fact stringlengths 17 6.18k | type stringclasses 17 values | library stringclasses 3 values | imports listlengths 0 12 | filename stringclasses 115 values | symbolic_name stringlengths 1 30 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
uip_prop_trans (dec : forall x y : A, x = y \/ x <> y) {x : A} : forall {y : A} (pf pf' : x = y), pf = pf' := let comp := fun (x y y' : A) (eq1 : x = y) (eq2 : x = y') => eq_ind x (fun a : A => a = y') eq2 y eq1 in let eq_dec := dec x in let nu := fun (y : A) (u : x = y) => match eq_dec y with | or_introl eqxy => eqxy | or_intror neqxy => False_ind (x = y) (neqxy u) end in let nu_constant := fun (y : A) (u v : x = y) => let o := eq_dec y in match o as o0 return (match o0 with | or_introl eqxy => eqxy | or_intror neqxy => False_ind (x = y) (neqxy u) end = match o0 with | or_introl eqxy => eqxy | or_intror neqxy => False_ind (x = y) (neqxy v) end) with | or_introl Heq => eq_refl | or_intror Hneq => match Hneq u as f return (False_ind (x = y) f = False_ind (x = y) (Hneq v)) with end end in let nu_inv := fun (y : A) (v : x = y) => comp x x y (nu x eq_refl) v in let trans_sym_eq := fun (x y : A) (u : x = y) => match u as e in (_ = y0) return (comp x y0 y0 e e = eq_refl) with | eq_refl => eq_refl end in let nu_left_inv_on := fun (y : A) (u : x = y) => match u as e in (_ = y0) return (nu_inv y0 (nu y0 e) = e) with | eq_refl => trans_sym_eq _ _ (nu _ eq_refl) end in fun (y : A) (p1 p2 : x = y) => eq_ind (nu_inv y (nu y p1)) (fun p3 : x = y => p3 = p2) (eq_ind (nu_inv y (nu y p2)) (fun p3 : x = y => nu_inv y (nu y p1) = p3) (eq_ind (nu y p1) (fun e : x = y => nu_inv y (nu y p1) = nu_inv y e) eq_refl (nu y p2) (nu_constant y p1 p2)) p2 (nu_left_inv_on _ p2)) p1 (nu_left_inv_on _ p1). | Definition | theories | [] | theories/Data/Eq/UIP_trans.v | uip_prop_trans | |
uip_trans (dec : forall x y : A, {x = y} + {x <> y}) := @uip_prop_trans (fun a b => match dec a b with | left pf => or_introl pf | right pf' => or_intror pf' end). | Definition | theories | [] | theories/Data/Eq/UIP_trans.v | uip_trans | |
BuildGraph : Type := { emptyGraph : G ; addVertex : V -> G -> G ; addEdge : V -> V -> G -> G }. | Class | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/BuildGraph.v | BuildGraph | |
GraphBuilderT (T : Type) : Type := stateT G m T. Global Instance Monad_GraphBuilder : Monad GraphBuilderT := Monad_stateT _ _. Global Instance MonadT_GraphBuilder : MonadT GraphBuilderT m := MonadT_stateT _ _. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/BuildGraph.v | GraphBuilderT | |
State_GraphBuilder : MonadState G GraphBuilderT := MonadState_stateT _ _. Import MonadNotation. Local Open Scope monad_scope. | Instance | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/BuildGraph.v | State_GraphBuilder | |
newEdge (f t : V) : GraphBuilderT unit := g <- get ;; put (addEdge f t g). | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/BuildGraph.v | newEdge | |
newVertex (v : V) : GraphBuilderT unit := g <- get ;; put (addVertex v g). | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/BuildGraph.v | newVertex | |
buildGraph {v} (c : GraphBuilderT v) (g : G) : m G := execStateT c g. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/BuildGraph.v | buildGraph | |
Graph : Type := { verticies : G -> list V ; successors : G -> V -> list V }. | Class | theories | [] | theories/Data/Graph/Graph.v | Graph | |
adj_graph : Type := map. | Definition | theories | [
"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.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/GraphAdjList.v | adj_graph | |
verts (g : adj_graph) : list V := let c := foldM (m := writerT (Monoid_list_app) ident) (fun k_v _ => let k := fst k_v in tell (k :: nil)) (ret tt) g in psnd (unIdent (runWriterT c)). | Definition | theories | [
"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.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/GraphAdjList.v | verts | |
succs (g : adj_graph) (v : V) : list V := match lookup v g with | None => nil | Some vs => vs end. Global Instance Graph_adj_graph : Graph V adj_graph := { verticies := verts ; successors := succs }. | Definition | theories | [
"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.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/GraphAdjList.v | succs | |
add_vertex (v : V) (g : adj_graph) : adj_graph := if contains v g then g else add v nil g. (** TODO: Move this **) | Definition | theories | [
"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.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/GraphAdjList.v | add_vertex | |
list_in_dec v (ls : list V) : bool := match ls with | nil => false | l :: ls => if eq_dec l v then true else list_in_dec v ls end. | Fixpoint | theories | [
"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.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/GraphAdjList.v | list_in_dec | |
add_edge (f t : V) (g : adj_graph) : adj_graph := match lookup f g with | None => add f (t :: nil) g | Some vs => if list_in_dec t vs then g else add f (t :: vs) g end. Global Instance GraphBuilder_adj_graph : BuildGraph V adj_graph := { emptyGraph := empty ; addVertex := add_vertex ; addEdge := add_edge }. | Definition | theories | [
"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.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Graph/GraphAdjList.v | add_edge | |
list_in_dec v (ls : list V) : bool := match ls with | nil => false | l :: ls => if eq_dec l v then true else list_in_dec v ls end. | Fixpoint | theories | [
"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."
