--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/list/sort/".
+
+include "datatypes/bool.ma".
+include "datatypes/constructors.ma".
+include "list/list.ma".
+
+let rec mem (A:Set) (eq: A → A → bool) x (l: list A) on l ≝
+ match l with
+ [ nil ⇒ false
+ | (cons a l') ⇒
+ match eq x a with
+ [ true ⇒ true
+ | false ⇒ mem A eq x l'
+ ]
+ ].
+
+let rec ordered (A:Set) (le: A → A → bool) (l: list A) on l ≝
+ match l with
+ [ nil ⇒ true
+ | (cons x l') ⇒
+ match l' with
+ [ nil ⇒ true
+ | (cons y l'') ⇒
+ le x y \land ordered A le l'
+ ]
+ ].
+
+let rec insert (A:Set) (le: A → A → bool) x (l: list A) on l ≝
+ match l with
+ [ nil ⇒ [x]
+ | (cons he l') ⇒
+ match le x he with
+ [ true ⇒ x::l
+ | false ⇒ he::(insert A le x l')
+ ]
+ ].
+
+lemma insert_ind :
+ ∀A:Set. ∀le: A → A → bool. ∀x.
+ ∀P:(list A → list A → Prop).
+ ∀H:(∀l: list A. l=[] → P [] [x]).
+ ∀H2:
+ (∀l: list A. ∀he. ∀l'. P l' (insert ? le x l') →
+ le x he = false → l=he::l' → P (he::l') (he::(insert ? le x l'))).
+ ∀H3:
+ (∀l: list A. ∀he. ∀l'. P l' (insert ? le x l') →
+ le x he = true → l=he::l' → P (he::l') (x::he::l')).
+ ∀l:list A. P l (insert ? le x l).
+ intros.
+ apply (
+ let rec insert_ind (l: list A) \def
+ match l in list
+ return
+ λli.
+ l = li → P li (insert ? le x li)
+ with
+ [ nil ⇒ H l
+ | (cons he l') ⇒
+ match le x he
+ return
+ λb. le x he = b → l = he::l' →
+ P (he::l')
+ (match b with
+ [ true ⇒ x::he::l'
+ | false ⇒ he::(insert ? le x l') ])
+ with
+ [ true ⇒ H2 l he l' (insert_ind l')
+ | false ⇒ H1 l he l' (insert_ind l')
+ ]
+ (refl_eq ? (le x he))
+ ] (refl_eq ? l) in insert_ind l).
+qed.
+
+
+let rec insertionsort (A:Set) (le: A → A → bool) (l: list A) on l ≝
+ match l with
+ [ nil ⇒ []
+ | (cons he l') ⇒
+ let l'' ≝ insertionsort A le l' in
+ insert A le he l''
+ ].
+
+lemma ordered_injective:
+ ∀A:Set. ∀le:A → A → bool.
+ ∀l:list A. ordered A le l = true → ordered A le (tail A l) = true.
+ intros 3 (A le l).
+ elim l
+ [ simplify; reflexivity;
+ | simplify;
+ generalize in match H1;
+ clear H1;
+ elim l1;
+ [ simplify; reflexivity;
+ | cut ((le s s1 \land ordered A le (s1::l2)) = true);
+ [ generalize in match Hcut;
+ apply andb_elim;
+ elim (le s s1);
+ [ simplify;
+ fold simplify (ordered ? le (s1::l2));
+ intros; assumption;
+ | simplify;
+ intros (Habsurd);
+ apply False_ind;
+ apply (not_eq_true_false);
+ symmetry;
+ assumption
+ ]
+ | exact H2;
+ ]
+ ]
+ ].
+qed.
+
+lemma insert_sorted:
+ \forall A:Set. \forall le:A\to A\to bool.
+ (\forall a,b:A. le a b = false \to le b a = true) \to
+ \forall l:list A. \forall x:A.
+ ordered A le l = true \to ordered A le (insert A le x l) = true.
+ intros 5 (A le H l x).
+ apply (insert_ind ? ? ? (λl,il. ordered ? le l = true → ordered ? le il = true));
+ clear l; intros; simplify; intros;
+ [2: rewrite > H1;
+ [ generalize in match (H ? ? H2); clear H2; intro;
+ generalize in match H4; clear H4;
+ elim l'; simplify;
+ [ rewrite > H5;
+ reflexivity
+ | elim (le x s); simplify;
+ [ rewrite > H5;
+ reflexivity
+ | simplify in H4;
+ rewrite > (andb_true_true ? ? H4);
+ reflexivity
+ ]
+ ]
+ | apply (ordered_injective ? ? ? H4)
+ ]
+ | reflexivity
+ | rewrite > H2;
+ rewrite > H4;
+ reflexivity
+ ].
+qed.
+
+theorem insertionsort_sorted:
+ ∀A:Set.
+ ∀le:A → A → bool.∀eq:A → A → bool.
+ (∀a,b:A. le a b = false → le b a = true) \to
+ ∀l:list A.
+ ordered A le (insertionsort A le l) = true.
+ intros 5 (A le eq le_tot l).
+ elim l;
+ [ simplify;
+ reflexivity;
+ | apply (insert_sorted ? ? le_tot (insertionsort ? le l1) s);
+ assumption;
+ ]
+qed.
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projects / helm.git / blobdiff