Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: StaticLibsImage.gmk Sprache: Coq

Original von: Coq^©

(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

(* Some code from mathcomp needed in order to run ssr_* tests *)

Require Import ssreflect ssrfun ssrbool.

Global Set SsrOldRewriteGoalsOrder.
Global Set Asymmetric Patterns.
Global Set Bullet Behavior "None".

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

(* eqtype ---------------------------------------------------------- *)

Module Equality.

Definition axiom T (e : rel T) := forall x y, reflect (x = y) (e x y).

Structure mixin_of T := Mixin {op : rel T; _ : axiom op}.
Notation class_of := mixin_of (only parsing).

Section ClassDef.

Structure type := Pack {sort; _ : class_of sort; _ : Type}.
Local Coercion sort : type >-> Sortclass.
Variables (T : Type) (cT : type).

Definition class := let: Pack _ c _ := cT return class_of cT in c.

Definition pack c := @Pack T c T.
Definition clone := fun c & cT -> T & phant_id (pack c) cT => pack c.

End ClassDef.

Module Exports.
Coercion sort : type >-> Sortclass.
Notation eqType := type.
Notation EqMixin := Mixin.
Notation EqType T m := (@pack T m).
Notation "[ 'eqMixin' 'of' T ]" := (class _ : mixin_of T)
  (at level 0, format "[ 'eqMixin' 'of' T ]") : form_scope.
Notation "[ 'eqType' 'of' T 'for' C ]" := (@clone T C _ idfun id)
  (at level 0, format "[ 'eqType' 'of' T 'for' C ]") : form_scope.
Notation "[ 'eqType' 'of' T ]" := (@clone T _ _ id id)
  (at level 0, format "[ 'eqType' 'of' T ]") : form_scope.
End Exports.

End Equality.
Export Equality.Exports.

Definition eq_op T := Equality.op (Equality.class T).

Lemma eqE T x : eq_op x = Equality.op (Equality.class T) x.
Proof. by []. Qed.

Lemma eqP T : Equality.axiom (@eq_op T).
Proof. by case: T => ? []. Qed.
Arguments eqP [T x y].

Delimit Scope eq_scope with EQ.
Open Scope eq_scope.

Notation "x == y" := (eq_op x y)
  (at level 70, no associativity) : bool_scope.
Notation "x == y :> T" := ((x : T) == (y : T))
  (at level 70, y at next level) : bool_scope.
Notation "x != y" := (~~ (x == y))
  (at level 70, no associativity) : bool_scope.
Notation "x != y :> T" := (~~ (x == y :> T))
  (at level 70, y at next level) : bool_scope.
Notation "x =P y" := (eqP : reflect (x = y) (x == y))
  (at level 70, no associativity) : eq_scope.
Notation "x =P y :> T" := (eqP : reflect (x = y :> T) (x == y :> T))
  (at level 70, y at next level, no associativity) : eq_scope.

Prenex Implicits eq_op eqP.

Lemma eq_refl (T : eqType) (x : T) : x == x. Proof. exact/eqP. Qed.
Notation eqxx := eq_refl.

Lemma eq_sym (T : eqType) (x y : T) : (x == y) = (y == x).
Proof. exact/eqP/eqP. Qed.

Hint Resolve eq_refl eq_sym.

Definition eqb b := addb (~~ b).

Lemma eqbP : Equality.axiom eqb.
Proof. by do 2!case; constructor. Qed.

Canonical bool_eqMixin := EqMixin eqbP.
Canonical bool_eqType := Eval hnf in EqType bool bool_eqMixin.

Section ProdEqType.

Variable T1 T2 : eqType.

Definition pair_eq := [rel u v : T1 * T2 | (u.1 == v.1) && (u.2 == v.2)].

Lemma pair_eqP : Equality.axiom pair_eq.
Proof.
move=> [x1 x2] [y1 y2] /=; apply: (iffP andP) => [[]|[<- <-]] //=.
by do 2!move/eqP->.
Qed.