] | theories/Data/Graph/GraphAlgos.v | list_in_dec | |
dfs' : V -> list V -> m (list V) := mfix_multi (V :: list V :: nil) (list V) (fun rec from seen => if list_in_dec from seen then ret (m:=m) seen else foldM (fun v acc => if list_in_dec v acc then ret (m:=m) acc else rec v acc) (ret (m:=m) seen) (successors g from)). | Definition | theories | [
"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."
] | theories/Data/Graph/GraphAlgos.v | dfs' | |
dfs (from : V) : list V := let count := Npos (List.fold_left (fun acc _ => Pos.succ acc) (verticies g) 1%positive) in let res := runGFix (dfs' from nil) count in match res with | Diverge => (** This should never happen! **) verticies g | Term v => v end. | Definition | theories | [
"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."
] | theories/Data/Graph/GraphAlgos.v | dfs | |
alist : Type := list (K * V). | Definition | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | alist | |
alist_remove (k : K) (m : alist) : alist := List.filter (fun x => negb (k ?[ R ] (fst x))) m. | Definition | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | alist_remove | |
alist_add (k : K) (v : V) (m : alist) : alist := (k, v) :: alist_remove k m. | Definition | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | alist_add | |
alist_find (k : K) (m : alist) : option V := match m with | nil => None | (k',v) :: ms => if k ?[ R ] k' then Some v else alist_find k ms end. | Fixpoint | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | alist_find | |
alist_find' (k: K) : alist -> option V := fmap snd ∘ find (rel_dec k ∘ fst). | Definition | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | alist_find' | |
alist_find_alt (m: alist) : forall k: K, alist_find k m = alist_find' k m. Proof. induction m; intuition. unfold alist_find', compose. simpl. destruct (k ?[ R ] a0) eqn:Heq; [intuition|]. rewrite IHm. reflexivity. Qed. | Lemma | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | alist_find_alt | |
fold_alist (acc : T) (map : alist) : T := match map with | nil => acc | (k,v) :: m => let acc := f k v acc in fold_alist acc m end. | Fixpoint | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | fold_alist | |
fold_alist' : T -> alist -> T := flip $ fold_left (flip $ uncurry f). | Definition | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | fold_alist' | |
fold_alist_alt (map: alist) : forall acc: T, fold_alist acc map = fold_alist' acc map. Proof. induction map; intuition. simpl. rewrite IHmap. reflexivity. Qed. | Lemma | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | fold_alist_alt | |
alist_union (m1 m2 : alist) : alist := fold_alist alist_add m2 m1. Global Instance Map_alist : Map K V alist := { empty := nil ; add := @alist_add ; remove := alist_remove ; lookup := alist_find ; union := @alist_union }. | Definition | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | alist_union | |
mapsto_alist (m : alist) k v : Prop := alist_find k m = Some v. | Definition | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | mapsto_alist | |
mapsto_alist_cons : forall k v m k' v', mapsto_alist ((k',v') :: m) k v <-> ( (mapsto_alist m k v /\ ~R k k') \/ (R k k' /\ v = v')). Proof. unfold mapsto_alist; intuition; simpl in *. { consider (k ?[ R ] k'); intros. { right. inversion H0; auto. } { left. auto. } } { consider (k ?[ R ] k'); intros; intuition. } { consider (k ?[ R ] k'); intros; intuition. congruence. } Qed. | Lemma | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | mapsto_alist_cons | |
mapsto_lookup_alist : forall (k : K) (v : V) (m : list (K * V)), lookup k m = Some v <-> mapsto_alist m k v. Proof. reflexivity. Qed. | Theorem | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | mapsto_lookup_alist | |
mapsto_remove_eq_alist : forall (m : list (K * V)) (k : K) (v : V), ~mapsto_alist (remove k m) k v. Proof. unfold mapsto_alist, remove, alist_remove; simpl. intros. induction m; simpl; auto. { congruence. } { destruct a; simpl in *. consider (k ?[ R ] k0); auto; intros. simpl. consider (k ?[ R ] k0); auto. } Qed. | Theorem | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | mapsto_remove_eq_alist | |
