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phanerozoic
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Coq-ExtLib

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hlist_get ls a (m : member a ls) : hlist ls -> F a := match m in member _ ls return hlist ls -> F a with | MZ _ => hlist_hd | MN _ _ r => fun hl => hlist_get r (hlist_tl hl) end.
Fixpoint
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_get
hlist_nth_error {ls} (hs : hlist ls) (n : nat) : option (match nth_error ls n with | None => unit | Some x => F x end) := match hs in hlist ls return option (match nth_error ls n with | None => unit | Some x => F x end) with | Hnil => None | Hcons l ls h hs => match n as n return option (match nth_error (l :: ls) n with | None => unit | Some x => F x end) with | 0 => Some h | S n => hlist_nth_error hs n end end. Polymorphic Fixpoint hlist_nth ls (h : hlist ls) (n : nat) : match nth_error ls n return Type with | None => unit | Some t => F t end := match h in hlist ls , n as n return match nth_error ls n with | None => unit | Some t => F t end with | Hnil , 0 => tt | Hnil , S _ => tt | Hcons _ _ x _ , 0 => x | Hcons _ _ _ h , S n => hlist_nth h n end.
Fixpoint
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_nth_error
nth_error_hlist_nth ls (n : nat) : option (hlist ls -> match nth_error ls n with | None => Empty_set | Some x => F x end) := match ls as ls return option (hlist ls -> match nth_error ls n with | None => Empty_set | Some x => F x end) with | nil => None | l :: ls => match n as n return option (hlist (l :: ls) -> match nth_error (l :: ls) n with | None => Empty_set | Some x => F x end) with | 0 => Some hlist_hd | S n => match nth_error_hlist_nth ls n with | None => None | Some f => Some (fun h => f (hlist_tl h)) end end end.
Fixpoint
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
nth_error_hlist_nth
cast1 T l : forall (l' : list T) n v, nth_error l n = Some v -> Some v = nth_error (l ++ l') n. Proof. induction l. intros. { exfalso. destruct n; inversion H. } { destruct n; simpl; intros; auto. } Defined.
Definition
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
cast1
cast2 T l : forall (l' : list T) n, nth_error l n = None -> nth_error l' (n - length l) = nth_error (l ++ l') n. Proof. induction l; simpl. { destruct n; simpl; auto. } { destruct n; simpl; auto. inversion 1. } Defined.
Definition
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
cast2
hlist_nth_hlist_app : forall l l' (h : hlist l) (h' : hlist l') n, hlist_nth (hlist_app h h') n = match nth_error l n as k return nth_error l n = k -> match nth_error (l ++ l') n return Type with | None => unit | Some t => F t end with | Some _ => fun pf => match cast1 _ _ _ pf in _ = z , eq_sym pf in _ = w return match w return Type with | None => unit | Some t => F t end -> match z return Type with | None => unit | Some t => F t end with | eq_refl , eq_refl => fun x => x end (hlist_nth h n) | None => fun pf => match cast2 _ _ _ pf in _ = z return match z with | Some t => F t | None => unit end with | eq_refl => hlist_nth h' (n - length l) end end eq_refl. Proof. induction h; simpl; intros. { destruct n; simpl in *; reflexivity. } { destruct n; simpl. { reflexivity. } { rewrite IHh. reflexivity. } } Qed.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_nth_hlist_app
hlist_app_assoc : forall ls ls' ls'' (a : hlist ls) (b : hlist ls') (c : hlist ls''), hlist_app (hlist_app a b) c = match eq_sym (app_ass_trans ls ls' ls'') in _ = t return hlist t with | eq_refl => hlist_app a (hlist_app b c) end. Proof. intros ls ls' ls''. generalize (eq_sym (app_assoc_reverse ls ls' ls'')). induction ls; simpl; intros. { rewrite (hlist_eta a); simpl. reflexivity. } { rewrite (hlist_eta a0). simpl. inversion H. erewrite (IHls H1). unfold f_equal. unfold eq_trans. unfold eq_sym. generalize (app_ass_trans ls ls' ls''). rewrite <- H1. clear. intro. generalize dependent (hlist_app (hlist_tl a0) (hlist_app b c)). destruct e. reflexivity. } Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_app_assoc
hlist_app_assoc' : forall (ls ls' ls'' : list iT) (a : hlist ls) (b : hlist ls') (c : hlist ls''), hlist_app a (hlist_app b c) = match app_ass_trans ls ls' ls'' in (_ = t) return (hlist t) with | eq_refl => hlist_app (hlist_app a b) c end. Proof. clear. intros. generalize (hlist_app_assoc a b c). generalize (hlist_app (hlist_app a b) c). generalize (hlist_app a (hlist_app b c)). destruct (app_ass_trans ls ls' ls''). simpl. auto. Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_app_assoc'
hlist_split ls ls' : hlist (ls ++ ls') -> hlist ls * hlist ls' := match ls as ls return hlist (ls ++ ls') -> hlist ls * hlist ls' with | nil => fun h => (Hnil, h) | l :: ls => fun h => let (a,b) := @hlist_split ls ls' (hlist_tl h) in (Hcons (hlist_hd h) a, b) end.
