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+(*
+    ||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".
+
+definition compare_states ≝ initN 3.
+
+definition comp0 : compare_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition comp1 : compare_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition comp2 : compare_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
+definition trans_compare_step ≝ 
+ λi,j.λsig:FinSet.λn.
+ λp:compare_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ match nth i ? a (None ?) with
+   [ None ⇒ 〈comp2,null_action sig n〉
+   | Some ai ⇒ match nth j ? a (None ?) with 
+     [ None ⇒ 〈comp2,null_action ? n〉
+     | Some aj ⇒ if ai == aj 
+         then 〈comp1,change_vec ? (S n) 
+                      (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) i)
+                        (〈None ?,R〉) j〉
+         else 〈comp2,null_action ? n〉 ]
+   ]
+ | S q ⇒ match q with 
+   [ O ⇒ (* 1 *) 〈comp1,null_action ? n〉
+   | S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ].
+
+definition compare_step ≝ 
+  λi,j,sig,n.
+  mk_mTM sig n compare_states (trans_compare_step i j sig n) 
+    comp0 (λq.q == comp1 ∨ q == comp2).
+
+definition R_comp_step_true ≝ 
+  λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
+  ∃x.
+   current ? (nth i ? int (niltape ?)) = Some ? x ∧
+   current ? (nth j ? int (niltape ?)) = Some ? x ∧
+   outt = change_vec ?? 
+            (change_vec ?? int
+              (tape_move_right ? (nth i ? int (niltape ?))) i)
+            (tape_move_right ? (nth j ? int (niltape ?))) j.
+
+definition R_comp_step_false ≝ 
+  λi,j:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
+   (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
+    current ? (nth i ? int (niltape ?)) = None ? ∨
+    current ? (nth j ? int (niltape ?)) = None ?) ∧ outt = int.
+
+lemma comp_q0_q2_null :
+  ∀i,j,sig,n,v.i < S n → j < S n → 
+  (nth i ? (current_chars ?? v) (None ?) = None ? ∨
+   nth j ? (current_chars ?? v) (None ?) = None ?) → 
+  step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) 
+  = mk_mconfig ??? comp2 v.
+#i #j #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 i ?? (None sig)) //
+  | whd in ⊢ (??(????(???%))?); >Hcurrent
+    cases (nth i ?? (None sig)) [|#x] @tape_move_null_action ] ]
+qed.
+
+lemma comp_q0_q2_neq :
+  ∀i,j,sig,n,v.i < S n → j < S n → 
+  (nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?)) → 
+  step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) 
+  = mk_mconfig ??? comp2 v.
+#i #j #sig #n #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
+cases (nth i ?? (None ?)) in ⊢ (???%→?);
+[ #Hnth #_ @comp_q0_q2_null // % //
+| #ai #Hai lapply (refl ? (nth j ?(current_chars ?? v)(None ?)))
+  cases (nth j ?? (None ?)) in ⊢ (???%→?);
+  [ #Hnth #_ @comp_q0_q2_null // %2 //
+  | #aj #Haj * #Hneq
+    whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+    [ whd in match (trans ????); >Hai >Haj
+      whd in ⊢ (??(???%)?); cut ((ai==aj)=false)
+      [>(\bf ?) /2 by not_to_not/ % #Haiaj @Hneq
+       >Hai >Haj //
+      | #Haiaj >Haiaj % ]
+    | whd in match (trans ????); >Hai >Haj
+      whd in ⊢ (??(????(???%))?); cut ((ai==aj)=false)
+      [>(\bf ?) /2 by not_to_not/ % #Haiaj @Hneq
+       >Hai >Haj //
+      |#Hcut >Hcut @tape_move_null_action
+      ]
+    ]
+  ]
+]
+qed.
