(* Copyright (c) 2012-2013, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This file collects general purpose definitions and theorems on lists that are not in the Coq standard library. *) Require Export Permutation. Require Export numbers base decidable option. Arguments length {_} _. Arguments cons {_} _ _. Arguments app {_} _ _. Arguments Permutation {_} _ _. Arguments Forall_cons {_} _ _ _ _ _. Notation tail := tl. Notation take := firstn. Notation drop := skipn. Arguments take {_} !_ !_ /. Arguments drop {_} !_ !_ /. Notation "(::)" := cons (only parsing) : C_scope. Notation "( x ::)" := (cons x) (only parsing) : C_scope. Notation "(:: l )" := (λ x, cons x l) (only parsing) : C_scope. Notation "(++)" := app (only parsing) : C_scope. Notation "( l ++)" := (app l) (only parsing) : C_scope. Notation "(++ k )" := (λ l, app l k) (only parsing) : C_scope. Infix "≡ₚ" := Permutation (at level 70, no associativity) : C_scope. Notation "(≡ₚ)" := Permutation (only parsing) : C_scope. Notation "( x ≡ₚ)" := (Permutation x) (only parsing) : C_scope. Notation "(≡ₚ x )" := (λ y, y ≡ₚ x) (only parsing) : C_scope. Notation "(≢ₚ)" := (λ x y, ¬x ≡ₚ y) (only parsing) : C_scope. Notation "x ≢ₚ y":= (¬x ≡ₚ y) (at level 70, no associativity) : C_scope. Notation "( x ≢ₚ)" := (λ y, x ≢ₚ y) (only parsing) : C_scope. Notation "(≢ₚ x )" := (λ y, y ≢ₚ x) (only parsing) : C_scope. (** * Definitions *) (** The operation [l !! i] gives the [i]th element of the list [l], or [None] in case [i] is out of bounds. *) Instance list_lookup {A} : Lookup nat A (list A) := fix go (i : nat) (l : list A) {struct l} : option A := match l with | [] => None | x :: l => match i with 0 => Some x | S i => @lookup _ _ _ go i l end end. (** The operation [alter f i l] applies the function [f] to the [i]th element of [l]. In case [i] is out of bounds, the list is returned unchanged. *) Instance list_alter {A} (f : A → A) : AlterD nat A (list A) f := fix go (i : nat) (l : list A) {struct l} := match l with | [] => [] | x :: l => match i with 0 => f x :: l | S i => x :: @alter _ _ _ f go i l end end. (** The operation [<[i:=x]> l] overwrites the element at position [i] with the value [x]. In case [i] is out of bounds, the list is returned unchanged. *) Instance list_insert {A} : Insert nat A (list A) := λ i x, alter (λ _, x) i. (** The operation [delete i l] removes the [i]th element of [l] and moves all consecutive elements one position ahead. In case [i] is out of bounds, the list is returned unchanged. *) Instance list_delete {A} : Delete nat (list A) := fix go (i : nat) (l : list A) {struct l} : list A := match l with | [] => [] | x :: l => match i with 0 => l | S i => x :: @delete _ _ go i l end end. (** The function [option_list o] converts an element [Some x] into the singleton list [[x]], and [None] into the empty list [[]]. *) Definition option_list {A} : option A → list A := option_rect _ (λ x, [x]) []. (** The function [filter P l] returns the list of elements of [l] that satisfies [P]. The order remains unchanged. *) Instance list_filter {A} : Filter A (list A) := fix go P _ l := match l with | [] => [] | x :: l => if decide (P x) then x :: @filter _ _ (@go) _ _ l else @filter _ _ (@go) _ _ l end. Fixpoint filter_Some {A} (l : list (option A)) : list A := match l with | [] => [] | Some x :: l => x :: filter_Some l | None :: l => filter_Some l end. (** The function [list_find P l] returns the first index [i] whose element satisfies the predicate [P]. *) Definition list_find {A} P `{∀ x, Decision (P x)} : list A → option nat := fix go l := match l with | [] => None | x :: l => if decide (P x) then Some 0 else S <\$> go l end. (** The function [replicate n x] generates a list with length [n] of elements with value [x]. *) Fixpoint replicate {A} (n : nat) (x : A) : list A := match n with 0 => [] | S n => x :: replicate n x end. (** The function [reverse l] returns the elements of [l] in reverse order. *) Definition reverse {A} (l : list A) : list A := rev_append l []. Fixpoint last' {A} (x : A) (l : list A) : A := match l with [] => x | x :: l => last' x l end. Definition last {A} (l : list A) : option A := match l with [] => None | x :: l => Some (last' x l) end. (** The function [resize n y l] takes the first [n] elements of [l] in case [length l ≤ n], and otherwise appends elements with value [x] to [l] to obtain a list of length [n]. *) Fixpoint resize {A} (n : nat) (y : A) (l : list A) : list A := match l with | [] => replicate n y | x :: l => match n with 0 => [] | S n => x :: resize n y l end end. Arguments resize {_} !_ _ !_. (* The function [reshape k l] transforms [l] into a list of lists whose sizes are specified by [k]. In case [l] is too short, the resulting list will end with a a certain number of empty lists. In case [l] is too long, it will be truncated. *) Fixpoint reshape {A} (szs : list nat) (l : list A) : list (list A) := match szs with | [] => [] | sz :: szs => take sz l :: reshape szs (drop sz l) end. Definition sublist_lookup {A} (i n : nat) (l : list A) : option (list A) := guard (i + n ≤ length l); Some \$ take n \$ drop i l. Definition sublist_insert {A} (i : nat) (k l : list A) : list A := take i l ++ take (length l - i) k ++ drop (i + length k) l. (** Functions to fold over a list. We redefine [foldl] with the arguments in the same order as in Haskell. *) Notation foldr := fold_right. Definition foldl {A B} (f : A → B → A) : A → list B → A := fix go a l := match l with | [] => a | x :: l => go (f a x) l end. (** The monadic operations. *) Instance list_ret: MRet list := λ A x, x :: @nil A. Instance list_fmap {A B} (f : A → B) : FMapD list f := fix go (l : list A) := match l with [] => [] | x :: l => f x :: @fmap _ _ _ f go l end. Instance list_bind {A B} (f : A → list B) : MBindD list f := fix go (l : list A) := match l with [] => [] | x :: l => f x ++ @mbind _ _ _ f go l end. Instance list_join: MJoin list := fix go A (ls : list (list A)) : list A := match ls with [] => [] | l :: ls => l ++ @mjoin _ go _ ls end. Definition mapM `{!MBind M} `{!MRet M} {A B} (f : A → M B) : list A → M (list B) := fix go l := match l with | [] => mret [] | x :: l => y ← f x; k ← go l; mret (y :: k) end. (** We define stronger variants of map and fold that allow the mapped function to use the index of the elements. *) Definition imap_go {A B} (f : nat → A → B) : nat → list A → list B := fix go (n : nat) (l : list A) := match l with [] => [] | x :: l => f n x :: go (S n) l end. Definition imap {A B} (f : nat → A → B) : list A → list B := imap_go f 0. Definition ifoldr {A B} (f : nat → B → A → A) (a : nat → A) : nat → list B → A := fix go n l := match l with [] => a n | b :: l => f n b (go (S n) l) end. Definition zipped_map {A B} (f : list A → list A → A → B) : list A → list A → list B := fix go l k := match k with [] => [] | x :: k => f l k x :: go (x :: l) k end. Inductive zipped_Forall {A} (P : list A → list A → A → Prop) : list A → list A → Prop := | zipped_Forall_nil l : zipped_Forall P l [] | zipped_Forall_cons l k x : P l k x → zipped_Forall P (x :: l) k → zipped_Forall P l (x :: k). Arguments zipped_Forall_nil {_ _} _. Arguments zipped_Forall_cons {_ _} _ _ _ _ _. (** Zipping lists. *) Definition zip_with {A B C} (f : A → B → C) : list A → list B → list C := fix go l1 l2 := match l1, l2 with x1 :: l1, x2 :: l2 => f x1 x2 :: go l1 l2 | _ , _ => [] end. Notation zip := (zip_with pair). (** The function [permutations l] yields all permutations of [l]. *) Fixpoint interleave {A} (x : A) (l : list A) : list (list A) := match l with | [] => [ [x] ] | y :: l => (x :: y :: l) :: ((y ::) <\$> interleave x l) end. Fixpoint permutations {A} (l : list A) : list (list A) := match l with | [] => [ [] ] | x :: l => permutations l ≫= interleave x end. (** The predicate [suffix_of] holds if the first list is a suffix of the second. The predicate [prefix_of] holds if the first list is a prefix of the second. *) Definition suffix_of {A} : relation (list A) := λ l1 l2, ∃ k, l2 = k ++ l1. Definition prefix_of {A} : relation (list A) := λ l1 l2, ∃ k, l2 = l1 ++ k. Infix "`suffix_of`" := suffix_of (at level 70) : C_scope. Infix "`prefix_of`" := prefix_of (at level 70) : C_scope. Section prefix_suffix_ops. Context `{∀ x y : A, Decision (x = y)}. Definition max_prefix_of : list A → list A → list A * list A * list A := fix go l1 l2 := match l1, l2 with | [], l2 => ([], l2, []) | l1, [] => (l1, [], []) | x1 :: l1, x2 :: l2 => if decide_rel (=) x1 x2 then snd_map (x1 ::) (go l1 l2) else (x1 :: l1, x2 :: l2, []) end. Definition max_suffix_of (l1 l2 : list A) : list A * list A * list A := match max_prefix_of (reverse l1) (reverse l2) with | (k1, k2, k3) => (reverse k1, reverse k2, reverse k3) end. Definition strip_prefix (l1 l2 : list A) := snd \$ fst \$ max_prefix_of l1 l2. Definition strip_suffix (l1 l2 : list A) := snd \$ fst \$ max_suffix_of l1 l2. End prefix_suffix_ops. (** A list [l1] is a sublist of [l2] if [l2] is obtained by removing elements from [l1] without changing the order. *) Inductive sublist {A} : relation (list A) := | sublist_nil : sublist [] [] | sublist_skip x l1 l2 : sublist l1 l2 → sublist (x :: l1) (x :: l2) | sublist_cons x l1 l2 : sublist l1 l2 → sublist l1 (x :: l2). Infix "`sublist`" := sublist (at level 70) : C_scope. (** A list [l2] contains a list [l1] if [l2] is obtained by removing elements from [l1] without changing the order. *) Inductive contains {A} : relation (list A) := | contains_nil : contains [] [] | contains_skip x l1 l2 : contains l1 l2 → contains (x :: l1) (x :: l2) | contains_swap x y l : contains (y :: x :: l) (x :: y :: l) | contains_cons x l1 l2 : contains l1 l2 → contains l1 (x :: l2) | contains_trans l1 l2 l3 : contains l1 l2 → contains l2 l3 → contains l1 l3. Infix "`contains`" := contains (at level 70) : C_scope. Section contains_dec_help. Context {A} {dec : ∀ x y : A, Decision (x = y)}. Fixpoint list_remove (x : A) (l : list A) : option (list A) := match l with | [] => None | y :: l => if decide (x = y) then Some l else (y ::) <\$> list_remove x l end. Fixpoint list_remove_list (k : list A) (l : list A) : option (list A) := match k with | [] => Some l | x :: k => list_remove x l ≫= list_remove_list k end. End contains_dec_help. (** The [same_length] view allows convenient induction over two lists with the same length. *) Inductive same_length {A B} : list A → list B → Prop := | same_length_nil : same_length [] [] | same_length_cons x1 x2 l1 l2 : same_length l1 l2 → same_length (x1 :: l1) (x2 :: l2). Infix "`same_length`" := same_length (at level 70) : C_scope. (** Set operations on lists *) Section list_set. Context {A} {dec : ∀ x y : A, Decision (x = y)}. Global Instance elem_of_list_dec {dec : ∀ x y : A, Decision (x = y)} (x : A) : ∀ l, Decision (x ∈ l). Proof. refine ( fix go l := match l return Decision (x ∈ l) with | [] => right _ | y :: l => cast_if_or (decide (x = y)) (go l) end); clear go dec; subst; try (by constructor); abstract by inversion 1. Defined. Fixpoint remove_dups (l : list A) : list A := match l with | [] => [] | x :: l => if decide_rel (∈) x l then remove_dups l else x :: remove_dups l end. Fixpoint list_difference (l k : list A) : list A := match l with | [] => [] | x :: l => if decide_rel (∈) x k then list_difference l k else x :: list_difference l k end. Fixpoint list_intersection (l k : list A) : list A := match l with | [] => [] | x :: l => if decide_rel (∈) x k then x :: list_intersection l k else list_intersection l k end. Definition list_intersection_with (f : A → A → option A) : list A → list A → list A := fix go l k := match l with | [] => [] | x :: l => foldr (λ y, match f x y with None => id | Some z => (z ::) end) (go l k) k end. End list_set. (** * Basic tactics on lists *) (** The tactic [discriminate_list_equality] discharges a goal if it contains a list equality involving [(::)] and [(++)] of two lists that have a different length as one of its hypotheses. *) Tactic Notation "discriminate_list_equality" hyp(H) := apply (f_equal length) in H; repeat (simpl in H || rewrite app_length in H); exfalso; lia. Tactic Notation "discriminate_list_equality" := match goal with | H : @eq (list _) _ _ |- _ => discriminate_list_equality H end. (** The tactic [simplify_list_equality] simplifies hypotheses involving equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies lookups in singleton lists. *) Ltac simplify_list_equality := repeat match goal with | _ => progress simplify_equality | H : _ ++ _ = _ ++ _ |- _ => first [ apply app_inj_tail in H; destruct H | apply app_inv_head in H | apply app_inv_tail in H ] | H : [?x] !! ?i = Some ?y |- _ => destruct i; [change (Some x = Some y) in H | discriminate] end; try discriminate_list_equality. Ltac simplify_list_equality' := repeat (progress simpl in * || simplify_list_equality). (** * General theorems *) Section general_properties. Context {A : Type}. Implicit Types x y z : A. Implicit Types l k : list A. Global Instance: Injective2 (=) (=) (=) (@cons A). Proof. by injection 1. Qed. Global Instance: ∀ k, Injective (=) (=) (k ++). Proof. intros ???. apply app_inv_head. Qed. Global Instance: ∀ k, Injective (=) (=) (++ k). Proof. intros ???. apply app_inv_tail. Qed. Global Instance: Associative (=) (@app A). Proof. intros ???. apply app_assoc. Qed. Global Instance: LeftId (=) [] (@app A). Proof. done. Qed. Global Instance: RightId (=) [] (@app A). Proof. intro. apply app_nil_r. Qed. Lemma app_nil l1 l2 : l1 ++ l2 = [] ↔ l1 = [] ∧ l2 = []. Proof. split. apply app_eq_nil. by intros [??]; subst. Qed. Lemma app_singleton l1 l2 x : l1 ++ l2 = [x] ↔ l1 = [] ∧ l2 = [x] ∨ l1 = [x] ∧ l2 = []. Proof. split. apply app_eq_unit. by intros [[??]