Fixpoint
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_split
hlist_app_hlist_split : forall ls' ls (h : hlist (ls ++ ls')), hlist_app (fst (hlist_split ls ls' h)) (snd (hlist_split ls ls' h)) = h. Proof. induction ls; simpl; intros; auto. rewrite (hlist_eta h); simpl. specialize (IHls (hlist_tl h)). destruct (hlist_split ls ls' (hlist_tl h)); simpl in *; auto. f_equal. auto. Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_app_hlist_split
hlist_split_hlist_app : forall ls' ls (h : hlist ls) (h' : hlist ls'), hlist_split _ _ (hlist_app h h') = (h,h'). Proof. induction ls; simpl; intros. { rewrite (hlist_eta h); simpl; auto. } { rewrite (hlist_eta h); simpl. rewrite IHls. reflexivity. } Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_split_hlist_app
hlist_hd_fst_hlist_split : forall t (xs ys : list _) (h : hlist (t :: xs ++ ys)), hlist_hd (fst (hlist_split (t :: xs) ys h)) = hlist_hd h. Proof. simpl. intros. match goal with | |- context [ match ?X with _ => _ end ] => destruct X end. reflexivity. Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_hd_fst_hlist_split
hlist_tl_fst_hlist_split : forall t (xs ys : list _) (h : hlist (t :: xs ++ ys)), hlist_tl (fst (hlist_split (t :: xs) ys h)) = fst (hlist_split xs ys (hlist_tl h)). Proof. simpl. intros. match goal with | |- context [ match ?X with _ => _ end ] => remember X end. destruct p. simpl. change h0 with (fst (h0, h1)). f_equal; trivial. Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_tl_fst_hlist_split
hlist_tl_snd_hlist_split : forall t (xs ys : list _) (h : hlist (t :: xs ++ ys)), snd (hlist_split xs ys (hlist_tl h)) = snd (hlist_split (t :: xs) ys h). Proof. simpl. intros. match goal with | |- context [ match ?X with _ => _ end ] => remember X end. destruct p. simpl. change h1 with (snd (h0, h1)). rewrite Heqp. reflexivity. Qed. Polymorphic Fixpoint nth_error_get_hlist_nth (ls : list iT) (n : nat) {struct ls} : option {t : iT & hlist ls -> F t} := match ls as ls0 return option {t : iT & hlist ls0 -> F t} with | nil => None | l :: ls0 => match n as n0 return option {t : iT & hlist (l :: ls0) -> F t} with | 0 => Some (@existT _ (fun t => hlist (l :: ls0) -> F t) l (@hlist_hd _ _)) | S n0 => match nth_error_get_hlist_nth ls0 n0 with | Some (existT x f) => Some (@existT _ (fun t => hlist _ -> F t) x (fun h : hlist (l :: ls0) => f (hlist_tl h))) | None => None end end end.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_tl_snd_hlist_split
nth_error_get_hlist_nth_Some : forall ls n s, nth_error_get_hlist_nth ls n = Some s -> exists pf : nth_error ls n = Some (projT1 s), forall h, projT2 s h = match pf in _ = t return match t return Type with | Some t => F t | None => unit end with | eq_refl => hlist_nth h n end. Proof. induction ls; simpl; intros; try congruence. { destruct n. { inv_all; subst; simpl. exists (eq_refl). intros. rewrite (hlist_eta h). reflexivity. } { forward. inv_all; subst. destruct (IHls _ _ H0); clear IHls. simpl in *. exists x0. intros. rewrite (hlist_eta h). simpl. auto. } } Qed.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
nth_error_get_hlist_nth_Some
nth_error_get_hlist_nth_None : forall ls n, nth_error_get_hlist_nth ls n = None <-> nth_error ls n = None. Proof. induction ls; simpl; intros; try congruence. { destruct n; intuition. } { destruct n; simpl; try solve [ intuition congruence ]. specialize (IHls n). forward. } Qed.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
nth_error_get_hlist_nth_None
nth_error_get_hlist_nth_weaken : forall ls ls' n x, nth_error_get_hlist_nth ls n = Some x -> exists z, nth_error_get_hlist_nth (ls ++ ls') n = Some (@existT iT (fun t => hlist (ls ++ ls') -> F t) (projT1 x) z) /\ forall h h', projT2 x h = z (hlist_app h h'). Proof. intros ls ls'. revert ls. induction ls; simpl; intros; try congruence. { destruct n; inv_all; subst. { simpl. eexists; split; eauto. intros. rewrite (hlist_eta h). reflexivity. } { forward. inv_all; subst. simpl. apply IHls in H0. forward_reason. rewrite H. eexists; split; eauto. intros. rewrite (hlist_eta h). simpl in *. auto. } } Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
nth_error_get_hlist_nth_weaken
