definition STape ≝ FinProd … FSUnialpha FinBool.
+definition only_bits ≝ λl.
+ ∀c.memb STape c l = true → is_bit (\fst c) = true.
+
definition only_bits_or_nulls ≝ λl.
∀c.memb STape c l = true → bit_or_null (\fst c) = true.
* // normalize #H destruct
qed.
+definition mk_tuple ≝ λqin,cin,qout,cout,mv.
+ qin @ cin :: 〈comma,false〉:: qout @ cout :: 〈comma,false〉 :: [mv].
+
(* by definition, a tuple is not marked *)
definition tuple_TM : nat → list STape → Prop ≝
- λn,t.∃qin,qout,mv.
- no_marks t ∧
- only_bits_or_nulls qin ∧ only_bits_or_nulls qout ∧ bit_or_null mv = true ∧
- |qin| = n ∧ |qout| = n (* ∧ |mv| = ? *) ∧
- t = qin@〈comma,false〉::qout@〈comma,false〉::[〈mv,false〉].
+ λn,t.∃qin,cin,qout,cout,mv.
+ no_marks qin ∧ no_marks qout ∧
+ only_bits qin ∧ only_bits qout ∧
+ bit_or_null cin = true ∧ bit_or_null cout = true ∧ bit_or_null mv = true ∧
+ (cout = null → mv = null) ∧
+ |qin| = n ∧ |qout| = n ∧
+ t = mk_tuple qin 〈cin,false〉 qout 〈cout,false〉 〈mv,false〉.
-inductive table_TM : nat → list STape → Prop ≝
-| ttm_nil : ∀n.table_TM n []
-| ttm_cons : ∀n,t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@〈bar,false〉::T).
+inductive table_TM (n:nat) : list STape → Prop ≝
+| ttm_nil : table_TM n []
+| ttm_cons : ∀t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@〈bar,false〉::T).
+
+inductive match_in_table (qin:list STape) (cin: STape)
+ (qout:list STape) (cout:STape) (mv:STape)
+: list STape → Prop ≝
+| mit_hd :
+ ∀tb.
+ match_in_table qin cin qout cout mv
+ (mk_tuple qin cin qout cout mv @〈bar,false〉::tb)
+| mit_tl :
+ ∀qin0,cin0,qout0,cout0,mv0,tb.
+ match_in_table qin cin qout cout mv tb →
+ match_in_table qin cin qout cout mv
+ (mk_tuple qin0 cin0 qout0 cout0 mv0@〈bar,false〉::tb).
+
+axiom append_l1_injective :
+ ∀A.∀l1,l2,l3,l4:list A. |l1| = |l2| → l1@l3 = l2@l4 → l1 = l2.
+axiom append_l2_injective :
+ ∀A.∀l1,l2,l3,l4:list A. |l1| = |l2| → l1@l3 = l2@l4 → l3 = l4.
+axiom cons_injective_l : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → a1 = a2.
+axiom cons_injective_r : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → l1 = l2.
+axiom tuple_len : ∀n,t.tuple_TM n t → |t| = 2*n+5.
+axiom append_eq_tech1 :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → ∃la:list A.l1@a::la = l3.
+axiom append_eq_tech2 :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false → ∃la:list A.l3 = l1@a::la.
+(*axiom list_decompose_cases :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → ∃la,lb:list A.l3 = la@a::lb ∨ l4 = la@a::lb.
+axiom list_decompose_l :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false →
+ ∃la,lb.l2 = la@lb ∧ l3 = l1@a::la.
+axiom list_decompose_r :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l3 = false →
+ ∃la,lb.l1 = la@lb ∧ l4 = lb@a::l2.
+axiom list_decompose_memb :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → memb A a l3 = true.*)
+
+lemma generic_match_to_match_in_table :
+ ∀n,T.table_TM n T →
+ ∀qin,cin,qout,cout,mv.|qin| = n → |qout| = n →
+ only_bits qin → only_bits qout →
+ bit_or_null (\fst cin) = true → bit_or_null (\fst cout) = true →
+ bit_or_null (\fst mv) = true →
+ ∀t1,t2.
