mk_TM alpha write_states
(λp.let 〈q,a〉 ≝ p in
match pi1 … q with
- [ O ⇒ 〈wr1,Some ? 〈c,N〉〉
- | S _ ⇒ 〈wr1,None ?〉 ])
+ [ O ⇒ 〈wr1,Some ? c,N〉
+ | S _ ⇒ 〈wr1,None ?,N〉 ])
wr0 (λx.x == wr1).
definition R_write ≝ λalpha,c,t1,t2.
#alpha #c #t @(ex_intro … 2) @ex_intro
[|% [% |#ls #c #rs #Ht >Ht % ] ]
qed.
-
+
+definition R_write_strong ≝ λalpha,c,t1,t2.
+ t2 = midtape alpha (left ? t1) c (right ? t1).
+
+lemma sem_write_strong : ∀alpha,c.Realize ? (write alpha c) (R_write_strong alpha c).
+#alpha #c #t @(ex_intro … 2) @ex_intro
+ [|% [% |cases t normalize // ] ]
+qed.
+
+(***************************** replace a with f a *****************************)
+
+definition writef ≝ λalpha,f.
+ mk_TM alpha write_states
+ (λp.let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ 〈wr1,Some ? (f a),N〉
+ | S _ ⇒ 〈wr1,None ?,N〉 ])
+ wr0 (λx.x == wr1).
+
+definition R_writef ≝ λalpha,f,t1,t2.
+ ∀c. current ? t1 = c →
+ t2 = midtape alpha (left ? t1) (f c) (right ? t1).
+
+lemma sem_writef : ∀alpha,f.
+ writef alpha f ⊨ R_writef alpha f.
+#alpha #f #t @(ex_intro … 2) @ex_intro
+ [|% [% |cases t normalize // ] ]
+qed.
+
(******************** moves the head one step to the right ********************)
definition move_states ≝ initN 2.
λalpha:FinSet.mk_TM alpha move_states
(λp.let 〈q,a〉 ≝ p in
match a with
- [ None ⇒ 〈move1,None ?〉
+ [ None ⇒ 〈move1,None ?,N〉
| Some a' ⇒ match (pi1 … q) with
- [ O ⇒ 〈move1,Some ? 〈a',R〉〉
- | S q ⇒ 〈move1,None ?〉 ] ])
+ [ O ⇒ 〈move1,Some ? a',R〉
+ | S q ⇒ 〈move1,None ?,N〉 ] ])
move0 (λq.q == move1).
definition R_move_r ≝ λalpha,t1,t2.
λalpha:FinSet.mk_TM alpha move_states
(λp.let 〈q,a〉 ≝ p in
match a with
- [ None ⇒ 〈move1,None ?〉
+ [ None ⇒ 〈move1,None ?,N〉
| Some a' ⇒ match pi1 … q with
- [ O ⇒ 〈move1,Some ? 〈a',L〉〉
- | S q ⇒ 〈move1,None ?〉 ] ])
+ [ O ⇒ 〈move1,Some ? a',L〉
+ | S q ⇒ 〈move1,None ?,N〉 ] ])
move0 (λq.q == move1).
definition R_move_l ≝ λalpha,t1,t2.
#ls1 #c1 #rs1 #H destruct cases ls1 // ] ] ]
qed.
+(* a slightly different move machine. *)
+definition smove_states ≝ initN 2.
+
+definition smove0 : smove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
+definition smove1 : smove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
+
+definition trans_smove ≝
+ λsig,D.
+ λp:smove_states × (option sig).
+ let 〈q,a〉 ≝ p in match (pi1 … q) with
+ [ O ⇒ 〈smove1,None sig, D〉
+ | S _ ⇒ 〈smove1,None sig, N〉 ].
+
+definition move ≝
+ λsig,D.mk_TM sig smove_states (trans_smove sig D) smove0 (λq.q == smove1).
+
+definition Rmove ≝
+ λalpha,D,t1,t2. t2 = tape_move alpha t1 D.
+
+lemma sem_move_single :
+ ∀alpha,D.move alpha D ⊨ Rmove alpha D.
+#alpha #D #int %{2} %{(mk_config ? smove_states smove1 ?)} [| % % ]
+qed.