+
+lemma comp_q0_q1 :
+  ∀i,j,sig,n,v,a.i ≠ j → i < S n → j < S n → 
+  nth i ? (current_chars ?? v) (None ?) = Some ? a →
+  nth j ? (current_chars ?? v) (None ?) = Some ? a → 
+  step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) =
+    mk_mconfig ??? comp1 
+     (change_vec ? (S n) 
+       (change_vec ?? v
+         (tape_move_right ? (nth i ? v (niltape ?))) i)
+       (tape_move_right ? (nth j ? v (niltape ?))) j).
+#i #j #sig #n #v #a #Heq #Hi #Hj #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 j (niltape ?)) in ⊢ (??%?);
+  <(change_vec_same … v i (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_comp_step :
+  ∀i,j,sig,n.i ≠ j → i < S n → j < S n → 
+  compare_step i j sig n ⊨ 
+    [ comp1: R_comp_step_true i j sig n, 
+             R_comp_step_false i j sig n ].
+#i #j #sig #n #Hneq #Hi #Hj #int
+lapply (refl ? (current ? (nth i ? int (niltape ?))))
+cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?);
+[ #Hcuri %{2} %
+  [| % [ %
+    [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ 
+    | normalize in ⊢ (%→?); #H destruct (H) ]
+  | #_ % // % %2 // ] ]
+| #a #Ha lapply (refl ? (current ? (nth j ? int (niltape ?))))
+  cases (current ? (nth j ? int (niltape ?))) in ⊢ (???%→?);
+  [ #Hcurj %{2} %
+    [| % [ %
+       [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2
+       | normalize in ⊢ (%→?); #H destruct (H) ]
+       | #_ % // >Ha >Hcurj % % % #H destruct (H) ] ]
+  | #b #Hb %{2} cases (true_or_false (a == b)) #Hab
+    [ %
+      [| % [ % 
+        [whd in ⊢  (??%?);  >(comp_q0_q1 … a Hneq Hi Hj) //
+         >(\P Hab) <Hb @sym_eq @nth_vec_map
+        | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) // ]
+        | * #H @False_ind @H %
+      ] ]
+    | %
+      [| % [ % 
+        [whd in ⊢  (??%?);  >comp_q0_q2_neq //
+         <(nth_vec_map ?? (current …) i ? int (niltape ?))
+         <(nth_vec_map ?? (current …) j ? int (niltape ?)) >Ha >Hb
+         @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
+        | normalize in ⊢ (%→?); #H destruct (H) ]
+      | #_ % // % % >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
+    ]
+  ]
+]
+qed.
+(* copy a character from src tape to dst tape without moving them *)
+
+definition copy_states ≝ initN 3.
+
+definition cc0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition cc1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+
+definition trans_copy_char_N ≝ 
+ λsrc,dst.λsig:FinSet.λn.
+ λp:copy_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ 〈cc1,change_vec ? (S n) 
+           (change_vec ? (S n) (null_action ? n) (〈None ?,N〉) src)
+           (〈nth src ? a (None ?),N〉) dst〉
+ | S _ ⇒ 〈cc1,null_action ? n〉 ].
+
+definition copy_char_N ≝ 
+  λsrc,dst,sig,n.
+  mk_mTM sig n copy_states (trans_copy_char_N src dst sig n) 
+    cc0 (λq.q == cc1).
+
+definition R_copy_char_N ≝ 
+  λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+  outt = change_vec ?? int
+          (tape_write  ? (nth dst ? int (niltape ?))
+            (current ? (nth src ? int (niltape ?)))) dst.
+
+lemma copy_char_N_q0_q1 :
+  ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n → 
+  step sig n (copy_char_N src dst sig n) (mk_mconfig ??? cc0 v) =
+    mk_mconfig ??? cc1 
+     (change_vec ?? v
+       (tape_write  ? (nth dst ? v (niltape ?))
+          (current ? (nth src ? v (niltape ?)))) dst).
+#src #dst #sig #n #v #Heq #Hsrc #Hdst
+whd in ⊢ (??%?); @eq_f
+<(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_f3 //
+[ >change_vec_same %
+| >change_vec_same >change_vec_same >nth_current_chars // ]
+qed.