mapsto_remove_neq_alist : forall (m : list (K * V)) (k : K), forall k', ~ R k k' -> forall v', (mapsto_alist m k' v' <-> mapsto_alist (remove k m) k' v'). Proof. unfold mapsto_alist, add; simpl. intros. induction m; simpl in *. { intuition. } { destruct a. simpl in *. consider (k' ?[ R ] k0); intros. { consider (k ?[ R ] k0); intros. { exfalso. eauto. } { simpl. consider (k' ?[ R ] k0); intros. { intuition. } { exfalso; auto. } } } { rewrite IHm. consider (k ?[ R ] k0); simpl; intros. { intuition. } { consider (k' ?[ R ] k0); intros. { exfalso; auto. } { intuition. } } } } Qed. | Theorem | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | mapsto_remove_neq_alist | |
mapsto_add_eq_alist : forall (m : list (K * V)) (k : K) (v : V), mapsto_alist (add k v m) k v. Proof. unfold mapsto_alist, add, alist_add; simpl. intros. consider (k ?[ R ] k); auto. intro. exfalso. apply H. reflexivity. Qed. | Theorem | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | mapsto_add_eq_alist | |
mapsto_add_neq_alist : forall (m : list (K * V)) (k : K) (v : V), forall k', ~ R k k' -> forall v', (mapsto_alist m k' v' <-> mapsto_alist (add k v m) k' v'). Proof. unfold mapsto_alist, add; simpl. intros. consider (k' ?[ R ] k); try solve [ intros; exfalso; auto ]. intros. eapply mapsto_remove_neq_alist in H. eapply H. Qed. | Theorem | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | mapsto_add_neq_alist | |
remove_eq_alist : forall (m : alist) (k : K), alist_find k (alist_remove k m) = None. Proof. unfold mapsto_alist. induction m; simpl; eauto; try congruence. intros; consider (k ?[ R ] fst a); simpl; intros; eauto. destruct a; simpl in *. consider (k ?[ R ] k0); eauto. tauto. Qed. | Theorem | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | remove_eq_alist | |
remove_neq_alist : forall (m : alist) (k k' : K), ~R k' k -> alist_find k (alist_remove k' m) = alist_find k m. Proof. unfold mapsto_alist. induction m; simpl; eauto; try congruence. destruct a; simpl. intros; consider (k' ?[ R ] k); simpl; intros; eauto. { consider (k0 ?[ R ] k); intros; eauto. exfalso. eapply H. etransitivity; eauto. } { consider (k0 ?[ R ] k); eauto. } Qed. Global Instance MapLaws_alist : MapOk R Map_alist. Proof. refine {| mapsto := fun k v m => mapsto_alist m k v |}; eauto using mapsto_lookup_alist, mapsto_add_eq_alist, mapsto_add_neq_alist. { intros; intro. inversion H. } { unfold mapsto_alist; simpl. intros. rewrite remove_eq_alist. congruence. } { unfold mapsto_alist. simpl; intros. erewrite (@remove_neq_alist m _ _ H). reflexivity. } Defined. | Theorem | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | remove_neq_alist | |
m := alist nat nat. | Definition | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | m | |
Map_m : Map nat (alist nat). apply Map_alist. eauto with typeclass_instances. Defined. | Instance | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | Map_m | |
z : m := (fix fill n acc : m := let acc := add n n acc in match n with | 0 => acc | S n => fill n acc end) 500 empty. Time Eval compute in let z := z in (fix find_all n : unit := let _ := lookup n z in match n with | 0 => tt | S n => find_all n end) 500. | Definition | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import\n Basics."
] | theories/Data/Map/FMapAList.v | z | |
pmap : Type := | Empty | Branch : option T -> pmap -> pmap -> pmap. | Inductive | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap | |
pmap_here (m : pmap) : option T := match m with | Empty => None | Branch d _ _ => d end. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_here | |
pmap_left (m : pmap) : pmap := match m with | Empty => Empty | Branch _ l _ => l end. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_left | |
pmap_right (m : pmap) : pmap := match m with | Empty => Empty | Branch _ _ r => r end. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_right | |
pmap_lookup (p : positive) (m : pmap) {struct p} : option T := match m with | Empty => None | Branch d l r => match p with | xH => d | xO p => pmap_lookup p l | xI p => pmap_lookup p r end end. | Fixpoint | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_lookup | |
pmap_insert (p : positive) (v : T) (m : pmap) {struct p} : pmap := match p with | xH => Branch (Some v) (pmap_left m) (pmap_right m) | xO p => Branch (pmap_here m) (pmap_insert p v (pmap_left m)) (pmap_right m) | xI p => Branch (pmap_here m) (pmap_left m) (pmap_insert p v (pmap_right m)) end. | Fixpoint | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_insert | |
branch (o : option T) (l r : pmap) : pmap := match o , l , r with | None , Empty , Empty => Empty | _ , _ , _ => Branch o l r end. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | branch | |
