seqZ: rename, by, cbn -> simpl

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.