definition R_copy_step_subcase ≝
λalpha,c,RelseM,t1,t2.
- ∀ls,x,rs.t1 = midtape (FinProd … alpha FinBool) ls 〈x,true〉 rs →
- (x = c ∧
- ∀a,l1,x0,a0,l2,l3. (∀c.memb ? c l1 = true → is_marked ? c = false) →
- ls = l1@〈a0,false〉::〈x0,true〉::l2 →
- rs = 〈a,false〉::l3 →
- t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨
+ ∀a,l1,x0,a0,l2,x,l3.
+ t1 = midtape (FinProd … alpha FinBool) (l1@〈a0,false〉::〈x0,true〉::l2)
+ 〈x,true〉 (〈a,false〉::l3) →
+ (∀c.memb ? c l1 = true → is_marked ? c = false) →
+ (x = c ∧ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨
(x ≠ c ∧ RelseM t1 t2).
axiom sem_copy_step_subcase :
mark;
move_r;
advance_to_mark_r;
-else nop
+else if current = null
+ then advance_mark_r;
+ move_l;
+ advance_to_mark_l
+ adv_mark_r;
+ move_r;
+ advance_to_mark_r
*)
+definition nocopy_subcase ≝
+ ifTM STape (test_char ? (λx:STape.x == 〈null,true〉))
+ (seq ? (adv_mark_r …)
+ (seq ? (move_l …)
+ (seq ? (adv_to_mark_l … (is_marked ?))
+ (seq ? (adv_mark_r …)
+ (seq ? (move_r …) (adv_to_mark_r … (is_marked ?)))))))
+ (nop ?) tc_true.
+
+definition R_nocopy_subcase ≝
+ λt1,t2.
+ ∀a,l1,x0,a0,l2,x,l3.
+ t1 = midtape STape (l1@〈a0,false〉::〈x0,true〉::l2)
+ 〈x,true〉 (〈a,false〉::l3) →
+ (∀c.memb ? c l1 = true → is_marked ? c = false) →
+ (x = null ∧
+ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3) ∨
+ (x ≠ null ∧ t2 = t1).
+
+axiom sem_nocopy_subcase : Realize ? nocopy_subcase R_nocopy_subcase.
+(* #intape
+cases (sem_if ? (test_char ? (λx:STape.x == 〈null,true〉)) ?????? tc_true
+ (sem_test_char ? (λx:STape.x == 〈null,true〉))
+ (sem_seq … (sem_adv_mark_r …)
+ (sem_seq … (sem_move_l …)
+ (sem_seq … (sem_adv_to_mark_l … (is_marked ?))
+ (sem_seq … (sem_adv_mark_r …)
+ (sem_seq … (sem_move_r …) (sem_adv_to_mark_r … (is_marked ?))
+ ))))) (sem_nop ?) intape)
+#k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+cases HR -HR
+[| * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
+ #ls #x #rs #Hintape %2 >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hx #Hta %
+ [ lapply (\Pf Hx) @not_to_not #Hx' >Hx' %
+ | <Hta @Houtc ] ]
+@daemon
+qed. *)
+
definition copy_step ≝
ifTM ? (test_char STape (λc.is_bit (\fst c)))
(single_finalTM ? (copy_step_subcase FSUnialpha (bit false)
- (copy_step_subcase FSUnialpha (bit true) (nop ?))))
+ (copy_step_subcase FSUnialpha (bit true) nocopy_subcase)))
(nop ?)
tc_true.
definition R_copy_step_true ≝
λt1,t2.
- ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls 〈c,true〉 rs →
- ∃x. c = bit x ∧
- (∀a,l1,c0,a0,l2,l3. (∀y.memb ? y l1 = true → is_marked ? y = false) →
- ls = l1@〈a0,false〉::〈c0,true〉::l2 →
- rs = 〈a,false〉::l3 →
- t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3).
-
+ ∀a,l1,x0,a0,l2,c,l3.
+ t1 = midtape STape (l1@〈a0,false〉::〈x0,true〉::l2)
+ 〈c,true〉 (〈a,false〉::l3) →
+ (∀c.memb ? c l1 = true → is_marked ? c = false) →
+ (∃x. c = bit x ∧
+ t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3) ∨
+ (c = null ∧
+ t2 = midtape ? (〈null,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3).