pmap_remove (p : positive) (m : pmap) {struct p} : pmap := match m with | Empty => Empty | Branch d l r => match p with | xH => branch None l r | xO p => branch d (pmap_remove p l) r | xI p => branch d l (pmap_remove p r) end end. | Fixpoint | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_remove | |
pmap_empty : pmap := Empty. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_empty | |
pmap_union (f m : pmap) : pmap := match f with | Empty => m | Branch d l r => Branch (match d with | Some x => Some x | None => pmap_here m end) (pmap_union l (pmap_left m)) (pmap_union r (pmap_right m)) end. Global Instance Map_pmap : Map positive T pmap := { empty := pmap_empty ; add := pmap_insert ; remove := pmap_remove ; lookup := pmap_lookup ; union := pmap_union }. | Fixpoint | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_union | |
tilde_1_inj_neg : forall k k', (k~1)%positive <> (k'~1)%positive -> k <> k'. Proof. induction k; destruct k'; intuition; try match goal with | H : _ = _ |- _ => inversion H; clear H; subst end; intuition eauto. Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | tilde_1_inj_neg | |
tilde_0_inj_neg : forall k k', (k~0)%positive <> (k'~0)%positive -> k <> k'. Proof. induction k; destruct k'; intuition; try match goal with | H : _ = _ |- _ => inversion H; clear H; subst end; intuition eauto. Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | tilde_0_inj_neg | |
pmap_lookup_insert_empty : forall k k' v, k <> k' -> pmap_lookup k' (pmap_insert k v Empty) = None. Proof. induction k; destruct k'; simpl; intros; eauto using tilde_0_inj_neg, tilde_1_inj_neg. destruct k'; simpl; auto. destruct k'; simpl; auto. destruct k'; simpl; auto. destruct k'; simpl; auto. congruence. Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_lookup_insert_empty | |
lookup_empty : forall k, pmap_lookup k Empty = None. Proof. destruct k; reflexivity. Qed. Hint Rewrite lookup_empty pmap_lookup_insert_empty using (eauto using tilde_1_inj_neg, tilde_1_inj_neg) : pmap_rw. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | lookup_empty | |
pmap_lookup_insert_eq : forall (m : pmap) (k : positive) (v : T), pmap_lookup k (pmap_insert k v m) = Some v. Proof. intros m k; revert m. induction k; simpl; intros; forward; Cases.rewrite_all_goal; eauto. Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_lookup_insert_eq | |
pmap_lookup_insert_Some_neq : forall (m : pmap) (k : positive) (v : T) (k' : positive), k <> k' -> forall v' : T, pmap_lookup k' m = Some v' <-> pmap_lookup k' (pmap_insert k v m) = Some v'. Proof. intros m k; revert m. induction k; simpl; intros; forward; Cases.rewrite_all_goal; autorewrite with pmap_rw; eauto. { destruct k'; simpl; destruct m; simpl; autorewrite with pmap_rw; Cases.rewrite_all_goal; try reflexivity. erewrite IHk; eauto using tilde_1_inj_neg. reflexivity. } { destruct k'; simpl; destruct m; simpl; autorewrite with pmap_rw; Cases.rewrite_all_goal; try reflexivity; try congruence. erewrite IHk. reflexivity. eauto using tilde_0_inj_neg. } { destruct k'; simpl; destruct m; simpl; autorewrite with pmap_rw; Cases.rewrite_all_goal; try reflexivity; try congruence. } Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_lookup_insert_Some_neq | |
pmap_lookup_insert_None_neq : forall (m : pmap) (k : positive) (v : T) (k' : positive), k <> k' -> pmap_lookup k' m = None <-> pmap_lookup k' (pmap_insert k v m) = None. Proof. intros m k; revert m. induction k; simpl; intros; forward; Cases.rewrite_all_goal; autorewrite with pmap_rw; eauto. { destruct k'; simpl; destruct m; simpl; autorewrite with pmap_rw; Cases.rewrite_all_goal; try reflexivity. erewrite IHk; eauto using tilde_1_inj_neg. reflexivity. } { destruct k'; simpl; destruct m; simpl; autorewrite with pmap_rw; Cases.rewrite_all_goal; try reflexivity; try congruence. erewrite IHk. reflexivity. eauto using tilde_0_inj_neg. } { destruct k'; simpl; destruct m; simpl; autorewrite with pmap_rw; Cases.rewrite_all_goal; try reflexivity; try congruence. } Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_lookup_insert_None_neq | |
pmap_lookup_insert_neq : forall (m : pmap) (k : positive) (v : T) (k' : positive), k <> k' -> forall v' : T, pmap_lookup k' (pmap_insert k v m) = pmap_lookup k' m. Proof. intros. remember (pmap_lookup k' m). destruct o; [ apply pmap_lookup_insert_Some_neq; intuition | apply pmap_lookup_insert_None_neq; intuition]. Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_lookup_insert_neq | |