+
+lemma sem_copy_char_N:
+  ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → 
+  copy_char_N src dst sig n ⊨ R_copy_char_N src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst #int
+%{2} % [| % [ % | whd >copy_char_N_q0_q1 // ]]
+qed.
+
+(* copy a character from src tape to dst tape and advance both tape to
+   the right - useful for copying stings 
+
+definition copy_char_states ≝ initN 3.
+
+definition trans_copy_char ≝ 
+ λsrc,dst.λsig:FinSet.λn.
+ λp:copy_char_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ 〈cc1,change_vec ? (S n) 
+           (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
+           (〈nth src ? a (None ?),R〉) dst〉
+ | S _ ⇒ 〈cc1,null_action ? n〉 ].
+
+definition copy_char ≝ 
+  λsrc,dst,sig,n.
+  mk_mTM sig n copy_char_states (trans_copy_char src dst sig n) 
+    cc0 (λq.q == cc1).
+
+definition R_copy_char ≝ 
+  λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+  outt = change_vec ?? 
+         (change_vec ?? int
+          (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
+          (tape_move_mono ? (nth dst ? int (niltape ?)) 
+           〈current ? (nth src ? int (niltape ?)), R〉) dst.
+
+lemma copy_char_q0_q1 :
+  ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n → 
+  step sig n (copy_char src dst sig n) (mk_mconfig ??? cc0 v) =
+    mk_mconfig ??? cc1 
+     (change_vec ? (S n) 
+       (change_vec ?? v
+         (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
+            (tape_move_mono ? (nth dst ? v (niltape ?)) 〈current ? (nth src ? v (niltape ?)), R〉) dst).
+#src #dst #sig #n #v #Heq #Hsrc #Hdst
+whd in ⊢ (??%?);
+<(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
+<(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
+>tape_move_multi_def @eq_f2 //
+>pmap_change >pmap_change <tape_move_multi_def
+>tape_move_null_action @eq_f2 // @eq_f2
+[ >change_vec_same %
+| >change_vec_same >change_vec_same // ]
+qed.
+
+lemma sem_copy_char:
+  ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → 
+  copy_char src dst sig n ⊨ R_copy_char src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst #int
+%{2} % [| % [ % | whd >copy_char_q0_q1 // ]]
+qed.*)
+
+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_copy_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_copy_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_copy_step_true src dst sig n, 
+            R_copy_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.
+
+
+(* advance in parallel on tapes src and dst; stops if one of the
+   two tapes is in oveflow *)
+
+definition parmove_states ≝ initN 3.
+
+definition parmove0 : parmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition parmove1 : parmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition parmove2 : parmove_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
+(*
+   src: a b c ... z ---→ a b c ... z 
+        ^                            ^
+   dst: _ _ _ ... _ ---→ a b c ... z 
+        ^                            ^
+
+   0) (x,_) → (x,_)(R,R) → 1
+      (None,_) → None 2
+   1) (_,_) → None 1
+   2) (_,_) → None 2
+*)
+
+definition trans_parmove_step ≝ 
+ λsrc,dst,sig,n,D.
+ λp:parmove_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ match nth src ? a (None ?) with
+   [ None ⇒ 〈parmove2,null_action sig n〉
+   | Some a0 ⇒ match nth dst ? a (None ?) with
+     [ None ⇒ 〈parmove2,null_action ? n〉
+     | Some a1 ⇒ 〈parmove1,change_vec ? (S n)
+                          (change_vec ?(S n)
+                           (null_action ? n) (〈None ?,D〉) src)
+                          (〈None ?,D〉) dst〉 ] ]
+ | S q ⇒ match q with 
+   [ O ⇒ (* 1 *) 〈parmove1,null_action ? n〉
+   | S _ ⇒ (* 2 *) 〈parmove2,null_action ? n〉 ] ].
+
+definition parmove_step ≝ 
+  λsrc,dst,sig,n,D.
+  mk_mTM sig n parmove_states (trans_parmove_step src dst sig n D) 
+    parmove0 (λq.q == parmove1 ∨ q == parmove2).