+
definition R_copy_step_false ≝
λt1,t2.
∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
- is_bit (\fst c) = false ∧ t2 = t1.
+ bit_or_null (\fst c) = false ∧ t2 = t1.
axiom sem_comp_step :
accRealize ? copy_step (inr … (inl … (inr … 0))) R_copy_step_true R_copy_step_false.
+
+(*
+1) il primo carattere è marcato
+2) l'ultimo carattere è l'unico che può essere null, gli altri sono bit
+3) il terminatore non è né bit, né null
+*)
-definition copy ≝ whileTM ? copy_step (inr … (inl … (inr … 0))).
+definition copy0 ≝ whileTM ? copy_step (inr … (inl … (inr … 0))).
+
+let rec merge_config (l1,l2:list STape) ≝
+ match l1 with
+ [ nil ⇒ nil ?
+ | cons p1 l1' ⇒ match l2 with
+ [ nil ⇒ nil ?
+ | cons p2 l2' ⇒
+ let 〈c1,b1〉 ≝ p1 in let 〈c2,b2〉 ≝ p2 in
+ match c2 with
+ [ null ⇒ p1 :: merge_config l1' l2'
+ | _ ⇒ p2 :: merge_config l1' l2' ] ] ].
+
+definition R_copy0 ≝ λt1,t2.
+ ∀ls,c,c0,rs,l1,l3,l4.
+ t1 = midtape STape (l3@l4@〈c0,true〉::ls) 〈c,true〉 (l1@rs) →
+ no_marks l1 → no_marks (l3@l4) → |l1| = |l4| →
+ ∀l1',bv.〈c,false〉::l1 = l1'@[〈comma,bv〉] → only_bits_or_nulls l1' →
+ ∀l4',bg.l4@[〈c0,true〉] = 〈grid,bg〉::l4' → only_bits_or_nulls l4' →
+ (c = comma ∧ t2 = t1) ∨
+ (c ≠ comma ∧
+ t2 = midtape ? (reverse ? l1'@l3@〈grid,true〉::
+ merge_config l4' (reverse ? l1')@ls)
+ 〈comma,true〉 rs).
+
+axiom sem_copy0 : Realize ? copy0 R_copy0.
+
+definition copy ≝
+ seq STape (move_l …) (seq ? (adv_to_mark_l … (is_marked ?))
+ (seq ? (clear_mark …) (seq ? (adv_to_mark_r … (is_marked ?)) (clear_mark …)))).
definition R_copy ≝ λt1,t2.
- ∀ls,c,rs. ∀l1,d,l2,l3,l4,l5,c0.
- t1 = midtape ? ls 〈c,true〉 rs →
- (* l1 è la stringa sorgente da copiare, l4@〈c0,true〉 è la sua destinazione *)
- 〈c,false〉::rs = l1@〈d,false〉::l2 → only_bits l1 → is_bit d = false →
- ls = l3@l4@〈c0,true〉::l5 → |l4@[〈c0,true〉]| = |l1| →
- no_marks (l3@l4) →
- (∀l4',d'.l4@[〈c0,false〉] = 〈d',false〉::l4' →
- t2 = midtape STape (reverse ? l1@l3@〈d',true〉::l4'@l5) 〈d,true〉 l2).
\ No newline at end of file
+ ∀ls,c,c0,rs,l1,l3,l4.
+ t1 = midtape STape (l3@〈grid,false〉::l4@〈c0,true〉::ls) 〈c,true〉 (l1@〈comma,false〉::rs) →
+ no_marks l1 → no_marks l3 → no_marks l4 → |l1| = |l4| →
+ only_bits_or_nulls (〈c0,true〉::l4) → only_bits_or_nulls (〈c,true〉::l1) →
+ t2 = midtape STape (reverse ? l1@l3@〈grid,false〉::
+ merge_config (l4@[〈c0,false〉]) (reverse ? (〈c,false〉::l1))@ls)
+ 〈comma,false〉 rs.
+
+axiom sem_copy : Realize ? copy R_copy.
projects / helm.git / commitdiff