|[??]]; subst. Qed. Lemma cons_middle x l1 l2 : l1 ++ x :: l2 = l1 ++ [x] ++ l2. Proof. done. Qed. Lemma app_inj l1 k1 l2 k2 : length l1 = length k1 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2. Proof. revert k1. induction l1; intros [|??]; naive_solver. Qed. Lemma list_eq l1 l2 : (∀ i, l1 !! i = l2 !! i) → l1 = l2. Proof. revert l2. induction l1; intros [|??] H. * done. * discriminate (H 0). * discriminate (H 0). * f_equal; [by injection (H 0) |]. apply IHl1. intro. apply (H (S _)). Qed. Lemma list_eq_nil l : (∀ i, l !! i = None) → l = nil. Proof. intros. by apply list_eq. Qed. Global Instance list_eq_dec {dec : ∀ x y, Decision (x = y)} : ∀ l k, Decision (l = k) := list_eq_dec dec. Definition list_singleton_dec l : { x | l = [x] } + { length l ≠ 1 }. Proof. by refine match l with [x] => inleft (x↾_) | _ => inright _ end. Defined. Lemma nil_or_length_pos l : l = [] ∨ length l ≠ 0. Proof. destruct l; simpl; auto with lia. Qed. Lemma nil_length l : length l = 0 → l = []. Proof. by destruct l. Qed. Lemma lookup_nil i : @nil A !! i = None. Proof. by destruct i. Qed. Lemma lookup_tail l i : tail l !! i = l !! S i. Proof. by destruct l. Qed. Lemma lookup_lt_Some l i x : l !! i = Some x → i < length l. Proof. revert i. induction l; intros [|?] ?; simplify_equality'; auto with arith. Qed. Lemma lookup_lt_is_Some_1 l i : is_Some (l !! i) → i < length l. Proof. intros [??]; eauto using lookup_lt_Some. Qed. Lemma lookup_lt_is_Some_2 l i : i < length l → is_Some (l !! i). Proof. revert i. induction l; intros [|?] ?; simplify_equality'; eauto with lia. Qed. Lemma lookup_lt_is_Some l i : is_Some (l !! i) ↔ i < length l. Proof. split; auto using lookup_lt_is_Some_1, lookup_lt_is_Some_2. Qed. Lemma lookup_ge_None l i : l !! i = None ↔ length l ≤ i. Proof. rewrite eq_None_not_Some, lookup_lt_is_Some. lia. Qed. Lemma lookup_ge_None_1 l i : l !! i = None → length l ≤ i. Proof. by rewrite lookup_ge_None. Qed. Lemma lookup_ge_None_2 l i : length l ≤ i → l !! i = None. Proof. by rewrite lookup_ge_None. Qed. Lemma list_eq_length l1 l2 : length l2 = length l1 → (∀ i x y, l1 !! i = Some x → l2 !! i = Some y → x = y) → l1 = l2. Proof. intros Hl ?. apply list_eq. intros i. destruct (l2 !! i) as [x|] eqn:Hx. * destruct (lookup_lt_is_Some_2 l1 i) as [y ?]; subst. + rewrite <-Hl. eauto using lookup_lt_Some. + naive_solver. * by rewrite lookup_ge_None, <-Hl, <-lookup_ge_None. Qed. Lemma lookup_app_l l1 l2 i : i < length l1 → (l1 ++ l2) !! i = l1 !! i. Proof. revert i. induction l1; intros [|?]; simpl; auto with lia. Qed. Lemma lookup_app_l_Some l1 l2 i x : l1 !! i = Some x → (l1 ++ l2) !! i = Some x. Proof. intros. rewrite lookup_app_l; eauto using lookup_lt_Some. Qed. Lemma lookup_app_r l1 l2 i : (l1 ++ l2) !! (length l1 + i) = l2 !! i. Proof. revert i. induction l1; intros [|i]; simplify_equality'; auto. Qed. Lemma lookup_app_r_alt l1 l2 i j : j = length l1 → (l1 ++ l2) !! (j + i) = l2 !! i. Proof. intros ->. by apply lookup_app_r. Qed. Lemma lookup_app_r_Some l1 l2 i x : l2 !! i = Some x → (l1 ++ l2) !! (length l1 + i) = Some x. Proof. by rewrite lookup_app_r. Qed. Lemma lookup_app_minus_r l1 l2 i : length l1 ≤ i → (l1 ++ l2) !! i = l2 !! (i - length l1). Proof. intros. rewrite <-(lookup_app_r l1 l2). f_equal. lia. Qed. Lemma lookup_app_inv l1 l2 i x : (l1 ++ l2) !! i = Some x → l1 !! i = Some x ∨ l2 !! (i - length l1) = Some x. Proof. revert i. induction l1; intros [|i] ?; simplify_equality'; auto. Qed. Lemma list_lookup_middle l1 l2 x : (l1 ++ x :: l2) !! length l1 = Some x. Proof. by induction l1; simpl. Qed. Lemma alter_length f l i : length (alter f i l) = length l. Proof. revert i. induction l; intros [|?]; simpl; auto with lia. Qed. Lemma insert_length l i x : length (<[i:=x]>l) = length l. Proof. apply alter_length. Qed. Lemma list_lookup_alter f l i : alter f i l !! i = f <\$> l !! i. Proof. revert i. induction l. done. intros [|i]. done. apply (IHl i). Qed. Lemma list_lookup_alter_ne f l i j : i ≠ j → alter f i l !! j = l !! j. Proof. revert i j. induction l; [done|]. intros [|i] [|j] ?; try done. apply (IHl i). congruence. Qed. Lemma list_lookup_insert l i x : i < length l → <[i:=x]>l !! i = Some x. Proof. intros Hi. unfold insert, list_insert. rewrite list_lookup_alter. by destruct (lookup_lt_is_Some_2 l i); simplify_option_equality. Qed. Lemma list_lookup_insert_ne l i j x : i ≠ j → <[i:=x]>l !! j = l !! j. Proof. apply list_lookup_alter_ne. Qed. Lemma list_lookup_other l i x : length l ≠ 1 → l !! i = Some x → ∃ j y, j ≠ i ∧ l !! j = Some y. Proof. intros. destruct i, l as [|x0 [|x1 l]]; simplify_equality'. * by exists 1 x1. * by exists 0 x0. Qed. Lemma alter_app_l f l1 l2 i : i < length l1 → alter f i (l1 ++ l2) = alter f i l1 ++ l2. Proof. revert i. induction l1; intros [|?] ?; simpl in *; f_equal; auto with lia. Qed. Lemma alter_app_r f l1 l2 i : alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2. Proof. revert i. induction l1; intros [|?]; simpl in *; f_equal; auto. Qed. Lemma alter_app_r_alt f l1 l2 i : length l1 ≤ i → alter f i (l1 ++ l2) = l1 ++ alter f (i - length l1) l2. Proof. intros. assert (i = length l1 + (i - length l1)) as Hi by lia. rewrite Hi at 1. by apply alter_app_r. Qed. Lemma list_alter_ext f g l i : (∀ x, l !! i = Some x → f x = g x) → alter f i l = alter g i l. Proof. revert i. induction l; intros [|?] ?; simpl; f_equal; auto. Qed. Lemma list_alter_compose f g l i : alter (f ∘ g) i l = alter f i (alter g i l). Proof. revert i. induction l; intros [|?]; simpl; f_equal; auto. Qed. Lemma list_alter_commute f g l i j : i ≠ j → alter f i (alter g j l) = alter g j (alter f i l). Proof. revert i j. induction l; intros [|?] [|?]; simpl; auto with f_equal congruence. Qed. Lemma insert_app_l l1 l2 i x : i < length l1 → <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2. Proof. apply alter_app_l. Qed. Lemma insert_app_r l1 l2 i x : <[length l1+i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2. Proof. apply alter_app_r. Qed. Lemma insert_app_r_alt l1 l2 i x : length l1 ≤ i → <[i:=x]>(l1 ++ l2) = l1 ++ <[i - length l1:=x]>l2. Proof. apply alter_app_r_alt. Qed. Lemma delete_middle l1 l2 x : delete (length l1) (l1 ++ x :: l2) = l1 ++ l2. Proof. induction l1; simpl; f_equal; auto. Qed. (** ** Properties of the [elem_of] predicate *) Lemma not_elem_of_nil x : x ∉ []. Proof. by inversion 1. Qed. Lemma elem_of_nil x : x ∈ [] ↔ False. Proof. intuition. by destruct (not_elem_of_nil x). Qed. Lemma elem_of_nil_inv l : (∀ x, x ∉ l) → l = []. Proof. destruct l. done. by edestruct 1; constructor. Qed. Lemma elem_of_cons l x y : x ∈ y :: l ↔ x = y ∨ x ∈ l. Proof. split. * inversion 1; subst. by left. by right. * intros [?|?]; subst. by left. by right. Qed. Lemma not_elem_of_cons l x y : x ∉ y :: l ↔ x ≠ y ∧ x ∉ l. Proof. rewrite elem_of_cons. tauto. Qed. Lemma elem_of_app l1 l2 x : x ∈ l1 ++ l2 ↔ x ∈ l1 ∨ x ∈ l2. Proof. induction l1. * split; [by right|]. intros [Hx|]; [|done]. by destruct (elem_of_nil x). * simpl. rewrite !elem_of_cons, IHl1. tauto. Qed. Lemma not_elem_of_app l1 l2 x : x ∉ l1 ++ l2 ↔ x ∉ l1 ∧ x ∉ l2. Proof. rewrite elem_of_app. tauto. Qed. Lemma elem_of_list_singleton x y : x ∈ [y] ↔ x = y. Proof. rewrite elem_of_cons, elem_of_nil. tauto. Qed. Global Instance elem_of_list_permutation_proper x : Proper ((≡ₚ) ==> iff) (x ∈). Proof. induction 1; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed. Lemma elem_of_list_split l x : x ∈ l → ∃ l1 l2, l = l1 ++ x :: l2. Proof. induction 1 as [x l|x y l ? [l1 [l2 ?]]]. * by eexists [], l. * subst. by exists (y :: l1) l2. Qed. Lemma elem_of_list_lookup_1 l x : x ∈ l → ∃ i, l !! i = Some x. Proof. induction 1 as [|???? IH]; [by exists 0 |]. destruct IH as [i ?]; auto. by exists (S i). Qed. Lemma elem_of_list_lookup_2 l i x : l !! i = Some x → x ∈ l. Proof. revert i. induction l; intros [|i] ?; simplify_equality'; constructor; eauto. Qed. Lemma elem_of_list_lookup l x : x ∈ l ↔ ∃ i, l !! i = Some x. Proof. firstorder eauto using elem_of_list_lookup_1, elem_of_list_lookup_2. Qed. (** ** Set operations on lists *) Section list_set. Context {dec : ∀ x y, Decision (x = y)}. Lemma elem_of_list_difference l k x : x ∈ list_difference l k ↔ x ∈ l ∧ x ∉ k. Proof. split; induction l; simpl; try case_decide; rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence. Qed. Lemma list_difference_nodup l k : NoDup l → NoDup (list_difference l k). Proof. induction 1; simpl; try case_decide. * constructor. * done. * constructor. rewrite elem_of_list_difference; intuition. done. Qed. Lemma elem_of_list_intersection l k x : x ∈ list_intersection l k ↔ x ∈ l ∧ x ∈ k. Proof. split; induction l; simpl; repeat case_decide; rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence. Qed. Lemma list_intersection_nodup l k : NoDup l → NoDup (list_intersection l k). Proof. induction 1; simpl; try case_decide. * constructor. * constructor. rewrite elem_of_list_intersection; intuition. done. * done. Qed. Lemma elem_of_list_intersection_with f l k x : x ∈ list_intersection_with f l k ↔ ∃ x1 x2, x1 ∈ l ∧ x2 ∈ k ∧ f x1 x2 = Some x. Proof. split. * induction l as [|x1 l IH]; simpl. + by rewrite elem_of_nil. + intros Hx. setoid_rewrite elem_of_cons. cut ((∃ x2, x2 ∈ k ∧ f x1 x2 = Some x) ∨ x ∈ list_intersection_with f l k); [naive_solver|]. clear IH. revert Hx. generalize (list_intersection_with f l k). induction k; simpl; [by auto|]. case_match; setoid_rewrite elem_of_cons; naive_solver. * intros (x1 & x2 & Hx1 & Hx2 & Hx). induction Hx1 as [x1 | x1 ? l ? IH]; simpl. + generalize (list_intersection_with f l k). induction Hx2; simpl; [by rewrite Hx; left |]. case_match; simpl; try setoid_rewrite elem_of_cons; auto. + generalize (IH Hx). clear Hx IH Hx2. generalize (list_intersection_with f l k). induction k; simpl; intros; [done |]. case_match; simpl; rewrite ?elem_of_cons; auto. Qed. End list_set. (** ** Properties of the [NoDup] predicate *) Lemma NoDup_nil : NoDup (@nil A) ↔ True. Proof. split; constructor. Qed. Lemma NoDup_cons x l : NoDup (x :: l) ↔ x ∉ l ∧ NoDup l. Proof. split. by inversion 1. intros [??]. by constructor. Qed. Lemma NoDup_cons_11 x l : NoDup (x :: l) → x ∉ l. Proof. rewrite NoDup_cons. by intros [??]. Qed. Lemma NoDup_cons_12 x l : NoDup (x :: l) → NoDup l. Proof. rewrite NoDup_cons. by intros [??]. Qed. Lemma NoDup_singleton x : NoDup [x]. Proof. constructor. apply not_elem_of_nil. constructor. Qed. Lemma NoDup_app l k : NoDup (l ++ k) ↔ NoDup l ∧ (∀ x, x ∈ l → x ∉ k) ∧ NoDup k. Proof. induction l; simpl. * rewrite NoDup_nil. setoid_rewrite elem_of_nil. naive_solver. * rewrite !NoDup_cons. setoid_rewrite elem_of_cons. setoid_rewrite elem_of_app. naive_solver. Qed. Global Instance NoDup_proper: Proper ((≡ₚ) ==> iff) (@NoDup A). Proof. induction 1 as [|x l k Hlk IH | |]. * by rewrite !NoDup_nil. * by rewrite !NoDup_cons, IH, Hlk. * rewrite !NoDup_cons, !elem_of_cons. intuition. * intuition. Qed. Lemma NoDup_lookup l i j x : NoDup l → l !! i = Some x → l !! j = Some x → i = j. Proof. intros Hl. revert i j. induction Hl as [|x' l Hx Hl IH]. { intros; simplify_equality. } intros [|i] [|j] ??; simplify_equality'; eauto with f_equal; exfalso; eauto using elem_of_list_lookup_2. Qed. Lemma NoDup_alt l : NoDup l ↔ ∀ i j x, l !! i = Some x → l !! j = Some x → i = j. Proof. split; eauto using NoDup_lookup. induction l as [|x l IH]; intros Hl; constructor. * rewrite elem_of_list_lookup. intros [i ?]. by feed pose proof (Hl (S i) 0 x); auto. * apply IH. intros i j x' ??. by apply (injective S), (Hl (S i) (S j) x'). Qed. Section no_dup_dec. Context `{!∀ x y, Decision (x = y)}. Global Instance NoDup_dec: ∀ l, Decision (NoDup l) := fix NoDup_dec l := match l return Decision (NoDup l) with | [] => left NoDup_nil_2 | x :: l => match decide_rel (∈) x l with | left Hin => right (λ H, NoDup_cons_11 _ _ H Hin) | right Hin => match NoDup_dec l with | left H => left (NoDup_cons_2 _ _ Hin H) | right H => right (H ∘ NoDup_cons_12 _ _) end end end. Lemma elem_of_remove_dups l x : x ∈ remove_dups l ↔ x ∈ l. Proof. split; induction l; simpl; repeat case_decide; rewrite ?elem_of_cons; intuition (simplify_equality; auto). Qed. Lemma remove_dups_nodup l : NoDup (remove_dups l). Proof. induction l; simpl; repeat case_decide; try constructor; auto. by rewrite elem_of_remove_dups. Qed. End no_dup_dec. (** ** Properties of the [filter] function *) Section filter. Context (P : A → Prop) `{∀ x, Decision (P x)}. Lemma elem_of_list_filter l x : x ∈ filter P l ↔ P x ∧ x ∈ l. Proof. unfold filter. induction l; simpl; repeat case_decide; rewrite ?elem_of_nil, ?elem_of_cons; naive_solver. Qed. Lemma filter_nodup l : NoDup l → NoDup (filter P l). Proof. unfold filter. induction 1; simpl; repeat case_decide; rewrite ?NoDup_nil, ?NoDup_cons, ?elem_of_list_filter; tauto. Qed. End filter. (** ** Properties of the [find] function *) Section find. Context (P : A → Prop) `{∀ x, Decision (P x)}. Lemma list_find_Some l i : list_find P l = Some i → ∃ x, l !! i = Some x ∧ P x. Proof. revert i. induction l; simpl; repeat case_decide; eauto with simplify_option_equality. Qed. Lemma list_find_elem_of l x : x ∈ l → P x → ∃ i, list_find P l = Some i. Proof. induction 1; simpl; repeat case_decide; naive_solver (by eauto with simplify_option_equality). Qed. End find. Section find_eq. Context `{∀ x y, Decision (x = y)}. Lemma list_find_eq_Some l i x : list_find (x =) l = Some i → l !! i = Some x. Proof. intros. destruct (list_find_Some (x =) l i) as (?&?&?); auto with congruence. Qed. Lemma list_find_eq_elem_of l x : x ∈ l → ∃ i, list_find (x=) l = Some i. Proof. eauto using list_find_elem_of. Qed. End find_eq. (** ** Properties of the [reverse] function *) Lemma reverse_nil : reverse [] = @nil A. Proof. done. Qed. Lemma reverse_singleton x : reverse [x] = [x]. Proof. done. Qed. Lemma reverse_cons l x : reverse (x :: l) = reverse l ++ [x]. Proof. unfold reverse. by rewrite <-!rev_alt. Qed. Lemma reverse_snoc l x : reverse (l ++ [x]) = x :: reverse l. Proof. unfold reverse. by rewrite <-!rev_alt, rev_unit. Qed. Lemma reverse_app l1 l2 : reverse (l1 ++ l2) = reverse l2 ++ reverse l1. Proof. unfold reverse. rewrite <-!rev_alt. apply rev_app_distr. Qed. Lemma reverse_length l : length (reverse l) = length l. Proof. unfold reverse. rewrite <-!rev_alt. apply rev_length. Qed. Lemma reverse_involutive l : reverse (reverse l) = l. Proof. unfold reverse. rewrite <-!rev_alt. apply rev_involutive. Qed. Lemma elem_of_reverse_2 x l : x ∈ l → x ∈ reverse l. Proof. induction 1; rewrite reverse_cons, elem_of_app, ?elem_of_list_singleton; intuition. Qed. Lemma elem_of_reverse x l : x ∈ reverse l ↔ x ∈ l. Proof. split; auto using elem_of_reverse_2. intros. rewrite <-(reverse_involutive l). by apply elem_of_reverse_2. Qed. Global Instance: Injective (=) (=) (@reverse A). Proof. intros l1 l2 Hl. by rewrite <-(reverse_involutive l1), <-(reverse_involutive l2), Hl. Qed. (** ** Properties of the [take] function *) Definition take_drop := @firstn_skipn A. Lemma take_nil n : take n (@nil A) = []. Proof. by destruct n. Qed. Lemma take_app l k : take (length l) (l ++ k) = l. Proof. induction l; simpl; f_equal; auto. Qed. Lemma take_app_alt l k n : n = length l → take n (l ++ k) = l. Proof. intros Hn. by rewrite Hn, take_app. Qed. Lemma take_app_le l k n : n ≤ length l → take n (l ++ k) = take n l. Proof. revert n. induction l; intros [|?] ?; simpl in *; f_equal; auto with lia. Qed. Lemma take_app_ge l k n : length l ≤ n → take n (l ++ k) = l ++ take (n - length l) k. Proof. revert n. induction l; intros [|?] ?; simpl in *; f_equal; auto with lia. Qed. Lemma take_ge l n : length l ≤ n → take n l = l. Proof. revert n. induction l; intros [|?] ?; simpl in *; f_equal; auto with lia. Qed. Lemma take_take l n m : take n (take m l) = take (min n m) l. Proof. revert n m. induction l; intros [|?] [|?]; simpl; f_equal; auto. Qed. Lemma take_idempotent l n : take n (take n l) = take n l. Proof. by rewrite take_take, Min.min_idempotent. Qed. Lemma take_length l n : length (take n l) = min n (length l). Proof. revert n. induction l; intros [|?]; simpl; f_equal; done. Qed. Lemma take_length_le l n : n ≤ length l → length (take n l) = n. Proof. rewrite take_length. apply Min.min_l. Qed. Lemma take_length_ge l n : length l ≤ n → length (take n l) = length l. Proof. rewrite take_length. apply Min.min_r. Qed. Lemma lookup_take l n i : i < n → take n l !! i = l !! i. Proof. revert n i. induction l; intros [|n] i ?; trivial. * auto with lia. * destruct i; simpl; auto with arith. Qed. Lemma lookup_take_ge l n i : n ≤ i → take n l !! i = None. Proof. revert n i. induction l; intros [|?] [|?] ?; simpl; auto with lia. Qed. Lemma take_alter f l n i : n ≤ i → take n (alter f i l) = take n l. Proof. intros. apply list_eq. intros j. destruct (le_lt_dec n j). * by rewrite !lookup_take_ge. * by rewrite !lookup_take, !list_lookup_alter_ne by lia. Qed. Lemma take_insert l n i x : n ≤ i → take n (<[i:=x]>l) = take n l. Proof. apply take_alter. Qed. (** ** Properties of the [drop] function *) Lemma drop_nil n : drop n (@nil A) = []. Proof. by destruct n. Qed. Lemma drop_app l k : drop (length l) (l ++ k) = k. Proof. induction l; simpl; f_equal; auto. Qed. Lemma drop_app_alt l k n : n = length l → drop n (l ++ k) = k. Proof. intros Hn. by rewrite Hn, drop_app. Qed. Lemma drop_length l n : length (drop n l) = length l - n. Proof. revert n. by induction