From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Tue, 15 Jan 2013 09:45:58 +0000 (+0000)
Subject: copy.ma
X-Git-Tag: make_still_working~1349
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=249addcfcf2681df796236b0f39a71260dddaa79;p=helm.git

copy.ma
---

diff --git a/matita/matita/lib/turing/multi_universal/copy.ma b/matita/matita/lib/turing/multi_universal/copy.ma
index 284349ec5..18438c034 100644
--- a/matita/matita/lib/turing/multi_universal/copy.ma
+++ b/matita/matita/lib/turing/multi_universal/copy.ma
@@ -1,20 +1,18 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
+(*
+    ||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/turing.ma".
+include "turing/multi_universal/moves.ma".
+include "turing/if_multi.ma".
 include "turing/inject.ma".
-include "turing/while_multi.ma".
+include "turing/basic_machines.ma".
 
 definition copy_states ≝ initN 3.
 
@@ -22,246 +20,243 @@ 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 …)).
 
-(*
-
-src: a b c ... z # ---→ a b c ... z #
-     ^                              ^
-
-dst: _ _ _ ... _ d ---→ a b c ... z d
-     ^                              ^
-
-0) (x ≠ sep,_) → (x,x)(R,R) → 1
-   (sep,d) → None 2
-1) (_,_) → None 1
-2) (_,_) → None 2
-
-*)
 
 definition trans_copy_step ≝ 
- λsrc,dst,sig,n,is_sep.
+ 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 ? n〉
-   | Some a0 ⇒ if is_sep a0 then 〈copy2,null_action ? n〉
-                  else 〈copy1,change_vec ? (S n)
-                          (change_vec ?(S n)
-                           (null_action ? n) (〈Some ? a0,R〉) src)
-                          (〈Some ? a0,R〉) dst〉 ]
+   [ 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,is_sep.
-  mk_mTM sig n copy_states (trans_copy_step src dst sig n is_sep) 
+  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_copy_step_true ≝ 
-  λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n).
-  ∃x1.
-   current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
-   is_sep x1 = false ∧
+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 ?)) (〈Some ? x1,R〉)) src)
-            (tape_move_mono ? (nth dst ? int (niltape ?)) (〈Some ? x1,R〉)) dst.
+              (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
+            (tape_move_mono ? (nth dst ? int (niltape ?)) 〈Some ? x, R〉) dst.
 
-definition R_copy_step_false ≝ 
-  λsrc,dst:nat.λsig,n,is_sep.λint,outt: Vector (tape sig) (S n).
-  (∃x1.
-   current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
-   is_sep x1 = true ∧ outt = int) ∨
-   current ? (nth src ? int (niltape ?)) = None ? ∧
-   outt = int.
+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,is_sep,v,t.src < S n → dst < S n → 
-  current ? t = None ? → 
-  step sig n (copy_step src dst sig n is_sep)
-    (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) =
-    mk_mconfig ??? copy2 (change_vec ? (S n) v t src).
-#src #dst #sig #n #is_sep #v #t #Hsrc #Hdst #Hcurrent
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ >current_chars_change_vec // whd in match (trans ????);
-  >nth_change_vec // >Hcurrent %
-| >current_chars_change_vec // whd in match (trans ????);
-  >nth_change_vec // >Hcurrent @tape_move_null_action
-]
-qed.
-
-lemma copy_q0_q2_sep :
-  ∀src,dst,sig,n,is_sep,v,t.src < S n → dst < S n → 
-  ∀s.current ? t = Some ? s → is_sep s = true → 
-  step sig n (copy_step src dst sig n is_sep)
-    (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) =
-    mk_mconfig ??? copy2 (change_vec ? (S n) v t src).
-#src #dst #sig #n #is_sep #v #t #Hsrc #Hdst #s #Hcurrent #Hsep
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ >current_chars_change_vec // whd in match (trans ????);
-  >nth_change_vec // >Hcurrent whd in ⊢ (??(???%)?); >Hsep %
-| >current_chars_change_vec // whd in match (trans ????);
-  >nth_change_vec // >Hcurrent whd in ⊢ (??(????(???%))?);
-  >Hsep @tape_move_null_action
-]
+  ∀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.
 