+
(********************************* test char **********************************)
(* the test_char machine ends up in two different states q1 and q2 wether or not
mk_TM alpha tc_states
(λp.let 〈q,a〉 ≝ p in
match a with
- [ None ⇒ 〈tc_false, None ?〉
+ [ None ⇒ 〈tc_false, None ?,N〉
| Some a' ⇒
match test a' with
- [ true ⇒ 〈tc_true,None ?〉
- | false ⇒ 〈tc_false,None ?〉 ]])
+ [ true ⇒ 〈tc_true,None ?,N〉
+ | false ⇒ 〈tc_false,None ?,N〉 ]])
tc_start (λx.notb (x == tc_start)).
definition Rtc_true ≝
]
qed.
+lemma test_char_inv :
+ ∀sig.∀P:tape sig → Prop.∀f,t,t0.P t → Rtc_true sig f t t0 → P t0.
+#sig #P #f #t #t0 #HPt * #_ //
+qed.
+
+definition test_null ≝ λalpha.test_char alpha (λ_.true).
+
+definition R_test_null_true ≝ λalpha,t1,t2.
+ current alpha t1 ≠ None ? ∧ t1 = t2.
+
+definition R_test_null_false ≝ λalpha,t1,t2.
+ current alpha t1 = None ? ∧ t1 = t2.
+
+lemma sem_test_null : ∀alpha.
+ test_null alpha ⊨ [ tc_true : R_test_null_true alpha, R_test_null_false alpha].
+#alpha #t1 cases (sem_test_char alpha (λ_.true) t1) #k * #outc * * #Hloop #Htrue
+#Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #_ #Houtc1 %
+ [ >Hcurt1 % #H destruct (H) | <Houtc1 % ] ]
+| #Houtc cases (Hfalse ?) [| @Houtc] #Habsurd #Houtc %
+ [ cases (current alpha t1) in Habsurd; // #c1 #Habsurd
+ lapply (Habsurd ? (refl ??)) #H destruct (H)
+ | <Houtc % ] ]
+qed.
+
(************************************* swap ***********************************)
definition swap_states : FinSet → FinSet ≝
λalpha:FinSet.FinProd (initN 4) alpha.
let 〈q',b〉 ≝ q in
let q' ≝ pi1 nat (λi.i<4) q' in
match a with
- [ None ⇒ 〈〈swap3,foo〉,None ?〉 (* if tape is empty then stop *)
+ [ None ⇒ 〈〈swap3,foo〉,None ?,N〉 (* if tape is empty then stop *)
| Some a' ⇒
match q' with
- [ O ⇒ (* q0 *) 〈〈swap1,a'〉,Some ? 〈a',R〉〉 (* save in register and move R *)
+ [ O ⇒ (* q0 *) 〈〈swap1,a'〉,Some ? a',R〉 (* save in register and move R *)
| S q' ⇒ match q' with
- [ O ⇒ (* q1 *) 〈〈swap2,a'〉,Some ? 〈b,L〉〉 (* swap with register and move L *)
+ [ O ⇒ (* q1 *) 〈〈swap2,a'〉,Some ? b,L〉 (* swap with register and move L *)
| S q' ⇒ match q' with
- [ O ⇒ (* q2 *) 〈〈swap3,foo〉,Some ? 〈b,N〉〉 (* copy from register and stay *)
- | S q' ⇒ (* q3 *) 〈〈swap3,foo〉,None ?〉 (* final state *)
+ [ O ⇒ (* q2 *) 〈〈swap3,foo〉,Some ? b,N〉 (* copy from register and stay *)
+ | S q' ⇒ (* q3 *) 〈〈swap3,foo〉,None ?,N〉 (* final state *)
]
]
]])
let 〈q',b〉 ≝ q in
let q' ≝ pi1 nat (λi.i<4) q' in
match a with
- [ None ⇒ 〈〈swap3,foo〉,None ?〉 (* if tape is empty then stop *)
+ [ None ⇒ 〈〈swap3,foo〉,None ?,N〉 (* if tape is empty then stop *)
| Some a' ⇒
match q' with
- [ O ⇒ (* q0 *) 〈〈swap1,a'〉,Some ? 〈a',L〉〉 (* save in register and move L *)
+ [ O ⇒ (* q0 *) 〈〈swap1,a'〉,Some ? a',L〉 (* save in register and move L *)
| S q' ⇒ match q' with
- [ O ⇒ (* q1 *) 〈〈swap2,a'〉,Some ? 〈b,R〉〉 (* swap with register and move R *)
+ [ O ⇒ (* q1 *) 〈〈swap2,a'〉,Some ? b,R〉 (* swap with register and move R *)
| S q' ⇒ match q' with
- [ O ⇒ (* q2 *) 〈〈swap3,foo〉,Some ? 〈b,N〉〉 (* copy from register and stay *)
- | S q' ⇒ (* q3 *) 〈〈swap3,foo〉,None ?〉 (* final state *)
+ [ O ⇒ (* q2 *) 〈〈swap3,foo〉,Some ? b,N〉 (* copy from register and stay *)
+ | S q' ⇒ (* q3 *) 〈〈swap3,foo〉,None ?,N〉 (* final state *)
]
]
]])
]
]
qed.