l; intros [|i]; simpl; f_equal. Qed. Lemma drop_ge l n : length l ≤ n → drop n l = []. Proof. revert n. induction l; intros [|??]; simpl in *; auto with lia. Qed. Lemma drop_all l : drop (length l) l = []. Proof. by apply drop_ge. Qed. Lemma drop_drop l n1 n2 : drop n1 (drop n2 l) = drop (n2 + n1) l. Proof. revert n2. induction l; intros [|?]; simpl; rewrite ?drop_nil; auto. Qed. Lemma lookup_drop l n i : drop n l !! i = l !! (n + i). Proof. revert n i. induction l; intros [|i] ?; simpl; auto. Qed. Lemma drop_alter f l n i : i < n → drop n (alter f i l) = drop n l. Proof. intros. apply list_eq. intros j. by rewrite !lookup_drop, !list_lookup_alter_ne by lia. Qed. Lemma drop_insert l n i x : i < n → drop n (<[i:=x]>l) = drop n l. Proof. apply drop_alter. Qed. Lemma delete_take_drop l i : delete i l = take i l ++ drop (S i) l. Proof. revert i. induction l; intros [|?]; simpl; auto using f_equal. Qed. (** ** Properties of the [replicate] function *) Lemma replicate_length n x : length (replicate n x) = n. Proof. induction n; simpl; auto. Qed. Lemma lookup_replicate n x i : i < n → replicate n x !! i = Some x. Proof. revert i. induction n; intros [|?]; naive_solver auto with lia. Qed. Lemma lookup_replicate_inv n x y i : replicate n x !! i = Some y → y = x ∧ i < n. Proof. revert i. induction n; intros [|?]; naive_solver auto with lia. Qed. Lemma lookup_replicate_None n x i : n ≤ i ↔ replicate n x !! i = None. Proof. rewrite eq_None_not_Some, Nat.le_ngt. split. * intros Hin [x' Hx']; destruct Hin. by destruct (lookup_replicate_inv n x x' i). * intros Hx ?. destruct Hx. exists x; auto using lookup_replicate. Qed. Lemma elem_of_replicate_inv x n y : x ∈ replicate n y → x = y. Proof. induction n; simpl; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed. Lemma replicate_S n x : replicate (S n) x = x :: replicate n x. Proof. done. Qed. Lemma replicate_plus n m x : replicate (n + m) x = replicate n x ++ replicate m x. Proof. induction n; simpl; f_equal; auto. Qed. Lemma take_replicate n m x : take n (replicate m x) = replicate (min n m) x. Proof. revert m. by induction n; intros [|?]; simpl; f_equal. Qed. Lemma take_replicate_plus n m x : take n (replicate (n + m) x) = replicate n x. Proof. by rewrite take_replicate, min_l by lia. Qed. Lemma drop_replicate n m x : drop n (replicate m x) = replicate (m - n) x. Proof. revert m. by induction n; intros [|?]; simpl; f_equal. Qed. Lemma drop_replicate_plus n m x : drop n (replicate (n + m) x) = replicate m x. Proof. rewrite drop_replicate. f_equal. lia. Qed. Lemma replicate_as_elem_of x n l : l = replicate n x ↔ length l = n ∧ ∀ y, y ∈ l → y = x. Proof. split. * intros; subst. eauto using elem_of_replicate_inv, replicate_length. * intros [? Hl]; subst. induction l as [|y l IH]; simpl; f_equal. + apply Hl. by left. + apply IH. intros ??. apply Hl. by right. Qed. Lemma reverse_replicate n x : reverse (replicate n x) = replicate n x. Proof. apply replicate_as_elem_of. rewrite reverse_length, replicate_length. split; auto. intros y. rewrite elem_of_reverse. by apply elem_of_replicate_inv. Qed. (** ** Properties of the [resize] function *) Lemma resize_spec l n x : resize n x l = take n l ++ replicate (n - length l) x. Proof. revert n. induction l; intros [|?]; simpl; f_equal; auto. Qed. Lemma resize_0 l x : resize 0 x l = []. Proof. by destruct l. Qed. Lemma resize_nil n x : resize n x [] = replicate n x. Proof. rewrite resize_spec. rewrite take_nil. simpl. f_equal. lia. Qed. Lemma resize_ge l n x : length l ≤ n → resize n x l = l ++ replicate (n - length l) x. Proof. intros. by rewrite resize_spec, take_ge. Qed. Lemma resize_le l n x : n ≤ length l → resize n x l = take n l. Proof. intros. rewrite resize_spec, (proj2 (Nat.sub_0_le _ _)) by done. simpl. by rewrite (right_id_L [] (++)). Qed. Lemma resize_all l x : resize (length l) x l = l. Proof. intros. by rewrite resize_le, take_ge. Qed. Lemma resize_all_alt l n x : n = length l → resize n x l = l. Proof. intros. subst. by rewrite resize_all. Qed. Lemma resize_plus l n m x : resize (n + m) x l = resize n x l ++ resize m x (drop n l). Proof. revert n m. induction l; intros [|?] [|?]; simpl; f_equal; auto. * by rewrite Nat.add_0_r, (right_id_L [] (++)). * by rewrite replicate_plus. Qed. Lemma resize_plus_eq l n m x : length l = n → resize (n + m) x l = l ++ replicate m x. Proof. intros. subst. by rewrite resize_plus, resize_all, drop_all, resize_nil. Qed. Lemma resize_app_le l1 l2 n x : n ≤ length l1 → resize n x (l1 ++ l2) = resize n x l1. Proof. intros. by rewrite !resize_le, take_app_le by (rewrite ?app_length; lia). Qed. Lemma resize_app_ge l1 l2 n x : length l1 ≤ n → resize n x (l1 ++ l2) = l1 ++ resize (n - length l1) x l2. Proof. intros. rewrite !resize_spec, take_app_ge, (associative_L (++)) by done. do 2 f_equal. rewrite app_length. lia. Qed. Lemma resize_length l n x : length (resize n x l) = n. Proof. rewrite resize_spec, app_length, replicate_length, take_length. lia. Qed. Lemma resize_replicate x n m : resize n x (replicate m x) = replicate n x. Proof. revert m. induction n; intros [|?]; simpl; f_equal; auto. Qed. Lemma lookup_resize l n x i : i < n → i < length l → resize n x l !! i = l !! i. Proof. intros ??. destruct (decide (n < length l)). * by rewrite resize_le, lookup_take by lia. * by rewrite resize_ge, lookup_app_l by lia. Qed. Lemma lookup_resize_new l n x i : length l ≤ i → i < n → resize n x l !! i = Some x. Proof. intros ??. rewrite resize_ge by lia. replace i with (length l + (i - length l)) by lia. by rewrite lookup_app_r, lookup_replicate by lia. Qed. Lemma lookup_resize_old l n x i : n ≤ i → resize n x l !! i = None. Proof. intros ?. apply lookup_ge_None_2. by rewrite resize_length. Qed. Lemma resize_resize l n m x : n ≤ m → resize n x (resize m x l) = resize n x l. Proof. revert n m. induction l; simpl. * intros. by rewrite !resize_nil, resize_replicate. * intros [|?] [|?] ?; simpl; f_equal; auto with lia. Qed. Lemma resize_idempotent l n x : resize n x (resize n x l) = resize n x l. Proof. by rewrite resize_resize. Qed. Lemma resize_take_le l n m x : n ≤ m → resize n x (take m l) = resize n x l. Proof. revert n m. induction l; intros [|?] [|?] ?; simpl; f_equal; auto with lia. Qed. Lemma resize_take_eq l n x : resize n x (take n l) = resize n x l. Proof. by rewrite resize_take_le. Qed. Lemma take_resize l n m x : take n (resize m x l) = resize (min n m) x l. Proof. revert n m. induction l; intros [|?] [|?]; simpl; f_equal; auto using take_replicate. Qed. Lemma take_resize_le l n m x : n ≤ m → take n (resize m x l) = resize n x l. Proof. intros. by rewrite take_resize, Min.min_l. Qed. Lemma take_resize_eq l n x : take n (resize n x l) = resize n x l. Proof. intros. by rewrite take_resize, Min.min_l. Qed. Lemma take_length_resize l n x : length l ≤ n → take (length l) (resize n x l) = l. Proof. intros. by rewrite take_resize_le, resize_all. Qed. Lemma take_length_resize_alt l n m x : m = length l → m ≤ n → take m (resize n x l) = l. Proof. intros. subst. by apply take_length_resize. Qed. Lemma take_resize_plus l n m x : take n (resize (n + m) x l) = resize n x l. Proof. by rewrite take_resize, min_l by lia. Qed. Lemma drop_resize_le l n m x : n ≤ m → drop n (resize m x l) = resize (m - n) x (drop n l). Proof. revert n m. induction l; simpl. * intros. by rewrite drop_nil, !resize_nil, drop_replicate. * intros [|?] [|?] ?; simpl; try case_match; auto with lia. Qed. Lemma drop_resize_plus l n m x : drop n (resize (n + m) x l) = resize m x (drop n l). Proof. rewrite drop_resize_le by lia. f_equal. lia. Qed. End general_properties. Section more_general_properties. Context {A : Type}. Implicit Types x y z : A. Implicit Types l k : list A. (** ** Properties of the [reshape] function *) Lemma reshape_length szs l : length (reshape szs l) = length szs. Proof. revert l. induction szs; simpl; auto with f_equal. Qed. Lemma sublist_lookup_reshape l i n m : 0 < n → length l = m * n → reshape (replicate m n) l !! i = sublist_lookup (i * n) n l. Proof. intros Hn Hl. unfold sublist_lookup. apply option_eq; intros x; split. * intros Hx. case_option_guard as Hi. { f_equal. clear Hi. revert i l Hl Hx. induction m as [|m IH]; intros [|i] l ??; simplify_equality'; auto. rewrite <-drop_drop. apply IH; rewrite ?drop_length; auto with lia. } destruct Hi. rewrite Hl, <-Nat.mul_succ_l. apply Nat.mul_le_mono_r, Nat.le_succ_l. apply lookup_lt_Some in Hx. by rewrite reshape_length, replicate_length in Hx. * intros Hx. case_option_guard as Hi; simplify_equality'. revert i l Hl Hi. induction m as [|m IH]; auto with lia. intros [|i] l ??; simpl; [done|]. rewrite <-drop_drop. rewrite IH; rewrite ?drop_length; auto with lia. Qed. Lemma join_reshape szs l : sum_list szs = length l → mjoin (reshape szs l) = l. Proof. revert l. induction szs as [|sz szs IH]; simpl; intros l Hl. { by destruct l. } by rewrite IH, take_drop by (rewrite drop_length; lia). Qed. Lemma sum_list_replicate n m : sum_list (replicate m n) = m * n. Proof. induction m; simpl; auto. Qed. (** ** Properties of [sublist_lookup] and [sublist_insert] *) Lemma sublist_lookup_Some l i n k : sublist_lookup i n l = Some k ↔ i + n ≤ length l ∧ length k = n ∧ ∀ j, j < n → l !! (i + j) = k !! j. Proof. unfold sublist_lookup in *. split. * intros Hk. simplify_option_equality. split_ands. + done. + by rewrite take_length_le by (rewrite drop_length; lia). + intros j ?. by rewrite lookup_take, lookup_drop by done. * intros (?&?&Hlookup). case_option_guard; [|lia]. f_equal; apply list_eq; intros j. destruct (decide (j < n)). + by rewrite <-Hlookup, lookup_take, lookup_drop by done. + by rewrite lookup_take_ge, lookup_ge_None_2 by lia. Qed. Lemma sublist_lookup_length l i n k : sublist_lookup i n l = Some k → length k = n. Proof. rewrite sublist_lookup_Some. intuition. Qed. Lemma sublist_insert_length l i k : length (sublist_insert i k l) = length l. Proof. unfold sublist_insert. intros. rewrite !app_length, drop_length. destruct (decide (i + length k ≤ length l)). * rewrite !take_length_le, !take_length_ge by lia. lia. * destruct (decide (i < length l)); rewrite ?take_length_ge, ?take_length_le by lia; lia. Qed. Lemma sublist_insert_ge l i k : length l ≤ i → sublist_insert i k l = l. Proof. unfold sublist_insert. intros ?. rewrite drop_ge by lia. rewrite take_ge, (proj2 (Nat.sub_0_le _ _)) by done; simpl. by rewrite (right_id [] (++)). Qed. Lemma lookup_sublist_insert l i k j : j < length l → i ≤ j < i + length k → sublist_insert i k l !! j = k !! (j - i). Proof. unfold sublist_insert. intros ? [??]. rewrite lookup_app_minus_r by (rewrite take_length; lia). rewrite take_length_le by lia. by rewrite lookup_app_l, lookup_take by (rewrite ?take_length; lia). Qed. Lemma lookup_sublist_insert_ne l i k j : j < i ∨ i + length k ≤ j → sublist_insert i k l !! j = l !! j. Proof. destruct (decide (length l ≤ j)). { by rewrite !lookup_ge_None_2 by (by rewrite ?sublist_insert_length). } unfold sublist_insert. intros [?|?]. { rewrite lookup_app_l by (rewrite take_length; apply Nat.min_glb_lt; lia). by rewrite lookup_take. } rewrite lookup_app_minus_r by (rewrite take_length_le; lia). rewrite take_length_le by lia. rewrite lookup_app_minus_r by (rewrite take_length_ge; lia). rewrite lookup_drop, take_length_ge by lia. f_equal. lia. Qed. Lemma lookup_sublist_proper l1 l2 i k j : l1 !! j = l2 !! j → sublist_insert i k l1 !! j = sublist_insert i k l2 !! j. Proof. destruct (l2 !! j) as [x|] eqn:Hx2; intros Hx1. * destruct (decide (j < i ∨ i + length k ≤ j)). { by rewrite !lookup_sublist_insert_ne, Hx1, Hx2 by done. } by rewrite !lookup_sublist_insert by eauto using lookup_lt_Some with lia. * by rewrite !lookup_ge_None_2 by (rewrite ?sublist_insert_length; eauto using lookup_ge_None_1). Qed. Lemma lookup_sublist_all l n : length l = n → sublist_lookup 0 n l = Some l. Proof. intros. unfold sublist_lookup; case_option_guard; [|lia]. by rewrite take_ge by (rewrite drop_length; lia). Qed. Lemma insert_sublist_all l k : length l = length k → sublist_insert 0 k l = k. Proof. intros Hlk. unfold sublist_insert. simpl. by rewrite <-Hlk, drop_all, take_ge, (right_id_L [] (++)) by lia. Qed. Lemma sublist_insert_join_aux (ls : list (list A)) l i : let g k f j := sublist_insert j k (f (length k + j)) in length l = i + sum_list (length <\$> ls) → foldr g (λ _, l) ls i = take i l ++ mjoin ls. Proof. intros g. revert i l. induction ls as [|l' ls IH]; simpl; intros i l ?. { by rewrite (right_id_L [] (++)), take_ge by lia. } unfold g at 1. rewrite IH by lia. unfold sublist_insert. rewrite (take_app_le _ _ i) by (rewrite take_length_le; lia). rewrite take_take, Nat.min_l by lia. rewrite app_length, take_length_le, (take_ge l') by lia. by rewrite drop_app_alt by (rewrite take_length_le; lia). Qed. Lemma sublist_insert_join (ls : list (list A)) l : let g k f i := sublist_insert i k (f (length k + i)) in length l = sum_list (length <\$> ls) → foldr g (λ _, l) ls 0 = mjoin ls. Proof. intros. apply (sublist_insert_join_aux _ _ 0); lia. Qed. (** ** Properties of the [seq] function *) Lemma fmap_seq j n : S <\$> seq j n = seq (S j) n. Proof. revert j. induction n; simpl; auto with f_equal. Qed. Lemma lookup_seq j n i : i < n → seq j n !! i = Some (j + i). Proof. revert j i. induction n as [|n IH]; intros j [|i] ?; simpl; auto with lia. rewrite IH; auto with lia. Qed. Lemma lookup_seq_ge j n i : n ≤ i → seq j n !! i = None. Proof. revert j i. induction n as [|n IH]; intros j [|i] ?; simpl; auto with lia. Qed. Lemma lookup_seq_inv j n i j' : seq j n !! i = Some j' → j' = j + i ∧ i < n. Proof. destruct (le_lt_dec n i). * by rewrite lookup_seq_ge. * 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 intro; subst. Qed. Lemma Permutation_singleton l x : l ≡ₚ [x] ↔ l = [x]. Proof. split. by intro; apply Permutation_length_1_inv. by intro; subst. Qed. Definition Permutation_skip := @perm_skip A. Definition Permutation_swap := @perm_swap A. Definition Permutation_singleton_inj := @Permutation_length_1 A. Global Existing Instance Permutation_app'_Proper. Global Instance: Proper ((≡ₚ) ==> (=)) (@length A). Proof. induction 1; simpl; auto with lia. Qed. Global Instance: Commutative (≡ₚ) (@app A). Proof. intros l1. induction l1 as [|x l1 IH]; intros l2; simpl. * by rewrite (right_id_L [] (++)). * rewrite Permutation_middle, IH. simpl. by rewrite Permutation_middle. Qed. Global Instance: ∀ x : A, Injective (≡ₚ) (≡ₚ) (x ::). Proof. red. eauto using Permutation_cons_inv. Qed. Global Instance: ∀ k : list A, Injective (≡ₚ) (≡ₚ) (k ++). Proof. red. induction k as [|x k IH]; intros l1 l2; simpl; auto. intros. by apply IH, (injective (x ::)). Qed. Global Instance: ∀ k : list A, Injective (≡ₚ) (≡ₚ) (++ k). Proof. intros k l1 l2. rewrite !(commutative (++) _ k). by apply (injective (k ++)). Qed. Lemma replicate_Permutation n x l : l ≡ₚ replicate n x → l = replicate n x. Proof. intros Hl. apply replicate_as_elem_of. split. * by rewrite Hl, replicate_length. * intros y. rewrite Hl. by apply elem_of_replicate_inv. Qed. Lemma reverse_Permutation l : reverse l ≡ₚ l. Proof. induction l as [|x l IH]; [done|]. by rewrite reverse_cons, (commutative (++)), IH. Qed. (** ** Properties of the [prefix_of] and [suffix_of] predicates *) Global Instance: PreOrder (@prefix_of A). Proof. split. * intros ?. eexists []. by rewrite (right_id_L [] (++)). * intros ??? [k1 ?] [k2 ?]