+ T = (t1@qin@cin::〈comma,false〉::qout@cout::〈comma,false〉::[mv])@t2 →
+ match_in_table qin cin qout cout mv T.
+#n #T #Htable #qin #cin #qout #cout #mv #Hlenqin #Hlenqout
+#Hqinbits #Hqoutbits #Hcin #Hcout #Hmv
+elim Htable
+[ #t1 #t2 <associative_append cases (t1@qin) normalize
+ [ #Hfalse destruct (Hfalse) | #c0 #t0 #Hfalse destruct (Hfalse) ]
+| #tuple #T0 * #qin0 * #cin0 * #qout0 * #cout0 * #mv0
+ * * * * * * * * * *
+ #Hqin0marks #Hqout0marks #Hqin0bits #Hqout0bits #Hcin0 #Hcout0 #Hmv0 #Hcout0mv0
+ #Hlenqin0 #Hlenqout0 #Htuple #Htable0 #IH #t1 #t2 #HT
+ cases t1 in HT;
+ [ >Htuple normalize in ⊢ (??%%→?);
+ >associative_append >associative_append #HT
+ cut (qin0 = qin ∧ (〈cin0,false〉 = cin ∧ (qout0 = qout ∧
+ (〈cout0,false〉 = cout ∧ (〈mv0,false〉 = mv ∧ 〈bar,false〉::T0 = t2)))))
+ [ lapply (append_l1_injective … HT) [ >Hlenqin @Hlenqin0 ]
+ #Hqin % [ @Hqin ] -Hqin
+ lapply (append_l2_injective … HT) [ >Hlenqin @Hlenqin0 ] -HT #HT
+ lapply (cons_injective_l ????? HT) #Hcin % [ @Hcin ] -Hcin
+ lapply (cons_injective_r ????? HT) -HT #HT
+ lapply (cons_injective_r ????? HT) -HT
+ >associative_append >associative_append #HT
+ lapply (append_l1_injective … HT) [ >Hlenqout @Hlenqout0 ]
+ #Hqout % [ @Hqout ] -Hqout
+ lapply (append_l2_injective … HT) [ >Hlenqout @Hlenqout0 ] -HT normalize #HT
+ lapply (cons_injective_l ????? HT) #Hcout % [ @Hcout ] -Hcout
+ lapply (cons_injective_r ????? HT) -HT #HT
+ lapply (cons_injective_r ????? HT) -HT #HT
+ lapply (cons_injective_l ????? HT) #Hmv % [ @Hmv ] -Hmv
+ @(cons_injective_r ????? HT) ]
+ -HT * #Hqin * #Hcin * #Hqout * #Hcout * #Hmv #HT0
+ >(?:qin0@(〈cin0,false〉::〈comma,false〉::qout0@[〈cout0,false〉;〈comma,false〉;〈mv0,false〉])@〈bar,false〉::T0
+ = mk_tuple qin cin qout cout mv@〈bar,false〉::T0)
+ [|>Hqin >Hqout >Hcin >Hcout >Hmv normalize >associative_append >associative_append
+ normalize >associative_append % ]
+ %
+ | #c0 #cs0 #HT cut (∃cs1.c0::cs0 = tuple@〈bar,false〉::cs1)
+ [ cases (append_eq_tech1 ?????? HT ?)
+ [ -HT #ta #Hta cases (append_eq_tech2 … Hta ?)
+ [ -Hta #tb #Htb %{tb} @Htb
+ | @daemon ]
+ | @le_S_S >length_append >(plus_n_O (|tuple|)) >commutative_plus @le_plus
+ [ @le_O_n
+ | >Htuple normalize >length_append >length_append @le_plus [ >Hlenqin >Hlenqin0 % ]
+ @le_S_S @le_S_S >length_append >length_append @le_plus [ >Hlenqout >Hlenqout0 % ] %] ]
+ ]
+ * #cs1 #Hcs1 >Hcs1 in HT; >associative_append >associative_append #HT
+ lapply (append_l2_injective … HT) // -HT #HT
+ lapply (cons_injective_r ????? HT) -HT
+ <associative_append #HT >Htuple %2 @(IH ?? HT)
+ ]
+]
+qed.
