include "datatypes/constructors.ma".
include "list/list.ma".
-let rec mem (A:Set) (eq: A → A → bool) x (l: list A) on l ≝
+let rec mem (A:Type) (eq: A → A → bool) x (l: list A) on l ≝
match l with
[ nil ⇒ false
| (cons a l') ⇒
]
].
-let rec ordered (A:Set) (le: A → A → bool) (l: list A) on l ≝
+let rec ordered (A:Type) (le: A → A → bool) (l: list A) on l ≝
match l with
[ nil ⇒ true
| (cons x l') ⇒
]
].
-let rec insert (A:Set) (le: A → A → bool) x (l: list A) on l ≝
+let rec insert (A:Type) (le: A → A → bool) x (l: list A) on l ≝
match l with
[ nil ⇒ [x]
| (cons he l') ⇒
].
lemma insert_ind :
- ∀A:Set. ∀le: A → A → bool. ∀x.
+ ∀A:Type. ∀le: A → A → bool. ∀x.
∀P:(list A → list A → Prop).
∀H:(∀l: list A. l=[] → P [] [x]).
∀H2:
qed.
-let rec insertionsort (A:Set) (le: A → A → bool) (l: list A) on l ≝
+let rec insertionsort (A:Type) (le: A → A → bool) (l: list A) on l ≝
match l with
[ nil ⇒ []
| (cons he l') ⇒
].
lemma ordered_injective:
- ∀A:Set. ∀le:A → A → bool.
+ ∀A:Type. ∀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
clear H1;
elim l1;
[ simplify; reflexivity;
- | cut ((le s s1 \land ordered A le (s1::l2)) = true);
+ | cut ((le t t1 \land ordered A le (t1::l2)) = true);
[ generalize in match Hcut;
apply andb_elim;
- elim (le s s1);
+ elim (le t t1);
[ simplify;
- fold simplify (ordered ? le (s1::l2));
+ fold simplify (ordered ? le (t1::l2));
intros; assumption;
| simplify;
intros (Habsurd);
qed.
lemma insert_sorted:
- \forall A:Set. \forall le:A\to A\to bool.
+ \forall A:Type. \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.
elim l'; simplify;
[ rewrite > H5;
reflexivity
- | elim (le x s); simplify;
+ | elim (le x t); simplify;
[ rewrite > H5;
reflexivity
| simplify in H4;
qed.
theorem insertionsort_sorted:
- ∀A:Set.
+ ∀A:Type.
∀le:A → A → bool.∀eq:A → A → bool.
(∀a,b:A. le a b = false → le b a = true) \to
∀l:list A.
elim l;
[ simplify;
reflexivity;
- | apply (insert_sorted ? ? le_tot (insertionsort ? le l1) s);
+ | apply (insert_sorted ? ? le_tot (insertionsort ? le l1) t);
assumption;
]
qed.
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