. exists (k1 ++ k2). subst. by rewrite (associative_L (++)). Qed. Lemma prefix_of_nil l : [] `prefix_of` l. Proof. by exists l. Qed. Lemma prefix_of_nil_not x l : ¬x :: l `prefix_of` []. Proof. by intros [k E]. Qed. Lemma prefix_of_cons x l1 l2 : l1 `prefix_of` l2 → x :: l1 `prefix_of` x :: l2. Proof. intros [k E]. exists k. by subst. Qed. Lemma prefix_of_cons_alt x y l1 l2 : x = y → l1 `prefix_of` l2 → x :: l1 `prefix_of` y :: l2. Proof. intro. subst. apply prefix_of_cons. Qed. Lemma prefix_of_cons_inv_1 x y l1 l2 : x :: l1 `prefix_of` y :: l2 → x = y. Proof. intros [k E]. by injection E. Qed. Lemma prefix_of_cons_inv_2 x y l1 l2 : x :: l1 `prefix_of` y :: l2 → l1 `prefix_of` l2. Proof. intros [k E]. exists k. by injection E. Qed. Lemma prefix_of_app k l1 l2 : l1 `prefix_of` l2 → k ++ l1 `prefix_of` k ++ l2. Proof. intros [k' ?]. subst. exists k'. by rewrite (associative_L (++)). Qed. Lemma prefix_of_app_alt k1 k2 l1 l2 : k1 = k2 → l1 `prefix_of` l2 → k1 ++ l1 `prefix_of` k2 ++ l2. Proof. intro. subst. apply prefix_of_app. Qed. Lemma prefix_of_app_l l1 l2 l3 : l1 ++ l3 `prefix_of` l2 → l1 `prefix_of` l2. Proof. intros [k ?]. red. exists (l3 ++ k). subst. by rewrite <-(associative_L (++)). Qed. Lemma prefix_of_app_r l1 l2 l3 : l1 `prefix_of` l2 → l1 `prefix_of` l2 ++ l3. Proof. intros [k ?]. exists (k ++ l3). subst. by rewrite (associative_L (++)). Qed. Lemma prefix_of_length l1 l2 : l1 `prefix_of` l2 → length l1 ≤ length l2. Proof. intros [??]. subst. rewrite app_length. lia. Qed. Lemma prefix_of_snoc_not l x : ¬l ++ [x] `prefix_of` l. Proof. intros [??]. discriminate_list_equality. Qed. Global Instance: PreOrder (@suffix_of A). Proof. split. * intros ?. by eexists []. * intros ??? [k1 ?] [k2 ?]. exists (k2 ++ k1). subst. by rewrite (associative_L (++)). Qed. Global Instance prefix_of_dec `{∀ x y, Decision (x = y)} : ∀ l1 l2, Decision (l1 `prefix_of` l2) := fix go l1 l2 := match l1, l2 return { l1 `prefix_of` l2 } + { ¬l1 `prefix_of` l2 } with | [], _ => left (prefix_of_nil _) | _, [] => right (prefix_of_nil_not _ _) | x :: l1, y :: l2 => match decide_rel (=) x y with | left Exy => match go l1 l2 with | left Hl1l2 => left (prefix_of_cons_alt _ _ _ _ Exy Hl1l2) | right Hl1l2 => right (Hl1l2 ∘ prefix_of_cons_inv_2 _ _ _ _) end | right Exy => right (Exy ∘ prefix_of_cons_inv_1 _ _ _ _) end end. Section prefix_ops. Context `{∀ x y, Decision (x = y)}. Lemma max_prefix_of_fst l1 l2 : l1 = snd (max_prefix_of l1 l2) ++ fst (fst (max_prefix_of l1 l2)). Proof. revert l2. induction l1; intros [|??]; simpl; repeat case_decide; simpl; f_equal; auto. Qed. Lemma max_prefix_of_fst_alt l1 l2 k1 k2 k3 : max_prefix_of l1 l2 = (k1, k2, k3) → l1 = k3 ++ k1. Proof. intro. pose proof (max_prefix_of_fst l1 l2). by destruct (max_prefix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_prefix_of_fst_prefix l1 l2 : snd (max_prefix_of l1 l2) `prefix_of` l1. Proof. eexists. apply max_prefix_of_fst. Qed. Lemma max_prefix_of_fst_prefix_alt l1 l2 k1 k2 k3 : max_prefix_of l1 l2 = (k1, k2, k3) → k3 `prefix_of` l1. Proof. eexists. eauto using max_prefix_of_fst_alt. Qed. Lemma max_prefix_of_snd l1 l2 : l2 = snd (max_prefix_of l1 l2) ++ snd (fst (max_prefix_of l1 l2)). Proof. revert l2. induction l1; intros [|??]; simpl; repeat case_decide; simpl; f_equal; auto. Qed. Lemma max_prefix_of_snd_alt l1 l2 k1 k2 k3 : max_prefix_of l1 l2 = (k1, k2, k3) → l2 = k3 ++ k2. Proof. intro. pose proof (max_prefix_of_snd l1 l2). by destruct (max_prefix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_prefix_of_snd_prefix l1 l2 : snd (max_prefix_of l1 l2) `prefix_of` l2. Proof. eexists. apply max_prefix_of_snd. Qed. Lemma max_prefix_of_snd_prefix_alt l1 l2 k1 k2 k3 : max_prefix_of l1 l2 = (k1,k2,k3) → k3 `prefix_of` l2. Proof. eexists. eauto using max_prefix_of_snd_alt. Qed. Lemma max_prefix_of_max l1 l2 k : k `prefix_of` l1 → k `prefix_of` l2 → k `prefix_of` snd (max_prefix_of l1 l2). Proof. intros [l1' ?] [l2' ?]. subst. by induction k; simpl; repeat case_decide; simpl; auto using prefix_of_nil, prefix_of_cons. Qed. Lemma max_prefix_of_max_alt l1 l2 k1 k2 k3 k : max_prefix_of l1 l2 = (k1,k2,k3) → k `prefix_of` l1 → k `prefix_of` l2 → k `prefix_of` k3. Proof. intro. pose proof (max_prefix_of_max l1 l2 k). by destruct (max_prefix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_prefix_of_max_snoc l1 l2 k1 k2 k3 x1 x2 : max_prefix_of l1 l2 = (x1 :: k1, x2 :: k2, k3) → x1 ≠ x2. Proof. intros Hl ?. subst. destruct (prefix_of_snoc_not k3 x2). eapply max_prefix_of_max_alt; eauto. * rewrite (max_prefix_of_fst_alt _ _ _ _ _ Hl). apply prefix_of_app, prefix_of_cons, prefix_of_nil. * rewrite (max_prefix_of_snd_alt _ _ _ _ _ Hl). apply prefix_of_app, prefix_of_cons, prefix_of_nil. Qed. End prefix_ops. Lemma prefix_suffix_reverse l1 l2 : l1 `prefix_of` l2 ↔ reverse l1 `suffix_of` reverse l2. Proof. split; intros [k E]; exists (reverse k). * by rewrite E, reverse_app. * by rewrite <-(reverse_involutive l2), E, reverse_app, reverse_involutive. Qed. Lemma suffix_prefix_reverse l1 l2 : l1 `suffix_of` l2 ↔ reverse l1 `prefix_of` reverse l2. Proof. by rewrite prefix_suffix_reverse, !reverse_involutive. Qed. Lemma suffix_of_nil l : [] `suffix_of` l. Proof. exists l. by rewrite (right_id_L [] (++)). Qed. Lemma suffix_of_nil_inv l : l `suffix_of` [] → l = []. Proof. by intros [[|?] ?]; simplify_list_equality. Qed. Lemma suffix_of_cons_nil_inv x l : ¬x :: l `suffix_of` []. Proof. by intros [[] ?]. Qed. Lemma suffix_of_snoc l1 l2 x : l1 `suffix_of` l2 → l1 ++ [x] `suffix_of` l2 ++ [x]. Proof. intros [k E]. exists k. subst. by rewrite (associative_L (++)). Qed. Lemma suffix_of_snoc_alt x y l1 l2 : x = y → l1 `suffix_of` l2 → l1 ++ [x] `suffix_of` l2 ++ [y]. Proof. intro. subst. apply suffix_of_snoc. Qed. Lemma suffix_of_app l1 l2 k : l1 `suffix_of` l2 → l1 ++ k `suffix_of` l2 ++ k. Proof. intros [k' E]. exists k'. subst. by rewrite (associative_L (++)). Qed. Lemma suffix_of_app_alt l1 l2 k1 k2 : k1 = k2 → l1 `suffix_of` l2 → l1 ++ k1 `suffix_of` l2 ++ k2. Proof. intro. subst. apply suffix_of_app. Qed. Lemma suffix_of_snoc_inv_1 x y l1 l2 : l1 ++ [x] `suffix_of` l2 ++ [y] → x = y. Proof. intros [k' E]. rewrite (associative_L (++)) in E. by simplify_list_equality. Qed. Lemma suffix_of_snoc_inv_2 x y l1 l2 : l1 ++ [x] `suffix_of` l2 ++ [y] → l1 `suffix_of` l2. Proof. intros [k' E]. exists k'. rewrite (associative_L (++)) in E. by simplify_list_equality. Qed. Lemma suffix_of_app_inv l1 l2 k : l1 ++ k `suffix_of` l2 ++ k → l1 `suffix_of` l2. Proof. intros [k' E]. exists k'. rewrite (associative_L (++)) in E. by simplify_list_equality. Qed. Lemma suffix_of_cons_l l1 l2 x : x :: l1 `suffix_of` l2 → l1 `suffix_of` l2. Proof. intros [k ?]. exists (k ++ [x]). subst. by rewrite <-(associative_L (++)). Qed. Lemma suffix_of_app_l l1 l2 l3 : l3 ++ l1 `suffix_of` l2 → l1 `suffix_of` l2. Proof. intros [k ?]. exists (k ++ l3). subst. by rewrite <-(associative_L (++)). Qed. Lemma suffix_of_cons_r l1 l2 x : l1 `suffix_of` l2 → l1 `suffix_of` x :: l2. Proof. intros [k ?]. exists (x :: k). by subst. Qed. Lemma suffix_of_app_r l1 l2 l3 : l1 `suffix_of` l2 → l1 `suffix_of` l3 ++ l2. Proof. intros [k ?]. exists (l3 ++ k). subst. by rewrite (associative_L _). Qed. Lemma suffix_of_cons_inv l1 l2 x y : x :: l1 `suffix_of` y :: l2 → x :: l1 = y :: l2 ∨ x :: l1 `suffix_of` l2. Proof. intros [[|? k] E]; [by left |]. right. simplify_equality. by apply suffix_of_app_r. Qed. Lemma suffix_of_length l1 l2 : l1 `suffix_of` l2 → length l1 ≤ length l2. Proof. intros [??]. subst. rewrite app_length. lia. Qed. Lemma suffix_of_cons_not x l : ¬x :: l `suffix_of` l. Proof. intros [??]. discriminate_list_equality. Qed. Global Instance suffix_of_dec `{∀ x y, Decision (x = y)} l1 l2 : Decision (l1 `suffix_of` l2). Proof. refine (cast_if (decide_rel prefix_of (reverse l1) (reverse l2))); abstract (by rewrite suffix_prefix_reverse). Defined. Section max_suffix_of. Context `{∀ x y, Decision (x = y)}. Lemma max_suffix_of_fst l1 l2 : l1 = fst (fst (max_suffix_of l1 l2)) ++ snd (max_suffix_of l1 l2). Proof. rewrite <-(reverse_involutive l1) at 1. rewrite (max_prefix_of_fst (reverse l1) (reverse l2)). unfold max_suffix_of. destruct (max_prefix_of (reverse l1) (reverse l2)) as ((?&?)&?); simpl. by rewrite reverse_app. Qed. Lemma max_suffix_of_fst_alt l1 l2 k1 k2 k3 : max_suffix_of l1 l2 = (k1, k2, k3) → l1 = k1 ++ k3. Proof. intro. pose proof (max_suffix_of_fst l1 l2). by destruct (max_suffix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_suffix_of_fst_suffix l1 l2 : snd (max_suffix_of l1 l2) `suffix_of` l1. Proof. eexists. apply max_suffix_of_fst. Qed. Lemma max_suffix_of_fst_suffix_alt l1 l2 k1 k2 k3 : max_suffix_of l1 l2 = (k1, k2, k3) → k3 `suffix_of` l1. Proof. eexists. eauto using max_suffix_of_fst_alt. Qed. Lemma max_suffix_of_snd l1 l2 : l2 = snd (fst (max_suffix_of l1 l2)) ++ snd (max_suffix_of l1 l2). Proof. rewrite <-(reverse_involutive l2) at 1. rewrite (max_prefix_of_snd (reverse l1) (reverse l2)). unfold max_suffix_of. destruct (max_prefix_of (reverse l1) (reverse l2)) as ((?&?)&?); simpl. by rewrite reverse_app. Qed. Lemma max_suffix_of_snd_alt l1 l2 k1 k2 k3 : max_suffix_of l1 l2 = (k1,k2,k3) → l2 = k2 ++ k3. Proof. intro. pose proof (max_suffix_of_snd l1 l2). by destruct (max_suffix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_suffix_of_snd_suffix l1 l2 : snd (max_suffix_of l1 l2) `suffix_of` l2. Proof. eexists. apply max_suffix_of_snd. Qed. Lemma max_suffix_of_snd_suffix_alt l1 l2 k1 k2 k3 : max_suffix_of l1 l2 = (k1,k2,k3) → k3 `suffix_of` l2. Proof. eexists. eauto using max_suffix_of_snd_alt. Qed. Lemma max_suffix_of_max l1 l2 k : k `suffix_of` l1 → k `suffix_of` l2 → k `suffix_of` snd (max_suffix_of l1 l2). Proof. generalize (max_prefix_of_max (reverse l1) (reverse l2)). rewrite !suffix_prefix_reverse. unfold max_suffix_of. destruct (max_prefix_of (reverse l1) (reverse l2)) as ((?&?)&?); simpl. rewrite reverse_involutive. auto. Qed. Lemma max_suffix_of_max_alt l1 l2 k1 k2 k3 k : max_suffix_of l1 l2 = (k1, k2, k3) → k `suffix_of` l1 → k `suffix_of` l2 → k `suffix_of` k3. Proof. intro. pose proof (max_suffix_of_max l1 l2 k). by destruct (max_suffix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_suffix_of_max_snoc l1 l2 k1 k2 k3 x1 x2 : max_suffix_of l1 l2 = (k1 ++ [x1], k2 ++ [x2], k3) → x1 ≠ x2. Proof. intros Hl ?. subst. destruct (suffix_of_cons_not x2 k3). eapply max_suffix_of_max_alt; eauto. * rewrite (max_suffix_of_fst_alt _ _ _ _ _ Hl). by apply (suffix_of_app [x2]), suffix_of_app_r. * rewrite (max_suffix_of_snd_alt _ _ _ _ _ Hl). by apply (suffix_of_app [x2]), suffix_of_app_r. Qed. End max_suffix_of. (** ** Properties of the [sublist] predicate *) Lemma sublist_length l1 l2 : l1 `sublist` l2 → length l1 ≤ length l2. Proof. induction 1; simpl; auto with arith. Qed. Lemma sublist_nil_l l : [] `sublist` l. Proof. induction l; try constructor; auto. Qed. Lemma sublist_nil_r l : l `sublist` [] ↔ l = []. Proof. split. by inversion 1. intros. subst. constructor. Qed. Lemma sublist_app l1 l2 k1 k2 : l1 `sublist` l2 → k1 `sublist` k2 → l1 ++ k1 `sublist` l2 ++ k2. Proof. induction 1; simpl; try constructor; auto. Qed. Lemma sublist_inserts_l k l1 l2 : l1 `sublist` l2 → l1 `sublist` k ++ l2. Proof. induction k; try constructor; auto. Qed. Lemma sublist_inserts_r k l1 l2 : l1 `sublist` l2 → l1 `sublist` l2 ++ k. Proof. induction 1; simpl; try constructor; auto using sublist_nil_l. Qed. Lemma sublist_cons_r x l k : l `sublist` x :: k ↔ l `sublist` k ∨ ∃ l', l = x :: l' ∧ l' `sublist` k. Proof. split. inversion 1; eauto. intros [?|(?&?&?)]; subst; constructor; auto. Qed. Lemma sublist_cons_l x l k : x :: l `sublist` k ↔ ∃ k1 k2, k = k1 ++ x :: k2 ∧ l `sublist` k2. Proof. split. * intros Hlk. induction k as [|y k IH]; inversion Hlk. + eexists [], k. by repeat constructor. + destruct IH as (k1&k2&?&?); subst; auto. by exists (y :: k1) k2. * intros (k1&k2&?&?). subst. by apply sublist_inserts_l, sublist_skip. Qed. Lemma sublist_app_r l k1 k2 : l `sublist` k1 ++ k2 ↔ ∃ l1 l2, l = l1 ++ l2 ∧ l1 `sublist` k1 ∧ l2 `sublist` k2. Proof. split. * revert l k2. induction k1 as [|y k1 IH]; intros l k2; simpl. { eexists [], l. by repeat constructor. } rewrite sublist_cons_r. intros [?|(l' & ? &?)]; subst. + destruct (IH l k2) as (l1&l2&?&?&?); trivial; subst. exists l1 l2. auto using sublist_cons. + destruct (IH l' k2) as (l1&l2&?&?&?); trivial; subst. exists (y :: l1) l2. auto using sublist_skip. * intros (?&?&?&?&?); subst. auto using sublist_app. Qed. Lemma sublist_app_l l1 l2 k : l1 ++ l2 `sublist` k ↔ ∃ k1 k2, k = k1 ++ k2 ∧ l1 `sublist` k1 ∧ l2 `sublist` k2. Proof. split. * revert l2 k. induction l1 as [|x l1 IH]; intros l2 k; simpl. { eexists [], k. by repeat constructor. } rewrite sublist_cons_l. intros (k1 & k2 &?&?); subst. destruct (IH l2 k2) as (h1 & h2 &?&?&?); trivial; subst. exists (k1 ++ x :: h1) h2. rewrite <-(associative_L (++)). auto using sublist_inserts_l, sublist_skip. * intros (?&?&?&?&?); subst. auto using sublist_app. Qed. Lemma sublist_app_inv_l k l1 l2 : k ++ l1 `sublist` k ++ l2 → l1 `sublist` l2. Proof. induction k as [|y k IH]; simpl; [done |]. rewrite sublist_cons_r. intros [Hl12|(?&?&?)]; [|simplify_equality; eauto]. rewrite sublist_cons_l in Hl12. destruct Hl12 as (k1&k2&Hk&?). apply IH. rewrite Hk. eauto using sublist_inserts_l, sublist_cons. Qed. Lemma sublist_app_inv_r k l1 l2 : l1 ++ k `sublist` l2 ++ k → l1 `sublist` l2. Proof. revert l1 l2. induction k as [|y k IH]; intros l1 l2. { by rewrite !(right_id_L [] (++)). } intros. feed pose proof (IH (l1 ++ [y]) (l2 ++ [y])) as Hl12. { by rewrite <-!(associative_L (++)). } rewrite sublist_app_l in Hl12. destruct Hl12 as (k1&k2&E&?&Hk2). destruct k2 as [|z k2] using rev_ind; [inversion Hk2|]. rewrite (associative_L (++)) in E. simplify_list_equality. eauto using sublist_inserts_r. Qed. Global Instance: PartialOrder (@sublist A). Proof. split; [split|]. * intros l. induction l; constructor; auto. * intros l1 l2 l3 Hl12. revert l3. induction Hl12. + auto using sublist_nil_l. + intros ?. rewrite sublist_cons_l. intros (?&?&?&?); subst. eauto using sublist_inserts_l, sublist_skip. + intros ?. rewrite sublist_cons_l. intros (?&?&?&?); subst. eauto using sublist_inserts_l, sublist_cons. * intros l1 l2 Hl12 Hl21. apply sublist_length in Hl21. induction Hl12; simpl in *; f_equal; auto with arith. apply sublist_length in Hl12. lia. Qed. Lemma sublist_take l i : take i l `sublist` l. Proof. rewrite <-(take_drop i l) at 2. by apply sublist_inserts_r. Qed. Lemma sublist_drop l i : drop i l `sublist` l. Proof. rewrite <-(take_drop i l) at 2. by apply sublist_inserts_l. Qed. Lemma sublist_delete l i : delete i l `sublist` l. Proof. revert i. by induction l; intros [|?]; simpl; constructor. Qed. Lemma sublist_delete_list l is : delete_list is l `sublist` l. Proof. induction is as [|i is IH]; simpl; [done |]. transitivity (delete_list is l); auto using sublist_delete. Qed. Lemma sublist_alt l1 l2 : l1 `sublist` l2 ↔ ∃ is, l1 = delete_list is l2. Proof. split. * intros Hl12. cut (∀ k, ∃ is, k ++ l1 = delete_list is (k ++ l2)). { intros help. apply (help []). } induction Hl12 as [|x l1 l2 _ IH|x l1 l2 _ IH]; intros k. + by eexists []. + destruct (IH (k ++ [x])) as [is His]. exists is. by rewrite <-!(associative_L (++)) in His. + destruct (IH k) as [is His]. exists (is ++ [length k]). unfold delete_list. rewrite fold_right_app. simpl. by rewrite delete_middle. * intros [is ?]. subst. apply sublist_delete_list. Qed. Lemma Permutation_sublist l1 l2 l3 : l1 ≡ₚ l2 → l2 `sublist` l3 → ∃ l4, l1 `sublist` l4 ∧ l4 ≡ₚ l3. Proof. intros Hl1l2. revert l3. induction Hl1l2 as [|x l1 l2 ? IH|x y l1|l1 l1' l2 ? IH1 ? IH2]. * intros l3. by exists l3. * intros l3. rewrite sublist_cons_l. intros (l3'&l3''&?