nth_error_get_hlist_nth_appL : forall tvs' tvs n, n < length tvs -> exists x, nth_error_get_hlist_nth (tvs ++ tvs') n = Some x /\ exists y, nth_error_get_hlist_nth tvs n = Some (@existT _ _ (projT1 x) y) /\ forall vs vs', (projT2 x) (hlist_app vs vs') = y vs. Proof. clear. induction tvs; simpl; intros. { exfalso; inversion H. } { destruct n. { clear H IHtvs. eexists; split; eauto. eexists; split; eauto. simpl. intros. rewrite (hlist_eta vs). reflexivity. } { apply Nat.succ_lt_mono in H. { specialize (IHtvs _ H). forward_reason. rewrite H0. rewrite H1. forward. subst. simpl in *. eexists; split; eauto. eexists; split; eauto. simpl. intros. rewrite (hlist_eta vs). simpl. auto. } } } Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
nth_error_get_hlist_nth_appL
nth_error_get_hlist_nth_appR : forall tvs' tvs n x, n >= length tvs -> nth_error_get_hlist_nth (tvs ++ tvs') n = Some x -> exists y, nth_error_get_hlist_nth tvs' (n - length tvs) = Some (@existT _ _ (projT1 x) y) /\ forall vs vs', (projT2 x) (hlist_app vs vs') = y vs'. Proof. clear. induction tvs; simpl; intros. { rewrite PeanoNat.Nat.sub_0_r. rewrite H0. destruct x. simpl. eexists; split; eauto. intros. rewrite (hlist_eta vs). reflexivity. } { destruct n. { inversion H. } { assert (n >= length tvs) by (eapply le_S_n; eassumption). clear H. { forward. inv_all; subst. simpl in *. specialize (IHtvs _ _ H1 H0). simpl in *. forward_reason. rewrite H. eexists; split; eauto. intros. rewrite (hlist_eta vs). simpl. auto. } } } Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
nth_error_get_hlist_nth_appR
hlist_map (ls : list A) (hl : hlist F ls) {struct hl} : hlist G ls := match hl in @hlist _ _ ls return hlist G ls with | Hnil => Hnil | Hcons _ _ hd tl => Hcons (ff hd) (hlist_map tl) end.
Fixpoint
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_map
hlist_app_hlist_map : forall ls ls' (a : hlist F ls) (b : hlist F ls'), hlist_map (hlist_app a b) = hlist_app (hlist_map a) (hlist_map b). Proof. induction a. simpl; auto. simpl. intros. f_equal. auto. Qed.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_app_hlist_map
hlist_map_hlist_map : forall ls (hl : hlist F ls), hlist_map gg (hlist_map ff hl) = hlist_map (fun _ x => gg (ff x)) hl. Proof. induction hl; simpl; f_equal. assumption. Defined.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_map_hlist_map
hlist_get_hlist_map : forall ls t (hl : hlist F ls) (m : member t ls), hlist_get m (hlist_map ff hl) = ff (hlist_get m hl). Proof. induction m; simpl. { rewrite (hlist_eta hl). reflexivity. } { rewrite (hlist_eta hl). simpl. auto. } Defined.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_get_hlist_map
hlist_map_ext : forall (ff gg : forall x, F x -> G x), (forall x t, ff x t = gg x t) -> forall ls (hl : hlist F ls), hlist_map ff hl = hlist_map gg hl. Proof. induction hl; simpl; auto. intros. f_equal; auto. Defined.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_map_ext
equiv_hlist_map : forall T U (F : T -> Type) (R : forall t, F t -> F t -> Prop) (R' : forall t, U t -> U t -> Prop) (f g : forall t, F t -> U t), (forall t (x y : F t), R t x y -> R' t (f t x) (g t y)) -> forall ls (a b : hlist F ls), equiv_hlist R a b -> equiv_hlist R' (hlist_map f a) (hlist_map g b). Proof. clear. induction 2; simpl; intros. - constructor. - constructor; eauto. Qed. (** Linking Heterogeneous Lists and Lists **)
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
equiv_hlist_map
hlist_gen ls : hlist F ls := match ls with | nil => Hnil | cons x ls' => Hcons (f x) (hlist_gen ls') end.
Fixpoint
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_gen
hlist_get_hlist_gen : forall ls t (m : member t ls), hlist_get m (hlist_gen ls) = f t. Proof. induction m; simpl; auto. Qed. (** This function is a generalisation of [hlist_gen] in which the function [f] takes the additional parameter [member a ls]. **)
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_get_hlist_gen
hlist_gen_member ls : (forall a, member a ls -> F a) -> hlist F ls := match ls as ls return ((forall a : A, member a ls -> F a) -> hlist F ls) with | nil => fun _ => Hnil | a :: ls' => fun fm => Hcons (fm a (MZ a ls')) (hlist_gen_member (fun a' (M : member a' ls') => fm a' (MN a M))) end.
