From 3887c4f745607a8e216901d252e44653cf88a900 Mon Sep 17 00:00:00 2001 From: Simon Spies Date: Wed, 26 Jun 2019 14:56:46 +0200 Subject: [PATCH] seqZ: rename, by, cbn -> simpl --- theories/list.v | 54 ++++++++++++++++++++++++------------------------- 1 file changed, 26 insertions(+), 28 deletions(-) diff --git a/theories/list.v b/theories/list.v index ba66d50..35586aa 100644 --- a/theories/list.v +++ b/theories/list.v @@ -396,11 +396,9 @@ Definition positives_unflatten (p : positive) : option (list positive) := (** [seqZ m n] generates the sequence [m], [m + 1], ..., [m + n - 1] over integers, provided [n >= 0]. If n < 0, then the range is empty. **) -Definition seqZ (m len: Z) : list Z := (λ x: nat, Z.add x m) <$> (seq 0 (Z.to_nat len)). +Definition seqZ (m len: Z) : list Z := (λ i: nat, Z.add i m) <$> (seq 0 (Z.to_nat len)). Arguments seqZ : simpl never. - - (** * Basic tactics on lists *) (** The tactic [discriminate_list] discharges a goal if it submseteq a list equality involving [(::)] and [(++)] of two lists that have a different @@ -1470,63 +1468,63 @@ Qed. (** ** Properties of the [seqZ] function *) Section seqZ. Implicit Types (m n : Z) (i j: nat). - Open Scope Z. + Local Open Scope Z. - Lemma empty_seqZ m n: n ≤ 0 → seqZ m n = []. - Proof. intros H; destruct n; cbn; eauto; lia. Qed. + Lemma seqZ_nil m n: n ≤ 0 → seqZ m n = []. + Proof. intros H; destruct n; simpl; eauto; lia. Qed. - Lemma step_seqZ m n: n > 0 → seqZ m n = m :: seqZ (m + 1) (n - 1). + Lemma seqZ_cons m n: n > 0 → seqZ m n = m :: seqZ (m + 1) (n - 1). Proof. intros H. unfold seqZ. replace (Z.to_nat n) with (S (Z.to_nat (n - 1))) by (rewrite <-Z2Nat.inj_succ; [f_equal|]; lia). - cbn; f_equal; try lia. + simpl; f_equal; try lia. erewrite <-fmap_seq, map_map, map_ext; eauto. intros; lia. Qed. - Lemma length_seqZ m n: length (seqZ m n) = Z.to_nat n. + Lemma seqZ_length m n: length (seqZ m n) = Z.to_nat n. Proof. - unfold seqZ; now rewrite map_length, seq_length. + unfold seqZ; by rewrite map_length, seq_length. Qed. - Lemma fmap_seqZ m m' n: (Z.add m) <$> seqZ m' n = seqZ (m + m') n. + Lemma seqZ_fmap m m' n: Z.add m <$> seqZ m' n = seqZ (m + m') n. Proof. assert (0 ≤ n ∨ n < 0) as [H|H] by lia. - revert m'. pattern n. eapply natlike_ind; auto; clear n H. - intros n H1 IH j. rewrite step_seqZ; try lia. - symmetry. rewrite step_seqZ; try lia. + intros n H1 IH j. rewrite seqZ_cons; try lia. + symmetry. rewrite seqZ_cons; try lia. replace (Z.succ n - 1) with n by lia. - cbn; rewrite IH. + simpl; rewrite IH. f_equal; try lia; f_equal; lia. - - rewrite !empty_seqZ; auto; lia. + - rewrite !seqZ_nil; auto; lia. Qed. - Lemma lookup_seqZ m n i: i < n → seqZ m n !! i = Some (m + i). + Lemma seqZ_lookup m n i: i < n → seqZ m n !! i = Some (m + i). Proof. assert (0 ≤ n ∨ n < 0) as [H|H] by lia. - revert m i. pattern n. eapply natlike_ind; auto; clear n H. intros; lia. - intros n H1 IH. intros j [|i] ?; rewrite step_seqZ. 2, 4: lia. - cbn; f_equal; lia. replace (Z.succ n - 1) with n by lia. - cbn; rewrite IH; f_equal; lia. - - rewrite !empty_seqZ; auto; lia. + intros n H1 IH. intros j [|i] ?; rewrite seqZ_cons. 2, 4: lia. + simpl; f_equal; lia. replace (Z.succ n - 1) with n by lia. + simpl; rewrite IH; f_equal; lia. + - rewrite !seqZ_nil; auto; lia. Qed. - Lemma lookup_seqZ_ge m n i: n ≤ i → seqZ m n !! i = None. + Lemma seqZ_lookup_ge m n i: n ≤ i → seqZ m n !! i = None. Proof. assert (0 ≤ n ∨ n < 0) as [H|H] by lia. - revert m i. pattern n. eapply natlike_ind; auto; clear n H. - intros n H1 IH. intros j [|i] ?; rewrite step_seqZ. 2, 4: lia. - cbn; f_equal; lia. replace (Z.succ n - 1) with n by lia. - cbn; rewrite IH; f_equal; lia. - - rewrite !empty_seqZ; auto; lia. + intros n H1 IH. intros j [|i] ?; rewrite seqZ_cons. 2, 4: lia. + simpl; f_equal; lia. replace (Z.succ n - 1) with n by lia. + simpl; rewrite IH; f_equal; lia. + - rewrite !seqZ_nil; auto; lia. Qed. - Lemma lookup_seqZ_inv m n i m' : seqZ m n !! i = Some m' → m' = m + i ∧ i < n. + Lemma seqZ_lookup_inv m n i m' : seqZ m n !! i = Some m' → m' = m + i ∧ i < n. Proof. - destruct (Z_le_gt_dec n i); [by rewrite lookup_seqZ_ge|]. - rewrite lookup_seqZ by lia. intuition; [congruence|lia]. + destruct (Z_le_gt_dec n i); [by rewrite seqZ_lookup_ge|]. + rewrite seqZ_lookup by lia. intuition; [congruence|lia]. Qed. End seqZ. -- 2.26.2