2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
8 \ / This file is distributed under the terms of the
9 \ / GNU General Public License Version 2
10 V_____________________________________________________________*)
17 include "turing/universal/tuples.ma".
19 definition write_states ≝ initN 2.
21 definition write ≝ λalpha,c.
22 mk_TM alpha write_states
25 [ O ⇒ 〈1,Some ? 〈c,N〉〉
29 definition R_write ≝ λalpha,c,t1,t2.
30 ∀ls,x,rs.t1 = midtape alpha ls x rs → t2 = midtape alpha ls c rs.
32 axiom sem_write : ∀alpha,c.Realize ? (write alpha c) (R_write alpha c).
34 definition copy_step_subcase ≝
35 λalpha,c,elseM.ifTM ? (test_char ? (λx.x == 〈c,true〉))
36 (seq (FinProd alpha FinBool) (adv_mark_r …)
38 (seq ? (adv_to_mark_l … (is_marked alpha))
39 (seq ? (write ? 〈c,false〉)
42 (seq ? (move_r …) (adv_to_mark_r … (is_marked alpha)))))))))
45 definition R_copy_step_subcase ≝
46 λalpha,c,RelseM,t1,t2.
48 t1 = midtape (FinProd … alpha FinBool) (l1@〈a0,false〉::〈x0,true〉::l2)
49 〈x,true〉 (〈a,false〉::l3) →
50 (∀c.memb ? c l1 = true → is_marked ? c = false) →
51 (x = c ∧ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨
52 (x ≠ c ∧ RelseM t1 t2).
54 axiom sem_copy_step_subcase :
55 ∀alpha,c,elseM,RelseM.
56 Realize ? (copy_step_subcase alpha c elseM) (R_copy_step_subcase alpha c RelseM).
68 else if current = 1,tt
77 else if current = null
86 definition nocopy_subcase ≝
87 ifTM STape (test_char ? (λx:STape.x == 〈null,true〉))
90 (seq ? (adv_to_mark_l … (is_marked ?))
92 (seq ? (move_r …) (adv_to_mark_r … (is_marked ?)))))))
95 definition R_nocopy_subcase ≝
98 t1 = midtape STape (l1@〈a0,false〉::〈x0,true〉::l2)
99 〈x,true〉 (〈a,false〉::l3) →
100 (∀c.memb ? c l1 = true → is_marked ? c = false) →
102 t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3) ∨
103 (x ≠ null ∧ t2 = t1).
105 axiom sem_nocopy_subcase : Realize ? nocopy_subcase R_nocopy_subcase.
107 cases (sem_if ? (test_char ? (λx:STape.x == 〈null,true〉)) ?????? tc_true
108 (sem_test_char ? (λx:STape.x == 〈null,true〉))
109 (sem_seq … (sem_adv_mark_r …)
110 (sem_seq … (sem_move_l …)
111 (sem_seq … (sem_adv_to_mark_l … (is_marked ?))
112 (sem_seq … (sem_adv_mark_r …)
113 (sem_seq … (sem_move_r …) (sem_adv_to_mark_r … (is_marked ?))
114 ))))) (sem_nop ?) intape)
115 #k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
117 [| * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
118 #ls #x #rs #Hintape %2 >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hx #Hta %
119 [ lapply (\Pf Hx) @not_to_not #Hx' >Hx' %
124 definition copy_step ≝
125 ifTM ? (test_char STape (λc.is_bit (\fst c)))
126 (single_finalTM ? (copy_step_subcase FSUnialpha (bit false)
127 (copy_step_subcase FSUnialpha (bit true) nocopy_subcase)))
131 definition R_copy_step_true ≝
134 t1 = midtape STape (l1@〈a0,false〉::〈x0,true〉::l2)
135 〈c,true〉 (〈a,false〉::l3) →
136 (∀c.memb ? c l1 = true → is_marked ? c = false) →
138 t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3) ∨
140 t2 = midtape ? (〈null,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3).
142 definition R_copy_step_false ≝
144 ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
145 bit_or_null (\fst c) = false ∧ t2 = t1.
147 axiom sem_comp_step :
148 accRealize ? copy_step (inr … (inl … (inr … 0))) R_copy_step_true R_copy_step_false.
151 1) il primo carattere è marcato
152 2) l'ultimo carattere è l'unico che può essere null, gli altri sono bit
153 3) il terminatore non è né bit, né null
156 definition copy0 ≝ whileTM ? copy_step (inr … (inl … (inr … 0))).
158 let rec merge_config (l1,l2:list STape) ≝
161 | cons p1 l1' ⇒ match l2 with
164 let 〈c1,b1〉 ≝ p1 in let 〈c2,b2〉 ≝ p2 in
166 [ null ⇒ p1 :: merge_config l1' l2'
167 | _ ⇒ p2 :: merge_config l1' l2' ] ] ].
169 definition R_copy0 ≝ λt1,t2.
170 ∀ls,c,c0,rs,l1,l3,l4.
171 t1 = midtape STape (l3@l4@〈c0,true〉::ls) 〈c,true〉 (l1@rs) →
172 no_marks l1 → no_marks (l3@l4) → |l1| = |l4| →
173 ∀l1',bv.〈c,false〉::l1 = l1'@[〈comma,bv〉] → only_bits_or_nulls l1' →
174 ∀l4',bg.l4@[〈c0,true〉] = 〈grid,bg〉::l4' → only_bits_or_nulls l4' →
175 (c = comma ∧ t2 = t1) ∨
177 t2 = midtape ? (reverse ? l1'@l3@〈grid,true〉::
178 merge_config l4' (reverse ? l1')@ls)
181 axiom sem_copy0 : Realize ? copy0 R_copy0.
184 seq STape (move_l …) (seq ? (adv_to_mark_l … (is_marked ?))
185 (seq ? (clear_mark …) (seq ? (adv_to_mark_r … (is_marked ?)) (clear_mark …)))).
187 definition R_copy ≝ λt1,t2.
188 ∀ls,c,c0,rs,l1,l3,l4.
189 t1 = midtape STape (l3@〈grid,false〉::l4@〈c0,true〉::ls) 〈c,true〉 (l1@〈comma,false〉::rs) →
190 no_marks l1 → no_marks l3 → no_marks l4 → |l1| = |l4| →
191 only_bits_or_nulls (〈c0,true〉::l4) → only_bits_or_nulls (〈c,true〉::l1) →
192 t2 = midtape STape (reverse ? l1@l3@〈grid,false〉::
193 merge_config (l4@[〈c0,false〉]) (reverse ? (〈c,false〉::l1))@ls)
196 axiom sem_copy : Realize ? copy R_copy.