+
+(********************************** combine ***********************************)
+(* replace the content x of a cell with a combiation f(x,y) of x and the content
+y of the adiacent cell *)
+
+definition combf_states : FinSet → FinSet ≝
+ λalpha:FinSet.FinProd (initN 4) alpha.
+
+definition combf0 : initN 4 ≝ mk_Sig ?? 0 (leb_true_to_le 1 4 (refl …)).
+definition combf1 : initN 4 ≝ mk_Sig ?? 1 (leb_true_to_le 2 4 (refl …)).
+definition combf2 : initN 4 ≝ mk_Sig ?? 2 (leb_true_to_le 3 4 (refl …)).
+definition combf3 : initN 4 ≝ mk_Sig ?? 3 (leb_true_to_le 4 4 (refl …)).
+
+definition combf_r ≝
+ λalpha:FinSet.λf.λfoo:alpha.
+ mk_TM alpha (combf_states alpha)
+ (λp.let 〈q,a〉 ≝ p in
+ let 〈q',b〉 ≝ q in
+ let q' ≝ pi1 nat (λi.i<4) q' in
+ match a with
+ [ None ⇒ 〈〈combf3,foo〉,None ?,N〉 (* if tape is empty then stop *)
+ | Some a' ⇒
+ match q' with
+ [ O ⇒ (* q0 *) 〈〈combf1,a'〉,Some ? a',R〉 (* save in register and move R *)
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q1 *) 〈〈combf2,f b a'〉,Some ? a',L〉
+ (* combine in register and move L *)
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q2 *) 〈〈combf3,foo〉,Some ? b,R〉
+ (* copy from register and move R *)
+ | S q' ⇒ (* q3 *) 〈〈combf3,foo〉,None ?,N〉 (* final state *)
+ ]
+ ]
+ ]])
+ 〈combf0,foo〉
+ (λq.\fst q == combf3).
+
+definition Rcombf_r ≝
+ λalpha,f,t1,t2.
+ (∀b,ls.
+ t1 = midtape alpha ls b [ ] →
+ t2 = rightof ? b ls) ∧
+ (∀a,b,ls,rs.
+ t1 = midtape alpha ls b (a::rs) →
+ t2 = midtape alpha ((f b a)::ls) a rs).
+
+lemma sem_combf_r : ∀alpha,f,foo.
+ combf_r alpha f foo ⊨ Rcombf_r alpha f.
+#alpha #f #foo *
+ [@(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (niltape ?)))
+ % [% |% [#b #ls | #a #b #ls #rs] #H destruct]
+ |#l0 #lt0 @(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (leftof ? l0 lt0)))
+ % [% | % [#b #ls | #a #b #ls #rs] #H destruct]
+ |#r0 #rt0 @(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (rightof ? r0 rt0)))
+ % [% |% [#b #ls | #a #b #ls #rs] #H destruct]
+ | #lt #c #rt @(ex_intro ?? 4) cases rt
+ [@ex_intro [|% [ % | %
+ [#b #ls #H destruct normalize // |#a #b #ls #rs #H destruct]]]
+ |#r0 #rt0 @ex_intro [| % [ % | %
+ [#b #ls #H destruct | #a #b #ls #rs #H destruct normalize //
+ ]
+ ]
+qed.