Definition prod_eqMixin := EqMixin pair_eqP.
Canonical prod_eqType := Eval hnf in EqType (T1 * T2) prod_eqMixin.

End ProdEqType.

Section OptionEqType.

Variable T : eqType.

Definition opt_eq (u v : option T) : bool :=
  oapp (fun x => oapp (eq_op x) false v) (~~ v) u.

Lemma opt_eqP : Equality.axiom opt_eq.
Proof.
case=> [x|] [y|] /=; by [constructor | apply: (iffP eqP) => [|[]] ->].
Qed.

Canonical option_eqMixin := EqMixin opt_eqP.
Canonical option_eqType := Eval hnf in EqType (option T) option_eqMixin.

End OptionEqType.

Notation xpred1 := (fun a1 x => x == a1).
Notation xpredU1 := (fun a1 (p : pred _) x => (x == a1) || p x).

Section EqPred.

Variable T : eqType.

Definition pred1 (a1 : T) := SimplPred (xpred1 a1).
Definition predU1 (a1 : T) p := SimplPred (xpredU1 a1 p).

End EqPred.

Section TransferEqType.

Variables (T : Type) (eT : eqType) (f : T -> eT).

Lemma inj_eqAxiom : injective f -> Equality.axiom (fun x y => f x == f y).
Proof. by move=> f_inj x y; apply: (iffP eqP) => [|-> //]; apply: f_inj. Qed.

Definition InjEqMixin f_inj := EqMixin (inj_eqAxiom f_inj).

Definition PcanEqMixin g (fK : pcancel f g) := InjEqMixin (pcan_inj fK).

Definition CanEqMixin g (fK : cancel f g) := InjEqMixin (can_inj fK).

End TransferEqType.

(* We use the module system to circumvent a silly limitation that  *)
(* forbids using the same constant to coerce to different targets. *)
Module Type EqTypePredSig.
Parameter sort : eqType -> predArgType.
End EqTypePredSig.
Module MakeEqTypePred (eqmod : EqTypePredSig).
Coercion eqmod.sort : eqType >-> predArgType.
End MakeEqTypePred.
Module Export EqTypePred := MakeEqTypePred Equality.

Section SubType.

Variables (T : Type) (P : pred T).

Structure subType : Type := SubType {
  sub_sort :> Type;
  val : sub_sort -> T;
  Sub : forall x, P x -> sub_sort;
  _ : forall K (_ : forall x Px, K (@Sub x Px)) u, K u;
  _ : forall x Px, val (@Sub x Px) = x
}.

Arguments Sub [s].
Lemma vrefl : forall x, P x -> x = x. Proof. by []. Qed.
Definition vrefl_rect := vrefl.

Definition clone_subType U v :=
  fun sT & sub_sort sT -> U =>
  fun c Urec cK (sT' := @SubType U v c Urec cK) & phant_id sT' sT => sT'.

Variable sT : subType.

CoInductive Sub_spec : sT -> Type := SubSpec x Px : Sub_spec (Sub x Px).

Lemma SubP u : Sub_spec u.
Proof. by case: sT Sub_spec SubSpec u => T' _ C rec /= _. Qed.

Lemma SubK x Px : @val sT (Sub x Px) = x.
Proof. by case: sT. Qed.

Definition insub x :=
  if @idP (P x) is ReflectT Px then @Some sT (Sub x Px) else None.

Definition insubd u0 x := odflt u0 (insub x).

CoInductive insub_spec x : option sT -> Type :=
  | InsubSome u of P x & val u = x : insub_spec x (Some u)
  | InsubNone   of ~~ P x          : insub_spec x None.

Lemma insubP x : insub_spec x (insub x).
Proof.
by rewrite /insub; case: {-}_ / idP; [left; rewrite ?SubK | right; apply/negP].
Qed.

Lemma insubT x Px : insub x = Some (Sub x Px).
Admitted.

Lemma insubF x : P x = false -> insub x = None.
Proof. by move/idP; case: insubP. Qed.

Lemma insubN x : ~~ P x -> insub x = None.
Proof. by move/negPf/insubF. Qed.