&?); subst. destruct (IH l3'') as (l4&?&Hl4); auto. exists (l3' ++ x :: l4). split. by apply sublist_inserts_l, sublist_skip. by rewrite Hl4. * intros l3. rewrite sublist_cons_l. intros (l3'&l3''&?& Hl3); subst. rewrite sublist_cons_l in Hl3. destruct Hl3 as (l5'&l5''&?& Hl5); subst. exists (l3' ++ y :: l5' ++ x :: l5''). split. - by do 2 apply sublist_inserts_l, sublist_skip. - by rewrite !Permutation_middle, Permutation_swap. * intros l3 ?. destruct (IH2 l3) as (l3'&?&?); trivial. destruct (IH1 l3') as (l3'' &?&?); trivial. exists l3''. split. done. etransitivity; eauto. Qed. Lemma sublist_Permutation l1 l2 l3 : l1 `sublist` l2 → l2 ≡ₚ l3 → ∃ l4, l1 ≡ₚ l4 ∧ l4 `sublist` l3. Proof. intros Hl1l2 Hl2l3. revert l1 Hl1l2. induction Hl2l3 as [|x l2 l3 ? IH|x y l2|l2 l2' l3 ? IH1 ? IH2]. * intros l1. by exists l1. * intros l1. rewrite sublist_cons_r. intros [?|(l1'&l1''&?)]; subst. { destruct (IH l1) as (l4&?&?); trivial. exists l4. split. done. by constructor. } destruct (IH l1') as (l4&?&Hl4); auto. exists (x :: l4). split. by constructor. by constructor. * intros l1. rewrite sublist_cons_r. intros [Hl1|(l1'&l1''&Hl1)]; subst. { exists l1. split; [done|]. rewrite sublist_cons_r in Hl1. destruct Hl1 as [?|(l1'&?&?)]; subst; by repeat constructor. } rewrite sublist_cons_r in Hl1. destruct Hl1 as [?|(l1''&?&?)]; subst. + exists (y :: l1'). by repeat constructor. + exists (x :: y :: l1''). by repeat constructor. * intros l1 ?. destruct (IH1 l1) as (l3'&?&?); trivial. destruct (IH2 l3') as (l3'' &?&?); trivial. exists l3''. split; [|done]. etransitivity; eauto. Qed. (** Properties of the [contains] predicate *) Lemma contains_length l1 l2 : l1 `contains` l2 → length l1 ≤ length l2. Proof. induction 1; simpl; auto with lia. Qed. Lemma contains_nil_l l : [] `contains` l. Proof. induction l; constructor; auto. Qed. Lemma contains_nil_r l : l `contains` [] ↔ l = []. Proof. split; [|intros; subst; constructor]. intros Hl. apply contains_length in Hl. destruct l; simpl in *; auto with lia. Qed. Global Instance: PreOrder (@contains A). Proof. split. * intros l. induction l; constructor; auto. * red. apply contains_trans. Qed. Lemma Permutation_contains l1 l2 : l1 ≡ₚ l2 → l1 `contains` l2. Proof. induction 1; econstructor; eauto. Qed. Lemma sublist_contains l1 l2 : l1 `sublist` l2 → l1 `contains` l2. Proof. induction 1; constructor; auto. Qed. Lemma contains_Permutation l1 l2 : l1 `contains` l2 → ∃ k, l2 ≡ₚ l1 ++ k. Proof. induction 1 as [|x y l ? [k Hk]| |x l1 l2 ? [k Hk]|l1 l2 l3 ? [k Hk] ? [k' Hk']]. * by eexists []. * exists k. by rewrite Hk. * eexists []. rewrite (right_id_L [] (++)). by constructor. * exists (x :: k). by rewrite Hk, Permutation_middle. * exists (k ++ k'). by rewrite Hk', Hk, (associative_L (++)). Qed. Lemma contains_Permutation_length_le l1 l2 : length l2 ≤ length l1 → l1 `contains` l2 → l1 ≡ₚ l2. Proof. intros Hl21 Hl12. destruct (contains_Permutation l1 l2) as [[|??] Hk]; auto. * by rewrite Hk, (right_id_L [] (++)). * rewrite Hk, app_length in Hl21; simpl in Hl21; lia. Qed. Lemma contains_Permutation_length_eq l1 l2 : length l2 = length l1 → l1 `contains` l2 → l1 ≡ₚ l2. Proof. intro. apply contains_Permutation_length_le. lia. Qed. Global Instance: Proper ((≡ₚ) ==> (≡ₚ) ==> iff) (@contains A). Proof. intros l1 l2 ? k1 k2 ?. split; intros. * transitivity l1. by apply Permutation_contains. transitivity k1. done. by apply Permutation_contains. * transitivity l2. by apply Permutation_contains. transitivity k2. done. by apply Permutation_contains. Qed. Global Instance: AntiSymmetric (≡ₚ) (@contains A). Proof. red. auto using contains_Permutation_length_le, contains_length. Qed. Lemma contains_take l i : take i l `contains` l. Proof. auto using sublist_take, sublist_contains. Qed. Lemma contains_drop l i : drop i l `contains` l. Proof. auto using sublist_drop, sublist_contains. Qed. Lemma contains_delete l i : delete i l `contains` l. Proof. auto using sublist_delete, sublist_contains. Qed. Lemma contains_delete_list l is : delete_list is l `sublist` l. Proof. auto using sublist_delete_list, sublist_contains. Qed. Lemma contains_sublist_l l1 l3 : l1 `contains` l3 ↔ ∃ l2, l1 `sublist` l2 ∧ l2 ≡ₚ l3. Proof. split. { intros Hl13. elim Hl13; clear l1 l3 Hl13. * by eexists []. * intros x l1 l3 ? (l2&?&?). exists (x :: l2). by repeat constructor. * intros x y l. exists (y :: x :: l). by repeat constructor. * intros x l1 l3 ? (l2&?&?). exists (x :: l2). by repeat constructor. * intros l1 l3 l5 ? (l2&?&?) ? (l4&?&?). destruct (Permutation_sublist l2 l3 l4) as (l3'&?&?); trivial. exists l3'. split; etransitivity; eauto. } intros (l2&?&?). transitivity l2; auto using sublist_contains, Permutation_contains. Qed. Lemma contains_sublist_r l1 l3 : l1 `contains` l3 ↔ ∃ l2, l1 ≡ₚ l2 ∧ l2 `sublist` l3. Proof. rewrite contains_sublist_l. split; intros (l2&?&?); eauto using sublist_Permutation, Permutation_sublist. Qed. Lemma contains_inserts_l k l1 l2 : l1 `contains` l2 → l1 `contains` k ++ l2. Proof. induction k; try constructor; auto. Qed. Lemma contains_inserts_r k l1 l2 : l1 `contains` l2 → l1 `contains` l2 ++ k. Proof. rewrite (commutative (++)). apply contains_inserts_l. Qed. Lemma contains_skips_l k l1 l2 : l1 `contains` l2 → k ++ l1 `contains` k ++ l2. Proof. induction k; try constructor; auto. Qed. Lemma contains_skips_r k l1 l2 : l1 `contains` l2 → l1 ++ k `contains` l2 ++ k. Proof. rewrite !(commutative (++) _ k). apply contains_skips_l. Qed. Lemma contains_app l1 l2 k1 k2 : l1 `contains` l2 → k1 `contains` k2 → l1 ++ k1 `contains` l2 ++ k2. Proof. transitivity (l1 ++ k2); auto using contains_skips_l, contains_skips_r. Qed. Lemma contains_cons_r x l k : l `contains` x :: k ↔ l `contains` k ∨ ∃ l', l ≡ₚ x :: l' ∧ l' `contains` k. Proof. split. * rewrite contains_sublist_r. intros (l'&E&Hl'). rewrite sublist_cons_r in Hl'. destruct Hl' as [?|(?&?&?)]; subst. + left. rewrite E. eauto using sublist_contains. + right. eauto using sublist_contains. * intros [?|(?&E&?)]; [|rewrite E]; by constructor. Qed. Lemma contains_cons_l x l k : x :: l `contains` k ↔ ∃ k', k ≡ₚ x :: k' ∧ l `contains` k'. Proof. split. * rewrite contains_sublist_l. intros (l'&Hl'&E). rewrite sublist_cons_l in Hl'. destruct Hl' as (k1&k2&?&?); subst. exists (k1 ++ k2). split; eauto using contains_inserts_l, sublist_contains. by rewrite Permutation_middle. * intros (?&E&?). rewrite E. by constructor. Qed. Lemma contains_app_r l k1 k2 : l `contains` k1 ++ k2 ↔ ∃ l1 l2, l ≡ₚ l1 ++ l2 ∧ l1 `contains` k1 ∧ l2 `contains` k2. Proof. split. * rewrite contains_sublist_r. intros (l'&E&Hl'). rewrite sublist_app_r in Hl'. destruct Hl' as (l1&l2&?&?&?); subst. exists l1 l2. eauto using sublist_contains. * intros (?&?&E&?&?). rewrite E. eauto using contains_app. Qed. Lemma contains_app_l l1 l2 k : l1 ++ l2 `contains` k ↔ ∃ k1 k2, k ≡ₚ k1 ++ k2 ∧ l1 `contains` k1 ∧ l2 `contains` k2. Proof. split. * rewrite contains_sublist_l. intros (l'&Hl'&E). rewrite sublist_app_l in Hl'. destruct Hl' as (k1&k2&?&?&?); subst. exists k1 k2. split. done. eauto using sublist_contains. * intros (?&?&E&?&?). rewrite E. eauto using contains_app. Qed. Lemma contains_app_inv_l l1 l2 k : k ++ l1 `contains` k ++ l2 → l1 `contains` l2. Proof. induction k as [|y k IH]; simpl; [done |]. rewrite contains_cons_l. intros (?&E&?). apply Permutation_cons_inv in E. apply IH. by rewrite E. Qed. Lemma contains_app_inv_r l1 l2 k : l1 ++ k `contains` l2 ++ k → l1 `contains` l2. Proof. revert l1 l2. induction k as [|y k IH]; intros l1 l2. { by rewrite !(right_id_L [] (++)). } intros. feed pose proof (IH (l1 ++ [y]) (l2 ++ [y])) as Hl12. { by rewrite <-!(associative_L (++)). } rewrite contains_app_l in Hl12. destruct Hl12 as (k1&k2&E1&?&Hk2). rewrite contains_cons_l in Hk2. destruct Hk2 as (k2'&E2&?). rewrite E2, (Permutation_cons_append k2'), (associative_L (++)) in E1. apply Permutation_app_inv_r in E1. rewrite E1. eauto using contains_inserts_r. Qed. Lemma contains_cons_middle x l k1 k2 : l `contains` k1 ++ k2 → x :: l `contains` k1 ++ x :: k2. Proof. rewrite <-Permutation_middle. by apply contains_skip. Qed. Lemma contains_app_middle l1 l2 k1 k2 : l2 `contains` k1 ++ k2 → l1 ++ l2 `contains` k1 ++ l1 ++ k2. Proof. rewrite !(associative (++)), (commutative (++) k1 l1), <-(associative_L (++)). by apply contains_skips_l. Qed. Lemma contains_middle l k1 k2 : l `contains` k1 ++ l ++ k2. Proof. by apply contains_inserts_l, contains_inserts_r. Qed. Lemma Permutation_alt l1 l2 : l1 ≡ₚ l2 ↔ length l1 = length l2 ∧ l1 `contains` l2. Proof. split. * by intros Hl; rewrite Hl. * intros [??]; auto using contains_Permutation_length_eq. Qed. Lemma NoDup_contains l k : NoDup l → (∀ x, x ∈ l → x ∈ k) → l `contains` k. Proof. intros Hl. revert k. induction Hl as [|x l Hx ? IH]. { intros k Hk. by apply contains_nil_l. } intros k Hlk. destruct (elem_of_list_split k x) as (l1&l2&?); subst. { apply Hlk. by constructor. } rewrite <-Permutation_middle. apply contains_skip, IH. intros y Hy. rewrite elem_of_app. specialize (Hlk y). rewrite elem_of_app, !elem_of_cons in Hlk. by destruct Hlk as [?|[?|?]]; subst; eauto. Qed. Lemma NoDup_Permutation l k : NoDup l → NoDup k → (∀ x, x ∈ l ↔ x ∈ k) → l ≡ₚ k. Proof. intros. apply (anti_symmetric contains); apply NoDup_contains; naive_solver. Qed. Section contains_dec. Context `{∀ x y, Decision (x = y)}. Lemma list_remove_Permutation l1 l2 k1 x : l1 ≡ₚ l2 → list_remove x l1 = Some k1 → ∃ k2, list_remove x l2 = Some k2 ∧ k1 ≡ₚ k2. Proof. intros Hl. revert k1. induction Hl as [|y l1 l2 ? IH|y1 y2 l|l1 l2 l3 ? IH1 ? IH2]; simpl; intros k1 Hk1. * done. * case_decide; simplify_equality; eauto. destruct (list_remove x l1) as [l|] eqn:?; simplify_equality. destruct (IH l) as (?&?&?); simplify_option_equality; eauto. * repeat case_decide; simplify_option_equality; eauto using Permutation_swap. * destruct (IH1 k1) as (k2&?&?); trivial. destruct (IH2 k2) as (k3&?&?); trivial. exists k3. split; eauto. by transitivity k2. Qed. Lemma list_remove_Some l k x : list_remove x l = Some k → l ≡ₚ x :: k. Proof. revert k. induction l as [|y l IH]; simpl; intros k ?; [done |]. case_decide; simplify_option_equality; [done|]. by rewrite Permutation_swap, <-IH. Qed. Lemma list_remove_Some_inv l k x : l ≡ₚ x :: k → ∃ k', list_remove x l = Some k' ∧ k ≡ₚ k'. Proof. intros. destruct (list_remove_Permutation (x :: k) l k x) as (k'&?&?). * done. * simpl; by case_decide. * by exists k'. Qed. Lemma list_remove_list_contains l1 l2 : l1 `contains` l2 ↔ is_Some (list_remove_list l1 l2). Proof. split. * revert l2. induction l1 as [|x l1 IH]; simpl. { intros l2 _. by exists l2. } intros l2. rewrite contains_cons_l. intros (k&Hk&?). destruct (list_remove_Some_inv l2 k x) as (k2&?&Hk2); trivial. simplify_option_equality. apply IH. by rewrite <-Hk2. * intros [k Hk]. revert l2 k Hk. induction l1 as [|x l1 IH]; simpl; intros l2 k. { intros. apply contains_nil_l. } destruct (list_remove x l2) as [k'|] eqn:?; intros; simplify_equality. rewrite contains_cons_l. eauto using list_remove_Some. Qed. Global Instance contains_dec l1 l2 : Decision (l1 `contains` l2). Proof. refine (cast_if (decide (is_Some (list_remove_list l1 l2)))); abstract (rewrite list_remove_list_contains; tauto). Defined. Global Instance Permutation_dec l1 l2 : Decision (l1 ≡ₚ l2). Proof. refine (cast_if_and (decide (length l1 = length l2)) (decide (l1 `contains` l2))); abstract (rewrite Permutation_alt; tauto). Defined. End contains_dec. End more_general_properties. (** ** Properties of the [same_length] predicate *) Instance: ∀ A, Reflexive (@same_length A A). Proof. intros A l. induction l; constructor; auto. Qed. Instance: ∀ A, Symmetric (@same_length A A). Proof. induction 1; constructor; auto. Qed. Section same_length. Context {A B : Type}. Implicit Types l : list A. Implicit Types k : list B. Lemma same_length_length_1 l k : l `same_length` k → length l = length k. Proof. induction 1; simpl; auto. Qed. Lemma same_length_length_2 l k : length l = length k → l `same_length` k. Proof. revert k. induction l; intros [|??]; try discriminate; constructor; auto with arith. Qed. Lemma same_length_length l k : l `same_length` k ↔ length l = length k. Proof. split; auto using same_length_length_1, same_length_length_2. Qed. Lemma same_length_lookup l k i : l `same_length` k → is_Some (l !! i) → is_Some (k !! i). Proof. rewrite same_length_length. rewrite !lookup_lt_is_Some. lia. Qed. Lemma same_length_take l k n : l `same_length` k → take n l `same_length` take n k. Proof. intros Hl. revert n; induction Hl; intros [|n]; constructor; auto. Qed. Lemma same_length_drop l k n : l `same_length` k → drop n l `same_length` drop n k. Proof. intros Hl. revert n; induction Hl; intros [|]; simpl; try constructor; auto. Qed. Lemma same_length_resize l k x y n : resize n x l `same_length` resize n y k. Proof. apply same_length_length. by rewrite !resize_length. Qed. End same_length. (** ** Properties of the [Forall] and [Exists] predicate *) Section Forall_Exists. Context {A} (P : A → Prop). Definition Forall_nil_2 := @Forall_nil A. Definition Forall_cons_2 := @Forall_cons A. Lemma Forall_forall l : Forall P l ↔ ∀ x, x ∈ l → P x. Proof. split. * induction 1; inversion 1; subst; auto. * intros Hin. induction l; constructor. + apply Hin. constructor. + apply IHl. intros ??. apply Hin. by constructor. Qed. Lemma Forall_nil : Forall P [] ↔ True. Proof. done. Qed. Lemma Forall_cons_1 x l : Forall P (x :: l) → P x ∧ Forall P l. Proof. by inversion 1. Qed. Lemma Forall_cons x l : Forall P (x :: l) ↔ P x ∧ Forall P l. Proof. split. by inversion 1. intros [??]. by constructor. Qed. Lemma Forall_singleton x : Forall P [x] ↔ P x. Proof. rewrite Forall_cons, Forall_nil; tauto. Qed. Lemma Forall_app_2 l1 l2 : Forall P l1 → Forall P l2 → Forall P (l1 ++ l2). Proof. induction 1; simpl; auto. Qed. Lemma Forall_app l1 l2 : Forall P (l1 ++ l2) ↔ Forall P l1 ∧ Forall P l2. Proof. split. * induction l1; inversion 1; intuition. * intros [??]; auto using Forall_app_2. Qed. Lemma Forall_true l : (∀ x, P x) → Forall P l. Proof. induction l; auto. Qed. Lemma Forall_impl l (Q : A → Prop) : Forall P l → (∀ x, P x → Q x) → Forall Q l. Proof. intros H ?. induction H; auto. Defined. Global Instance Forall_proper: Proper (pointwise_relation _ (↔) ==> (=) ==> (↔)) (@Forall A). Proof. split; subst; induction 1; constructor; firstorder. Qed. Lemma Forall_iff l (Q : A → Prop) : (∀ x, P x ↔ Q x) → Forall P l ↔ Forall Q l. Proof. intros H. apply Forall_proper. red. apply H. done. Qed. Lemma Forall_delete l i : Forall P l → Forall P (delete i l). Proof. intros H. revert i. by induction H; intros [|i]; try constructor. Qed. Lemma Forall_lookup l : Forall P l ↔ ∀ i x, l !! i = Some x → P x. Proof. rewrite Forall_forall. setoid_rewrite elem_of_list_lookup. naive_solver. Qed. Lemma Forall_lookup_1 l i x : Forall P l → l !! i = Some x → P x. Proof. rewrite Forall_lookup. eauto. Qed. Lemma Forall_lookup_2 l : (∀ i x, l !! i = Some x → P x) → Forall P l. Proof. by rewrite Forall_lookup. Qed. Lemma Forall_alter f l i : Forall P l → (∀ x, l !! i = Some x → P x → P (f x)) → Forall P (alter f i l). Proof. intros Hl. revert i. induction Hl; simpl; intros [|i]; constructor; auto. Qed. Lemma Forall_replicate n x : P x → Forall P (replicate n x). Proof. induction n; simpl; constructor; auto. Qed. Lemma Forall_replicate_eq n (x : A) : Forall (=x) (replicate n x). Proof. induction n; simpl; constructor; auto. Qed. Lemma Forall_take n l : Forall P l → Forall P (take n l). Proof. intros Hl. revert n. induction Hl; intros [|?]; simpl; auto. Qed. Lemma Forall_drop n l : Forall P l → Forall P (drop n l). Proof. intros Hl. revert n. induction Hl; intros [|?]; simpl; auto. Qed. Lemma Forall_resize n x l : P x → Forall P l → Forall P (resize n x l). Proof. intros ? Hl. revert n. induction Hl; intros [|?]; simpl; auto using Forall_replicate. Qed. Lemma Forall_sublist_lookup l i n k : sublist_lookup i n l = Some k → Forall P l → Forall P k. Proof. unfold sublist_lookup. intros; simplify_option_equality. auto using Forall_take, Forall_drop. Qed. Lemma Forall_sublist_insert l i k : Forall P l → Forall P k → Forall P (sublist_insert i k l). Proof. unfold sublist_insert. auto using Forall_app_2, Forall_drop, Forall_take. Qed. Lemma Forall_reshape l szs : Forall P l → Forall (Forall P) (reshape szs l). Proof. revert l. induction szs; simpl; auto using Forall_take, Forall_drop. Qed. Lemma Forall_rev_ind (Q : list A → Prop) : Q [] → (∀ x l, P x → Forall P l → Q l → Q (l ++ [x])) → ∀ l, Forall P l → Q l. Proof. intros ?? l. induction l using rev_ind; auto. rewrite Forall_app, Forall_singleton; intros [??]; auto. Qed. Lemma Exists_exists l : Exists P l ↔ ∃ x, x ∈ l ∧ P x. Proof. split. * induction 1 as [x|y ?? IH]. + exists x. split. constructor. done. + destruct IH as [x [??]]