Fixpoint
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_gen_member
hlist_gen_member_hlist_gen : forall ls, hlist_gen_member (fun a _ => f a) = hlist_gen ls. Proof. induction ls; simpl; f_equal; auto. Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_gen_member_hlist_gen
hlist_gen_member_ext : forall ls (f g : forall a, member a ls -> F a), (forall x M, f x M = g x M) -> hlist_gen_member f = hlist_gen_member g. Proof. intros. induction ls; simpl; f_equal; auto. Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_gen_member_ext
hlist_gen_member_hlist_map : forall A (F G : A -> Type) (ff : forall t, F t -> G t) ls f, hlist_map ff (hlist_gen_member F (ls := ls) f) = hlist_gen_member G (fun a M => ff _ (f _ M)). Proof. intros. induction ls; simpl; f_equal; auto. Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_gen_member_hlist_map
hlist_gen_hlist_map : forall A (F G : A -> Type) (ff : forall t, F t -> G t) f ls, hlist_map ff (hlist_gen f ls) = hlist_gen (fun a => ff _ (f a)) ls. Proof. intros. do 2 rewrite <- hlist_gen_member_hlist_gen. apply hlist_gen_member_hlist_map. Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_gen_hlist_map
hlist_gen_ext : forall A F (f g : forall a, F a), (forall x, f x = g x) -> forall ls : list A, hlist_gen f ls = hlist_gen g ls. Proof. intros. do 2 rewrite <- hlist_gen_member_hlist_gen. apply hlist_gen_member_ext. auto. Qed. Global Instance Proper_hlist_gen : forall A F, Proper (forall_relation (fun _ => eq) ==> forall_relation (fun _ => eq)) (@hlist_gen A F). Proof. repeat intro. apply hlist_gen_ext. auto. Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_gen_ext
equiv_hlist_gen : forall T (F : T -> Type) (f : forall t, F t) f' (R : forall t, F t -> F t -> Prop), (forall t, R t (f t) (f' t)) -> forall ls, equiv_hlist R (hlist_gen f ls) (hlist_gen f' ls). Proof. induction ls; simpl; constructor; auto. Qed. Global Instance Proper_equiv_hlist_gen : forall A (F : A -> Type) R, Proper (forall_relation R ==> forall_relation (@equiv_hlist _ _ R)) (@hlist_gen A F). Proof. repeat intro. apply equiv_hlist_gen. auto. Qed.
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
equiv_hlist_gen
hlist_erase {A B} {ls : list A} (hs : hlist (fun _ => B) ls) : list B := match hs with | Hnil => nil | Hcons _ _ x hs' => cons x (hlist_erase hs') end.
Fixpoint
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_erase
hlist_erase_hlist_gen : forall A B ls (f : A -> B), hlist_erase (hlist_gen f ls) = map f ls. Proof. induction ls; simpl; intros; f_equal; auto. Qed. (** Linking Heterogeneous Lists and Predicates **)
Lemma
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_erase_hlist_gen
hlist_Forall ls (hs : hlist P ls) : Forall P ls := match hs with | Hnil => Forall_nil _ | Hcons _ _ H hs' => Forall_cons _ H (hlist_Forall hs') end.
Fixpoint
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_Forall
hlist_hrel : forall ls, hlist F ls -> hlist G ls -> Prop := | hrel_Hnil : hlist_hrel Hnil Hnil | hrel_Hcons : forall t ts x y xs ys, @R t x y -> @hlist_hrel ts xs ys -> @hlist_hrel (t :: ts) (Hcons x xs) (Hcons y ys).
Inductive
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_hrel
hlist_hrel_map : forall ls xs ys, @hlist_hrel A F G R ls xs ys -> @hlist_hrel A F' G' R' ls (hlist_map ff xs) (hlist_map gg ys). Proof. induction 1; simpl; constructor; eauto. Qed.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_hrel_map
hlist_hrel_cons : forall l ls x xs y ys, @hlist_hrel A F G R (l :: ls) (Hcons x xs) (Hcons y ys) -> @R l x y /\ @hlist_hrel A F G R ls xs ys. Proof. intros. refine match H in @hlist_hrel _ _ _ _ ls' xs' ys' return match ls' as ls' return hlist F ls' -> hlist G ls' -> Prop with | nil => fun _ _ => True | l' :: ls' => fun x y => R (hlist_hd x) (hlist_hd y) /\ hlist_hrel R (hlist_tl x) (hlist_tl y) end xs' ys' with | hrel_Hnil => I | hrel_Hcons _ _ _ _ _ _ pf pf' => conj pf pf' end. Qed.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_hrel_cons
hlist_hrel_app : forall l ls x xs y ys, @hlist_hrel A F G R (l ++ ls) (hlist_app x xs) (hlist_app y ys) -> @hlist_hrel A F G R l x y /\ @hlist_hrel A F G R ls xs ys. Proof. induction x. + intros xs y ys. rewrite (hlist_eta y). simpl; intros; split; auto. constructor. + intros xs y ys. rewrite (hlist_eta y). intros. eapply hlist_hrel_cons in H. destruct H. apply IHx in H0. intuition. constructor; auto. Qed.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_hrel_app
hlist_hrel_equiv : forall T (F : T -> Type) (R : forall t, F t -> F t -> Prop) ls (h h' : hlist F ls), hlist_hrel R h h' -> equiv_hlist R h h'. Proof. induction 1; constructor; auto. Qed.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_hrel_equiv
hlist_hrel_flip : forall T (F G : T -> Type) (R : forall t, F t -> G t -> Prop) ls (h : hlist F ls) (h' : hlist G ls), hlist_hrel R h h' -> hlist_hrel (fun t a b => R t b a) h' h. Proof. induction 1; constructor; auto. Qed.
