--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "turing/turing.ma".
+include "turing/inject.ma".
+include "turing/while_multi.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 …)).
+
+(*
+
+0) (x,x) → (x,x)(R,R) → 1
+ (x,y≠x) → None 2
+1) (_,_) → None 1
+2) (_,_) → None 2
+
+*)
+
+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 ? 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) (Some ? 〈ai,R〉) i)
+ (Some ? 〈aj,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 ? (nth i ? int (niltape ?)) (Some ? 〈x,R〉)) i)
+ (tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) 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 ⊢ (??(???%)?); >(\bf ?) // @(not_to_not … Hneq) //
+ | whd in match (trans ????); >Hai >Haj
+ whd in ⊢ (??(???????(???%))?); >(\bf ?) /2 by not_to_not/
+ @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 ? (nth i ? v (niltape ?)) (Some ? 〈a,R〉)) i)
+ (tape_move ? (nth j ? v (niltape ?)) (Some ? 〈a,R〉)) 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 ⊢ (??%?);
+ >pmap_change >pmap_change >tape_move_null_action
+ @eq_f2 // @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/ % <Hcuri in ⊢ (???%);
+ @sym_eq @nth_vec_map
+ | 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 <Hcurj in ⊢ (???%);
+ @sym_eq @nth_vec_map
+ | 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
+ |<Ha @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.
+
+definition compare ≝ λi,j,sig,n.
+ whileTM … (compare_step i j sig n) comp1.
+
+definition R_compare ≝
+ λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
+ (current sig (nth i (tape sig) int (niltape sig))
+ ≠current sig (nth j (tape sig) int (niltape sig)) →
+ outt = int) ∧
+ (∀ls,x,xs,ci,rs,ls0,cj,rs0.
+ nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+ nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → ci ≠ cj →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
+ (midtape sig (reverse ? xs@x::ls0) cj rs0) j).
+
+lemma wsem_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 #ta #k #outc #Hloop
+lapply (sem_while … (sem_comp_step i j sig n Hneq Hi Hj) … Hloop) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ #tc whd in ⊢ (%→?); * * [ *
+ [ #Hcicj #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj
+ >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
+ ]
+ | #Hci #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci;
+ normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | #Hcj #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj;
+ normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | #tc #td #te * #x * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
+ #IH1 #IH2 %
+ [ >Hci >Hcj * #H @False_ind @H %
+ | #ls #c0 #xs #ci #rs #ls0 #cj #rs0 cases xs
+ [ #Hnthi #Hnthj #Hcicj >IH1
+ [ >Hd @eq_f3 //
+ [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
+ >Hnthi in Hci;normalize #H destruct (H) %
+ | >(?:c0=x) [ >Hnthj % ]
+ >Hnthi in Hci;normalize #H destruct (H) % ]
+ | >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // >Hnthi >Hnthj normalize @(not_to_not ??? Hcicj)
+ #H destruct (H) % ]
+ | #x0 #xs0 #Hnthi #Hnthj #Hcicj
+ >(IH2 (c0::ls) x0 xs0 ci rs (c0::ls0) cj rs0 … Hcicj)
+ [ >Hd >change_vec_commute in ⊢ (??%?); //
+ >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
+ @sym_not_eq //
+ | >Hd >nth_change_vec // >Hnthj normalize
+ >Hnthi in Hci;normalize #H destruct (H) %
+ | >Hd >nth_change_vec_neq [|@sym_not_eq //] >Hnthi
+ >nth_change_vec // normalize
+ >Hnthi in Hci;normalize #H destruct (H) %
+ ]
+]]]
+qed.
+
+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 %
+ #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) #Hcur
+ >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
+ ]
+]
+qed.
+
+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/
+qed.
+
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