Commit 78315828 authored by Robbert's avatar Robbert

Merge branch 'master' into 'master'

SeqZ Function

See merge request !72
parents 8dfc9983 e3c9027e
Pipeline #18053 passed with stage
in 8 minutes and 35 seconds
......@@ -394,6 +394,13 @@ used by [positives_flatten]. *)
Definition positives_unflatten (p : positive) : option (list positive) :=
positives_unflatten_go p [] 1.
(** [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 i: nat, Z.add i m) <$> (seq 0 (Z.to_nat len)).
Arguments seqZ : simpl never.
(** * Basic tactics on lists *)
(** The tactic [discriminate_list] discharges a goal if it submseteq
a list equality involving [(::)] and [(++)] of two lists that have a different
......@@ -1460,6 +1467,7 @@ Proof.
rewrite lookup_seq by done. intuition congruence.
Qed.
(** ** Properties of the [Permutation] predicate *)
Lemma Permutation_nil l : l ≡ₚ [] l = [].
Proof. split. by intro; apply Permutation_nil. by intros ->. Qed.
......@@ -3345,6 +3353,55 @@ Section mapM.
Proof. eauto using mapM_fmap_Forall2_Some_inv, Forall2_true, mapM_length. Qed.
End mapM.
(** ** Properties of the [seqZ] function *)
Section seqZ.
Implicit Types (m n: Z) (i j: nat).
Local Open Scope Z.
Lemma seqZ_nil m n: n 0 seqZ m n = [].
Proof. by destruct n. Qed.
Lemma seqZ_cons m n: 0 < n seqZ m n = m :: seqZ (Z.succ m) (Z.pred n).
Proof.
intros H. unfold seqZ. replace n with (Z.succ (Z.pred n)) at 1 by lia.
rewrite Z2Nat.inj_succ by lia. f_equal/=.
rewrite <-fmap_seq, <-list_fmap_compose.
apply map_ext; naive_solver lia.
Qed.
Lemma seqZ_length m n: length (seqZ m n) = Z.to_nat n.
Proof. unfold seqZ; by rewrite fmap_length, seq_length. Qed.
Lemma seqZ_fmap m m' n: Z.add m <$> seqZ m' n = seqZ (m + m') n.
Proof.
revert m'. induction n as [|n ? IH|] using (Z_succ_pred_induction 0); intros m'.
- by rewrite seqZ_nil.
- rewrite (seqZ_cons m') by lia. rewrite (seqZ_cons (m + m')) by lia.
f_equal/=. rewrite Z.pred_succ, IH; simpl. f_equal; lia.
- by rewrite !seqZ_nil by lia.
Qed.
Lemma seqZ_lookup_lt m n i: i < n seqZ m n !! i = Some (m + i).
Proof.
revert m i.
induction n as [|n ? IH|] using (Z_succ_pred_induction 0); intros m i Hi; try lia.
rewrite seqZ_cons by lia. destruct i as [|i]; simpl.
- f_equal; lia.
- rewrite Z.pred_succ, IH by lia. f_equal; lia.
Qed.
Lemma seqZ_lookup_ge m n i : n i seqZ m n !! i = None.
Proof.
revert m i.
induction n as [|n ? IH|] using (Z_succ_pred_induction 0); intros m i Hi; try lia.
- by rewrite seqZ_nil.
- rewrite seqZ_cons by lia.
destruct i as [|i]; simpl; [lia|]. by rewrite Z.pred_succ, IH by lia.
- by rewrite seqZ_nil by lia.
Qed.
Lemma seqZ_lookup m n i m' : seqZ m n !! i = Some m' m' = m + i i < n.
Proof.
destruct (Z_le_gt_dec n i).
{ rewrite seqZ_lookup_ge by lia. naive_solver lia. }
rewrite seqZ_lookup_lt by lia. naive_solver lia.
Qed.
End seqZ.
(** ** Properties of the [permutations] function *)
Section permutations.
Context {A : Type}.
......
......@@ -467,6 +467,13 @@ Proof.
apply Nat.mod_bound_pos; lia. }
by rewrite <-Nat2Z.inj_mul, <-Nat2Z.inj_add, <-Nat.div_mod.
Qed.
Lemma Z_succ_pred_induction y (P : Z Prop) :
P y
( x, y x P x P (Z.succ x))
( x, x y P x P (Z.pred x))
( x, P x).
Proof. intros H0 HS HP. by apply (Z.order_induction' _ _ y). Qed.
Close Scope Z_scope.
(** * Injectivity of casts *)
......
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