--- /dev/null
+(*
+ ||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.
\ No newline at end of file
projects / helm.git / blobdiff