Lemma isSome_insub : ([eta insub] : pred T) =1 P.
Proof. by apply: fsym => x; case: insubP => // /negPf. Qed.

Lemma insubK : ocancel insub (@val _).
Proof. by move=> x; case: insubP. Qed.

Lemma valP (u : sT) : P (val u).
Proof. by case/SubP: u => x Px; rewrite SubK. Qed.

Lemma valK : pcancel (@val _) insub.
Proof. by case/SubP=> x Px; rewrite SubK; apply: insubT. Qed.

Lemma val_inj : injective (@val sT).
Proof. exact: pcan_inj valK. Qed.

Lemma valKd u0 : cancel (@val _) (insubd u0).
Proof. by move=> u; rewrite /insubd valK. Qed.

Lemma val_insubd u0 x : val (insubd u0 x) = if P x then x else val u0.
Proof. by rewrite /insubd; case: insubP => [u -> | /negPf->]. Qed.

Lemma insubdK u0 : {in P, cancel (insubd u0) (@val _)}.
Proof. by move=> x Px; rewrite /= val_insubd [P x]Px. Qed.

Definition insub_eq x :=
  let Some_sub Px := Some (Sub x Px : sT) in
  let None_sub _ := None in
  (if P x as Px return P x = Px -> _ then Some_sub else None_sub) (erefl _).

Lemma insub_eqE : insub_eq =1 insub.
Proof.
rewrite /insub_eq /insub => x; case: {2 3}_ / idP (erefl _) => // Px Px'.
by congr (Some _); apply: val_inj; rewrite !SubK.
Qed.

End SubType.

Arguments SubType [T P].
Arguments Sub [T P s].
Arguments vrefl [T P].
Arguments vrefl_rect [T P].
Arguments clone_subType [T P] U v [sT] _ [c Urec cK].
Arguments insub [T P sT].
Arguments insubT [T] P [sT x].
Arguments val_inj [T P sT].
Prenex Implicits val Sub vrefl vrefl_rect insub insubd val_inj.

Local Notation inlined_sub_rect :=
  (fun K K_S u => let (x, Px) as u return K u := u in K_S x Px).

Local Notation inlined_new_rect :=
  (fun K K_S u => let (x) as u return K u := u in K_S x).

Notation "[ 'subType' 'for' v ]" := (SubType _ v _ inlined_sub_rect vrefl_rect)
(at level 0, only parsing) : form_scope.

Notation "[ 'sub' 'Type' 'for' v ]" := (SubType _ v _ _ vrefl_rect)
(at level 0, format "[ 'sub' 'Type' 'for' v ]") : form_scope.

Notation "[ 'subType' 'for' v 'by' rec ]" := (SubType _ v _ rec vrefl)
(at level 0, format "[ 'subType' 'for' v 'by' rec ]") : form_scope.

Notation "[ 'subType' 'of' U 'for' v ]" := (clone_subType U v id idfun)
(at level 0, format "[ 'subType' 'of' U 'for' v ]") : form_scope.

(*
Notation "[ 'subType' 'for' v ]" := (clone_subType _ v id idfun)
(at level 0, format "[ 'subType'  'for'  v ]") : form_scope.
*)
Notation "[ 'subType' 'of' U ]" := (clone_subType U _ id id)
(at level 0, format "[ 'subType' 'of' U ]") : form_scope.

Definition NewType T U v c Urec :=
  let Urec' P IH := Urec P (fun x : T => IH x isT : P _) in
  SubType U v (fun x _ => c x) Urec'.
Arguments NewType [T U].

Notation "[ 'newType' 'for' v ]" := (NewType v _ inlined_new_rect vrefl_rect)
(at level 0, only parsing) : form_scope.

Notation "[ 'new' 'Type' 'for' v ]" := (NewType v _ _ vrefl_rect)
(at level 0, format "[ 'new' 'Type' 'for' v ]") : form_scope.

