phanerozoic/Coq-Finmap · Datasets at Hugging Face

fact string	type string	library string	imports list	filename string	symbolic_name string
PredType : forall T pT, (pT -> pred T) -> predType T. exact PredType \|\| exact mkPredType. Defined. Arguments PredType [T pT] toP. Local Notation predOfType T := (pred_of_simpl (@pred_of_argType T)).	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	PredType
oextract (T : Type) (o : option T) : o -> T := if o is Some t return o -> T then fun=> t else False_rect T \o notF.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	oextract
oextractE (T : Type) (x : T) (xP : Some x) : oextract xP = x. Proof. by []. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	oextractE
Some_oextract T (x : option T) (x_ex : x) : Some (oextract x_ex) = x. Proof. by case: x x_ex. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	Some_oextract
ojoin T (x : option (option T)) := if x is Some y then y else None.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	ojoin
Some_ojoin T (x : option (option T)) : x -> Some (ojoin x) = x. Proof. by case : x. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	Some_ojoin
ojoinT T (x : option (option T)) : ojoin x -> x. Proof. by case: x. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	ojoinT
TaggedP (T1 : Type) (T2 : T1 -> Type) (P : forall x, T2 x -> Type) : (forall t : {x : T1 & T2 x}, P _ (tagged t)) -> forall (x : T1) (y : T2 x), P x y. Proof. by move=> /(_ (Tagged _ _)). Qed. Arguments TaggedP {T1} T2.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	TaggedP
f : seq K -> seq K.	Axiom	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	f
perm : forall s, perm_eq (f s) (undup s).	Axiom	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	perm
uniq : forall s, uniq (f s).	Axiom	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	uniq
E : forall (s : seq K), f s =i s.	Axiom	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	E
eq : forall (s s' : seq K), s =i s' <-> f s = f s'.	Axiom	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	eq
f (s : seq K) := choose (perm_eq (undup s)) (undup s).	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	f
perm s : perm_eq (f s) (undup s). Proof. by rewrite perm_sym chooseP. Qed.	Fact	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	perm
uniq s : uniq (f s). Proof. by rewrite (perm_uniq (perm _)) undup_uniq. Qed.	Fact	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	uniq
E (s : seq K) : f s =i s. Proof. by move=> x; rewrite (perm_mem (perm _)) mem_undup. Qed.	Fact	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	E
eq (s s' : seq K) : s =i s' <-> f s = f s'. Proof. split=> [eq_ss'\|eq_ss' k]; last by rewrite -E eq_ss' E. rewrite /f; have peq_ss' : perm_eq (undup s) (undup s'). by apply: uniq_perm; rewrite ?undup_uniq // => x; rewrite !mem_undup. rewrite (@choose_id _ _ _ (undup s')) //=; apply: eq_choose => x /=. by apply: sym_left_transitive; [exact: perm_sym \| exact: (@perm_trans)\|]. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	eq
sort_keys := SortKeys.f.	Notation	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	sort_keys
sort_keys_perm := SortKeys.perm.	Notation	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	sort_keys_perm
sort_keys_uniq := SortKeys.uniq.	Notation	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	sort_keys_uniq
sort_keysE := SortKeys.E.	Notation	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	sort_keysE
eq_sort_keys := SortKeys.eq.	Notation	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	eq_sort_keys
mem_sort_keys ks k : k \in ks -> k \in sort_keys ks. Proof. by rewrite sort_keysE. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	mem_sort_keys
mem_sort_keys_intro ks k : k \in sort_keys ks -> k \in ks. Proof. by rewrite sort_keysE. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	mem_sort_keys_intro
sort_keys_nil : sort_keys [::] = [::] :> seq K. Proof. have := sort_keysE ([::] : seq K). by case: sort_keys => //= a l /(_ a); rewrite mem_head. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	sort_keys_nil
sort_keys_id ks : sort_keys (sort_keys ks) = sort_keys ks. Proof. by have /eq_sort_keys := sort_keysE ks. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	sort_keys_id
canonical_keys ks := sort_keys ks == ks.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	canonical_keys
canonical_uniq ks : canonical_keys ks -> uniq ks. Proof. by move=> /eqP <-; exact: sort_keys_uniq. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	canonical_uniq