-axiom copy_q0_q1 :
-  ∀src,dst,sig,n,is_sep,v,t.src ≠ dst → src < S n → dst < S n → 
-  ∀s.current ? t = Some ? s → is_sep s = false → 
-  step sig n (copy_step src dst sig n is_sep)
-    (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) =
+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 ? t (〈Some ? s,R〉)) src)
-       (tape_move_mono ? (nth dst ? v (niltape ?)) (〈Some ? s,R〉)) dst).
-(*       
-#src #dst #sig #n #is_sep #v #t #Hneq #Hsrc #Hdst #s #Hcurrent #Hsep
+         (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
-[ >current_chars_change_vec // whd in match (trans ????);
-  >nth_change_vec // >Hcurrent whd in ⊢ (??(???%)?); >Hsep %
-| >current_chars_change_vec // whd in match (trans ????);
-  >nth_change_vec // >Hcurrent whd in ⊢ (??(????(???%))?);
-  >Hsep whd in ⊢ (??(????(???%))?); >change_vec_commute // >pmap_change
-  >change_vec_commute // @eq_f3 //
-  <(change_vec_same ?? v dst (niltape ?)) in ⊢(??%?);
-  >pmap_change @eq_f3 //
+[ 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.*)
+qed.
 
 lemma sem_copy_step :
-  ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst < S n → 
-  copy_step src dst sig n is_sep ⊨ 
-    [ copy1: R_copy_step_true src dst sig n is_sep, 
-             R_copy_step_false src dst sig n is_sep ].
-#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #int
+  ∀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 <(change_vec_same … int src (niltape ?)) %{2} %
-  [| % [ % 
+[ #Hcur_src %{2} %
+  [| % [ %
     [ whd in ⊢ (??%?); >copy_q0_q2_null /2/ 
     | normalize in ⊢ (%→?); #H destruct (H) ]
-    | #_ %2 >nth_change_vec >Hcur // % // ] ]
-| #c #Hcur cases (true_or_false (is_sep c)) #Hsep
-  [ <(change_vec_same … int src (niltape ?)) %{2} % 
+  | #_ % // % // ] ]
+| #a #Ha lapply (refl ? (current ? (nth dst ? int (niltape ?))))
+  cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?);
+  [ #Hcur_dst %{2} %
     [| % [ %
-      [ whd in ⊢ (??%?); >copy_q0_q2_sep /2/
-      | normalize in ⊢ (%→?); #H destruct (H) ]
-      | #_ % >nth_change_vec // %{c} % [ % /2/ | // ] ] ]
-  | %{2} % [| % [ %
-    [ whd in ⊢ (??%?);
-      <(change_vec_same … int src (niltape ?)) in ⊢ (??%?);
-      >Hcur in ⊢ (??%?); whd in ⊢ (??%?); >(copy_q0_q1 … Hsep) /2/
-    | #_ whd %{c} % % /2/ ]
-    | * #Hfalse @False_ind /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,is_sep.
-  whileTM … (copy_step src dst sig n is_sep) copy1.
+definition copy ≝ λsrc,dst,sig,n.
+  whileTM … (copy_step src dst sig n) copy1.
 