pmap_lookup_remove_eq : forall (m : pmap) (k : positive) (v : T), pmap_lookup k (pmap_remove k m) <> Some v. Proof. induction m; destruct k; simpl; intros; try congruence. { destruct o; simpl; eauto. destruct m1; simpl; eauto. destruct (pmap_remove k m2) eqn:?; try congruence. rewrite <- Heqp. eauto. } { destruct o; simpl; eauto. destruct (pmap_remove k m1) eqn:?; try congruence. - destruct m2; try congruence; eauto. destruct k; simpl; congruence. - rewrite <- Heqp. eauto. } { destruct m1; try congruence. destruct m2; try congruence. } Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_lookup_remove_eq | |
pmap_lookup_remove_neq : forall (m : pmap) (k k' : positive), k <> k' -> forall v' : T, pmap_lookup k' m = Some v' <-> pmap_lookup k' (pmap_remove k m) = Some v'. Proof. induction m. Local Ltac t := unfold branch; repeat match goal with | |- context [ match ?X with _ => _ end ] => lazymatch X with | match _ with _ => _ end => fail | _ => destruct X eqn:?; subst; try tauto end end. { destruct k; simpl; split; try congruence. } { destruct k', k; simpl; intros; try solve [ t; rewrite lookup_empty; tauto ]. { assert (k <> k') by congruence. rewrite IHm2; eauto. simpl. t. rewrite lookup_empty. tauto. } { assert (k <> k') by congruence. rewrite IHm1; eauto. simpl. t. rewrite lookup_empty. tauto. } } Qed. Global Instance MapOk_pmap : MapOk (@eq _) Map_pmap. Proof. refine {| mapsto := fun k v m => pmap_lookup k m = Some v |}. { abstract (induction k; simpl; congruence). } { abstract (induction k; simpl; intros; forward). } { eauto using pmap_lookup_insert_eq. } { eauto using pmap_lookup_insert_Some_neq. } { eauto using pmap_lookup_remove_eq. } { eauto using pmap_lookup_remove_neq. } Defined. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | pmap_lookup_remove_neq | |
from_list : list T -> pmap := (fix from_list acc i ls {struct ls} := match ls with | nil => acc | List.cons l ls => from_list (pmap_insert i l acc) (Pos.succ i) ls end) Empty 1%positive. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | from_list | |
fmap_pmap (m : pmap T) : pmap U := match m with | Empty => Empty | Branch h l r => Branch (fmap f h) (fmap_pmap l) (fmap_pmap r) end. | Fixpoint | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | fmap_pmap | |
fmap_lookup : forall a b m, mapsto a b m -> mapsto a (f b) (fmap_pmap m). Proof. induction a; destruct m; simpl; intros; try congruence. { eapply IHa. eapply H. } { eapply IHa; eapply H. } { subst. auto. } Qed. | Theorem | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | fmap_lookup | |
fmap_lookup_bk : forall a b m, mapsto a b (fmap_pmap m) -> exists b', mapsto a b' m /\ f b' = b. Proof. induction a; destruct m; simpl; intros; try congruence. { eapply IHa. eapply H. } { eapply IHa. eapply H. } { destruct o; try congruence. eexists; split; eauto. inversion H; auto. } Qed. | Theorem | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapPositive.v | fmap_lookup_bk | |
twothree (T : Type) : Type := | Leaf | Two : twothree T -> K -> T -> twothree T -> twothree T | Three : twothree T -> K -> T -> twothree T -> K -> T -> twothree T -> twothree T. Arguments Leaf {T}. Arguments Two {T} _ _ _ _. Arguments Three {T} _ _ _ _ _ _ _. | Inductive | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | twothree | |
remove_greatest (m : twothree V) {T} (k_oops : unit -> T) (k_ok : K -> V -> twothree V -> T) : T := match m with | Leaf => k_oops tt | Two l k v r => remove_greatest r (fun _ => k_ok k v l) (fun k' v' r' => k_ok k' v' (Two l k v r')) | Three l k v m k' v' r => remove_greatest r (fun _ => k_ok k' v' (Two l k v m)) (fun k'' v'' r'' => k_ok k'' v'' (Three l k v m k' v' r'')) end. | Fixpoint | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | remove_greatest | |
twothree_modify (m : twothree V) {T} (k_ok : twothree V -> T) (k_splice_up : twothree V -> K -> V -> twothree V -> T) {struct m} : T := match m with | Leaf => match def with | Some v => k_splice_up Leaf k v Leaf | None => k_ok Leaf end | Two l k' v' r => match RD_K k k' with | Eq => match upd v' with | Some v' => k_ok (Two l k v' r) | None => remove_greatest l (fun _ => k_ok r) (fun k v l => k_ok (Two l k v r)) end | Lt => twothree_modify l (fun l => k_ok (Two l k' v' r)) (fun l'' k'' v'' r'' => k_ok (Three l'' k'' v'' r'' k' v' r)) | Gt => twothree_modify r (fun r => k_ok (Two l k' v' r)) (fun l'' k'' v'' r'' => k_ok (Three l k' v' l'' k'' v'' r'')) end | Three l k1 v1 m k2 v2 r => match RD_K k k1 with | Eq => match upd v1 with | Some v' => k_ok (Three