whd in ⊢  (??%?);  >(copy_q0_q1 … a b Hneq Hsrc Hdst) //
+      | #_ %{a} %{b} % // % //]
+      | * #H @False_ind @H %
+      ]
+    ]
+  ]
+]
+qed.
+
+definition copy ≝ λsrc,dst,sig,n.
+  whileTM … (copy_step src dst sig n) copy1.
+
+definition R_copy ≝ 
+  λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+  ((current ? (nth src ? int (niltape ?)) = None ? ∨
+    current ? (nth dst ? int (niltape ?)) = None ?) → outt = int) ∧
+  (∀ls,x,x0,rs,ls0,rs0. 
+    nth src ? int (niltape ?) = midtape sig ls x rs →
+    nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+    (∃rs01,rs02.rs0 = rs01@rs02 ∧ |rs01| = |rs| ∧
+     outt = change_vec ?? 
+            (change_vec ?? int  
+              (mk_tape sig (reverse sig rs@x::ls) (None sig) []) src)
+            (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs02)
+            (tail sig rs02)) dst) ∨
+    (∃rs1,rs2.rs = rs1@rs2 ∧ |rs1| = |rs0| ∧
+     outt = change_vec ?? 
+            (change_vec ?? int  
+              (mk_tape sig (reverse sig rs1@x::ls) (option_hd sig rs2)
+            (tail sig rs2)) src)
+            (mk_tape sig (reverse sig rs1@x::ls0) (None sig) []) dst)).
+
+lemma wsem_copy : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → 
+  copy src dst sig n ⊫ R_copy src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
+lapply (sem_while … (sem_copy_step src dst sig n Hneq Hsrc Hdst) … Hloop) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ whd in ⊢ (%→?); * #Hnone #Hout %
+  [#_ @Hout
+  |#ls #x #x0 #rs #ls0 #rs0 #Hsrc1 #Hdst1 @False_ind cases Hnone
+    [>Hsrc1 normalize #H destruct (H) | >Hdst1 normalize #H destruct (H)]
+  ]
+|#tc #td * #x * #y * * #Hcx #Hcy #Htd #Hstar #IH #He lapply (IH He) -IH *
+ #IH1 #IH2 %
+  [* [>Hcx #H destruct (H) | >Hcy #H destruct (H)]
+  |#ls #x' #y' #rs #ls0 #rs0 #Hnth_src #Hnth_dst
+   >Hnth_src in Hcx; whd in ⊢ (??%?→?); #H destruct (H)
+   >Hnth_dst in Hcy; whd in ⊢ (??%?→?); #H destruct (H)
+   >Hnth_src in Htd; >Hnth_dst -Hnth_src -Hnth_dst
+   cases rs
+    [(* the source tape is empty after the move *)
+     #Htd lapply (IH1 ?) 
+      [%1 >Htd >nth_change_vec_neq [2:@(not_to_not … Hneq) //] >nth_change_vec //]
+     #Hout (* whd in match (tape_move ???); *) %1 %{([])} %{rs0} % 
+      [% [// | // ] 
+      |whd in match (reverse ??); whd in match (reverse ??);
+       >Hout >Htd @eq_f2 // cases rs0 //
+      ]
+    |#c1 #tl1 cases rs0
+      [(* the dst tape is empty after the move *)
+       #Htd lapply (IH1 ?) [%2 >Htd >nth_change_vec //] 
+       #Hout (* whd in match (tape_move ???); *) %2 %{[ ]} %{(c1::tl1)} % 
+        [% [// | // ] 
+        |whd in match (reverse ??); whd in match (reverse ??);
+         >Hout >Htd @eq_f2 // 
+        ]
+      |#c2 #tl2 whd in match (tape_move_mono ???); whd in match (tape_move_mono ???);
+       #Htd
+       cut (nth src (tape sig) td (niltape sig)=midtape sig (x::ls) c1 tl1)
+         [>Htd >nth_change_vec_neq [2:@(not_to_not … Hneq) //] @nth_change_vec //]
+       #Hsrc_td
+       cut (nth dst (tape sig) td (niltape sig)=midtape sig (x::ls0) c2 tl2)
+         [>Htd @nth_change_vec //]
+       #Hdst_td cases (IH2 … Hsrc_td Hdst_td) -Hsrc_td -Hdst_td
+        [* #rs01 * #rs02 * * #H1 #H2 #H3 %1
+         %{(c2::rs01)} %{rs02} % [% [@eq_f //|normalize @eq_f @H2]]
+         >Htd in H3; >change_vec_commute // >change_vec_change_vec
+         >change_vec_commute [2:@(not_to_not … Hneq) //] >change_vec_change_vec 
+         #H >reverse_cons >associative_append >associative_append @H 
+        |* #rs11 * #rs12 * * #H1 #H2 #H3 %2
+         %{(c1::rs11)} %{rs12} % [% [@eq_f //|normalize @eq_f @H2]]
+         >Htd in H3; >change_vec_commute // >change_vec_change_vec
+         >change_vec_commute [2:@(not_to_not … Hneq) //] >change_vec_change_vec 
+         #H >reverse_cons >associative_append >associative_append @H 
+        ]
+      ]
+    ]
+  ]
+qed.
+     
+ 
+lemma terminate_copy :  ∀src,dst,sig,n,t.
+  src ≠ dst → src < S n → dst < S n → copy src dst sig n ↓ t.
+#src #dst #sig #n #t #Hneq #Hsrc #Hdts
+@(terminate_while … (sem_copy_step …)) //
+<(change_vec_same … t src (niltape ?))
+cases (nth src (tape sig) t (niltape ?))
+[ % #t1 * #x * #y * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+|2,3: #a0 #al0 % #t1 * #x * #y * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+| #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
+  [#t #ls #c % #t1 * #x * #y * * >nth_change_vec // normalize in ⊢ (%→?);
+   #H1 destruct (H1) #_ >change_vec_change_vec #Ht1 % 
+   #t2 * #x0 * #y0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
+   >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
+  |#r0 #rs0 #IH #t #ls #c % #t1 * #x * #y * * >nth_change_vec //
+   normalize in ⊢ (%→?); #H destruct (H) #Hcur
+   >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
+  ]
+]
+qed.
+
+lemma sem_copy : ∀src,dst,sig,n.
+  src ≠ dst → src < S n → dst < S n → 
+  copy src dst sig n ⊨ R_copy src dst sig n.
+#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize [/2/| @wsem_copy // ]
+qed.