 definition R_copy ≝ 
-  λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n).
-  (∀ls,x,xs,rs,sep.
-    nth src ? int (niltape ?) = midtape sig ls x (xs@sep::rs) →
-    (∀c.memb ? c (x::xs) = true → is_sep c = false) → is_sep sep = true → 
-    ∀ls0,x0,target,c,rs0.|xs| = |target| → 
-    nth dst ? int (niltape ?) = midtape sig ls0 x0 (target@c::rs0) → 
-    outt = change_vec ?? 
-           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) sep rs) src)
-           (midtape sig (reverse ? xs@x::ls0) c rs0) dst) ∧
-  (∀c.current ? (nth src ? int (niltape ?)) = Some ? c → is_sep c = true → 
-   outt = int) ∧
-  (current ? (nth src ? int (niltape ?)) = None ? → outt = int).
-  
-lemma wsem_copy : ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst < S n → 
-  copy src dst sig n is_sep ⊫ R_copy src dst sig n is_sep.
-#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
-lapply (sem_while … (sem_copy_step src dst sig n is_sep Hneq Hsrc Hdst) … Hloop) //
--Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -ta
-[ whd in ⊢ (%→?); *
-  [ * #x * * #Hx #Hsep #Houtc % [ %
-    [ #ls #x0 #xs #rs #sep #Hsrctc #Hnosep >Hsrctc in Hx; normalize in ⊢ (%→?);
-      #Hx0 destruct (Hx0) lapply (Hnosep ? (memb_hd …)) >Hsep
-      #Hfalse destruct (Hfalse)
-    | #c #Hc #Hsepc @Houtc ]
-    | #_ @Houtc ]
-  | * #Hcur #Houtc % [ %
-    [ #ls #x0 #xs #rs #sep #Hsrctc >Hsrctc in Hcur; normalize in ⊢ (%→?); 
-      #Hcur destruct (Hcur)
-    | #c #Hc #Hsepc @Houtc ]
-    | #_ @Houtc ]
+  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)).
+          
+axiom daemon : ∀P:Prop.P.
+
+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)]
   ]
-| #td #te * #c0 * * #Hc0 #Hc0nosep #Hd #Hstar #IH #He 
-  lapply (IH He) -IH * * #IH1 #IH2 #IH3 % [ %
-  [ #ls #x #xs #rs #sep #Hsrc_tc #Hnosep #Hsep #ls0 #x0 #target
-    #c #rs0 #Hlen #Hdst_tc
-    >Hsrc_tc in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
-    <(change_vec_same … td src (niltape ?)) in Hd:(???(???(???%??)??));
-    <(change_vec_same … td dst (niltape ?)) in ⊢(???(???(???%??)??)→?);
-    >Hdst_tc >Hsrc_tc >(change_vec_change_vec ?) >change_vec_change_vec
-    >(change_vec_commute ?? td ?? dst src) [|@(sym_not_eq … Hneq)]
-    >change_vec_change_vec @(list_cases2 … Hlen)
-    [ #Hxsnil #Htargetnil #Hd>(IH2 … Hsep)
-      [ >Hd -Hd >Hxsnil >Htargetnil @(eq_vec … (niltape ?))
-        #i #Hi cases (decidable_eq_nat i src) #Hisrc
-        [ >Hisrc >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
-          >nth_change_vec // >nth_change_vec //
-          >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
-          >nth_change_vec // whd in ⊢ (??%?); %
-        | cases (decidable_eq_nat i dst) #Hidst
-          [ >Hidst >nth_change_vec // >nth_change_vec //
-            >nth_change_vec_neq // >Hdst_tc >Htargetnil %
-          | >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
-            >nth_change_vec_neq [|@(sym_not_eq … Hisrc)]
-            >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
-            >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] % ]
-         ]
-      | >Hd >nth_change_vec_neq [|@(sym_not_eq … Hneq)]
-        >nth_change_vec // >nth_change_vec // >Hxsnil % ]
-    |#hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd
-     >(IH1 (c0::ls) hd1 tl1 rs sep ?? Hsep (c0::ls0) hd2 tl2 c rs0)
-     [ >Hd >(change_vec_commute … ?? td ?? src dst) //
-       >change_vec_change_vec
-       >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //]
-       >change_vec_change_vec
-       >reverse_cons >associative_append >associative_append % 
-     | >Hd >nth_change_vec // >nth_change_vec_neq // >Hdst_tc >Htarget //
-     | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) //
-     | <Hxs #c1 #Hc1 @Hnosep @memb_cons //
-     | >Hd >nth_change_vec_neq [|@sym_not_eq //]
-       >nth_change_vec // >nth_change_vec // ]
-   ]
- | #c #Hc #Hsepc >Hc in Hc0; #Hcc0 destruct (Hcc0) >Hc0nosep in Hsepc;
-   #H destruct (H)
- ]
-| #HNone >HNone in Hc0; #Hc0 destruct (Hc0) ] ]
+|#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 *)
+     lapply (IH1 ?) [@daemon]
+     #Hout (* whd in match (tape_move ???); *) #Htemp %1 %{([])} %{rs0} % 
+      [% [// | // ] 
+      |whd in match (reverse ??); whd in match (reverse ??);
+       >Hout >Htemp @eq_f2 // cases rs0 //
+      ]
+    |#c1 #tl1 cases rs0
+      [(* the dst tape is empty after the move *)
+       lapply (IH1 ?) [@daemon]
+       #Hout (* whd in match (tape_move ???); *) #Htemp %2 %{[ ]} %{(c1::tl1)} % 
+        [% [// | // ] 
+        |whd in match (reverse ??); whd in match (reverse ??);
+         >Hout >Htemp @eq_f2 // 
+        ]
+      |#c2 #tl2 whd in match (tape_move_mono ???); whd in match (tape_move_mono ???);
+       #Htd
+    
+
+  
+  [ >Hci >Hcj * [ * 
+    [ * #H @False_ind @H % | #H destruct (H)] | #H destruct (H)] 
+  | #ls #c0 #rs #ls0 #rs0 cases rs
+    [ -IH2 #Hnthi #Hnthj % %2 %{rs0} % [%]
+      >Hnthi in Hd; #Hd >Hd in IH1; #IH1 >IH1
+      [| % %2 >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // % ]
+      >Hnthj cases rs0 [| #r1 #rs1 ] %
+    | #r1 #rs1 #Hnthi cases rs0
+      [ -IH2 #Hnthj % % %{(r1::rs1)} % [%]
+        >Hnthj in Hd; #Hd >Hd in IH1; #IH1 >IH1
+        [| %2 >nth_change_vec // ]
+        >nth_change_vec //
+      | #r2 #rs2 #Hnthj lapply IH2; >Hd in IH1; >Hnthi >Hnthj
+        >nth_change_vec //
+        >nth_change_vec_neq [| @sym_not_eq // ] >nth_change_vec //
+        cases (true_or_false (r1 == r2)) #Hr1r2
+        [ >(\P Hr1r2) #_ #IH2 cases (IH2 … (refl ??) (refl ??)) [ *
+          [ * #rs' * #Hrs1 #Hcurout_j % % %{rs'}
+            >Hrs1 >Hcurout_j normalize % //
+          | * #rs0' * #Hrs2 #Hcurout_i % %2 %{rs0'}
+            >Hrs2 >Hcurout_i % //
+            >change_vec_commute // >change_vec_change_vec
+            >change_vec_commute [|@sym_not_eq//] >change_vec_change_vec
+            >reverse_cons >associative_append >associative_append % ]
+          | * #xs * #ci * #cj * #rs' * #rs0' * * * #Hcicj #Hrs1 #Hrs2 
+            >change_vec_commute // >change_vec_change_vec 
+            >change_vec_commute [| @sym_not_eq ] // >change_vec_change_vec 
+            #Houtc %2 %{(r2::xs)} %{ci} %{cj} %{rs'} %{rs0'}
+            % [ % [ % [ // | >Hrs1 // ] | >Hrs2 // ] 
+              | >reverse_cons >associative_append >associative_append >Houtc % ] ]
+        | lapply (\Pf Hr1r2) -Hr1r2 #Hr1r2 #IH1 #_ %2
+          >IH1 [| % % normalize @(not_to_not … Hr1r2) #H destruct (H) % ]
+          %{[]} %{r1} %{r2} %{rs1} %{rs2} % [ % [ % /2/ | % ] | % ] ]]]]]
 qed.
  