l k v' m k2 v2 r) | None => remove_greatest l (fun _ => k_ok (Two m k2 v2 r)) (fun k1 v1 l => k_ok (Three l k1 v2 m k2 v2 r)) end | Lt => twothree_modify l (fun l' => k_ok (Three l' k1 v1 m k2 v2 r)) (fun l' k' v' r' => k_splice_up (Two l' k' v' r') k1 v1 (Two m k2 v2 r)) | Gt => match RD_K k k2 with | Eq => match upd v2 with | Some v2 => k_ok (Three l k1 v1 m k v2 r) | None => remove_greatest m (fun _ => k_ok (Two l k1 v1 r)) (fun k' v' m' => k_ok (Three l k1 v1 m' k' v' r)) end | Lt => twothree_modify m (fun m' => k_ok (Three l k1 v1 m' k2 v2 r)) (fun l' k' v' r' => k_splice_up (Two l k1 v1 l') k' v' (Two r' k2 v2 r)) | Gt => twothree_modify r (fun r' => k_ok (Three l k1 v1 m k2 v2 r')) (fun l' k' v' r' => k_splice_up (Two l k1 v1 m) k2 v2 (Two l' k' v' r')) end end end. | Fixpoint | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | twothree_modify | |
twothree_add {V} (k : K) (v : V) (m : twothree V) : twothree V := twothree_modify k (fun _ => Some v) (Some v) m (fun m => m) Two. | Definition | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | twothree_add | |
twothree_remove {V} (k : K) (m : twothree V) : twothree V := twothree_modify k (fun _ => None) None m (fun m => m) Two. | Definition | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | twothree_remove | |
twothree_find {V} (k : K) (m : twothree V) : option V := match m with | Leaf => None | Two l k' v' r => match RD_K k k' with | Eq => Some v' | Lt => twothree_find k l | Gt => twothree_find k r end | Three l k1 v1 m k2 v2 r => match RD_K k k1 with | Eq => Some v1 | Lt => twothree_find k l | Gt => match RD_K k k2 with | Eq => Some v2 | Lt => twothree_find k m | Gt => twothree_find k r end end end. | Fixpoint | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | twothree_find | |
twothree_fold (acc : T) (map : twothree V) : T := match map with | Leaf => acc | Two l k v r => let acc := twothree_fold acc l in let acc := f k v acc in twothree_fold acc r | Three l k1 v1 m k2 v2 r => let acc := twothree_fold acc l in let acc := f k1 v1 acc in let acc := twothree_fold acc m in let acc := f k2 v2 acc in twothree_fold acc r end. | Fixpoint | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | twothree_fold | |
twothree_union {V} (m1 m2 : twothree V) : twothree V := twothree_fold twothree_add m2 m1. Global Instance Map_twothree V : Map K V (twothree V) := { empty := Leaf ; add := twothree_add ; remove := twothree_remove ; lookup := twothree_find ; union := twothree_union }. Global Instance Foldable_twothree V : Foldable (twothree V) (K * V) := fun _ f b x => twothree_fold (fun k v => f (k,v)) b x. | Definition | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | twothree_union | |
m := twothree nat nat. | Definition | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | m | |
Map_m : Map nat (twothree nat). apply Map_twothree. apply Compare_dec.nat_compare. Defined. | Instance | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | Map_m | |
z : m := (fix fill n acc : m := let acc := add n n acc in match n with | 0 => acc | S n => fill n acc end) 500 empty. Time Eval vm_compute in let z := z in (fix find_all n : unit := let _ := lookup n z in match n with | 0 => tt | S n => find_all n end) 500. | Definition | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Map/FMapTwoThreeK.v | z | |
cont (t : Type) : Type := mkCont { runCont : (t -> Ans) -> Ans }. Global Instance Monad_cont : Monad cont := { ret := fun _ v => mkCont (fun k => k v) ; bind := fun _ c1 _ c2 => mkCont (fun k => runCont c1 (fun t => runCont (c2 t) k)) }. Global Instance Cont_cont : Cont cont := { callCC := fun _ _ f => mkCont (fun c => runCont (f (fun x => mkCont (fun _ => c x))) c) }. | Record | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/ContMonad.v | cont | |
mapCont (f : Ans -> Ans) {a} (c : cont a) : cont a := mkCont (fun x => f (runCont c x)). | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/ContMonad.v | mapCont | |
withCont {a b} (f : (b -> Ans) -> (a -> Ans)) (c : cont a) : cont b := mkCont (fun x => runCont c (f x)). *) Variable M : Type -> Type. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/ContMonad.v | withCont | |
contT (A : Type) : Type := mkContT { runContT : (A -> M R) -> M R }. Global Instance Monad_contT : Monad contT := { ret := fun _ x => mkContT (fun k => k x) ; bind := fun _ _ c1 c2 => mkContT (fun c => runContT c1 (fun a => runContT (c2 a) c)) }. Global Instance MonadT_contT {Monad_M : Monad M} : MonadT contT M := { lift := fun _ c => mkContT (bind c) }. (* | Record | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/ContMonad.v | contT | |