. exists x. split. by constructor. done. * intros [x [Hin ?]]. induction l. + by destruct (not_elem_of_nil x). + inversion Hin; subst. by left. right; auto. Qed. Lemma Exists_inv x l : Exists P (x :: l) → P x ∨ Exists P l. Proof. inversion 1; intuition trivial. Qed. Lemma Exists_app l1 l2 : Exists P (l1 ++ l2) ↔ Exists P l1 ∨ Exists P l2. Proof. split. * induction l1; inversion 1; intuition. * intros [H|H]; [induction H | induction l1]; simpl; intuition. Qed. Global Instance Exists_proper: Proper (pointwise_relation _ (↔) ==> (=) ==> (↔)) (@Exists A). Proof. split; subst; (induction 1; [left|right]; firstorder auto). Qed. Lemma Exists_not_Forall l : Exists (not ∘ P) l → ¬Forall P l. Proof. induction 1; inversion_clear 1; contradiction. Qed. Lemma Forall_not_Exists l : Forall (not ∘ P) l → ¬Exists P l. Proof. induction 1; inversion_clear 1; contradiction. Qed. Context {dec : ∀ x, Decision (P x)}. Fixpoint Forall_Exists_dec l : {Forall P l} + {Exists (not ∘ P) l}. Proof. refine ( match l with | [] => left _ | x :: l => cast_if_and (dec x) (Forall_Exists_dec l) end); clear Forall_Exists_dec; abstract intuition. Defined. Lemma not_Forall_Exists l : ¬Forall P l → Exists (not ∘ P) l. Proof. intro. destruct (Forall_Exists_dec l); intuition. Qed. Global Instance Forall_dec l : Decision (Forall P l) := match Forall_Exists_dec l with | left H => left H | right H => right (Exists_not_Forall _ H) end. Fixpoint Exists_Forall_dec l : {Exists P l} + {Forall (not ∘ P) l}. Proof. refine ( match l with | [] => right _ | x :: l => cast_if_or (dec x) (Exists_Forall_dec l) end); clear Exists_Forall_dec; abstract intuition. Defined. Lemma not_Exists_Forall l : ¬Exists P l → Forall (not ∘ P) l. Proof. intro. destruct (Exists_Forall_dec l); intuition. Qed. Global Instance Exists_dec l : Decision (Exists P l) := match Exists_Forall_dec l with | left H => left H | right H => right (Forall_not_Exists _ H) end. End Forall_Exists. Lemma replicate_as_Forall {A} (x : A) n l : l = replicate n x ↔ length l = n ∧ Forall (x =) l. Proof. rewrite replicate_as_elem_of, Forall_forall. naive_solver. Qed. Lemma Forall_swap {A B} (Q : A → B → Prop) l1 l2 : Forall (λ y, Forall (Q y) l1) l2 ↔ Forall (λ x, Forall (flip Q x) l2) l1. Proof. repeat setoid_rewrite Forall_forall. simpl. split; eauto. Qed. Lemma Forall_seq (P : nat → Prop) i n : Forall P (seq i n) ↔ ∀ j, i ≤ j < i + n → P j. Proof. rewrite Forall_lookup. split. * intros H j [??]. apply (H (j - i)). rewrite lookup_seq; auto with f_equal lia. * intros H j x Hj. apply lookup_seq_inv in Hj. destruct Hj; subst. auto with lia. Qed. (** ** Properties of the [Forall2] predicate *) Section Forall2. Context {A B} (P : A → B → Prop). Lemma Forall2_nil_inv_l k : Forall2 P [] k → k = []. Proof. by inversion 1. Qed. Lemma Forall2_nil_inv_r k : Forall2 P k [] → k = []. Proof. by inversion 1. Qed. Lemma Forall2_cons_inv l1 l2 x1 x2 : Forall2 P (x1 :: l1) (x2 :: l2) → P x1 x2 ∧ Forall2 P l1 l2. Proof. by inversion 1. Qed. Lemma Forall2_cons_inv_l l1 k x1 : Forall2 P (x1 :: l1) k → ∃ x2 l2, P x1 x2 ∧ Forall2 P l1 l2 ∧ k = x2 :: l2. Proof. inversion 1; subst; eauto. Qed. Lemma Forall2_cons_inv_r k l2 x2 : Forall2 P k (x2 :: l2) → ∃ x1 l1, P x1 x2 ∧ Forall2 P l1 l2 ∧ k = x1 :: l1. Proof. inversion 1; subst; eauto. Qed. Lemma Forall2_cons_nil_inv l1 x1 : Forall2 P (x1 :: l1) [] → False. Proof. by inversion 1. Qed. Lemma Forall2_nil_cons_inv l2 x2 : Forall2 P [] (x2 :: l2) → False. Proof. by inversion 1. Qed. Lemma Forall2_app_inv l1 l2 k1 k2 : l1 `same_length` k1 → Forall2 P (l1 ++ l2) (k1 ++ k2) → Forall2 P l1 k1 ∧ Forall2 P l2 k2. Proof. induction 1. done. inversion 1; naive_solver. Qed. Lemma Forall2_app_inv_l l1 l2 k : Forall2 P (l1 ++ l2) k → ∃ k1 k2, Forall2 P l1 k1 ∧ Forall2 P l2 k2 ∧ k = k1 ++ k2. Proof. revert k. induction l1; simpl; inversion 1; naive_solver. Qed. Lemma Forall2_app_inv_r l k1 k2 : Forall2 P l (k1 ++ k2) → ∃ l1 l2, Forall2 P l1 k1 ∧ Forall2 P l2 k2 ∧ l = l1 ++ l2. Proof. revert l. induction k1; simpl; inversion 1; naive_solver. Qed. Lemma Forall2_length l1 l2 : Forall2 P l1 l2 → length l1 = length l2. Proof. induction 1; simpl; auto. Qed. Lemma Forall2_length_l l1 l2 n : Forall2 P l1 l2 → length l1 = n → length l2 = n. Proof. intros ? <-; symmetry. by apply Forall2_length. Qed. Lemma Forall2_length_r l1 l2 n : Forall2 P l1 l2 → length l2 = n → length l1 = n. Proof. intros ? <-. by apply Forall2_length. Qed. Lemma Forall2_same_length l1 l2 : Forall2 P l1 l2 → l1 `same_length` l2. Proof. induction 1; constructor; auto. Qed. Lemma Forall2_flip l1 l2 : Forall2 P l1 l2 ↔ Forall2 (flip P) l2 l1. Proof. split; induction 1; constructor; auto. Qed. Lemma Forall2_impl (Q : A → B → Prop) l1 l2 : Forall2 P l1 l2 → (∀ x y, P x y → Q x y) → Forall2 Q l1 l2. Proof. intros H ?. induction H; auto. Defined. Lemma Forall2_unique l k1 k2 : Forall2 P l k1 → Forall2 P l k2 → (∀ x y1 y2, P x y1 → P x y2 → y1 = y2) → k1 = k2. Proof. intros H. revert k2. induction H; inversion_clear 1; intros; f_equal; eauto. Qed. Lemma Forall2_Forall_l (Q : A → Prop) l k : Forall2 P l k → Forall (λ y, ∀ x, P x y → Q x) k → Forall Q l. Proof. induction 1; inversion_clear 1; eauto. Qed. Lemma Forall2_Forall_r (Q : B → Prop) l k : Forall2 P l k → Forall (λ x, ∀ y, P x y → Q y) l → Forall Q k. Proof. induction 1; inversion_clear 1; eauto. Qed. Lemma Forall2_lookup_lr l1 l2 i x y : Forall2 P l1 l2 → l1 !! i = Some x → l2 !! i = Some y → P x y. Proof. intros H. revert i. induction H; [done|]. intros [|?] ??; simplify_equality'; eauto. Qed. Lemma Forall2_lookup_l l1 l2 i x : Forall2 P l1 l2 → l1 !! i = Some x → ∃ y, l2 !! i = Some y ∧ P x y. Proof. intros H. revert i. induction H; [done|]. intros [|?] ?; simplify_equality'; eauto. Qed. Lemma Forall2_lookup_r l1 l2 i y : Forall2 P l1 l2 → l2 !! i = Some y → ∃ x, l1 !! i = Some x ∧ P x y. Proof. intros H. revert i. induction H; [done|]. intros [|?] ?; simplify_equality'; eauto. Qed. Lemma Forall2_lookup_2 l1 l2 : l1 `same_length` l2 → (∀ i x y, l1 !! i = Some x → l2 !! i = Some y → P x y) → Forall2 P l1 l2. Proof. eauto using Forall2_same_length, Forall2_lookup_lr. intros Hl Hlookup. induction Hl as [|????? IH]; constructor. * by apply (Hlookup 0). * apply IH. intros i. apply (Hlookup (S i)). Qed. Lemma Forall2_lookup l1 l2 : Forall2 P l1 l2 ↔ l1 `same_length` l2 ∧ (∀ i x y, l1 !! i = Some x → l2 !! i = Some y → P x y). Proof. split. * eauto using Forall2_same_length, Forall2_lookup_lr. * intros [??]; eauto using Forall2_lookup_2. Qed. Lemma Forall2_alter_l f l1 l2 i : Forall2 P l1 l2 → (∀ x1 x2, l1 !! i = Some x1 → l2 !! i = Some x2 → P x1 x2 → P (f x1) x2) → Forall2 P (alter f i l1) l2. Proof. intros Hl. revert i. induction Hl; simpl; intros [|i]; constructor; auto. Qed. Lemma Forall2_alter_r f l1 l2 i : Forall2 P l1 l2 → (∀ x1 x2, l1 !! i = Some x1 → l2 !! i = Some x2 → P x1 x2 → P x1 (f x2)) → Forall2 P l1 (alter f i l2). Proof. intros Hl. revert i. induction Hl; simpl; intros [|i]; constructor; auto. Qed. Lemma Forall2_alter f g l1 l2 i : Forall2 P l1 l2 → (∀ x1 x2, l1 !! i = Some x1 → l2 !! i = Some x2 → P x1 x2 → P (f x1) (g x2)) → Forall2 P (alter f i l1) (alter g i l2). Proof. intros Hl. revert i. induction Hl; simpl; intros [|i]; constructor; auto. Qed. Lemma Forall2_delete l1 l2 i : Forall2 P l1 l2 → Forall2 P (delete i l1) (delete i l2). Proof. intros Hl12. revert i. induction Hl12; intros [|i]; simpl; intuition. Qed. Lemma Forall2_replicate_l l n x : Forall (P x) l → length l = n → Forall2 P (replicate n x) l. Proof. intros Hl. revert n. induction Hl; intros [|?] ?; simplify_equality; constructor; auto. Qed. Lemma Forall2_replicate_r l n x : Forall (flip P x) l → length l = n → Forall2 P l (replicate n x). Proof. intros Hl. revert n. induction Hl; intros [|?] ?; simplify_equality; constructor; auto. Qed. Lemma Forall2_replicate n x1 x2 : P x1 x2 → Forall2 P (replicate n x1) (replicate n x2). Proof. induction n; simpl; constructor; auto. Qed. Lemma Forall2_take l1 l2 n : Forall2 P l1 l2 → Forall2 P (take n l1) (take n l2). Proof. intros Hl1l2. revert n. induction Hl1l2; intros [|?]; simpl; auto. Qed. Lemma Forall2_drop l1 l2 n : Forall2 P l1 l2 → Forall2 P (drop n l1) (drop n l2). Proof. intros Hl1l2. revert n. induction Hl1l2; intros [|?]; simpl; auto. Qed. Lemma Forall2_resize l1 l2 x1 x2 n : P x1 x2 → Forall2 P l1 l2 → Forall2 P (resize n x1 l1) (resize n x2 l2). Proof. intros. rewrite !resize_spec, (Forall2_length l1 l2) by done. auto using Forall2_app, Forall2_take, Forall2_replicate. Qed. Lemma Forall2_resize_ge_l l1 l2 x1 x2 n m : P x1 x2 → Forall (flip P x2) l1 → n ≤ m → Forall2 P (resize n x1 l1) l2 → Forall2 P (resize m x1 l1) (resize m x2 l2). Proof. intros. assert (n = length l2). { by rewrite <-(Forall2_length (resize n x1 l1) l2), resize_length. } rewrite (le_plus_minus n m) by done. subst. rewrite !resize_plus, resize_all, drop_all, resize_nil. apply Forall2_app; [done |]. apply Forall2_replicate_r; [| by rewrite resize_length]. eauto using Forall_resize, Forall_drop. Qed. Lemma Forall2_resize_ge_r l1 l2 x1 x2 n m : P x1 x2 → Forall (P x1) l2 → n ≤ m → Forall2 P l1 (resize n x2 l2) → Forall2 P (resize m x1 l1) (resize m x2 l2). Proof. intros. assert (n = length l1). { by rewrite (Forall2_length l1 (resize n x2 l2)), resize_length. } rewrite (le_plus_minus n m) by done. subst. rewrite !resize_plus, resize_all, drop_all, resize_nil. apply Forall2_app; [done |]. apply Forall2_replicate_l; [| by rewrite resize_length]. eauto using Forall_resize, Forall_drop. Qed. Lemma Forall2_sublist_lookup_l l1 l2 n i k1 : Forall2 P l1 l2 → sublist_lookup n i l1 = Some k1 → ∃ k2, sublist_lookup n i l2 = Some k2 ∧ Forall2 P k1 k2. Proof. unfold sublist_lookup. intros Hl12 Hl1. exists (take i (drop n l2)); simplify_option_equality. * auto using Forall2_take, Forall2_drop. * apply Forall2_length in Hl12; lia. Qed. Lemma Forall2_sublist_lookup_r l1 l2 n i k2 : Forall2 P l1 l2 → sublist_lookup n i l2 = Some k2 → ∃ k1, sublist_lookup n i l1 = Some k1 ∧ Forall2 P k1 k2. Proof. unfold sublist_lookup. intros Hl12 Hl2. exists (take i (drop n l1)); simplify_option_equality. * auto using Forall2_take, Forall2_drop. * apply Forall2_length in Hl12; lia. Qed. Lemma Forall2_sublist_insert l1 l2 i k1 k2 : Forall2 P l1 l2 → Forall2 P k1 k2 → Forall2 P (sublist_insert i k1 l1) (sublist_insert i k2 l2). Proof. unfold sublist_insert. intros. erewrite !Forall2_length by eauto. auto using Forall2_app, Forall2_take, Forall2_drop. Qed. Lemma Forall2_transitive {C} (Q : B → C → Prop) (R : A → C → Prop) l1 l2 l3 : (∀ x1 x2 x3, P x1 x2 → Q x2 x3 → R x1 x3) → Forall2 P l1 l2 → Forall2 Q l2 l3 → Forall2 R l1 l3. Proof. intros ? Hl1l2. revert l3. induction Hl1l2; inversion_clear 1; eauto. Qed. Lemma Forall2_Forall (Q : A → A → Prop) l : Forall (λ x, Q x x) l → Forall2 Q l l. Proof. induction 1; constructor; auto. Qed. Global Instance Forall2_dec `{∀ x1 x2, Decision (P x1 x2)} : ∀ l1 l2, Decision (Forall2 P l1 l2). Proof. refine ( fix go l1 l2 : Decision (Forall2 P l1 l2) := match l1, l2 with | [], [] => left _ | x1 :: l1, x2 :: l2 => cast_if_and (decide (P x1 x2)) (go l1 l2) | _, _ => right _ end); clear go; abstract first [by constructor | by inversion 1]. Defined. End Forall2. Section Forall2_order. Context {A} (R : relation A). Global Instance: Reflexive R → Reflexive (Forall2 R). Proof. intros ? l. induction l; by constructor. Qed. Global Instance: Symmetric R → Symmetric (Forall2 R). Proof. intros. induction 1; constructor; auto. Qed. Global Instance: Transitive R → Transitive (Forall2 R). Proof. intros ????. apply Forall2_transitive. apply transitivity. Qed. Global Instance: Equivalence R → Equivalence (Forall2 R). Proof. split; apply _. Qed. Global Instance: PreOrder R → PreOrder (Forall2 R). Proof. split; apply _. Qed. Global Instance: AntiSymmetric (=) R → AntiSymmetric (=) (Forall2 R). Proof. induction 2; inversion_clear 1; f_equal; auto. Qed. Global Instance: Proper (R ==> Forall2 R ==> Forall2 R) (::). Proof. by constructor. Qed. Global Instance: Proper (Forall2 R ==> Forall2 R ==> Forall2 R) (++). Proof. repeat intro. eauto using Forall2_app. Qed. Global Instance: Proper (Forall2 R ==> Forall2 R) (delete i). Proof. repeat intro. eauto using Forall2_delete. Qed. Global Instance: Proper (R ==> Forall2 R) (replicate n). Proof. repeat intro. eauto using Forall2_replicate. Qed. Global Instance: Proper (Forall2 R ==> Forall2 R) (take n). Proof. repeat intro. eauto using Forall2_take. Qed. Global Instance: Proper (Forall2 R ==> Forall2 R) (drop n). Proof. repeat intro. eauto using Forall2_drop. Qed. Global Instance: Proper (R ==> Forall2 R ==> Forall2 R) (resize n). Proof. repeat intro. eauto using Forall2_resize. Qed. Global Instance: Proper ((=) ==> R ==> Forall2 R ==> Forall2 R) insert. Proof. repeat intro. subst. apply Forall2_alter; auto. Qed. End Forall2_order. (** * Properties of the monadic operations *) Section fmap. Context {A B : Type} (f : A → B). Lemma list_fmap_id (l : list A) : id <\$> l = l. Proof. induction l; simpl; f_equal; auto. Qed. Lemma list_fmap_compose {C} (g : B → C) l : g ∘ f <\$> l = g <\$> f <\$> l. Proof. induction l; simpl; f_equal; auto. Qed. Lemma list_fmap_ext (g : A → B) (l1 l2 : list A) : (∀ x, f x = g x) → l1 = l2 → fmap f l1 = fmap g l2. Proof. intros ? <-. induction l1; simpl; f_equal; auto. Qed. Global Instance: Injective (=) (=) f → Injective (=) (=) (fmap f). Proof. intros ? l1. induction l1 as [|x l1 IH]. * by intros [|??]