+
+definition combf_l ≝
+ λalpha:FinSet.λf.λfoo:alpha.
+ mk_TM alpha (combf_states alpha)
+ (λp.let 〈q,a〉 ≝ p in
+ let 〈q',b〉 ≝ q in
+ let q' ≝ pi1 nat (λi.i<4) q' in
+ match a with
+ [ None ⇒ 〈〈combf3,foo〉,None ?,N〉 (* if tape is empty then stop *)
+ | Some a' ⇒
+ match q' with
+ [ O ⇒ (* q0 *) 〈〈combf1,a'〉,Some ? a',L〉 (* save in register and move R *)
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q1 *) 〈〈combf2,f b a'〉,Some ? a',R〉
+ (* combine in register and move L *)
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q2 *) 〈〈combf3,foo〉,Some ? b,L〉
+ (* copy from register and move R *)
+ | S q' ⇒ (* q3 *) 〈〈combf3,foo〉,None ?,N〉 (* final state *)
+ ]
+ ]
+ ]])
+ 〈combf0,foo〉
+ (λq.\fst q == combf3).
+
+definition Rcombf_l ≝
+ λalpha,f,t1,t2.
+ (∀b,rs.
+ t1 = midtape alpha [ ] b rs →
+ t2 = leftof ? b rs) ∧
+ (∀a,b,ls,rs.
+ t1 = midtape alpha (a::ls) b rs →
+ t2 = midtape alpha ls a ((f b a)::rs)).
+
+lemma sem_combf_l : ∀alpha,f,foo.
+ combf_l alpha f foo ⊨ Rcombf_l alpha f.
+#alpha #f #foo *
+ [@(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (niltape ?)))
+ % [% |% [#b #ls | #a #b #ls #rs] #H destruct]
+ |#l0 #lt0 @(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (leftof ? l0 lt0)))
+ % [% | % [#b #ls | #a #b #ls #rs] #H destruct]
+ |#r0 #rt0 @(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (rightof ? r0 rt0)))
+ % [% |% [#b #ls | #a #b #ls #rs] #H destruct]
+ | #lt #c #rt @(ex_intro ?? 4) cases lt
+ [@ex_intro [|% [ % | %
+ [#b #ls #H destruct normalize // |#a #b #ls #rs #H destruct]]]
+ |#r0 #rt0 @ex_intro [| % [ % | %
+ [#b #ls #H destruct | #a #b #ls #rs #H destruct normalize //
+ ]
+ ]
+qed.
+
+(********************************* new_combine ********************************)
+(* replace the content x of a cell with a combiation f(x,y) of x and the content
+y of the adiacent cell; if there is no adjacent cell, combines with a default
+value foo *)
+
+definition ncombf_r ≝
+ λalpha:FinSet.λf.λfoo:alpha.
+ mk_TM alpha (combf_states alpha)
+ (λp.let 〈q,a〉 ≝ p in
+ let 〈q',b〉 ≝ q in
+ let q' ≝ pi1 nat (λi.i<4) q' in
+ match a with
+ [ None ⇒ if (eqb q' 1)then (* if on right cell, combine in register and move L *)
+ 〈〈combf2,f b foo〉,None ?,L〉
+ else 〈〈combf3,foo〉,None ?,N〉 (* else stop *)
+ | Some a' ⇒
+ match q' with
+ [ O ⇒ (* q0 *) 〈〈combf1,a'〉,Some ? a',R〉 (* save in register and move R *)
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q1 *) 〈〈combf2,f b a'〉,Some ? a',L〉
+ (* combine in register and move L *)
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q2 *) 〈〈combf3,foo〉,Some ? b,R〉
+ (* copy from register and move R *)
+ | S q' ⇒ (* q3 *) 〈〈combf3,foo〉,None ?,N〉 (* final state *)
+ ]
+ ]
+ ]])
+ 〈combf0,foo〉
+ (λq.\fst q == combf3).