Notation "[ 'newType' 'for' v 'by' rec ]" := (NewType v _ rec vrefl)
(at level 0, format "[ 'newType' 'for' v 'by' rec ]") : form_scope.

Definition innew T nT x := @Sub T predT nT x (erefl true).
Arguments innew [T nT].
Prenex Implicits innew.

Lemma innew_val T nT : cancel val (@innew T nT).
Proof. by move=> u; apply: val_inj; apply: SubK. Qed.

(* Prenex Implicits and renaming. *)
Notation sval := (@proj1_sig _ _).
Notation "@ 'sval'" := (@proj1_sig) (at level 10, format "@ 'sval'").

Section SubEqType.

Variables (T : eqType) (P : pred T) (sT : subType P).

Local Notation ev_ax := (fun T v => @Equality.axiom T (fun x y => v x == v y)).
Lemma val_eqP : ev_ax sT val. Proof. exact: inj_eqAxiom val_inj. Qed.

Definition sub_eqMixin := EqMixin val_eqP.
Canonical sub_eqType := Eval hnf in EqType sT sub_eqMixin.

Definition SubEqMixin :=
  (let: SubType _ v _ _ _ as sT' := sT
     return ev_ax sT' val -> Equality.class_of sT' in
   fun vP : ev_ax _ v => EqMixin vP
   ) val_eqP.

Lemma val_eqE (u v : sT) : (val u == val v) = (u == v).
Proof. by []. Qed.

End SubEqType.

Arguments val_eqP [T P sT x y].
Prenex Implicits val_eqP.

Notation "[ 'eqMixin' 'of' T 'by' <: ]" := (SubEqMixin _ : Equality.class_of T)
  (at level 0, format "[ 'eqMixin' 'of' T 'by' <: ]") : form_scope.

(* ssrnat ---------------------------------------------------------- *)

Notation succn := Datatypes.S.
Notation predn := Peano.pred.

Notation "n .+1" := (succn n) (at level 2, left associativity,
  format "n .+1") : nat_scope.
Notation "n .+2" := n.+1.+1 (at level 2, left associativity,
  format "n .+2") : nat_scope.
Notation "n .+3" := n.+2.+1 (at level 2, left associativity,
  format "n .+3") : nat_scope.
Notation "n .+4" := n.+2.+2 (at level 2, left associativity,
  format "n .+4") : nat_scope.

Notation "n .-1" := (predn n) (at level 2, left associativity,
  format "n .-1") : nat_scope.
Notation "n .-2" := n.-1.-1 (at level 2, left associativity,
  format "n .-2") : nat_scope.

Fixpoint eqn m n {struct m} :=
  match m, n with
  | 0, 0 => true
  | m'.+1, n'.+1 => eqn m' n'
  | _, _ => false
  end.

Lemma eqnP : Equality.axiom eqn.
Proof.
move=> n m; apply: (iffP idP) => [|<-]; last by elim n.
by elim: n m => [|n IHn] [|m] //= /IHn->.
Qed.

Canonical nat_eqMixin := EqMixin eqnP.
Canonical nat_eqType := Eval hnf in EqType nat nat_eqMixin.

Arguments eqnP [x y].
Prenex Implicits eqnP.

Coercion nat_of_bool (b : bool) := if b then 1 else 0.

Fixpoint odd n := if n is n'.+1 then ~~ odd n' else false.

Lemma oddb (b : bool) : odd b = b. Proof. by case: b. Qed.

Definition subn_rec := minus.
Notation "m - n" := (subn_rec m n) : nat_rec_scope.

Definition subn := nosimpl subn_rec.
Notation "m - n" := (subn m n) : nat_scope.

Definition leq m n := m - n == 0.

Notation "m <= n" := (leq m n) : nat_scope.
Notation "m < n"  := (m.+1 <= n) : nat_scope.
Notation "m >= n" := (n <= m) (only parsing) : nat_scope.
Notation "m > n"  := (n < m) (only parsing)  : nat_scope.