+
+definition R_parmove_step_true ≝ 
+  λsrc,dst,sig,n,D.λint,outt: Vector (tape sig) (S n).
+  ∃x1,x2.
+   current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
+   current ? (nth dst ? int (niltape ?)) = Some ? x2 ∧
+   outt = change_vec ?? 
+            (change_vec ?? int
+              (tape_move ? (nth src ? int (niltape ?)) D) src)
+            (tape_move ? (nth dst ? int (niltape ?)) D) dst.
+
+definition R_parmove_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 parmove_q0_q2_null_src :
+  ∀src,dst,sig,n,D,v.src < S n → dst < S n → 
+  nth src ? (current_chars ?? v) (None ?) = None ? → 
+  step sig n (parmove_step src dst sig n D)
+    (mk_mconfig ??? parmove0 v) =
+    mk_mconfig ??? parmove2 v.
+#src #dst #sig #n #D #v #Hsrc #Hdst #Hcurrent
+whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
+@eq_f2
+[ whd in ⊢ (??(???%)?); >Hcurrent %
+| whd in ⊢ (??(????(???%))?); >Hcurrent @tape_move_null_action ]
+qed.
+
+lemma parmove_q0_q2_null_dst :
+  ∀src,dst,sig,n,D,v,s.src < S n → dst < S n → 
+  nth src ? (current_chars ?? v) (None ?) = Some ? s → 
+  nth dst ? (current_chars ?? v) (None ?) = None ? → 
+  step sig n (parmove_step src dst sig n D)
+    (mk_mconfig ??? parmove0 v) =
+    mk_mconfig ??? parmove2 v.
+#src #dst #sig #n #D #v #s #Hsrc #Hdst #Hcursrc #Hcurdst
+whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
+@eq_f2
+[ whd in ⊢ (??(???%)?); >Hcursrc whd in ⊢ (??(???%)?); >Hcurdst %
+| whd in ⊢ (??(????(???%))?); >Hcursrc
+  whd in ⊢ (??(????(???%))?); >Hcurdst @tape_move_null_action ]
+qed.
+
+lemma parmove_q0_q1 :
+  ∀src,dst,sig,n,D,v.src ≠ dst → src < S n → dst < S n → 
+  ∀a1,a2.
+  nth src ? (current_chars ?? v) (None ?) = Some ? a1 →
+  nth dst ? (current_chars ?? v) (None ?) = Some ? a2 → 
+  step sig n (parmove_step src dst sig n D)
+    (mk_mconfig ??? parmove0 v) =
+    mk_mconfig ??? parmove1 
+     (change_vec ? (S n) 
+       (change_vec ?? v
+         (tape_move ? (nth src ? v (niltape ?)) D) src)
+       (tape_move ? (nth dst ? v (niltape ?)) D) dst).
+#src #dst #sig #n #D #v #Hneq #Hsrc #Hdst
+#a1 #a2 #Hcursrc #Hcurdst
+whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+[ whd in match (trans ????);
+  >Hcursrc >Hcurdst %
+| whd in match (trans ????);
+  >Hcursrc >Hcurdst whd in ⊢ (??(????(???%))?); 
+  >tape_move_multi_def <(change_vec_same ?? v dst (niltape ?)) in ⊢ (??%?);
+  >pmap_change <(change_vec_same ?? v src (niltape ?)) in ⊢(??%?);
+  >pmap_change <tape_move_multi_def >tape_move_null_action
+  @eq_f2 // >nth_change_vec_neq //
+]
+qed.
+
+lemma sem_parmove_step :
+  ∀src,dst,sig,n,D.src ≠ dst → src < S n → dst < S n → 
+  parmove_step src dst sig n D ⊨ 
+    [ parmove1: R_parmove_step_true src dst sig n D, 
+             R_parmove_step_false src dst sig n ].
+#src #dst #sig #n #D #Hneq #Hsrc #Hdst #int
+lapply (refl ? (current ? (nth src ? int (niltape ?))))
+cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?);
+[ #Hcursrc %{2} %
+  [| % [ %
+    [ whd in ⊢ (??%?); >parmove_q0_q2_null_src /2/
+    | normalize in ⊢ (%→?); #H destruct (H) ]
+    | #_ % // % // ] ]
+| #a #Ha lapply (refl ? (current ? (nth dst ? int (niltape ?))))
+  cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?);
+  [ #Hcurdst %{2} %
+    [| % [ %
+      [ whd in ⊢ (??%?); >(parmove_q0_q2_null_dst …) /2/ 
+      | normalize in ⊢ (%→?); #H destruct (H) ]
+      | #_ % // %2 // ] ]
+  | #b #Hb %{2} %
+    [| % [ % 
+      [whd in ⊢  (??%?); >(parmove_q0_q1 … Hneq Hsrc Hdst ? b ??)
+        [2: <(nth_vec_map ?? (current …) dst ? int (niltape ?)) //
+        |3: <(nth_vec_map ?? (current …) src ? int (niltape ?)) //
+        | // ]
+      | #_ %{a} %{b} % // % // ]
+      | * #H @False_ind @H % ]
+]]]
+qed.
+
+(* perform a symultaneous test on all tapes; ends up in state partest1 if
+   the test is succesfull and partest2 otherwise *)
+
+definition partest_states ≝ initN 3.
+
+definition partest0 : partest_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition partest1 : partest_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition partest2 : partest_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
+definition trans_partest ≝ 
+ λsig,n,test.
+ λp:partest_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ if test a then 〈partest1,null_action sig n〉 
+ else 〈partest2,null_action ? n〉.
+
+definition partest ≝ 
+  λsig,n,test.
+  mk_mTM sig n partest_states (trans_partest sig n test) 
+    partest0 (λq.q == partest1 ∨ q == partest2).
+
+definition R_partest_true ≝ 
+  λsig,n,test.λint,outt: Vector (tape sig) (S n).
+  test (current_chars ?? int) = true ∧ outt = int.
+  
+definition R_partest_false ≝ 
+  λsig,n,test.λint,outt: Vector (tape sig) (S n).
+  test (current_chars ?? int) = false ∧ outt = int.
+
+lemma partest_q0_q1:
+  ∀sig,n,test,v.
+  test (current_chars ?? v) = true → 
+  step sig n (partest sig n test)(mk_mconfig ??? partest0 v) 
+    = mk_mconfig ??? partest1 v.
+#sig #n #test #v #Htest
+whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
+@eq_f2
+[ whd in ⊢ (??(???%)?); >Htest %
+| whd in ⊢ (??(????(???%))?); >Htest @tape_move_null_action ]
+qed.
+
+lemma partest_q0_q2:
+  ∀sig,n,test,v.
+  test (current_chars ?? v) = false → 
+  step sig n (partest sig n test)(mk_mconfig ??? partest0 v) 
+    = mk_mconfig ??? partest2 v.
+#sig #n #test #v #Htest
+whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
+@eq_f2
+[ whd in ⊢ (??(???%)?); >Htest %
+| whd in ⊢ (??(????(???%))?); >Htest @tape_move_null_action ]
+qed.
+
+lemma sem_partest:
+  ∀sig,n,test.
+  partest sig n test ⊨ 
+    [ partest1: R_partest_true sig n test, R_partest_false sig n test ].
+#sig #n #test #int
+cases (true_or_false (test (current_chars ?? int))) #Htest
+[ %{2} %{(mk_mconfig ? partest_states n partest1 int)} %
+  [ % [ whd in ⊢ (??%?); >partest_q0_q1 /2/ | #_ % // ] 
+  | * #H @False_ind @H %
+  ]
+| %{2} %{(mk_mconfig ? partest_states n partest2 int)} %
+  [ % [ whd in ⊢ (??%?); >partest_q0_q2 /2/ 
+      | whd in ⊢ (??%%→?); #H destruct (H)]
+  | #_ % //]
+]
+qed.
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