+
+definition Rncombf_r ≝
+ λalpha,f,foo,t1,t2.
+ (∀b,ls.
+ t1 = midtape alpha ls b [ ] →
+ t2 = rightof ? (f b foo) ls) ∧
+ (∀a,b,ls,rs.
+ t1 = midtape alpha ls b (a::rs) →
+ t2 = midtape alpha ((f b a)::ls) a rs).
+
+lemma sem_ncombf_r : ∀alpha,f,foo.
+ ncombf_r alpha f foo ⊨ Rncombf_r alpha f foo.
+#alpha #f #foo *
+ [@(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (niltape ?)))
+ % [% |% [#b #ls | #a #b #ls #rs] #H destruct]
+ |#l0 #lt0 @(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (leftof ? l0 lt0)))
+ % [% | % [#b #ls | #a #b #ls #rs] #H destruct]
+ |#r0 #rt0 @(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (rightof ? r0 rt0)))
+ % [% |% [#b #ls | #a #b #ls #rs] #H destruct]
+ | #lt #c #rt @(ex_intro ?? 4) cases rt
+ [@ex_intro [|% [ % | %
+ [#b #ls #H destruct normalize // |#a #b #ls #rs #H destruct]]]
+ |#r0 #rt0 @ex_intro [| % [ % | %
+ [#b #ls #H destruct | #a #b #ls #rs #H destruct normalize //
+ ]
+ ]
+qed.
+
+definition ncombf_l ≝
+ λalpha:FinSet.λf.λfoo:alpha.
+ mk_TM alpha (combf_states alpha)
+ (λp.let 〈q,a〉 ≝ p in
+ let 〈q',b〉 ≝ q in
+ let q' ≝ pi1 nat (λi.i<4) q' in
+ match a with
+ [ None ⇒ if (eqb q' 1)then
+ (* if on left cell, combine in register and move R *)
+ 〈〈combf2,f b foo〉,None ?,R〉
+ else 〈〈combf3,foo〉,None ?,N〉 (* else stop *)
+ | Some a' ⇒
+ match q' with
+ [ O ⇒ (* q0 *) 〈〈combf1,a'〉,Some ? a',L〉 (* save in register and move R *)
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q1 *) 〈〈combf2,f b a'〉,Some ? a',R〉
+ (* combine in register and move L *)
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q2 *) 〈〈combf3,foo〉,Some ? b,L〉
+ (* copy from register and move R *)
+ | S q' ⇒ (* q3 *) 〈〈combf3,foo〉,None ?,N〉 (* final state *)
+ ]
+ ]
+ ]])
+ 〈combf0,foo〉
+ (λq.\fst q == combf3).
+
+definition Rncombf_l ≝
+ λalpha,f,foo,t1,t2.
+ (∀b,rs.
+ t1 = midtape alpha [ ] b rs →
+ t2 = leftof ? (f b foo) rs) ∧
+ (∀a,b,ls,rs.
+ t1 = midtape alpha (a::ls) b rs →
+ t2 = midtape alpha ls a ((f b a)::rs)).
+
+lemma sem_ncombf_l : ∀alpha,f,foo.
+ ncombf_l alpha f foo ⊨ Rncombf_l alpha f foo.
+#alpha #f #foo *
+ [@(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (niltape ?)))
+ % [% |% [#b #ls | #a #b #ls #rs] #H destruct]
+ |#l0 #lt0 @(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (leftof ? l0 lt0)))
+ % [% | % [#b #ls | #a #b #ls #rs] #H destruct]
+ |#r0 #rt0 @(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈combf3,foo〉 (rightof ? r0 rt0)))
+ % [% |% [#b #ls | #a #b #ls #rs] #H destruct]
+ | #lt #c #rt @(ex_intro ?? 4) cases lt
+ [@ex_intro [|% [ % | %
+ [#b #ls #H destruct normalize // |#a #b #ls #rs #H destruct]]]
+ |#r0 #rt0 @ex_intro [| % [ % | %
+ [#b #ls #H destruct | #a #b #ls #rs #H destruct normalize //
+ ]
+ ]
+qed.
projects / helm.git / blobdiff