. * intros [|??]; simpl; intros; f_equal; simplify_equality; auto. Qed. Lemma fmap_app l1 l2 : f <\$> l1 ++ l2 = (f <\$> l1) ++ (f <\$> l2). Proof. induction l1; simpl; by f_equal. Qed. Lemma fmap_nil_inv k : f <\$> k = [] → k = []. Proof. by destruct k. Qed. Lemma fmap_cons_inv y l k : f <\$> l = y :: k → ∃ x l', y = f x ∧ k = f <\$> l' ∧ l = x :: l'. Proof. intros. destruct l; simplify_equality'; eauto. Qed. Lemma fmap_app_inv l k1 k2 : f <\$> l = k1 ++ k2 → ∃ l1 l2, k1 = f <\$> l1 ∧ k2 = f <\$> l2 ∧ l = l1 ++ l2. Proof. revert l. induction k1 as [|y k1 IH]; simpl. * intros l ?. by eexists [], l. * intros [|x l] ?; simplify_equality'. destruct (IH l) as [l1 [l2 [? [??]]]]; subst; [done |]. by exists (x :: l1) l2. Qed. Lemma fmap_length l : length (f <\$> l) = length l. Proof. induction l; simpl; by f_equal. Qed. Lemma fmap_reverse l : f <\$> reverse l = reverse (f <\$> l). Proof. induction l; simpl; [done |]. by rewrite !reverse_cons, fmap_app, IHl. Qed. Lemma fmap_replicate n x : f <\$> replicate n x = replicate n (f x). Proof. induction n; simpl; f_equal; auto. Qed. Lemma list_lookup_fmap l i : (f <\$> l) !! i = f <\$> (l !! i). Proof. revert i. induction l; by intros [|]. Qed. Lemma list_lookup_fmap_inv l i x : (f <\$> l) !! i = Some x → ∃ y, x = f y ∧ l !! i = Some y. Proof. intros Hi. rewrite list_lookup_fmap in Hi. destruct (l !! i) eqn:?; simplify_equality; eauto. Qed. Lemma list_alter_fmap (g : A → A) (h : B → B) l i : Forall (λ x, f (g x) = h (f x)) l → f <\$> alter g i l = alter h i (f <\$> l). Proof. intros Hl. revert i. induction Hl; intros [|i]; simpl; f_equal; auto. Qed. Lemma elem_of_list_fmap_1 l x : x ∈ l → f x ∈ f <\$> l. Proof. induction 1; simpl; rewrite elem_of_cons; intuition. Qed. Lemma elem_of_list_fmap_1_alt l x y : x ∈ l → y = f x → y ∈ f <\$> l. Proof. intros. subst. by apply elem_of_list_fmap_1. Qed. Lemma elem_of_list_fmap_2 l x : x ∈ f <\$> l → ∃ y, x = f y ∧ y ∈ l. Proof. induction l as [|y l IH]; simpl; inversion_clear 1. + exists y. split; [done | by left]. + destruct IH as [z [??]]. done. exists z. split; [done | by right]. Qed. Lemma elem_of_list_fmap l x : x ∈ f <\$> l ↔ ∃ y, x = f y ∧ y ∈ l. Proof. firstorder eauto using elem_of_list_fmap_1_alt, elem_of_list_fmap_2. Qed. Lemma fmap_nodup_1 l : NoDup (f <\$> l) → NoDup l. Proof. induction l; simpl; inversion_clear 1; constructor; auto. rewrite elem_of_list_fmap in *. naive_solver. Qed. Lemma fmap_nodup_2 `{!Injective (=) (=) f} l : NoDup l → NoDup (f <\$> l). Proof. induction 1; simpl; constructor; trivial. rewrite elem_of_list_fmap. intros [y [Hxy ?]]. apply (injective f) in Hxy. by subst. Qed. Lemma fmap_nodup `{!Injective (=) (=) f} l : NoDup (f <\$> l) ↔ NoDup l. Proof. split; auto using fmap_nodup_1, fmap_nodup_2. Qed. Global Instance fmap_sublist: Proper (sublist ==> sublist) (fmap f). Proof. induction 1; simpl; econstructor; eauto. Qed. Global Instance fmap_contains: Proper (contains ==> contains) (fmap f). Proof. induction 1; simpl; econstructor; eauto. Qed. Global Instance fmap_Permutation: Proper ((≡ₚ) ==> (≡ₚ)) (fmap f). Proof. induction 1; simpl; econstructor; eauto. Qed. Lemma Forall_fmap_ext (g : A → B) (l : list A) : Forall (λ x, f x = g x) l ↔ fmap f l = fmap g l. Proof. split. * induction 1; simpl; f_equal; auto. * induction l; simpl; constructor; simplify_equality; auto. Qed. Lemma Forall_fmap (P : B → Prop) l : Forall P (f <\$> l) ↔ Forall (P ∘ f) l. Proof. split; induction l; inversion_clear 1; constructor; auto. Qed. Lemma Forall2_fmap_l {C} (P : B → C → Prop) l1 l2 : Forall2 P (f <\$> l1) l2 ↔ Forall2 (P ∘ f) l1 l2. Proof. split; revert l2; induction l1; inversion_clear 1; constructor; auto. Qed. Lemma Forall2_fmap_r {C} (P : C → B → Prop) l1 l2 : Forall2 P l1 (f <\$> l2) ↔ Forall2 (λ x, P x ∘ f) l1 l2. Proof. split; revert l1; induction l2; inversion_clear 1; constructor; auto. Qed. Lemma Forall2_fmap_1 {C D} (g : C → D) (P : B → D → Prop) l1 l2 : Forall2 P (f <\$> l1) (g <\$> l2) → Forall2 (λ x1 x2, P (f x1) (g x2)) l1 l2. Proof. revert l2; induction l1; intros [|??]; inversion_clear 1; auto. Qed. Lemma Forall2_fmap_2 {C D} (g : C → D) (P : B → D → Prop) l1 l2 : Forall2 (λ x1 x2, P (f x1) (g x2)) l1 l2 → Forall2 P (f <\$> l1) (g <\$> l2). Proof. induction 1; simpl; auto. Qed. Lemma Forall2_fmap {C D} (g : C → D) (P : B → D → Prop) l1 l2 : Forall2 P (f <\$> l1) (g <\$> l2) ↔ Forall2 (λ x1 x2, P (f x1) (g x2)) l1 l2. Proof. split; auto using Forall2_fmap_1, Forall2_fmap_2. Qed. Lemma list_fmap_bind {C} (g : B → list C) l : (f <\$> l) ≫= g = l ≫= g ∘ f. Proof. induction l; simpl; f_equal; auto. Qed. End fmap. Lemma NoDup_fmap_fst {A B} (l : list (A * B)) : (∀ x y1 y2, (x,y1) ∈ l → (x,y2) ∈ l → y1 = y2) → NoDup l → NoDup (fst <\$> l). Proof. intros Hunique. induction 1 as [|[x1 y1] l Hin Hnodup IH]; simpl; constructor. * rewrite elem_of_list_fmap. intros [[x2 y2] [??]]; simpl in *; subst. destruct Hin. rewrite (Hunique x2 y1 y2); rewrite ?elem_of_cons; auto. * apply IH. intros. eapply Hunique; rewrite ?elem_of_cons; eauto. Qed. Section bind. Context {A B : Type} (f : A → list B). Lemma list_bind_ext (g : A → list B) l1 l2 : (∀ x, f x = g x) → l1 = l2 → l1 ≫= f = l2 ≫= g. Proof. intros ? <-. induction l1; simpl; f_equal; auto. Qed. Lemma Forall_bind_ext (g : A → list B) (l : list A) : Forall (λ x, f x = g x) l → l ≫= f = l ≫= g. Proof. induction 1; simpl; f_equal; auto. Qed. Global Instance bind_sublist: Proper (sublist ==> sublist) (mbind f). Proof. induction 1; simpl; auto. * done. * by apply sublist_app. * by apply sublist_inserts_l. Qed. Global Instance bind_contains: Proper (contains ==> contains) (mbind f). Proof. induction 1; simpl; auto. * done. * by apply contains_app. * by rewrite !(associative_L (++)), (commutative (++) (f _)). * by apply contains_inserts_l. * etransitivity; eauto. Qed. Global Instance bind_Permutation: Proper ((≡ₚ) ==> (≡ₚ)) (mbind f). Proof. induction 1; simpl; auto. * by f_equiv. * by rewrite !(associative_L (++)), (commutative (++) (f _)). * etransitivity; eauto. Qed. Lemma bind_app (l1 l2 : list A) : (l1 ++ l2) ≫= f = (l1 ≫= f) ++ (l2 ≫= f). Proof. induction l1; simpl; [done|]. by rewrite <-(associative_L (++)), IHl1. Qed. Lemma elem_of_list_bind (x : B) (l : list A) : x ∈ l ≫= f ↔ ∃ y, x ∈ f y ∧ y ∈ l. Proof. split. * induction l as [|y l IH]; simpl; [inversion 1|]. rewrite elem_of_app. intros [?|?]. + exists y. split; [done | by left]. + destruct IH as [z [??]]. done. exists z. split; [done | by right]. * intros [y [Hx Hy]]. induction Hy; simpl; rewrite elem_of_app; intuition. Qed. Lemma Forall2_bind {C D} (g : C → list D) (P : B → D → Prop) l1 l2 : Forall2 (λ x1 x2, Forall2 P (f x1) (g x2)) l1 l2 → Forall2 P (l1 ≫= f) (l2 ≫= g). Proof. induction 1; simpl; auto using Forall2_app. Qed. End bind. Section ret_join. Context {A : Type}. Lemma list_join_bind (ls : list (list A)) : mjoin ls = ls ≫= id. Proof. induction ls; simpl; f_equal; auto. Qed. Global Instance mjoin_Permutation: Proper (@Permutation (list A) ==> (≡ₚ)) mjoin. Proof. intros ?? E. by rewrite !list_join_bind, E. Qed. Lemma elem_of_list_ret (x y : A) : x ∈ @mret list _ A y ↔ x = y. Proof. apply elem_of_list_singleton. Qed. Lemma elem_of_list_join (x : A) (ls : list (list A)) : x ∈ mjoin ls ↔ ∃ l, x ∈ l ∧ l ∈ ls. Proof. by rewrite list_join_bind, elem_of_list_bind. Qed. Lemma join_nil (ls : list (list A)) : mjoin ls = [] ↔ Forall (= []) ls. Proof. split. * by induction ls as [|[|??] ?]; constructor; auto. * by induction 1 as [|[|??] ?]. Qed. Lemma join_nil_1 (ls : list (list A)) : mjoin ls = [] → Forall (= []) ls. Proof. by rewrite join_nil. Qed. Lemma join_nil_2 (ls : list (list A)) : Forall (= []) ls → mjoin ls = []. Proof. by rewrite join_nil. Qed. Lemma join_length (ls : list (list A)) : length (mjoin ls) = foldr (plus ∘ length) 0 ls. Proof. by induction ls; simpl; rewrite ?app_length; f_equal. Qed. Lemma join_length_same (ls : list (list A)) n : Forall (λ l, length l = n) ls → length (mjoin ls) = length ls * n. Proof. rewrite join_length. by induction 1; simpl; f_equal. Qed. Lemma Forall_join (P : A → Prop) (ls: list (list A)) : Forall (Forall P) ls → Forall P (mjoin ls). Proof. induction 1; simpl; auto using Forall_app_2. Qed. Lemma lookup_join_same_length (ls : list (list A)) n i : n ≠ 0 → Forall (λ l, length l = n) ls → mjoin ls !! i = ls !! (i `div` n) ≫= (!! (i `mod` n)). Proof. intros Hn Hls. revert i. induction Hls as [|l ls ? Hls IH]; simpl; [done |]. intros i. destruct (decide (i < n)) as [Hin|Hin]. * rewrite <-(Nat.div_unique i n 0 i), <-(Nat.mod_unique i n 0 i) by lia. simpl. rewrite lookup_app_l; auto with lia. * replace i with ((i - n) + 1 * n) by lia. rewrite Nat.div_add, Nat.mod_add by done. replace (i - n + 1 * n) with (length l + (i - n)) by lia. by rewrite (Nat.add_comm _ 1), lookup_app_r, IH. Qed. (* This should be provable using the previous lemma in a shorter way *) Lemma alter_join_same_length f (ls : list (list A)) n i : n ≠ 0 → Forall (λ l, length l = n) ls → alter f i (mjoin ls) = mjoin (alter (alter f (i `mod` n)) (i `div` n) ls). Proof. intros Hn Hls. revert i. induction Hls as [|l ls ? Hls IH]; simpl; [done |]. intros i. destruct (decide (i < n)) as [Hin|Hin]. * rewrite <-(Nat.div_unique i n 0 i), <-(Nat.mod_unique i n 0 i) by lia. simpl. rewrite alter_app_l; auto with lia. * replace i with ((i - n) + 1 * n) by lia. rewrite Nat.div_add, Nat.mod_add by done. replace (i - n + 1 * n) with i by lia. rewrite (Nat.add_comm _ 1), alter_app_r_alt, IH by lia. by subst. Qed. Lemma insert_join_same_length (ls : list (list A)) n i x : n ≠ 0 → Forall (λ l, length l = n) ls → <[i:=x]>(mjoin ls) = mjoin (alter <[i `mod` n:=x]> (i `div` n) ls). Proof. apply alter_join_same_length. Qed. Lemma Forall2_join {B} (P : A → B → Prop) ls1 ls2 : Forall2 (Forall2 P) ls1 ls2 → Forall2 P (mjoin ls1) (mjoin ls2). Proof. induction 1; simpl; auto using Forall2_app. Qed. End ret_join. Section mapM. Context {A B : Type} (f : A → option B). Lemma mapM_ext (g : A → option B) l : (∀ x, f x = g x) → mapM f l = mapM g l. Proof. intros Hfg. by induction l; simpl; rewrite ?Hfg, ?IHl. Qed. Lemma Forall2_mapM_ext (g : A → option B) l k : Forall2 (λ x y, f x = g y) l k → mapM f l = mapM g k. Proof. induction 1 as [|???? Hfg ? IH]; simpl. done. by rewrite Hfg, IH. Qed. Lemma Forall_mapM_ext (g : A → option B) l : Forall (λ x, f x = g x) l → mapM f l = mapM g l. Proof. induction 1 as [|?? Hfg ? IH]; simpl. done. by rewrite Hfg, IH. Qed. Lemma mapM_Some_1 l k : mapM f l = Some k → Forall2 (λ x y, f x = Some y) l k. Proof. revert k. induction l as [|x l]; intros [|y k]; simpl; try done. * destruct (f x); simpl; [|discriminate]. by destruct (mapM f l). * destruct (f x) eqn:?; simpl; [|discriminate]. destruct (mapM f l); intros; simplify_equality. constructor; auto. Qed. Lemma mapM_Some_2 l k : Forall2 (λ x y, f x = Some y) l k → mapM f l = Some k. Proof. induction 1 as [|???? Hf ? IH]; simpl; [done |]. rewrite Hf. simpl. by rewrite IH. Qed. Lemma mapM_Some l k : mapM f l = Some k ↔ Forall2 (λ x y, f x = Some y) l k. Proof. split; auto using mapM_Some_1, mapM_Some_2. Qed. Lemma mapM_length l k : mapM f l = Some k → length l = length k. Proof. intros. by eapply Forall2_length, mapM_Some_1. Qed. Lemma mapM_None_1 l : mapM f l = None → Exists (λ x, f x = None) l. Proof. induction l as [|x l IH]; simpl; [done|]. destruct (f x) eqn:?; simpl; eauto. by destruct (mapM f l); eauto. Qed. Lemma mapM_None_2 l : Exists (λ x, f x = None) l → mapM f l = None. Proof. induction 1 as [x l Hx|x l ? IH]; simpl; [by rewrite Hx|]. by destruct (f x); simpl; rewrite ?IH. Qed. Lemma mapM_None l : mapM f l = None ↔ Exists (λ x, f x = None) l. Proof. split; auto using mapM_None_1, mapM_None_2. Qed. Lemma mapM_is_Some_1 l : is_Some (mapM f l) → Forall (is_Some ∘ f) l. Proof. unfold compose. setoid_rewrite <-not_eq_None_Some. rewrite mapM_None. apply (not_Exists_Forall _). Qed. Lemma mapM_is_Some_2 l : Forall (is_Some ∘ f) l → is_Some (mapM f l). Proof. unfold compose. setoid_rewrite <-not_eq_None_Some. rewrite mapM_None. apply (Forall_not_Exists _). Qed. Lemma mapM_is_Some l : is_Some (mapM f l) ↔ Forall (is_Some ∘ f) l. Proof. split; auto using mapM_is_Some_1, mapM_is_Some_2. Qed. Lemma mapM_fmap_Some (g : B → A) (l : list B) : (∀ x, f (g x) = Some x) → mapM f (g <\$> l) = Some l. Proof. intros. by induction l; simpl; simplify_option_equality. Qed. Lemma mapM_fmap_Some_inv (g : B → A) (l : list B) (k : list A) : (∀ x y, f y = Some x → y = g x) → mapM f k = Some l → k = g <\$> l. Proof. intros Hgf. revert l; induction k as [|??]; intros [|??] ?; simplify_option_equality; f_equiv; eauto. Qed. End mapM. (** ** Properties of the [permutations] function *) Section permutations. Context {A : Type}. Implicit Types x y z : A. Implicit Types l : list A. Lemma interleave_cons x l : x :: l ∈ interleave x l. Proof. destruct l; simpl; rewrite elem_of_cons; auto. Qed. Lemma interleave_Permutation x l l' : l' ∈ interleave x l → l' ≡ₚ x :: l. Proof. revert l'. induction l as [|y l IH]; intros l'; simpl. * rewrite elem_of_list_singleton. intros. by subst. * rewrite elem_of_cons, elem_of_list_fmap. intros [?|[? [? H]]]; subst. + by constructor. + rewrite (IH _ H). constructor. Qed. Lemma permutations_refl l : l ∈ permutations l. Proof. induction l; simpl. * by apply elem_of_list_singleton. * apply elem_of_list_bind. eauto using interleave_cons. Qed. Lemma permutations_skip x l l' : l ∈ permutations l' → x :: l ∈ permutations (x :: l'). Proof. intros Hl. simpl. apply elem_of_list_bind. eauto using interleave_cons. Qed. Lemma permutations_swap x y l : y :: x :: l ∈ permutations (x :: y :: l). Proof. simpl. apply elem_of_list_bind. exists (y :: l). split; simpl. * destruct l; simpl; rewrite !elem_of_cons; auto. * apply elem_of_list_bind. simpl. eauto using interleave_cons, permutations_refl. Qed. Lemma permutations_nil l : l ∈ permutations [] ↔ l = []. Proof. simpl. by rewrite elem_of_list_singleton. Qed. Lemma interleave_interleave_toggle x1 x2 l1 l2 l3 : l1 ∈ interleave x1 l2 → l2 ∈ interleave x2 l3 → ∃ l4, l1 ∈ interleave x2 l4 ∧ l4 ∈ interleave x1 l3. Proof. revert l1 l2. induction l3 as [|y l3 IH]; intros l1 l2; simpl. { intros Hl1 Hl2. rewrite elem_of_list_singleton in Hl2. subst. simpl in Hl1. rewrite elem_of_cons, elem_of_list_singleton in Hl1. exists [x1]. simpl. rewrite elem_of_cons, !elem_of_list_singleton. tauto. } rewrite elem_of_cons, elem_of_list_fmap. intros Hl1 [? | [l2' [??]]]; subst; simpl in *. * rewrite !elem_of_cons, elem_of_list_fmap in Hl1. destruct Hl1 as [? | [? | [l4 [??]]]]; subst. + exists (x1 :: y :: l3). simpl. rewrite !elem_of_cons. tauto. + exists (x1 :: y :: l3). simpl. rewrite !elem_of_cons. tauto. + exists l4. simpl. rewrite elem_of_cons. auto using interleave_cons. * rewrite elem_of_cons, elem_of_list_fmap in Hl1. destruct Hl1 as [? | [l1' [??]]]; subst. + exists (x1 :: y :: l3). simpl. rewrite !elem_of_cons, !elem_of_list_fmap. split; [| by auto]. right. right. exists (y :: l2'). rewrite elem_of_list_fmap. naive_solver. + destruct (IH l1' l2') as [l4 [??]]; auto. exists (y :: l4). simpl. rewrite !elem_of_cons, !elem_of_list_fmap. naive_solver. Qed. Lemma permutations_interleave_toggle x l1 l2 l3 : l1 ∈ permutations l2 → l2 ∈ interleave x l3 → ∃ l4, l1 ∈ interleave x l4 ∧ l4 ∈ permutations l3. Proof. revert l1 l2. induction l3 as [|y l3 IH]; intros l1 l2; simpl. { intros Hl1 Hl2. eexists []. simpl. split; [| by rewrite elem_of_list_singleton]. rewrite elem_of_list_singleton in Hl2. by rewrite Hl2 in Hl1. } rewrite elem_of_cons, elem_of_list_fmap. intros Hl1 [? | [l2' [? Hl2']]]; subst; simpl in *. * rewrite elem_of_list_bind in Hl1. destruct Hl1 as [l1' [??]]. by exists l1'. * rewrite elem_of_list_bind in Hl1. setoid_rewrite elem_of_list_bind. destruct Hl1 as [l1' [??]]. destruct (IH l1' l2') as (l1''&?&?); auto. destruct (interleave_interleave_toggle y x l1 l1' l1'') as (?&?&?); eauto. Qed. Lemma permutations_trans l1 l2 l3 : l1 ∈ permutations l2 → l2 ∈ permutations l3 → l1 ∈ permutations l3. Proof. revert l1 l2. induction l3 as [|x l3 IH]; intros l1 l2; simpl. * intros Hl1 Hl2. rewrite elem_of_list_singleton in Hl2. by rewrite Hl2 in Hl1. * rewrite !elem_of_list_bind. intros Hl1 [l2' [Hl2 Hl2']]. destruct (permutations_interleave_toggle x l1 l2 l2') as [? [??]]; eauto. Qed. Lemma permutations_Permutation l l' : l' ∈ permutations l ↔ l ≡ₚ l'. Proof. split. * revert l'. induction l; simpl; intros l''. + rewrite elem_of_list_singleton. intros. subst. constructor. + rewrite elem_of_list_bind. intros [l' [Hl'' ?]]