Theorem
theories
[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]
theories/Data/HList.v
hlist_hrel_flip
Lazy (t : Type) : Type := unit -> t. (** Note: in order for this to have the right behavior, it must be beta-delta reduced. **)
Definition
theories
[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Lazy.v
Lazy
_lazy {T : Type} (l : T) : Lazy T := fun _ => l.
Definition
theories
[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Lazy.v
_lazy
force {T : Type} (l : Lazy T) : T := l tt. Global Instance CoMonad_Lazy : CoMonad Lazy := { extract := @force ; extend _A _B b a := fun x : unit => b a }. Global Instance Functor_Lazy : Functor Lazy := { fmap _A _B f l := fun x => f (l x) }. Global Instance Monad_Lazy : Monad Lazy := { ret := @_lazy ; bind _A _B a b := fun x => b (a x) x }.
Definition
theories
[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Lazy.v
force
llist : Type := | lnil : llist | lcons : T -> (unit -> llist) -> llist.
Inductive
theories
[]
theories/Data/LazyList.v
llist
force (l : llist) : list T := match l with | lnil => nil | lcons a b => cons a (force (b tt)) end.
Fixpoint
theories
[]
theories/Data/LazyList.v
force
list_ind_singleton @{u} : forall {T : Type@{u}} (P : list T -> Prop) (Hnil : P nil) (Hsingle : forall t, P (t :: nil)) (Hcons : forall t u us, P (u :: us) -> P (t :: u :: us)), forall ls, P ls. Proof. induction ls; eauto. destruct ls. eauto. eauto. Qed.
Lemma
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
list_ind_singleton
list_rev_ind @{u} : forall (T : Type@{u}) (P : list T -> Prop), P nil -> (forall l ls, P ls -> P (ls ++ l :: nil)) -> forall ls, P ls. Proof. clear. intros. rewrite <- rev_involutive. induction (rev ls). apply H. simpl. auto. Qed.
Lemma
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
list_rev_ind
allb @{} (ls : list T) : bool := match ls with | nil => true | l :: ls => if p l then allb ls else false end.
Fixpoint
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
allb
anyb @{} (ls : list T) : bool := match ls with | nil => false | l :: ls => if p l then true else anyb ls end.
Fixpoint
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
anyb
Forall_map @{uT uU} : forall (T : Type@{uT}) (U : Type@{uU}) (f : T -> U) P ls, Forall P (List.map f ls) <-> Forall (fun x => P (f x)) ls. Proof. induction ls; simpl. { split; intros; constructor. } { split; inversion 1; intros; subst; constructor; auto. apply IHls. auto. apply IHls. auto. } Qed.
Lemma
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
Forall_map
Forall_cons_iff @{u} : forall (T : Type@{u}) (P : T -> Prop) a b, Forall P (a :: b) <-> (P a /\ Forall P b). Proof. clear. split. inversion 1; auto. destruct 1; constructor; auto. Qed.
Lemma
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
Forall_cons_iff
Forall_nil_iff @{u} : forall (T : Type@{u}) (P : T -> Prop), Forall P nil <-> True. Proof. clear. split; auto. Qed. Global Instance Foldable_list@{u} {T : Type@{u}} : Foldable (list T) T := fun _ f x ls => fold_right f x ls. Require Import ExtLib.Structures.Traversable. Require Import ExtLib.Structures.Functor. Require Import ExtLib.Structures.Monads. Require Import ExtLib.Structures.Applicative.
Lemma
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
Forall_nil_iff
mapT_list @{} (ls : list A) : F (list B) := match ls with | nil => pure nil | l :: ls => ap (ap (pure (@cons B)) (f l)) (mapT_list ls) end.
Fixpoint
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
mapT_list
R_list_len @{u} {T : Type@{u}} : list T -> list T -> Prop := | R_l_len : forall n m, length n < length m -> R_list_len n m.
Inductive
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
R_list_len
wf_R_list_len @{u} (T : Type@{u}) : well_founded (@R_list_len T). Proof. constructor. intros. refine (@Fix _ _ Nat.wf_R_lt (fun n : nat => forall ls : list T, n = length ls -> Acc R_list_len ls) (fun x rec ls pfls => Acc_intro _ _) _ _ refl_equal). refine ( match ls as ls return x = length ls -> forall z : list T, R_list_len z ls -> Acc R_list_len z with | nil => fun (pfls : x = 0) z pf => _ | cons l ls => fun pfls z pf => rec _ (match pf in R_list_len xs ys return x = length ys -> Nat.R_nat_lt (length xs) x with | R_l_len n m pf' => fun pf_eq => match eq_sym pf_eq in _ = x return Nat.R_nat_lt (length n) x with | refl_equal => Nat.R_lt pf' end end pfls) _ eq_refl end pfls). clear - pf; abstract (inversion pf; subst; simpl in *; inversion H). Defined.