canonical_sort_keys ks : canonical_keys (sort_keys ks). Proof. by rewrite /canonical_keys sort_keys_id. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	canonical_sort_keys
canonical_eq_keys ks ks' : canonical_keys ks -> canonical_keys ks' -> ks =i ks' -> ks = ks'. Proof. move=> /eqP; case: _ /; move=> /eqP; case: _ / => eq_ks_ks'. by apply/eq_sort_keys => x; rewrite -sort_keysE eq_ks_ks' sort_keysE. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	canonical_eq_keys
size_sort_keys ks : size (sort_keys ks) = size (undup ks). Proof. exact: perm_size. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	size_sort_keys
finset_of (_ : phant K) := finSet.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	finset_of
pred_of_finset (K : choiceType) (f : finSet K) : pred K := fun k => k \in (enum_fset f).	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	pred_of_finset
finSetPredType (K : choiceType) := PredType (@pred_of_finset K).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	finSetPredType
keys_canonical : canonical_keys (enum_fset A). Proof. by case: A. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	keys_canonical
fset_uniq : uniq (enum_fset A). Proof. by rewrite canonical_uniq // keys_canonical. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset_uniq
fset_sub_type : predArgType := FSetSub {fsval : K; fsvalP : in_mem fsval (@mem K _ A)}. HB.instance Definition _ := [isSub for fsval]. HB.instance Definition _ := [Choice of fset_sub_type by <:]. HB.instance Definition _ (T : countType) := [Countable of {fset T} by <:].	Record	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset_sub_type
fset_sub_enum : seq fset_sub_type := undup (pmap insub (enum_fset A)).	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset_sub_enum
mem_fset_sub_enum x : x \in fset_sub_enum. Proof. by rewrite mem_undup mem_pmap -valK map_f // fsvalP. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	mem_fset_sub_enum
val_fset_sub_enum : map val fset_sub_enum = enum_fset A. Proof. rewrite /fset_sub_enum undup_id ?pmap_sub_uniq ?fset_uniq//. rewrite (pmap_filter (@insubK _ _ _)); apply/all_filterP. by apply/allP => x; rewrite isSome_insub. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	val_fset_sub_enum
fset_sub_pickle x := index x fset_sub_enum.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset_sub_pickle
fset_sub_unpickle n := nth None (map some fset_sub_enum) n.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset_sub_unpickle
fset_sub_pickleK : pcancel fset_sub_pickle fset_sub_unpickle. Proof. rewrite /fset_sub_unpickle => x. by rewrite (nth_map x) ?nth_index ?index_mem ?mem_fset_sub_enum. Qed. HB.instance Definition _ := Countable.copy fset_sub_type (pcan_type fset_sub_pickleK). HB.instance Definition _ := isFinite.Build fset_sub_type (Finite.uniq_enumP (undup_uniq _) mem_fset_sub_enum).	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset_sub_pickleK
enum_fsetE : enum_fset A = [seq val i \| i <- enum fset_sub_type]. Proof. by rewrite enumT unlock val_fset_sub_enum. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	enum_fsetE
cardfE : size (enum_fset A) = #\|fset_sub_type\|. Proof. by rewrite cardE enum_fsetE size_map. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	cardfE
fset_sub_type : finSet >-> predArgType. #[global] Hint Resolve fsvalP fset_uniq mem_fset_sub_enum : core. Declare Scope fset_scope. Delimit Scope fset_scope with fset. Local Open Scope fset_scope.	Coercion	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset_sub_type
fsetsubE (T : choiceType) (A : {fset T}) (x : A) (xA : val x \in A) : [` xA] = x. Proof. by apply/val_inj => /=. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fsetsubE
fset_predT {T : choiceType} {A : {fset T}} : simpl_pred A := @predT A.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset_predT
set_of_fset (K : choiceType) (A : {fset K}) : {set A} := [set x in {: A}]. Arguments pred_of_finset : simpl never.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	set_of_fset
seq_fset : forall K : choiceType, seq K -> {fset K} := locked_with finset_key (fun K s => mkFinSet (@canonical_sort_keys K s)). Variable (K : choiceType) (s : seq K).	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	seq_fset
seq_fsetE : seq_fset s =i s. Proof. by move=> a; rewrite [seq_fset]unlock sort_keysE. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	seq_fsetE
