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author | jadep <jade.philipoom@gmail.com> | 2016-09-17 14:44:20 -0400 |
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committer | jadep <jade.philipoom@gmail.com> | 2016-09-17 14:50:49 -0400 |
commit | 1bacc083da890d7289f1ee54a41996db7a787a92 (patch) | |
tree | 472a1d5159578a923ee4543cf13b73a241541661 /src/Util | |
parent | 3959bc9986391882b3b73acd25e0fba04cdebbd9 (diff) |
Move side lemmas to appropriate files
Diffstat (limited to 'src/Util')
-rw-r--r-- | src/Util/ListUtil.v | 37 | ||||
-rw-r--r-- | src/Util/Tuple.v | 23 |
2 files changed, 60 insertions, 0 deletions
diff --git a/src/Util/ListUtil.v b/src/Util/ListUtil.v index 11c92c72c..4f544d2c8 100644 --- a/src/Util/ListUtil.v +++ b/src/Util/ListUtil.v @@ -1541,6 +1541,43 @@ Proof. intuition (congruence || eauto). Qed. +Lemma Forall2_forall_iff : forall {A} R (xs ys : list A) d, length xs = length ys -> + (Forall2 R xs ys <-> (forall i, (i < length xs)%nat -> R (nth_default d xs i) (nth_default d ys i))). +Proof. + split; intros. + + revert xs ys H H0 H1. + induction i; intros; destruct H0; distr_length; autorewrite with push_nth_default; auto. + eapply IHi; auto. omega. + + revert xs ys H H0; induction xs; intros; destruct ys; distr_length; econstructor. + - specialize (H0 0%nat). + autorewrite with push_nth_default in *; auto. + apply H0; omega. + - apply IHxs; try omega. + intros. + specialize (H0 (S i)). + autorewrite with push_nth_default in *; auto. + apply H0; omega. +Qed. + +Lemma nth_default_firstn : forall {A} (d : A) l i n, + nth_default d (firstn n l) i = if le_dec n (length l) + then if lt_dec i n then nth_default d l i else d + else nth_default d l i. +Proof. + induction n; intros; break_if; autorewrite with push_nth_default; auto; try omega. + + rewrite (firstn_succ d) by omega. + autorewrite with push_nth_default; repeat (break_if; distr_length); + rewrite Min.min_l in * by omega; try omega. + - apply IHn; omega. + - replace i with n in * by omega. + rewrite Nat.sub_diag. + autorewrite with push_nth_default; auto. + - rewrite nth_default_out_of_bounds; distr_length; auto. + + rewrite firstn_all2 by omega. + auto. +Qed. +Hint Rewrite @nth_default_firstn : push_nth_default. + Lemma nth_error_repeat {T} x n i v : nth_error (@repeat T x n) i = Some v -> v = x. Proof. revert n x v; induction i as [|i IHi]; destruct n; simpl in *; eauto; congruence. diff --git a/src/Util/Tuple.v b/src/Util/Tuple.v index 76064e7b4..dbd585cca 100644 --- a/src/Util/Tuple.v +++ b/src/Util/Tuple.v @@ -238,5 +238,28 @@ Definition apply {R T} (n:nat) : function R T n -> tuple T n -> R := | S n' => fun f x => apply' n' f x end. +Require Import Coq.Lists.SetoidList. + +Lemma fieldwise_to_list_iff : forall {T n} R (s t : tuple T n), + (fieldwise R s t <-> Forall2 R (to_list _ s) (to_list _ t)). +Proof. + induction n; split; intros. + + constructor. + + cbv [fieldwise]. auto. + + destruct n; cbv [tuple to_list fieldwise] in *. + - cbv [to_list']; auto. + - simpl in *. destruct s,t; cbv [fst snd] in *. + constructor; intuition auto. + apply IHn; auto. + + destruct n; cbv [tuple to_list fieldwise] in *. + - cbv [fieldwise']; auto. + cbv [to_list'] in *; inversion H; auto. + - simpl in *. destruct s,t; cbv [fst snd] in *. + inversion H; subst. + split; try assumption. + apply IHn; auto. +Qed. + + Require Import Crypto.Util.ListUtil. (* To initialize [distr_length] database *) Hint Rewrite length_to_list' @length_to_list : distr_length. |