Notation "m <= n <= p" := ((m <= n) && (n <= p)) : nat_scope.
Notation "m < n <= p" := ((m < n) && (n <= p)) : nat_scope.
Notation "m <= n < p" := ((m <= n) && (n < p)) : nat_scope.
Notation "m < n < p" := ((m < n) && (n < p)) : nat_scope.

Open Scope nat_scope.

Lemma ltnS m n : (m < n.+1) = (m <= n). Proof. by []. Qed.
Lemma leq0n n : 0 <= n.                 Proof. by []. Qed.
Lemma ltn0Sn n : 0 < n.+1.              Proof. by []. Qed.
Lemma ltn0 n : n < 0 = false.           Proof. by []. Qed.
Lemma leqnn n : n <= n.                 Proof. by elim: n. Qed.
Hint Resolve leqnn.
Lemma leqnSn n : n <= n.+1.             Proof. by elim: n. Qed.

Lemma leq_trans n m p : m <= n -> n <= p -> m <= p.
Admitted.
Lemma leq_ltn_trans n m p : m <= n -> n < p -> m < p.
Admitted.
Lemma leqW m n : m <= n -> m <= n.+1.
Admitted.
Hint Resolve leqnSn.
Lemma ltnW m n : m < n -> m <= n.
Proof. exact: leq_trans. Qed.
Hint Resolve ltnW.

Definition addn_rec := plus.
Notation "m + n" := (addn_rec m n) : nat_rec_scope.

Definition addn := nosimpl addn_rec.
Notation "m + n" := (addn m n) : nat_scope.

Lemma addn0 : right_id 0 addn. Proof. by move=> n; apply/eqP; elim: n. Qed.
Lemma add0n : left_id 0 addn.            Proof. by []. Qed.
Lemma addSn m n : m.+1 + n = (m + n).+1. Proof. by []. Qed.
Lemma addnS m n : m + n.+1 = (m + n).+1. Proof. by elim: m. Qed.

Lemma addnCA : left_commutative addn.
Proof. by move=> m n p; elim: m => //= m; rewrite addnS => <-. Qed.

Lemma addnC : commutative addn.
Proof. by move=> m n; rewrite -{1}[n]addn0 addnCA addn0. Qed.

Lemma addnA : associative addn.
Proof. by move=> m n p; rewrite (addnC n) addnCA addnC. Qed.

Lemma subnK m n : m <= n -> (n - m) + m = n.
Admitted.

Definition muln_rec := mult.
Notation "m * n" := (muln_rec m n) : nat_rec_scope.

Definition muln := nosimpl muln_rec.
Notation "m * n" := (muln m n) : nat_scope.

Lemma mul0n : left_zero 0 muln.          Proof. by []. Qed.
Lemma muln0 : right_zero 0 muln.         Proof. by elim. Qed.
Lemma mul1n : left_id 1 muln.            Proof. exact: addn0. Qed.

Lemma mulSn m n : m.+1 * n = n + m * n.  Proof. by []. Qed.
Lemma mulSnr m n : m.+1 * n = m * n + n. Proof. exact: addnC. Qed.

Lemma mulnS m n : m * n.+1 = m + m * n.
Proof. by elim: m => // m; rewrite !mulSn !addSn addnCA => ->. Qed.

Lemma mulnSr m n : m * n.+1 = m * n + m.
Proof. by rewrite addnC mulnS. Qed.

Lemma muln1 : right_id 1 muln.
Proof. by move=> n; rewrite mulnSr muln0. Qed.

Lemma mulnC : commutative muln.
Proof.
by move=> m n; elim: m => [|m]; rewrite (muln0, mulnS) // mulSn => ->.
Qed.

Lemma mulnDl : left_distributive muln addn.
Proof. by move=> m1 m2 n; elim: m1 => //= m1 IHm; rewrite -addnA -IHm. Qed.

Lemma mulnDr : right_distributive muln addn.
Proof. by move=> m n1 n2; rewrite !(mulnC m) mulnDl. Qed.

Lemma mulnA : associative muln.
Proof. by move=> m n p; elim: m => //= m; rewrite mulSn mulnDl => ->. Qed.