size_seq_fset : size (seq_fset s) = size (undup s). Proof. by rewrite [seq_fset]unlock /= size_sort_keys. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	size_seq_fset
seq_fset_uniq : uniq (seq_fset s). Proof. by rewrite [seq_fset]unlock /= sort_keys_uniq. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	seq_fset_uniq
seq_fset_perm : perm_eq (seq_fset s) (undup s). Proof. by rewrite [seq_fset]unlock //= sort_keys_perm. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	seq_fset_perm
finpredType_predType (T : eqType) (fpT : finpredType T) := @PredType T (finpred_sort fpT) (@tofinpred T fpT).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	finpredType_predType
enum_finpred (T : eqType) (fpT : finpredType T) : fpT -> seq T := let: FinPredType _ _ (exist s _) := fpT in s.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	enum_finpred
enum_finpred_uniq (T : eqType) (fpT : finpredType T) (p : fpT) : uniq (enum_finpred p). Proof. by case: fpT p => ?? [s sE] p; rewrite sE. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	enum_finpred_uniq
enum_finpredE (T : eqType) (fpT : finpredType T) (p : fpT) : enum_finpred p =i p. Proof. by case: fpT p => ?? [s sE] p x; rewrite sE -topredE. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	enum_finpredE
mkFinPredType_of_subproof (T : eqType) (pT : predType T) (fpred_seq : pT -> seq T) (pred_fsetE : forall p, fpred_seq p =i p) : forall p x, x \in fpred_seq p = topred p x. Proof. by move=> p x; rewrite topredE pred_fsetE. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	mkFinPredType_of_subproof
mkFinPredType_of (T : eqType) (U : Type) := fun (pT : predType T) & pred_sort pT -> U => fun a (pT' := @PredType T U a) & phant_id pT' pT => fun (fpred_seq : pT' -> seq T) (fpred_seq_uniq : forall p, uniq (fpred_seq p)) (fpred_seqE : forall p, fpred_seq p =i p) => @FinPredType T U a (exist _ fpred_seq (fpred_seq_uniq, (mkFinPredType_of_subproof fpred_seqE))).	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	mkFinPredType_of
clone_finpredType (T : eqType) (U : Type) := fun (pT : finpredType T) & finpred_sort pT -> U => fun a pP (pT' := @FinPredType T U a pP) & phant_id pT' pT => pT'. Structure is_finite (T : eqType) (P : pred T) := IsFinite { seq_of_is_finite :> seq T; _ : uniq seq_of_is_finite; _ : forall x, x \in seq_of_is_finite = P x; }.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	clone_finpredType
is_finite_uniq (T : eqType) (P : pred T) (p : is_finite P) : uniq p. Proof. by case: p. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	is_finite_uniq
is_finiteE (T : eqType) (P : pred T) (p : is_finite P) x : x \in (seq_of_is_finite p) = P x. Proof. by case: p. Qed. Structure finpred (T : eqType) (pT : predType T) := FinPred { pred_of_finpred :> pT; _ : is_finite [pred x in pred_of_finpred] }.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	is_finiteE
enum_fin (T : eqType) (pT : predType T) (p : finpred pT) : seq T := let: FinPred _ fp := p in fp.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	enum_fin
enum_fin_uniq (T : eqType) (pT : predType T) (p : finpred pT) : uniq (enum_fin p). Proof. by case: p => ?[]. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	enum_fin_uniq
enum_finE (T : eqType) (pT : predType T) (p : finpred pT) : enum_fin p =i (pred_of_finpred p). Proof. by case: p => ?[]. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	enum_finE
fin_finpred (T : eqType) (pT : finpredType T) (p : pT) := @FinPred _ _ p (@IsFinite _ _ (enum_finpred p) (enum_finpred_uniq p) (enum_finpredE p)).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fin_finpred
finpred_of (T : eqType) (pT : predType T) (p : pT) (fp : finpred pT) & phantom pT fp : finpred pT := fp. Structure finmempred (T : eqType) := FinMemPred { pred_of_finmempred :> mem_pred T; _ : is_finite (fun x => in_mem x pred_of_finmempred) }.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	finpred_of
enum_finmem (T : eqType) (p : finmempred T) : seq T := let: FinMemPred _ fp := p in fp.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	enum_finmem
enum_finmem_uniq (T : eqType) (p : finmempred T) : uniq (enum_finmem p). Proof. by case: p => ?[]. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	enum_finmem_uniq
enum_finmemE (T : eqType) (p : finmempred T) : enum_finmem p =i p. Proof. by case: p => ?[]. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	enum_finmemE
finmempred_of (T : eqType) (P : pred T) (mP : finmempred T) & phantom (mem_pred T) mP : finmempred T := mP.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	finmempred_of
