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author | Sylvain Boulmé <sylvain.boulme@univ-grenoble-alpes.fr> | 2020-06-19 11:27:27 +0200 |
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committer | Sylvain Boulmé <sylvain.boulme@univ-grenoble-alpes.fr> | 2020-06-19 11:27:27 +0200 |
commit | 98fd5a5cb11db3d6560d2c9dd41dbdecefc04e0a (patch) | |
tree | 5c1401bf5fc8ea0a6e1e7ac73414141a1e7701c9 /kvx/lib/IterList.v | |
parent | 558667608e8d5900447b9fc3c8f498948b296972 (diff) | |
parent | 01674e9c79d0ddf77f3a97f80267d3fd01d19774 (diff) | |
download | compcert-kvx-98fd5a5cb11db3d6560d2c9dd41dbdecefc04e0a.tar.gz compcert-kvx-98fd5a5cb11db3d6560d2c9dd41dbdecefc04e0a.zip |
Merge branch 'kvx-work' into PseudoAsmblock
Diffstat (limited to 'kvx/lib/IterList.v')
-rw-r--r-- | kvx/lib/IterList.v | 85 |
1 files changed, 85 insertions, 0 deletions
diff --git a/kvx/lib/IterList.v b/kvx/lib/IterList.v new file mode 100644 index 00000000..b9eb5922 --- /dev/null +++ b/kvx/lib/IterList.v @@ -0,0 +1,85 @@ +Require Import Coqlib. + +(** TODO: are these def and lemma already defined in the standard library ? + +In this case, it should be better to reuse those of the standard library ! + +*) + +Fixpoint iter {A} (n:nat) (f: A -> A) (x: A) {struct n}: A := + match n with + | O => x + | S n0 => iter n0 f (f x) + end. + +Lemma iter_S A (n:nat) (f: A -> A): forall x, iter (S n) f x = f (iter n f x). +Proof. + induction n; simpl; auto. + intros; erewrite <- IHn; simpl; auto. +Qed. + +Lemma iter_plus A (n m:nat) (f: A -> A): forall x, iter (n+m) f x = iter m f (iter n f x). +Proof. + induction n; simpl; auto. +Qed. + +Definition iter_tail {A} (n:nat) (l: list A) := iter n (@tl A) l. + +Lemma iter_tail_S {A} (n:nat) (l: list A): iter_tail (S n) l = tl (iter_tail n l). +Proof. + apply iter_S. +Qed. + +Lemma iter_tail_plus A (n m:nat) (l: list A): iter_tail (n+m) l = iter_tail m (iter_tail n l). +Proof. + apply iter_plus. +Qed. + +Lemma iter_tail_length A l1: forall (l2: list A), iter_tail (length l1) (l1 ++ l2) = l2. +Proof. + induction l1; auto. +Qed. + +Lemma iter_tail_nil A n: @iter_tail A n nil = nil. +Proof. + unfold iter_tail; induction n; simpl; auto. +Qed. + +Lemma iter_tail_reach_nil A (l: list A): iter_tail (length l) l = nil. +Proof. + rewrite (app_nil_end l) at 2. + rewrite iter_tail_length. + auto. +Qed. + +Lemma length_iter_tail {A} (n:nat): forall (l: list A), (n <= List.length l)%nat -> (List.length l = n + List.length (iter_tail n l))%nat. +Proof. + unfold iter_tail; induction n; auto. + intros l; destruct l. { simpl; omega. } + intros; simpl. erewrite IHn; eauto. + simpl in *; omega. +Qed. + +Lemma iter_tail_S_ex {A} (n:nat): forall (l: list A), (n < length l)%nat -> exists x, iter_tail n l = x::(iter_tail (S n) l). +Proof. + unfold iter_tail; induction n; simpl. + - intros l; destruct l; simpl; omega || eauto. + - intros l H; destruct (IHn (tl l)) as (x & H1). + + destruct l; simpl in *; try omega. + + rewrite H1; eauto. +Qed. + +Lemma iter_tail_inject1 {A} (n1 n2:nat) (l: list A): (n1 <= List.length l)%nat -> (n2 <= List.length l)%nat -> iter_tail n1 l = iter_tail n2 l -> n1=n2. +Proof. + intros H1 H2 EQ; exploit (length_iter_tail n1 l); eauto. + rewrite EQ. + rewrite (length_iter_tail n2 l); eauto. + omega. +Qed. + +Lemma iter_tail_nil_inject {A} (n:nat) (l: list A): iter_tail n l = nil -> (List.length l <= n)%nat. +Proof. + destruct (le_lt_dec n (List.length l)); try omega. + intros; exploit (iter_tail_inject1 n (length l) l); try omega. + rewrite iter_tail_reach_nil. auto. +Qed.
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