-lemma terminate_copy :  ∀src,dst,sig,n,is_sep,t.
-  src ≠ dst → src < S n → dst < S n → 
-  copy src dst sig n is_sep ↓ t.
-#src #dst #sig #n #is_sep #t #Hneq #Hsrc #Hdst
-@(terminate_while … (sem_copy_step …)) //
-<(change_vec_same … t src (niltape ?))
-cases (nth src (tape sig) t (niltape ?))
-[ % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct 
+lemma terminate_compare :  ∀i,j,sig,n,t.
+  i ≠ j → i < S n → j < S n → 
+  compare i j sig n ↓ t.
+#i #j #sig #n #t #Hneq #Hi #Hj
+@(terminate_while … (sem_comp_step …)) //
+<(change_vec_same … t i (niltape ?))
+cases (nth i (tape sig) t (niltape ?))
+[ % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
 |2,3: #a0 #al0 % #t1 * #x * * >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 * * >nth_change_vec // normalize in ⊢ (%→?);
-   #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 % 
+   #H1 destruct (H1) #_ >change_vec_change_vec #Ht1 % 
    #t2 * #x0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
    >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
   |#r0 #rs0 #IH #t #ls #c % #t1 * #x * * >nth_change_vec //
-   normalize in ⊢ (%→?); #H destruct (H) #Hxsep
+   normalize in ⊢ (%→?); #H destruct (H) #Hcur
    >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
   ]
 ]
 qed.
 
-lemma sem_copy : ∀src,dst,sig,n,is_sep.
-  src ≠ dst → src < S n → dst < S n → 
-  copy src dst sig n is_sep ⊨ R_copy src dst sig n is_sep.
-#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst @WRealize_to_Realize /2/
-qed.
\ No newline at end of file
+lemma sem_compare : ∀i,j,sig,n.
+  i ≠ j → i < S n → j < S n → 
+  compare i j sig n ⊨ R_compare i j sig n.
+#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize 
+  [/2/| @wsem_compare // ]
+qed.