mapContT (f : m Ans -> m Ans) {a} (c : contT a) : contT a := mkContT (fun x => f (runContT c x)). | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/ContMonad.v | mapContT | |
withContT {a b} (f : (b -> m Ans) -> (a -> m Ans)) (c : contT a) : contT b := mkContT (fun x => runContT c (f x)). *) | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/ContMonad.v | withContT | |
resetT {M} {Monad_M : Monad M} {R R'} (u : contT R M R) : contT R' M R := mkContT (fun k => bind (runContT u ret) k). | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/ContMonad.v | resetT | |
shiftT {M} {Monad_M : Monad M} {R A} (f : (A -> M R) -> contT R M R) : contT R M A := mkContT (fun k => runContT (f k) ret). | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/ContMonad.v | shiftT | |
eitherT A := mkEitherT { unEitherT : m (sum T A) }. Variable M : Monad m. Global Instance Monad_eitherT : Monad eitherT := { ret := fun _ x => mkEitherT (ret (inr x)) ; bind := fun _ _ c f => mkEitherT ( xM <- unEitherT c ;; match xM with | inl x => ret (inl x) | inr x => unEitherT (f x) end ) }. Global Instance Exception_eitherT : MonadExc T eitherT := { raise := fun _ v => mkEitherT (ret (inl v)) ; catch := fun _ c h => mkEitherT ( xM <- unEitherT c ;; match xM with | inl x => unEitherT (h x) | inr x => ret (inr x) end ) }. Global Instance MonadPlus_eitherT : MonadPlus eitherT := { mplus _A _B mA mB := mkEitherT ( x <- unEitherT mA ;; match x with | inl _ => y <- unEitherT mB ;; match y with | inl t => ret (inl t) | inr b => ret (inr (inr b)) end | inr a => ret (inr (inl a)) end ) }. Global Instance MonadT_eitherT : MonadT eitherT m := { lift := fun _ c => mkEitherT (liftM inr c) }. Global Instance MonadState_eitherT {T} (MS : MonadState T m) : MonadState T eitherT := { get := lift get ; put := fun v => lift (put v) }. Global Instance MonadReader_eitherT {T} (MR : MonadReader T m) : MonadReader T eitherT := { ask := lift ask ; local := fun _ f cmd => mkEitherT (local f (unEitherT cmd)) }. Global Instance MonadWriter_eitherT {T} (Mon : Monoid T) (MW : MonadWriter Mon m) : MonadWriter Mon eitherT := { tell := fun x => lift (tell x) ; listen := fun _ c => mkEitherT ( x <- listen (unEitherT c) ;; match x with | (inl l, _) => ret (inl l) | (inr a, t) => ret (inr (a, t)) end) ; pass := fun _ c => mkEitherT ( x <- unEitherT c ;; match x with | inl s => ret (inl s) | inr (a,f) => pass (ret (inr a, f)) end) }. Global Instance MonadFix_eitherT (MF : MonadFix m) : MonadFix eitherT := { mfix := fun _ _ r v => mkEitherT (mfix (fun f x => unEitherT (r (fun x => mkEitherT (f x)) x)) v) }. | Inductive | theories | [
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/EitherMonad.v | eitherT | |
FixResult (T : Type) := | Term : T -> FixResult T | Diverge : FixResult T. Arguments Diverge {_}. Arguments Term {_} _. (** The GFix monad is like monad fix except that it encapsulates the "gas" that is used as the measure **) | Inductive | theories | [
"Require Import ExtLib.",
"Require Import BinPos.",
"Require Import ExtLib."
] | theories/Data/Monads/FuelMonad.v | FixResult | |
GFix (T : Type) : Type := mkGFix { runGFix : N -> FixResult T }. Global Instance MonadFix_GFix : MonadFix GFix := { mfix := fun T U f v => mkGFix (fun n : N => match n with | N0 => Diverge | Npos g => let F := fix rec (acc : T -> FixResult U) (gas : positive) (x : T) : FixResult U := match gas return FixResult U with | xI p => runGFix (f (fun x => mkGFix (fun n => rec (fun x => rec acc p x) p x)) x) n | xO p => rec (fun x => rec acc p x) p x | xH => runGFix (f (fun x => mkGFix (fun _ => acc x)) x) n end in F (fun x => runGFix (f (fun _ => mkGFix (fun _ => Diverge)) x) n) g v end) }. Global Instance Monad_GFix : Monad GFix := { ret := fun _ v => mkGFix (fun _ => Term v) ; bind := fun _ _ c1 c2 => mkGFix (fun gas => match runGFix c1 gas with | Diverge => Diverge | Term v => runGFix (c2 v) gas end) }. | Inductive | theories | [
"Require Import ExtLib.",
"Require Import BinPos.",
"Require Import ExtLib."
] | theories/Data/Monads/FuelMonad.v | GFix | |
foo : nat -> GFix ident nat := mfix (fun recur n => match n with | 0 => ret 0 | S n => recur n end). Eval compute in runGFix (foo 10) 100000000000000000000000. **) | Definition | theories | [
"Require Import ExtLib.",
"Require Import BinPos.",
"Require Import ExtLib."