. rewrite (interleave_Permutation _ _ _ Hl''). constructor; auto. * induction 1; eauto using permutations_refl, permutations_skip, permutations_swap, permutations_trans. Qed. End permutations. (** ** Properties of the folding functions *) Definition foldr_app := @fold_right_app. Lemma foldl_app {A B} (f : A → B → A) (l k : list B) (a : A) : foldl f a (l ++ k) = foldl f (foldl f a l) k. Proof. revert a. induction l; simpl; auto. Qed. Lemma foldr_permutation {A B} (R : relation B) `{!Equivalence R} (f : A → B → B) (b : B) `{!Proper ((=) ==> R ==> R) f} (Hf : ∀ a1 a2 b, R (f a1 (f a2 b)) (f a2 (f a1 b))) : Proper ((≡ₚ) ==> R) (foldr f b). Proof. induction 1; simpl. * done. * by f_equiv. * apply Hf. * etransitivity; eauto. Qed. Lemma ifoldr_app {A B} (f : nat → B → A → A) (a : nat → A) (l1 l2 : list B) n : ifoldr f a n (l1 ++ l2) = ifoldr f (λ n, ifoldr f a n l2) n l1. Proof. revert n a. induction l1; intros; simpl; f_equal; auto. Qed. (** ** Properties of the [zip_with] and [zip] functions *) Section zip_with. Context {A B C : Type} (f : A → B → C). Lemma zip_with_ext (g : A → B → C) l1 l2 k1 k2 : (∀ x y, f x y = g x y) → l1 = l2 → k1 = k2 → zip_with f l1 k1 = zip_with g l2 k2. Proof. intros ? <- <-. revert k1. induction l1; intros [|??]; simpl; f_equal; auto. Qed. Lemma Forall_zip_with_ext_l (g : A → B → C) l k1 k2 : Forall (λ x, ∀ y, f x y = g x y) l → k1 = k2 → zip_with f l k1 = zip_with g l k2. Proof. intros Hl <-. revert k1. induction Hl; intros [|??]; simpl; f_equal; auto. Qed. Lemma Forall_zip_with_ext_r (g : A → B → C) l1 l2 k : l1 = l2 → Forall (λ y, ∀ x, f x y = g x y) k → zip_with f l1 k = zip_with g l2 k. Proof. intros <- Hk. revert l1. induction Hk; intros [|??]; simpl; f_equal; auto. Qed. Lemma zip_with_fmap_l {D} (g : D → A) l k : zip_with f (g <\$> l) k = zip_with (λ x, f (g x)) l k. Proof. revert k. induction l; intros [|??]; simpl; f_equal; auto. Qed. Lemma zip_with_fmap_r {D} (g : D → B) l k : zip_with f l (g <\$> k) = zip_with (λ x y, f x (g y)) l k. Proof. revert k. induction l; intros [|??]; simpl; f_equal; auto. Qed. Lemma zip_with_nil_inv l1 l2 : zip_with f l1 l2 = [] → l1 = [] ∨ l2 = []. Proof. destruct l1, l2; simpl; auto with congruence. Qed. Lemma zip_with_cons_inv y l1 l2 k : zip_with f l1 l2 = y :: k → ∃ x1 x2 l1' l2', y = f x1 x2 ∧ k = zip_with f l1' l2' ∧ l1 = x1 :: l1' ∧ l2 = x2 :: l2'. Proof. intros. destruct l1, l2; simplify_equality'; repeat eexists. Qed. Lemma zip_with_app_inv l1 l2 k' k'' : zip_with f l1 l2 = k' ++ k'' → ∃ l1' l1'' l2' l2'', k' = zip_with f l1' l2' ∧ k'' = zip_with f l1'' l2'' ∧ l1 = l1' ++ l1'' ∧ l2 = l2' ++ l2''. Proof. revert l1 l2. induction k' as [|y k' IH]; simpl. * intros l1 l2 ?. by eexists [], l1, [], l2. * intros [|x1 l1] [|x2 l2] ?; simplify_equality'. destruct (IH l1 l2) as (l1'&l1''&l2'&l2''&?&?&?&?); subst; [done |]. by exists (x1 :: l1') l1'' (x2 :: l2') l2''. Qed. Lemma zip_with_inj l1 l2 k1 k2 : (∀ x1 x2 y1 y2, f x1 x2 = f y1 y2 → x1 = y1 ∧ x2 = y2) → l1 `same_length` l2 → k1 `same_length` k2 → zip_with f l1 l2 = zip_with f k1 k2 → l1 = k1 ∧ l2 = k2. Proof. intros ? Hl. revert k1 k2. induction Hl; intros ?? [] ?; simplify_equality'; f_equal; naive_solver. Qed. Lemma zip_with_length l1 l2 : length l1 ≤ length l2 → length (zip_with f l1 l2) = length l1. Proof. revert l2. induction l1; intros [|??]; simpl; auto with lia. Qed. Lemma zip_with_fmap_fst_le (g : C → A) l1 l2 : (∀ x y, g (f x y) = x) → length l1 ≤ length l2 → g <\$> zip_with f l1 l2 = l1. Proof. revert l2. induction l1; intros [|??] ??; simpl in *; f_equal; auto with lia. Qed. Lemma zip_with_fmap_snd_le (g : C → B) l1 l2 : (∀ x y, g (f x y) = y) → length l2 ≤ length l1 → g <\$> zip_with f l1 l2 = l2. Proof. revert l1. induction l2; intros [|??] ??; simpl in *; f_equal; auto with lia. Qed. Lemma zip_with_fmap_fst (g : C → A) l1 l2 : (∀ x y, g (f x y) = x) → l1 `same_length` l2 → g <\$> zip_with f l1 l2 = l1. Proof. induction 2; simpl; f_equal; auto. Qed. Lemma zip_with_fmap_snd (g : C → B) l1 l2 : (∀ x y, g (f x y) = y) → l1 `same_length` l2 → g <\$> zip_with f l1 l2 = l2. Proof. induction 2; simpl; f_equal; auto. Qed. Lemma Forall_zip_with_fst (P : A → Prop) (Q : C → Prop) l1 l2 : Forall P l1 → Forall (λ y, ∀ x, P x → Q (f x y)) l2 → Forall Q (zip_with f l1 l2). Proof. intros Hl. revert l2. induction Hl; destruct 1; simpl in *; auto. Qed. Lemma Forall_zip_with_snd (P : B → Prop) (Q : C → Prop) l1 l2 : Forall (λ x, ∀ y, P y → Q (f x y)) l1 → Forall P l2 → Forall Q (zip_with f l1 l2). Proof. intros Hl. revert l2. induction Hl; destruct 1; simpl in *; auto. Qed. End zip_with. Section zip. Context {A B : Type}. Implicit Types l : list A. Implicit Types k : list B. Lemma zip_length l k : length l ≤ length k → length (zip l k) = length l. Proof. by apply zip_with_length. Qed. Lemma zip_fmap_fst_le l k : length l ≤ length k → fst <\$> zip l k = l. Proof. by apply zip_with_fmap_fst_le. Qed. Lemma zip_fmap_snd l k : length k ≤ length l → snd <\$> zip l k = k. Proof. by apply zip_with_fmap_snd_le. Qed. Lemma zip_fst l k : l `same_length` k → fst <\$> zip l k = l. Proof. by apply zip_with_fmap_fst. Qed. Lemma zip_snd l k : l `same_length` k → snd <\$> zip l k = k. Proof. by apply zip_with_fmap_snd. Qed. End zip. Lemma elem_of_zipped_map {A B} (f : list A → list A → A → B) l k x : x ∈ zipped_map f l k ↔ ∃ k' k'' y, k = k' ++ [y] ++ k'' ∧ x = f (reverse k' ++ l) k'' y. Proof. split. * revert l. induction k as [|z k IH]; simpl; intros l; inversion_clear 1. + by eexists [], k, z. + destruct (IH (z :: l)) as [k' [k'' [y [??]]]]; [done |]; subst. eexists (z :: k'), k'', y. split; [done |]. by rewrite reverse_cons, <-(associative_L (++)). * intros [k' [k'' [y [??]]]]; subst. revert l. induction k' as [|z k' IH]; intros l; [by left|]. right. by rewrite reverse_cons, <-!(associative_L (++)). Qed. Section zipped_list_ind. Context {A} (P : list A → list A → Prop). Context (Pnil : ∀ l, P l []) (Pcons : ∀ l k x, P (x :: l) k → P l (x :: k)). Fixpoint zipped_list_ind l k : P l k := match k with | [] => Pnil _ | x :: k => Pcons _ _ _ (zipped_list_ind (x :: l) k) end. End zipped_list_ind. Lemma zipped_Forall_app {A} (P : list A → list A → A → Prop) l k k' : zipped_Forall P l (k ++ k') → zipped_Forall P (reverse k ++ l) k'. Proof. revert l. induction k as [|x k IH]; simpl; [done |]. inversion_clear 1. rewrite reverse_cons, <-(associative_L (++)). by apply IH. Qed. (** * Relection over lists *) (** We define a simple data structure [rlist] to capture a syntactic representation of lists consisting of constants, applications and the nil list. Note that we represent [(x ::)] as [rapp (rnode [x])]. For now, we abstract over the type of constants, but later we use [nat]s and a list representing a corresponding environment. *) Inductive rlist (A : Type) := | rnil : rlist A | rnode : A → rlist A | rapp : rlist A → rlist A → rlist A. Arguments rnil {_}. Arguments rnode {_} _. Arguments rapp {_} _ _. Module rlist. Fixpoint to_list {A} (t : rlist A) : list A := match t with | rnil => [] | rnode l => [l] | rapp t1 t2 => to_list t1 ++ to_list t2 end. Notation env A := (list (list A)) (only parsing). Definition eval {A} (E : env A) : rlist nat → list A := fix go t := match t with | rnil => [] | rnode i => from_option [] (E !! i) | rapp t1 t2 => go t1 ++ go t2 end. (** A simple quoting mechanism using type classes. [QuoteLookup E1 E2 x i] means: starting in environment [E1], look up the index [i] corresponding to the constant [x]. In case [x] has a corresponding index [i] in [E1], the original environment is given back as [E2]. Otherwise, the environment [E2] is extended with a binding [i] for [x]. *) Section quote_lookup. Context {A : Type}. Class QuoteLookup (E1 E2 : list A) (x : A) (i : nat) := {}. Global Instance quote_lookup_here E x : QuoteLookup (x :: E) (x :: E) x 0. Global Instance quote_lookup_end x : QuoteLookup [] [x] x 0. Global Instance quote_lookup_further E1 E2 x i y : QuoteLookup E1 E2 x i → QuoteLookup (y :: E1) (y :: E2) x (S i) | 1000. End quote_lookup. Section quote. Context {A : Type}. Class Quote (E1 E2 : env A) (l : list A) (t : rlist nat) := {}. Global Instance quote_nil: Quote E1 E1 [] rnil. Global Instance quote_node E1 E2 l i: QuoteLookup E1 E2 l i → Quote E1 E2 l (rnode i) | 1000. Global Instance quote_cons E1 E2 E3 x l i t : QuoteLookup E1 E2 [x] i → Quote E2 E3 l t → Quote E1 E3 (x :: l) (rapp (rnode i) t). Global Instance quote_app E1 E2 E3 l1 l2 t1 t2 : Quote E1 E2 l1 t1 → Quote E2 E3 l2 t2 → Quote E1 E3 (l1 ++ l2) (rapp t1 t2). End quote. Section eval. Context {A} (E : env A). Lemma eval_alt t : eval E t = to_list t ≫= from_option [] ∘ (E !!). Proof. induction t; simpl. * done. * by rewrite (right_id_L [] (++)). * rewrite bind_app. by f_equal. Qed. Lemma eval_eq t1 t2 : to_list t1 = to_list t2 → eval E t1 = eval E t2. Proof. intros Ht. by rewrite !eval_alt, Ht. Qed. Lemma eval_Permutation t1 t2 : to_list t1 ≡ₚ to_list t2 → eval E t1 ≡ₚ eval E t2. Proof. intros Ht. by rewrite !eval_alt, Ht. Qed. Lemma eval_contains t1 t2 : to_list t1 `contains` to_list t2 → eval E t1 `contains` eval E t2. Proof. intros Ht. by rewrite !eval_alt, Ht. Qed. End eval. End rlist. (** * Tactics *) Ltac quote_Permutation := match goal with | |- ?l1 ≡ₚ ?l2 => match type of (_ : rlist.Quote [] _ l1 _) with rlist.Quote _ ?E2 _ ?t1 => match type of (_ : rlist.Quote E2 _ l2 _) with rlist.Quote _ ?E3 _ ?t2 => change (rlist.eval E3 t1 ≡ₚ rlist.eval E3 t2) end end end. Ltac solve_Permutation := quote_Permutation; apply rlist.eval_Permutation; apply (bool_decide_unpack _); by vm_compute. Ltac quote_contains := match goal with | |- ?l1 `contains` ?l2 => match type of (_ : rlist.Quote [] _ l1 _) with rlist.Quote _ ?E2 _ ?t1 => match type of (_ : rlist.Quote E2 _ l2 _) with rlist.Quote _ ?E3 _ ?t2 => change (rlist.eval E3 t1 `contains` rlist.eval E3 t2) end end end. Ltac solve_contains := quote_contains; apply rlist.eval_contains; apply (bool_decide_unpack _); by vm_compute. Ltac decompose_elem_of_list := repeat match goal with | H : ?x ∈ [] |- _ => by destruct (not_elem_of_nil x) | H : _ ∈ _ :: _ |- _ => apply elem_of_cons in H; destruct H | H : _ ∈ _ ++ _ |- _ => apply elem_of_app in H; destruct H end. Ltac simplify_list_fmap_equality := repeat match goal with | _ => progress simplify_equality | H : _ <\$> _ = [] |- _ => apply fmap_nil_inv in H | H : [] = _ <\$> _ |- _ => symmetry in H; apply fmap_nil_inv in H | H : _ <\$> _ = _ :: _ |- _ => apply fmap_cons_inv in H; destruct H as (?&?&?&?&?) | H : _ :: _ = _ <\$> _ |- _ => symmetry in H | H : _ <\$> _ = _ ++ _ |- _ => apply fmap_app_inv in H; destruct H as (?&?&?&?&?) | H : _ ++ _ = _ <\$> _ |- _ => symmetry in H end. Ltac simplify_zip_equality := repeat match goal with | _ => progress simplify_equality | H : zip_with _ _ _ = [] |- _ => apply zip_with_nil_inv in H; destruct H | H : [] = zip_with _ _ _ |- _ => symmetry in H | H : zip_with _ _ _ = _ :: _ |- _ => apply zip_with_cons_inv in H; destruct H as (?&?&?&?&?&?&?&?) | H : _ :: _ = zip_with _ _ _ |- _ => symmetry in H | H : zip_with _ _ _ = _ ++ _ |- _ => apply zip_with_app_inv in H; destruct H as (?&?&?&?&?&?&?&?) | H : _ ++ _ = zip_with _ _ _ |- _ => symmetry in H end. Ltac decompose_Forall_hyps := repeat match goal with | H : Forall _ [] |- _ => clear H | H : Forall _ (_ :: _) |- _ => rewrite Forall_cons in H; destruct H | H : Forall _ (_ ++ _) |- _ => rewrite Forall_app in H; destruct H | H : Forall _ (_ <\$> _) |- _ => rewrite Forall_fmap in H | H : Forall2 _ [] [] |- _ => clear H | H : Forall2 _ (_ :: _) [] |- _ => destruct (Forall2_cons_nil_inv _ _ _ H) | H : Forall2 _ [] (_ :: _) |- _ => destruct (Forall2_nil_cons_inv _ _ _ H) | H : Forall2 _ [] ?l |- _ => apply Forall2_nil_inv_l in H; subst l | H : Forall2 _ ?l [] |- _ => apply Forall2_nil_inv_r in H; subst l | H : Forall2 _ (_ :: _) (_ :: _) |- _ => apply Forall2_cons_inv in H; destruct H | H : Forall2 _ (_ :: _) ?l |- _ => apply Forall2_cons_inv_l in H; destruct H as (? & ? & ? & ? & ?); subst l | H : Forall2 _ ?l (_ :: _) |- _ => apply Forall2_cons_inv_r in H; destruct H as (? & ? & ? & ? & ?); subst l | H : Forall2 _ (_ ++ _) (_ ++ _) |- _ => destruct (Forall2_app_inv _ _ _ _ _ H); [eauto using Forall2_same_length |] | H : Forall2 _ (_ ++ _) ?l |- _ => apply Forall2_app_inv_l in H; destruct H as (? & ? & ? & ? & ?); subst l | H : Forall2 _ ?l (_ ++ _) |- _ => apply Forall2_app_inv_r in H; destruct H as (? & ? & ? & ? & ?); subst l | H : Forall ?P ?l, H1 : ?l !! _ = Some ?x |- _ => unless (P x) by done; let E := fresh in assert (P x) as E by (apply (Forall_lookup_1 P _ _ _ H H1)); lazy beta in E | H : Forall2 ?P ?l1 ?l2 |- _ => lazymatch goal with | H1 : l1 !! ?i = Some ?x, H2 : l2 !! ?i = Some ?y |- _ => unless (P x y) by done; let E := fresh in assert (P x y) as E by (apply (Forall2_lookup_lr P _ _ _ _ _ H H1 H2)); lazy beta in E | H1 : l1 !! _ = Some ?x |- _ => destruct (Forall2_lookup_l P _ _ _ _ H H1) as (?&?&?) | H2 : l2 !! _ = Some ?y |- _ => destruct (Forall2_lookup_r P _ _ _ _ H H2) as (?&?&?) end end. Ltac decompose_Forall := repeat match goal with | |- Forall _ _ => by apply Forall_true | |- Forall _ [] => constructor | |- Forall _ (_ :: _) => constructor | |- Forall _ (_ ++ _) => apply Forall_app | |- Forall _ (_ <\$> _) => apply Forall_fmap | |- Forall2 _ _ _ => apply Forall2_Forall | |- Forall2 _ [] [] => constructor | |- Forall2 _ (_ :: _) (_ :: _) => constructor | |- Forall2 _ (_ ++ _) (_ ++ _) => first [ apply Forall2_app; [by decompose_Forall |] | apply Forall2_app; [| by decompose_Forall]] | |- Forall2 _ (_ <\$> _) _ => apply Forall2_fmap_l | |- Forall2 _ _ (_ <\$> _) => apply Forall2_fmap_r | _ => progress decompose_Forall_hyps | |- Forall _ _ => apply Forall_lookup_2; intros ???; progress decompose_Forall_hyps | |- Forall2 _ _ _ => apply Forall2_lookup_2; [by eauto using Forall2_same_length|]; intros ?????; progress decompose_Forall_hyps end. (** The [simplify_suffix_of] tactic removes [suffix_of] hypotheses that are tautologies, and simplifies [suffix_of] hypotheses involving [(::)] and [(++)]. *) Ltac simplify_suffix_of := repeat match goal with | H : suffix_of (_ :: _) _ |- _ => destruct (suffix_of_cons_not _ _ H) | H : suffix_of (_ :: _) [] |- _ => apply suffix_of_nil_inv in H | H : suffix_of (_ ++ _) (_ ++ _) |- _ => apply suffix_of_app_inv in H | H : suffix_of (_ :: _) (_ :: _) |- _ => destruct (suffix_of_cons_inv _ _ _ _ H); clear H | H : suffix_of ?x ?x |- _ => clear H | H : suffix_of ?x (_ :: ?x) |- _ => clear H | H : suffix_of ?x (_ ++ ?x) |- _ => clear H | _ => progress simplify_equality end. (** The [solve_suffix_of] tactic tries to solve goals involving [suffix_of]. It uses [simplify_suffix_of] to simplify hypotheses and tries to solve [suffix_of] conclusions. This tactic either fails or proves the goal. *) Ltac solve_suffix_of := by intuition (repeat match goal with | _ => done | _ => progress simplify_suffix_of | |- suffix_of [] _ => apply suffix_of_nil | |- suffix_of _ _ => reflexivity | |- suffix_of _ (_ :: _) => apply suffix_of_cons_r | |- suffix_of _ (_ ++ _) => apply suffix_of_app_r | H : suffix_of _ _ → False |- _ => destruct H end). Hint Extern 0 (PropHolds (suffix_of _ _)) => unfold PropHolds; solve_suffix_of : typeclass_instances.