Theorem
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
wf_R_list_len
Monoid_list_app @{u v} {T : Type@{u}} : Monoid@{v} (list T) := {| monoid_plus := @List.app _ ; monoid_unit := @nil _ |}.
Definition
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
Monoid_list_app
list_eqb @{} (ls rs : list T) : bool := match ls , rs with | nil , nil => true | cons l ls , cons r rs => if l ?[ eq ] r then list_eqb ls rs else false | _ , _ => false end. (** Specialization for equality **) Global Instance RelDec_eq_list@{} : RelDec (@eq (list T)) := { rel_dec := list_eqb }. Variable EDCT : RelDec_Correct EDT. Global Instance RelDec_Correct_eq_list@{v} : RelDec_Correct RelDec_eq_list. Proof. constructor; induction x; destruct y; split; simpl in *; intros; try reflexivity + discriminate. - destruct (_ : Reflect (rel_dec a t) _ _); try discriminate. replace y with x by (apply IHx; auto); subst; auto. - inversion H; subst. rewrite (rel_dec_eq_true _) by auto. apply IHx; auto. Qed.
Fixpoint
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
list_eqb
eq_list_eq @{u v} : forall (T : Type@{u}) (a b : T) (pf : a = b) (F : T -> Type@{v}) val, match pf in _ = x return list (F x) with | eq_refl => val end = map (fun val => match pf in _ = x return F x with | eq_refl => val end) val. Proof. destruct pf. intros. rewrite map_id. reflexivity. Qed. Hint Rewrite eq_list_eq : eq_rw. *) Export Coq.Lists.List.
Lemma
theories
[ "From Coq Require Import List EquivDec.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/List.v
eq_list_eq
firstn_app_L : forall T n (a b : list T), n <= length a -> firstn n (a ++ b) = firstn n a. Proof. induction n; destruct a; simpl in *; intros; auto. exfalso; lia. f_equal. eapply IHn; eauto. lia. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.ZArith Require Import ZArith.", "From Coq.micromega Require Import Lia." ]
theories/Data/ListFirstnSkipn.v
firstn_app_L
firstn_app_R : forall T n (a b : list T), length a <= n -> firstn n (a ++ b) = a ++ firstn (n - length a) b. Proof. induction n; destruct a; simpl in *; intros; auto. exfalso; lia. f_equal. eapply IHn; eauto. lia. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.ZArith Require Import ZArith.", "From Coq.micromega Require Import Lia." ]
theories/Data/ListFirstnSkipn.v
firstn_app_R
firstn_all : forall T n (a : list T), length a <= n -> firstn n a = a. Proof. induction n; destruct a; simpl; intros; auto. exfalso; lia. simpl. f_equal. eapply IHn; lia. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.ZArith Require Import ZArith.", "From Coq.micromega Require Import Lia." ]
theories/Data/ListFirstnSkipn.v
firstn_all
firstn_0 : forall T n (a : list T), n = 0 -> firstn n a = nil. Proof. intros; subst; auto. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.ZArith Require Import ZArith.", "From Coq.micromega Require Import Lia." ]
theories/Data/ListFirstnSkipn.v
firstn_0
firstn_cons : forall T n a (b : list T), 0 < n -> firstn n (a :: b) = a :: firstn (n - 1) b. Proof. destruct n; intros. lia. simpl. replace (n - 0) with n; [ | lia ]. reflexivity. Qed. #[global] Hint Rewrite firstn_app_L firstn_app_R firstn_all firstn_0 firstn_cons using lia : list_rw.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.ZArith Require Import ZArith.", "From Coq.micromega Require Import Lia." ]
theories/Data/ListFirstnSkipn.v
firstn_cons
skipn_app_R : forall T n (a b : list T), length a <= n -> skipn n (a ++ b) = skipn (n - length a) b. Proof. induction n; destruct a; simpl in *; intros; auto. exfalso; lia. eapply IHn. lia. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.ZArith Require Import ZArith.", "From Coq.micromega Require Import Lia." ]
theories/Data/ListFirstnSkipn.v
skipn_app_R
skipn_app_L : forall T n (a b : list T), n <= length a -> skipn n (a ++ b) = (skipn n a) ++ b. Proof. induction n; destruct a; simpl in *; intros; auto. exfalso; lia. eapply IHn. lia. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.ZArith Require Import ZArith.", "From Coq.micromega Require Import Lia." ]
theories/Data/ListFirstnSkipn.v
skipn_app_L
skipn_0 : forall T n (a : list T), n = 0 -> skipn n a = a. Proof. intros; subst; auto. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.ZArith Require Import ZArith.", "From Coq.micromega Require Import Lia." ]
theories/Data/ListFirstnSkipn.v
skipn_0
skipn_all : forall T n (a : list T), length a <= n -> skipn n a = nil. Proof. induction n; destruct a; simpl in *; intros; auto. exfalso; lia. apply IHn; lia. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.ZArith Require Import ZArith.", "From Coq.micromega Require Import Lia." ]
theories/Data/ListFirstnSkipn.v