mem_fin (T : eqType) (pT : predType T) (p : finpred pT) := @FinMemPred _ (mem p) (@IsFinite _ _ (enum_fin p) (enum_fin_uniq p) (enum_finE p)).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	mem_fin
mkFinPredType T s s_uniq sE := (@mkFinPredType_of _ T _ id _ id s s_uniq sE).	Notation	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	mkFinPredType
fset_finpredType (T: choiceType) := mkFinPredType (finSet T) (@enum_fset T) (@fset_uniq T) (fun _ _ => erefl).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset_finpredType
pred_finpredType (T : finType) := mkFinPredType (pred T) (fun P => enum_mem (mem P)) (@enum_uniq T) (@mem_enum T).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	pred_finpredType
simpl_pred_finpredType (T : finType) := mkFinPredType (simpl_pred T) (fun P => enum_mem (mem P)) (@enum_uniq T) (@mem_enum T).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	simpl_pred_finpredType
fset_finpred (T: choiceType) (A : finSet T) := @FinPred _ _ (enum_fset A) (@IsFinite _ _ (enum_fset A) (fset_uniq _) (fun=> erefl)).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset_finpred
Canonical subfinset_finpred (T : choiceType) (A : finmempred T) (Q : pred T) := @FinPred _ _ [pred x in A \| Q x] (@IsFinite _ _ [seq x <- enum_finmem A \| Q x] _ _).	Program	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	Canonical
Obligation . by rewrite filter_uniq// enum_finmem_uniq. Qed.	Next	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	Obligation
Obligation . by rewrite !inE !mem_filter andbC enum_finmemE. Qed.	Next	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	Obligation
seq_finpredType (T : eqType) := mkFinPredType (seq T) undup (@undup_uniq T) (@mem_undup T).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	seq_finpredType
imfset : forall (key : unit) (T K : choiceType) (f : T -> K) (p : finmempred T), phantom (mem_pred T) p -> {fset K}.	Parameter	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset
imfset2 : forall (key : unit) (K T1 : choiceType) (T2 : T1 -> choiceType)(f : forall x : T1, T2 x -> K) (p1 : finmempred T1) (p2 : forall x : T1, finmempred (T2 x)), phantom (mem_pred T1) p1 -> phantom (forall x, mem_pred (T2 x)) p2 -> {fset K}.	Parameter	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset2
imfsetE : forall key, imfset key = imfset_def key.	Axiom	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfsetE
imfset2E : forall key, imfset2 key = imfset2_def key.	Axiom	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset2E
imfset key := imfset_def key.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset
imfset2 key := imfset2_def key.	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset2
imfsetE key : imfset key = imfset_def key. Proof. by []. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfsetE
imfset2E key : imfset2 key = imfset2_def key. Proof. by []. Qed.	Lemma	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset2E
imfset key f p := (Imfset.imfset key f (Phantom _ (pred_of_finmempred p))).	Notation	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset
imfset2 key f p q := (Imfset.imfset2 key f (Phantom _ (pred_of_finmempred p)) (Phantom _ (fun x => (pred_of_finmempred (q x))))).	Notation	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset2
imfset_unlock k := Unlockable (Imfset.imfsetE k).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset_unlock
imfset2_unlock k := Unlockable (Imfset.imfset2E k).	Canonical	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset2_unlock
imfset_key : unit. Proof. exact: tt. Qed.	Fact	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	imfset_key
fset0 : {fset K} := @mkFinSet K [::] (introT eqP (@sort_keys_nil K)). (* very transparent, to ensure x \in fset0 is convertible to false *)	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset0
fset1_key : unit. Proof. exact: tt. Qed.	Fact	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset1_key
fset1 a : {fset K} := [fset[fset1_key] x in [:: a]].	Definition	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fset1
fsetU_key : unit. Proof. exact: tt. Qed.	Fact	root	[ "From Corelib Require Import Setoid.", "From HB Require Import structures.", "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop." ]	finmap.v	fsetU_key

PredType : forall T pT, (pT -> pred T) -> predType T. exact PredType || exact mkPredType. Defined. Arguments PredType [T pT] toP. Local Notation predOfType T := (pred_of_simpl (@pred_of_argType T)).

Definition

root