+
+lemma no_grids_in_tuple : ∀n,l.tuple_TM n l → no_grids l.
+#n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * *
+#_ #_ #Hqin #Hqout #Hcin #Hcout #Hmv #_ #_ #_ #Hl >Hl
+#c #Hc normalize in Hc; cases (memb_append … Hc) -Hc #Hc
+[ @bit_not_grid @(Hqin … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid //
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| cases (memb_append …Hc) -Hc #Hc
+[ @bit_not_grid @(Hqout … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid //
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| >(memb_single … Hc) @bit_or_null_not_grid @Hmv
+]]]]]]
+qed.
+
+lemma no_marks_in_tuple : ∀n,l.tuple_TM n l → no_marks l.
+#n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * *
+#Hqin #Hqout #_ #_ #_ #_ #_ #_ #_ #_ #Hl >Hl
+#c #Hc normalize in Hc; cases (memb_append … Hc) -Hc #Hc
+[ @(Hqin … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) %
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| cases (memb_append … Hc) -Hc #Hc
+[ @(Hqout … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) %
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| >(memb_single … Hc) %
+]]]]]]
+qed.
lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
#n #l #t elim t
- [normalize #n #x #H destruct
- |#m #t1 #t2 * #qin * #qout * #mv * * * * * *
- #Hmarks #Hqin #Hqout #Hmv #_ #_ #Heq #Ht2 #Hind
- whd >Heq #x #membx
- cases (memb_append … membx) -membx #membx
- [cases (memb_append … membx) -membx #membx
- [@bit_or_null_not_grid @Hqin //
- |cases (orb_true_l … membx) -membx #membx
- [>(\P membx) //
- |cases (memb_append … membx) -membx #membx
- [@bit_or_null_not_grid @Hqout //
- |cases (orb_true_l … membx) -membx #membx
- [>(\P membx) //
- |@bit_or_null_not_grid >(memb_single … membx) @Hmv
- ]
- ]
- ]
- ]
- |cases (orb_true_l … membx) -membx #membx
- [>(\P membx) //
- |@Hind //
- ]
- ]
- ]
+ [normalize #c #H destruct
+ |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_grids_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
+ cases (memb_append … Hx) -Hx #Hx
+ [ @(Ht1 … Hx)
+ | cases (orb_true_l … Hx) -Hx #Hx
+ [ >(\P Hx) %
+ | @(IH … Hx) ] ] ]
qed.
lemma no_marks_in_table: ∀n.∀l.table_TM n l → no_marks l.
#n #l #t elim t
- [normalize #n #x #H destruct
- |#m #t1 #t2 * #qin * #qout * #mv * * * * * *
- #Hmarks #_ #_ #_ #_ #_ #_ #Ht2 #Hind
- #x #Hx cases (memb_append … Hx) -Hx #Hx
- [@Hmarks //
- |cases (orb_true_l … Hx) -Hx #Hx
- [>(\P Hx) //
- |@Hind //
- ]
- ]
- ]
+ [normalize #c #H destruct
+ |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_marks_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
+ cases (memb_append … Hx) -Hx #Hx
+ [ @(Ht1 … Hx)
+ | cases (orb_true_l … Hx) -Hx #Hx
+ [ >(\P Hx) %
+ | @(IH … Hx) ] ] ]
qed.
axiom last_of_table: ∀n,l,b.¬ table_TM n (l@[〈bar,b〉]).
(l1@〈grid,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
(* facciamo match *)
(∃l3,newc,mv,l4.
- 〈c1,false〉::l2 = l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4 ∧
+ 〈c1,false〉::l2 = l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4 ∧
t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
- (l3@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv@l4@〈grid,false〉::rs))
+ (l3@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l4@〈grid,false〉::rs))
∨
(* non facciamo match su nessuna tupla;
non specifichiamo condizioni sul nastro di output, perché
non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
(current ? t2 = Some ? 〈grid,true〉 ∧
∀l3,newc,mv,l4.
- 〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4).
+ 〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4).
projects / helm.git / blobdiff