only_bits qin ∧ cin ≠ bar ∧ |qin| = S n ∧ t = bar::qin@[cin].
lemma table_to_list: ∀n,l,h,c. is_config n c →
- (∃ll,lr.table_TM n l h = ll@c@lr) →
- ∃out,t. tuple_encoding n h t = (c@out) ∧ mem ? t l.
+ ∀ll,lr.table_TM n l h = ll@c@lr →
+ ∃out,lr0,t. lr = out@lr0 ∧ tuple_encoding n h t = (c@out) ∧ mem ? t l.
#n #l #h #c * #qin * #cin * * * #H1 #H2 #H3 #H4
- * #ll * #lr lapply ll -ll elim l
+#ll #lr lapply ll -ll elim l
[>H4 #ll cases ll normalize [|#hd #tl ] #Habs destruct
|#t1 #othert #Hind #ll >table_TM_cons #Htuple
cut (S n < |ll@c@lr|)
normalize in ⊢ (???%→?); whd in ⊢ (??%?→?); #Htemp
lapply (cons_injective_l ????? Htemp) #Hc1
lapply (cons_injective_r ????? Htemp) -Htemp #Heq2
- %{(q2@[c2;m])} %{t1} %
- [>Ht1 >Heq1 >Hc1 @eq_f >associative_append %
+ %{(q2@[c2;m])} %{(table_TM n othert h)} %{t1} %
+ [ %[ // |>Ht1 >Heq1 >Hc1 @eq_f >associative_append % ]
|%1 %
]
|(* ll not nil *)
whd in ⊢ (??%?→?); #Heq destruct (Heq)
cases (compare_append … e0) #l *
[* cases l
- [#_ #Htab cases (Hind [ ] (sym_eq … Htab)) #out * #t * #Ht #Hmemt
- %{out} %{t} % // %2 //
+ [#_ #Htab cases (Hind [ ] (sym_eq … Htab))
+ #out * #lr0 * #t * * #Hlr #Ht #Hmemt
+ %{out} %{lr0} %{t} % [% //| %2 //]
|(* this case is absurd *)
#al #tll #Heq1 >H4 #Heq2 @False_ind
lapply (cons_injective_l ? bar … Heq2) #Hbar <Hbar in Heq1; #Heq1
]
]
]
- |* #Htl #Htab cases (Hind … Htab) #out * #t * #Ht #Hmemt
- %{out} %{t} % // %2 //
+ |* #Htl #Htab cases (Hind … Htab) #out * #lr0 * #t * * #Hlr #Ht #Hmemt
+ %{out} %{lr0} %{t} % [% // | %2 //]
]
]
]
#Heqout <Heqout in tuplet2; <Heqq <Heqa >tuplet1
@append_l2_injective %
qed.
-
-lemma cfg_in_table_to_tuple: ∀n,l,h,c. is_config n c →
- ∀ll,lr.table_TM n l h = ll@c@lr →
- ∃out,m,lr0. lr = out@m::lr0 ∧ is_config n (bar::out) ∧ m ≠ bar.
-#n #l #h #c * #qin * #cin * * * #H1 #H2 #H3 #H4
-#ll #lr lapply ll -ll elim l
- [>H4 #ll cases ll normalize [|#hd #tl ] #Habs destruct
- |#t1 #othert #Hind #ll >table_TM_cons #Htuple
- cut (S n < |ll@c@lr|)
- [<Htuple >length_append >(length_of_tuple … (is_tuple … ))
- /2 by transitive_lt, le_n/] #Hsplit lapply Htuple -Htuple
- cases (is_tuple … n h t1) #q1 * #c1 * #q2 * #c2 * #m
- * * * * * * * #Hq1 #Hq2 #Hc1 #Hc2 #Hm #Hlen1 #Hlen2
- whd in ⊢ (???%→?); #Ht1
- (* if ll is empty we match the first tuple t1, otherwise
- we match inside othert *)
- cases ll
- [>H4 >Ht1 normalize in ⊢ (???%→?);
- >associative_append whd in ⊢ (??%?→?); #Heq destruct (Heq) -Heq
- >associative_append in e0; #e0
- lapply (append_l1_injective … e0) [>H3 @Hlen1] #Heq1
- lapply (append_l2_injective … e0) [>H3 @Hlen1]
- normalize in ⊢ (???%→?); whd in ⊢ (??%?→?); #Htemp
- lapply (cons_injective_l ????? Htemp) #Hc1
- lapply (cons_injective_r ????? Htemp) -Htemp #Heq2
- %{(q2@[c2])} %{m} %{(table_TM n othert h)} % // %
- [ <Heq2 >associative_append >associative_append % | %{q2} %{c2} % // % // % // ]
- |(* ll not nil *)
- #b #tl >Ht1 normalize in ⊢ (???%→?);
- whd in ⊢ (??%?→?); #Heq destruct (Heq)
- cases (compare_append … e0) #l *
- [* cases l
- [#_ #Htab cases (Hind [ ] (sym_eq … Htab)) #out * #m0 * #lr0 * * #Hlr #Hcfg #Hm0
- %{out} %{m0} %{lr0} % // % //
- |(* this case is absurd *)
- #al #tll #Heq1 >H4 #Heq2 @False_ind
- lapply (cons_injective_l ? bar … Heq2) #Hbar <Hbar in Heq1; #Heq1
- @(absurd (mem ? bar (q1@(c1::q2@[c2; m]))))
- [>Heq1 @mem_append_l2 %1 //
- |% #Hmembar cases (mem_append ???? Hmembar) -Hmembar
- [#Hmembar lapply(Hq1 bar Hmembar) normalize #Habs destruct (Habs)
- |* [#Habs @absurd //]
- #Hmembar cases (mem_append ???? Hmembar) -Hmembar
- [#Hmembar lapply(Hq2 bar Hmembar) normalize #Habs destruct (Habs)
- |* [#Habs @absurd //] #Hmembar @(absurd ?? Hm) @sym_eq @mem_single //
- ]
- ]
- ]
- ]
- |* #Htl #Htab cases (Hind … Htab) #out * #m0 * #lr0 * * #Hlr #Hcfg #Hm0
- %{out} %{m0} %{lr0} % // % //
- ]
- ]
- ]
-qed.
-