] | theories/Data/Monads/FuelMonad.v | foo | |
FixResult_eq (a b : FixResult T) : Prop := match a , b with | Diverge , Diverge => True | Term a , Term b => equal a b | _ , _ => False end. Global Instance type_FixResult : type (FixResult T) := type_from_equal FixResult_eq. Variable tokE : typeOk e. Global Instance typeOk_FixResult : typeOk type_FixResult. Proof. eapply typeOk_from_equal. { unfold proper; simpl. destruct x; destruct y; simpl; intros; auto; try contradiction. apply only_proper in H; auto. destruct H; split; apply tokE; assumption. } { red. destruct x; destruct y; simpl; auto; simpl. symmetry; auto. } { red. destruct x; destruct y; destruct z; simpl; intros; auto; try contradiction. etransitivity; eauto. } Qed. | Definition | theories | [
"Require Import RelationClasses.",
"Require Import Setoid.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/FuelMonadLaws.v | FixResult_eq | |
fix_meq (l r : GFix T) : Prop := equal (runGFix l) (runGFix r). Global Instance type_GFix : type (GFix T) := type_from_equal fix_meq. Global Instance typeOk_GFix : typeOk type_GFix. Proof. eapply typeOk_from_equal. { destruct x; destruct y; simpl. intros; split; intros. { red; simpl. red in H; red. simpl FuelMonad.runGFix in *. eapply only_proper in H; eauto with typeclass_instances. intros; subst. eapply preflexive with (wf := proper); eauto with typeclass_instances. eapply equiv_prefl; eauto with typeclass_instances. solve_proper; intuition. } { red; simpl. red in H; red; simpl FuelMonad.runGFix in *. eapply only_proper in H; eauto with typeclass_instances. intros; subst. eapply preflexive with (wf := proper); eauto with typeclass_instances. eapply equiv_prefl; eauto with typeclass_instances. solve_proper. intuition. } } { red. destruct x; destruct y; simpl; unfold fix_meq. simpl FuelMonad.runGFix in *. intros. symmetry; auto. } { red; destruct x; destruct y; destruct z; simpl; unfold fix_meq; simpl FuelMonad.runGFix in *. intros. etransitivity; eauto. } Qed. Global Instance proper_runGFix : proper (@runGFix T). Proof. repeat red; simpl; intros. eapply H. subst. reflexivity. Qed. Global Instance proper_mkGFix : proper (@mkGFix T). Proof. repeat red; simpl; intros. eapply H. subst. reflexivity. Qed. | Definition | theories | [
"Require Import RelationClasses.",
"Require Import Setoid.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/FuelMonadLaws.v | fix_meq | |
Diverge_minimal : forall C (eC : relation C) x, FixResult_leq eC Diverge x. Proof. destruct x; compute; auto. Qed. | Theorem | theories | [
"Require Import RelationClasses.",
"Require Import Setoid.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/FuelMonadLaws.v | Diverge_minimal | |
Term_maximal : forall C (eC : relation C) x y, FixResult_leq eC (Term x) y -> exists z, y = Term z /\ eC x z. Proof. destruct y; simpl; intros; try contradiction; eauto. Qed. | Theorem | theories | [
"Require Import RelationClasses.",
"Require Import Setoid.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/FuelMonadLaws.v | Term_maximal | |
leq_app : forall B C (eB : relation B) (eC : relation C) (pB : Proper eB) (pC : Proper eC) g (b b' : B) n n', proper g -> proper b -> proper b' -> eB b b' -> BinNat.N.le n n' -> FixResult_leq eC (runGFix (g b) n) (runGFix (g b') n'). Proof. intros. destruct H. specialize (H4 _ _ H0 H1 H2 _ _ H3). auto. Qed. *) | Lemma | theories | [
"Require Import RelationClasses.",
"Require Import Setoid.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/FuelMonadLaws.v | leq_app | |
ident A := mkIdent { unIdent : A }. Global Instance Monad_ident : Monad ident := { ret := fun _ v => mkIdent v ; bind := fun _ _ c f => f (unIdent c) }. | Inductive | theories | [
"Require Import ExtLib."
] | theories/Data/Monads/IdentityMonad.v | ident | |
equal_ident (a b : ident T) : Prop := equal (unIdent a) (unIdent b). Global Instance type_ident : type (ident T) := { equal := equal_ident ; proper := fun x => proper (unIdent x) }. Global Instance typeOk_ident (tT : typeOk e) : typeOk type_ident. Proof. constructor. { unfold equal, proper, type_ident, equal_ident; simpl; intros. apply only_proper; auto. } { red. destruct x. intros. red; simpl. red; simpl. eapply preflexive with (wf := proper); eauto with typeclass_instances. } { red. simpl. unfold equal_ident. intros. symmetry. assumption. } { red; simpl. unfold equal_ident. intros. etransitivity; eassumption. } Qed. Global Instance proper_unIdent : proper unIdent. Proof. red; simpl; red; simpl. destruct x; compute; auto. Qed. Global Instance proper_mkIdent : proper mkIdent. Proof. do 7 red. compute; auto. Qed. | Definition | theories | [
"Require Import Coq.",
"Require Import Setoid.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Monads/IdentityMonadLaws.v | equal_ident | |
istate (i s t: Type) : Type := mkIState { runIState : i -> t * s }. | Record | theories | [
"Require Import ExtLib."
] | theories/Data/Monads/IStateMonad.v | istate | |
evalState {i s t} (c : istate i s t) (s : i) : t := fst (runIState c s). | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Monads/IStateMonad.v | evalState | |
execState {i s t} (c : istate i s t) (st : i) : s := snd (runIState c st). Global Instance IMonad_Ixstate : IxMonad istate := { ret := fun _ _ v => mkIState (fun s => (v, s)) ; bind := fun _ _ _ _ _ c1 c2 => mkIState (fun s => let (v,s) := runIState c1 s in runIState (c2 v) s) }. | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Monads/IStateMonad.v | execState | |
get {i : Type} := @mkIState i i i (fun s => (s,s)). | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Monads/IStateMonad.v | get | |
put {i o : Type} := (fun v => @mkIState i o unit (fun _ => (tt, v))). | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Monads/IStateMonad.v | put | |
put_ {i o : Type} (s : o) : istate i o unit := (bind (put s) (fun _ => ret tt)). | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Monads/IStateMonad.v | put_ |
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