skipn_all
skipn_cons : forall T n a (b : list T), 0 < n -> skipn n (a :: b) = skipn (n - 1) b. Proof. destruct n; intros. lia. simpl. replace (n - 0) with n; [ | lia ]. reflexivity. Qed. #[global] Hint Rewrite skipn_app_L skipn_app_R skipn_0 skipn_all skipn_cons using lia : list_rw.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.ZArith Require Import ZArith.", "From Coq.micromega Require Import Lia." ]
theories/Data/ListFirstnSkipn.v
skipn_cons
nth_error_app_L : forall (A B : list T) n, n < length A -> nth_error (A ++ B) n = nth_error A n. Proof. induction A; destruct n; simpl; intros; auto. { inversion H. } { inversion H. } { eapply IHA. apply Nat.succ_lt_mono; assumption. } Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.Arith Require Import PeanoNat." ]
theories/Data/ListNth.v
nth_error_app_L
nth_error_app_R : forall (A B : list T) n, length A <= n -> nth_error (A ++ B) n = nth_error B (n - length A). Proof. induction A; destruct n; simpl; intros; auto. + inversion H. + apply IHA. apply Nat.succ_le_mono; assumption. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.Arith Require Import PeanoNat." ]
theories/Data/ListNth.v
nth_error_app_R
nth_error_weaken : forall ls' (ls : list T) n v, nth_error ls n = Some v -> nth_error (ls ++ ls') n = Some v. Proof. clear. induction ls; destruct n; simpl; intros; unfold value, error in *; try congruence; auto. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.Arith Require Import PeanoNat." ]
theories/Data/ListNth.v
nth_error_weaken
nth_error_nil : forall n, nth_error nil n = @None T. Proof. destruct n; reflexivity. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.Arith Require Import PeanoNat." ]
theories/Data/ListNth.v
nth_error_nil
nth_error_past_end : forall (ls : list T) n, length ls <= n -> nth_error ls n = None. Proof. clear. induction ls; destruct n; simpl; intros; auto. + inversion H. + apply IHls. apply Nat.succ_le_mono; assumption. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.Arith Require Import PeanoNat." ]
theories/Data/ListNth.v
nth_error_past_end
nth_error_length : forall (ls ls' : list T) n, nth_error (ls ++ ls') (n + length ls) = nth_error ls' n. Proof. induction ls; simpl; intros. rewrite Nat.add_0_r. auto. rewrite <-Nat.add_succ_comm. simpl. eapply IHls. Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.Arith Require Import PeanoNat." ]
theories/Data/ListNth.v
nth_error_length
nth_error_length_ge : forall T (ls : list T) n, nth_error ls n = None -> length ls <= n. Proof. induction ls; destruct n; simpl in *; auto; simpl in *. + intro. apply Nat.le_0_l. + inversion 1. + intros. apply ->Nat.succ_le_mono. auto. Qed.
Theorem
theories
[ "From Coq.Lists Require Import List.", "From Coq.Arith Require Import PeanoNat." ]
theories/Data/ListNth.v
nth_error_length_ge
nth_error_length_lt : forall {T} (ls : list T) n val, nth_error ls n = Some val -> n < length ls. Proof. induction ls; destruct n; simpl; intros; auto. + inversion H. + inversion H. + apply Nat.lt_0_succ. + apply ->Nat.succ_lt_mono. apply IHls with (1 := H). Qed.
Lemma
theories
[ "From Coq.Lists Require Import List.", "From Coq.Arith Require Import PeanoNat." ]
theories/Data/ListNth.v
nth_error_length_lt
nth_error_map : forall U (f : T -> U) ls n, nth_error (map f ls) n = match nth_error ls n with | None => None | Some x => Some (f x) end. Proof. induction ls; destruct n; simpl; auto. Qed.
Theorem
theories
[ "From Coq.Lists Require Import List.", "From Coq.Arith Require Import PeanoNat." ]
theories/Data/ListNth.v
nth_error_map
member (a : T) : list T -> Type := | MZ : forall ls, member a (a :: ls) | MN : forall l ls, member a ls -> member a (l :: ls).
Inductive
theories
[ "Require Import Coq.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Member.v
member
to_nat {ls} (m : member a ls) : nat := match m with | MZ _ => 0 | MN _ _ m => S (to_nat m) end.
Fixpoint
theories
[ "Require Import Coq.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Member.v
to_nat
member_eta : forall x ls (m : member x ls), m = match m in member _ ls return member x ls with | MZ ls => MZ x ls | MN _ _ n => MN _ n end. Proof. destruct m; auto. Qed.
Lemma
theories
[ "Require Import Coq.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Member.v
member_eta
member_case : forall x ls (m : member x ls), match ls as ls return member x ls -> Prop with | nil => fun _ => False | l :: ls' => fun m => (exists (pf : l = x), m = match pf in _ = z return member z (l :: ls') with | eq_refl => MZ _ ls' end) \/ exists m' : member x ls', m = MN _ m' end m. Proof. induction m. { left. exists eq_refl. reflexivity. } { right. eauto. } Qed.
