definition R_copy_step_true ≝
λt1,t2.
- ∀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).
+ ∀ls,c,rs. t1 = midtape STape ls 〈c,true〉 rs →
+ bit_or_null c = true ∧
+ (∀a,l1,x0,a0,l2,l3.
+ ls = (l1@〈a0,false〉::〈x0,true〉::l2) →
+ rs = (〈a,false〉::l3) →
+ no_marks l1 →
+ ((∃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 →
bit_or_null (\fst c) = false ∧ t2 = t1.
-axiom sem_comp_step :
+axiom sem_copy_step :
accRealize ? copy_step (inr … (inl … (inr … 0))) R_copy_step_true R_copy_step_false.
(*
| 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' ] ] ].
+ [ null ⇒ p1
+ | _ ⇒ p2 ] :: merge_config l1' l2' ] ].
+
+lemma merge_config_append :
+ ∀l1,l2,l3,l4.|l1| = |l2| →
+ merge_config (l1@l3) (l2@l4) = merge_config l1 l2@merge_config l3 l4.
+#l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen)
+[normalize //
+| #t1 #t2 * #c1 #b1 * #c2 #b2 #IH whd in ⊢ (??%%); >IH % ]
+qed.
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' →
+ ∀l4',bg.l4@[〈c0,false〉] = 〈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).
+
+lemma inj_append_singleton_l1 :
+ ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → l1 = l2.
+#A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
+>reverse_append >reverse_append normalize #H1 destruct
+lapply (eq_f … (reverse ?) … e0) >reverse_reverse >reverse_reverse //
+qed.
+
+lemma inj_append_singleton_l2 :
+ ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → a1 = a2.
+#A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
+>reverse_append >reverse_append normalize #H1 destruct %
+qed.
+
+lemma length_reverse : ∀A,l.|reverse A l| = |l|.
+#A #l elim l //
+#a0 #l0 #IH normalize >rev_append_def >length_append >IH normalize //
+qed.
+
+lemma wsem_copy0 : WRealize ? copy0 R_copy0.
+#intape #k #outc #Hloop
+lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop
+* #ta * #Hstar @(star_ind_l ??????? Hstar)
+[ #tb whd in ⊢ (%→?); #Hleft
+ #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
+ #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits
+ cases (Hleft … Htb) -Hleft #Hc #Houtc % %
+ [ generalize in match Hl1bits; -Hl1bits cases l1' in Hl1;
+ [ normalize #Hl1 #c1 destruct (Hl1) %
+ | * #c' #b' #l0 #Heq normalize in Heq; destruct (Heq)
+ #Hl1bits lapply (Hl1bits 〈c',false〉 ?) [ @memb_hd ]
+ >Hc #Hfalse destruct ]
+ | @Houtc ]
+| #tb #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd
+ lapply (Hind Htd) -Hind #Hind
+ #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
+ #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits %2
+ cases (Htc … Htb) -Htc #Hcbitnull #Htc
+ % [ % #Hc' >Hc' in Hcbitnull; normalize #Hfalse destruct (Hfalse) ]
+ cut (|l1| = |reverse ? l4|) [//] #Hlen1
+ @(list_cases_2 … Hlen1)
+ [ (* case l1 = [] is discriminated because l1 contains at least comma *)
+ #Hl1nil @False_ind >Hl1nil in Hl1; cases l1' normalize
+ [ #Hl1 destruct normalize in Hcbitnull; destruct (Hcbitnull)
+ | #p0 #l0 normalize #Hfalse destruct (Hfalse) cases l0 in e0;
+ [ normalize #Hfalse1 destruct (Hfalse1)
+ | #p0' #l0' normalize #Hfalse1 destruct (Hfalse1) ] ]
+ | (* case c::l1 = c::a::l1'' *)
+ * #a #ba * #a0 #ba0 #l1'' #l4'' #Hl1cons #Hl4cons
+ lapply (eq_f ?? (reverse ?) ?? Hl4cons) >reverse_reverse >reverse_cons -Hl4cons #Hl4cons
+ cut (ba = false)
+ [ >Hl1cons in Hl1nomarks; #Hl1nomarks lapply (Hl1nomarks 〈a,ba〉 ?)
+ [ @memb_hd | normalize // ] ] #Hba
+ cut (ba0 = false)
+ [ >Hl4cons in Hl3l4nomarks; #Hl3l4nomarks lapply (Hl3l4nomarks 〈a0,ba0〉 ?)
+ [ @memb_append_l2 @memb_append_l2 @memb_hd | normalize // ] ] #Hba0
+ >Hba0 in Hl4cons; >Hba in Hl1cons; -Hba0 -Hba #Hl1cons #Hl4cons
+ >Hl4cons in Htc; >Hl1cons #Htc
+ lapply (Htc a (l3@reverse ? l4'') c0 a0 ls (l1''@rs) ? (refl ??) ?)