Lemma
theories
[ "Require Import Coq.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Member.v
member_case
to_nat_eq_member_eq : forall {_ : EqDec T (@eq T)} x ls (a b : member x ls), to_nat a = to_nat b -> a = b. Proof. induction a; intros. { destruct (member_case b). { destruct H0. subst. rewrite (UIP_refl x0). reflexivity. } { destruct H0. subst. simpl in *. congruence. } } { destruct (member_case b). { exfalso. destruct H0. subst. simpl in *. congruence. } { destruct H0. subst. simpl in *. inversion H; clear H; subst. eapply IHa in H1. f_equal. assumption. } } Qed.
Lemma
theories
[ "Require Import Coq.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Member.v
to_nat_eq_member_eq
nth_member (ls : list T) (n : nat) {struct n} : option { x : T & member x ls } := match ls as ls return option { x : T & member x ls } with | nil => None | l :: ls => match n with | 0 => Some (@existT _ (fun x => member x (l :: ls)) l (MZ _ _)) | S n => match nth_member ls n with | Some (existT v m) => Some (@existT _ _ v (MN _ m)) | None => None end end end.
Fixpoint
theories
[ "Require Import Coq.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Member.v
nth_member
get_next ls y x (m : member x (y :: ls)) : option (member x ls) := match m in member _ ls' return match ls' with | nil => unit | l' :: ls' => option (member x ls') end with | MZ _ => None | MN _ _ m => Some m end.
Definition
theories
[ "Require Import Coq.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Member.v
get_next
Injective_MN x y ls m m' : Injective (@MN x y ls m = @MN x y ls m'). Proof. refine {| result := m = m' |}. intro. assert (get_next (MN y m) = get_next (MN y m')). { rewrite H. reflexivity. } { simpl in *. inversion H0. reflexivity. } Defined.
Instance
theories
[ "Require Import Coq.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Member.v
Injective_MN
nth_member_nth_error : forall ls p, nth_member ls (to_nat (projT2 p)) = Some p <-> nth_error ls (to_nat (projT2 p)) = Some (projT1 p). Proof. destruct p. simpl in *. induction m. { simpl. split; auto. } { simpl. split. { intros. destruct (nth_member ls (to_nat m)); try congruence. { destruct s. inv_all. subst. inv_all. subst. apply IHm. reflexivity. } } { intros. eapply IHm in H. rewrite H. reflexivity. } } Qed.
Lemma
theories
[ "Require Import Coq.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Member.v
nth_member_nth_error
member_In : forall ls (t : T), member t ls -> List.In t ls. Proof. induction 1; simpl; auto. Qed.
Lemma
theories
[ "Require Import Coq.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Member.v
member_In
R_nat_S : nat -> nat -> Prop := | R_S : forall n, R_nat_S n (S n).
Inductive
theories
[ "From Coq.Arith Require Arith.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Nat.v
R_nat_S
wf_R_S : well_founded R_nat_S. Proof. red; induction a; constructor; intros. inversion H. inversion H; subst; auto. Defined.
Theorem
theories
[ "From Coq.Arith Require Arith.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Nat.v
wf_R_S
R_nat_lt : nat -> nat -> Prop := | R_lt : forall n m, n < m -> R_nat_lt n m.
Inductive
theories
[ "From Coq.Arith Require Arith.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Nat.v
R_nat_lt
wf_R_lt : well_founded R_nat_lt. Proof. red; induction a; constructor; intros. { inversion H. exfalso. subst. inversion H0. } { inversion H; clear H; subst. inversion H0; clear H0; subst; auto. inversion IHa. eapply H. constructor. eapply H1. } Defined.
Theorem
theories
[ "From Coq.Arith Require Arith.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Nat.v
wf_R_lt
Monoid_nat_plus : Monoid nat := {| monoid_plus := plus ; monoid_unit := 0 |}.
Definition
theories
[ "From Coq.Arith Require Arith.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Nat.v
Monoid_nat_plus
Monoid_nat_mult : Monoid nat := {| monoid_plus := mult ; monoid_unit := 1 |}. Global Instance Injective_S (a b : nat) : Injective (S a = S b). refine {| result := a = b |}. abstract (inversion 1; auto). Defined.
Definition
theories
[ "From Coq.Arith Require Arith.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Nat.v
Monoid_nat_mult
nat_get_eq (n m : nat) (pf : unit -> n = m) : n = m := match PeanoNat.Nat.eq_dec n m with | left pf => pf | right bad => match bad (pf tt) with end end.
Definition
theories
[ "From Coq.Arith Require Arith.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Nat.v
nat_get_eq
Roption : Relation_Definitions.relation (option T) := | Roption_None : Roption None None | Roption_Some : forall x y, R x y -> Roption (Some x) (Some y).
Inductive
theories
[ "Require Import Coq.", "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Option.v
Roption
Reflexive_Roption : Reflexive R -> Reflexive Roption. Proof. clear. compute. destruct x; try constructor; auto. Qed.
Lemma
theories
[ "Require Import Coq.", "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Option.v
Reflexive_Roption
Symmetric_Roption : Symmetric R -> Symmetric Roption. Proof. clear. compute. intros. destruct H0; constructor. auto. Qed.
Lemma
theories
[ "Require Import Coq.", "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]
theories/Data/Option.v
Symmetric_Roption