+ [ #x #Hx @Hl3l4nomarks >Hl4cons <associative_append
+ @memb_append_l1 @Hx
+ | >associative_append >associative_append %
+ | -Htc
+ cut (∃la.l1' = 〈c,false〉::la)
+ [ >Hl1cons in Hl1; cases l1'
+ [normalize #Hfalse destruct (Hfalse)
+ | #p #la normalize #Hla destruct (Hla) @(ex_intro ?? la) % ] ]
+ * #la #Hla
+ cut (∃lb.l4' = lb@[〈c0,false〉])
+ [ >Hl4cons in Hl4;
+ @(list_elim_left … l4')
+ [ #Heq lapply (eq_f … (length ?) … Heq)
+ >length_append >length_append
+ >commutative_plus normalize >commutative_plus normalize
+ #Hfalse destruct
+ | #a1 #tl #_ #Heq
+ >(inj_append_singleton_l2 ? (reverse ? l4''@[〈a0,false〉]) (〈grid,bg〉::tl) 〈c0,false〉 a1 Heq)
+ @ex_intro //
+ ] ] * #lb #Hlb
+ cut (|lb| = |reverse ? la|)
+ [ >Hla in Hl1; >Hlb in Hl4; #Hl4 #Hl1
+ >(?:l1 = la@[〈comma,bv〉]) in Hlen;
+ [|normalize in Hl1; destruct (Hl1) %]
+ >(?:l4 = 〈grid,bg〉::lb)
+ [|@(inj_append_singleton_l1 ?? (〈grid,bg〉::lb) ?? Hl4) ]
+ >length_append >commutative_plus >length_reverse
+ normalize #Hlalb destruct (Hlalb) //
+ ] #Hlen2
+ *
+ (* by hyp on the first iteration step,
+ we consider whether c = bit x or c = null *)
+ (* c = bit x *)
+ [ * #x * #Hx #Htc
+ lapply (Hind (〈bit x,false〉::ls) a a0 rs l1''
+ (〈bit x,false〉::l3) (reverse ? l4'') ????)
+ [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus
+ normalize #Hlen destruct (Hlen) //
+ | #x0 #Hx0 cases (orb_true_l … Hx0)
+ [ #Hx0eq >(\P Hx0eq) %
+ | -Hx0 #Hx0 @Hl3l4nomarks >Hl4cons
+ <associative_append @memb_append_l1 // ]
+ | #x0 #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
+ | >Htc >associative_append %
+ | -Hind
+ <Hl1cons <Hl4cons #Hind lapply (Hind la bv ?? lb bg ??)
+ [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
+ | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
+ @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
+ | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
+ | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
+ destruct (Hl1) // ] -Hind
+ (* by IH, we proceed by cases, whether a = comma
+ (consequently several lists = []) or not *)
+ *
+ [ * #Ha #Houtc1
+(* cut (l1 = [〈a,false〉])
+ [ cases l1'' in Hl1cons; // #y #ly #Hly
+ >Hly in Hl1; cases l1' in Hl1bits;
+ [ #_ normalize #Hfalse destruct (Hfalse)
+ | #p #lp #Hl1bits normalize #Heq destruct (Heq)
+ @False_ind lapply (Hl1bits 〈a,false〉 ?)
+ [ cases lp in e0;
+ [ normalize #Hfalse destruct (Hfalse)
+ | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
+ @memb_cons @memb_hd ]
+ | >Ha normalize #Hfalse destruct (Hfalse) ]
+ ]
+ ] #Hl1a
+ cut (l4 = [〈a0,false〉])
+ [ generalize in match Hl4bits; cases l4' in Hl4;
+ [ >Hl4cons #Hfalse #_
+ lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
+ cases (reverse ? l4'') normalize
+ [ #Hfalse1 | #p0 #lp0 #Hfalse1 ] destruct (Hfalse1)
+ | #p #lp
+
+ cases l4'' in Hl4cons; // #y #ly #Hly
+ >Hly in Hl4; cases l4' in Hl4bits;
+ [ #_ >reverse_cons #Hfalse
+ lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
+ -Hfalse cases ly normalize
+ [ #Hfalse | #p #Hp #Hfalse ] destruct (Hfalse)
+
+ | #p #lp #Hl1bits normalize #Heq destruct (Heq)
+ @False_ind lapply (Hl1bits 〈a,false〉 ?)
+ [ cases lp in e0;
+ [ normalize #Hfalse destruct (Hfalse)
+ | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
+ @memb_cons @memb_hd ]
+ | >Ha normalize #Hfalse destruct (Hfalse) ]
+ ]
+ ] #Hl1a
+
+ >Hla normalize #Hl1 destruct (Hl1) lapply (inj_append_ @False_ind
+
+ cut (l1'' = [] ∧ l4'' = [])
+ [ % [ >Hla in Hl1; normalize #Hl1 destruct (Hl1)
+
+ cases l1'' in Hl1bits;
+
+ [ #_ normalize #H *)
+ cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
+ [ @daemon ] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
+ >Hl1cons in Hl1; >Hla
+ >Houtc1 >Htc #Hl1
+ >Hl4cons in Hl4; >Hlb #Hl4
+ >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hx
+ cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
+ normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
+ | * #Ha #Houtc1 >Houtc1 @eq_f3 //
+ >Hla >reverse_cons >associative_append @eq_f
+ >Hx whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
+ >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
+ ]
+ ]
+ | (* c = null *)
+ * #Hc #Htc
+ lapply (Hind (〈c0,false〉::ls) a a0 rs l1'' (〈null,false〉::l3) (reverse ? l4'') ????)
+ [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus normalize
+ #Hlen destruct (Hlen) @e0
+ | #x0 #Hx0 cases (memb_append STape ? [〈null,false〉] (l3@reverse ? l4'') … Hx0) -Hx0 #Hx0
+ [ >(memb_single … Hx0) %
+ | @Hl3l4nomarks cases (memb_append … Hx0) -Hx0 #Hx0
+ [ @memb_append_l1 //
+ | @memb_append_l2 >Hl4cons @memb_append_l1 // ]
+ ]
+ | >Hl1cons #x' #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
+ | >Htc @eq_f3 // >associative_append % ] -Hind <Hl1cons <Hl4cons #Hind
+ lapply (Hind la bv ?? lb bg ??)
+ [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
+ | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
+ @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
+ | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
+ | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
+ destruct (Hl1) // ] -Hind *
+ (* by IH, we proceed by cases, whether a = comma
+ (consequently several lists = []) or not *)
+ [ * #Ha #Houtc1 >Hl1cons in Hl1; >Hla
+ >Houtc1 >Htc #Hl1
+ >Hl4cons in Hl4; >Hlb #Hl4
+ cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
+ [@daemon] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
+ >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hc
+ cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
+ normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
+ | * #Ha #Houtc1 >Houtc1 @eq_f3 //
+ >Hla >reverse_cons >associative_append @eq_f
+ >Hc whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
+ >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
+ ]
+ ]
+]]]
+qed.
+
+definition merge_char ≝ λc1,c2.
+ match c2 with
+ [ null ⇒ c1
+ | _ ⇒ c2 ].
+
+lemma merge_cons :
+ ∀c1,c2,conf1,conf2.
+ merge_config (〈c1,false〉::conf1) (〈c2,false〉::conf2) =
+ 〈merge_char c1 c2,false〉::merge_config conf1 conf2.
+#c1 #c2 #conf1 #conf2 normalize @eq_f2 //
+cases c2 /2/
+qed.
-axiom sem_copy0 : Realize ? copy0 R_copy0.
+lemma merge_config_c_nil :
+ ∀c.merge_config c [] = [].
+#c cases c normalize //
+qed.
-definition copy ≝
+axiom reverse_merge_config :
+ ∀c1,c2.|c1| = |c2| → reverse ? (merge_config c1 c2) =
+ merge_config (reverse ? c1) (reverse ? c2).
+
+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 …)))).
+(*
+ s0, s1 = caratteri di testa dello stato
+ c0 = carattere corrente del nastro oggetto
+ c1 = carattere in scrittura sul nastro oggetto
+
+ questa dimostrazione sfrutta il fatto che
+ merge_config (l0@[c0]) (l1@[c1]) = l1@[merge_char c0 c1]
+ se l0 e l1 non contengono null
+*)
+
definition R_copy ≝ λt1,t2.
- ∀ls,c,c0,rs,l1,l3,l4.
- t1 = midtape STape (l3@〈grid,false〉::l4@〈c0,true〉::ls) 〈c,true〉 (l1@〈comma,false〉::rs) →
+ ∀ls,s0,s1,c0,c1,rs,l1,l3,l4.
+ t1 = midtape STape (l3@〈grid,false〉::〈c0,false〉::l4@〈s0,true〉::ls) 〈s1,true〉 (l1@〈c1,false〉::〈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.
-
+ only_bits (l4@[〈s0,true〉]) → only_bits (〈s1,true〉::l1) →
+ bit_or_null c0 = true → bit_or_null c1 = true →
+ t2 = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l3@〈grid,false〉::
+ 〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)
+ 〈comma,false〉 rs.
+
axiom sem_copy : Realize ? copy R_copy.
+i
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projects / helm.git / blobdiff