Lemma mulnCA : left_commutative muln.
Proof. by move=> m n1 n2; rewrite !mulnA (mulnC m). Qed.

Lemma mulnAC : right_commutative muln.
Proof. by move=> m n p; rewrite -!mulnA (mulnC n). Qed.

Lemma mulnACA : interchange muln muln.
Proof. by move=> m n p q; rewrite -!mulnA (mulnCA n). Qed.

(* seq ------------------------------------------------------------- *)

Delimit Scope seq_scope with SEQ.
Open Scope seq_scope.

(* Inductive seq (T : Type) : Type := Nil | Cons of T & seq T. *)
Notation seq := list.
Prenex Implicits cons.
Notation Cons T := (@cons T) (only parsing).
Notation Nil T := (@nil T) (only parsing).

Bind Scope seq_scope with list.
Arguments cons _%type _ _%SEQ.

(* As :: and ++ are (improperly) declared in Init.datatypes, we only rebind   *)
(* them here.                                                                 *)
Infix "::" := cons : seq_scope.

(* GG - this triggers a camlp4 warning, as if this Notation had been Reserved *)
Notation "[ :: ]" := nil (at level 0, format "[ :: ]") : seq_scope.

Notation "[ :: x1 ]" := (x1 :: [::])
  (at level 0, format "[ :: x1 ]") : seq_scope.

Notation "[ :: x & s ]" := (x :: s) (at level 0, only parsing) : seq_scope.

Notation "[ :: x1 , x2 , .. , xn & s ]" := (x1 :: x2 :: .. (xn :: s) ..)
  (at level 0, format
  "'[hv' [ :: '[' x1 , '/' x2 , '/' .. , '/' xn ']' '/ ' & s ] ']'"
  ) : seq_scope.

Notation "[ :: x1 ; x2 ; .. ; xn ]" := (x1 :: x2 :: .. [:: xn] ..)
  (at level 0, format "[ :: '[' x1 ; '/' x2 ; '/' .. ; '/' xn ']' ]"
  ) : seq_scope.

Section Sequences.

Variable n0 : nat.  (* numerical parameter for take, drop et al *)
Variable T : Type.  (* must come before the implicit Type     *)
Variable x0 : T.    (* default for head/nth *)

Implicit Types x y z : T.
Implicit Types m n : nat.
Implicit Type s : seq T.

Fixpoint size s := if s is _ :: s' then (size s').+1 else 0.

Fixpoint cat s1 s2 := if s1 is x :: s1' then x :: s1' ++ s2 else s2
where "s1 ++ s2" := (cat s1 s2) : seq_scope.

Lemma cat0s s : [::] ++ s = s. Proof. by []. Qed.

Lemma cats0 s : s ++ [::] = s.
Proof. by elim: s => //= x s ->. Qed.

Lemma catA s1 s2 s3 : s1 ++ s2 ++ s3 = (s1 ++ s2) ++ s3.
Proof. by elim: s1 => //= x s1 ->. Qed.

Fixpoint nth s n {struct n} :=
  if s is x :: s' then if n is n'.+1 then @nth s' n' else x else x0.

Fixpoint rcons s z := if s is x :: s' then x :: rcons s' z else [:: z].

CoInductive last_spec : seq T -> Type :=
  | LastNil        : last_spec [::]
  | LastRcons s x  : last_spec (rcons s x).

Lemma lastP s : last_spec s.
Proof using. Admitted.

Lemma last_ind P :
  P [::] -> (forall s x, P s -> P (rcons s x)) -> forall s, P s.
Proof using. Admitted.

Section Map.

Variables (T2 : Type) (f : T -> T2).

Fixpoint map s := if s is x :: s' then f x :: map s' else [::].

End Map.

Section SeqFind.

Variable a : pred T.

Fixpoint count s := if s is x :: s' then a x + count s' else 0.

Fixpoint filter s :=
  if s is x :: s' then if a x then x :: filter s' else filter s' else [::].

End SeqFind.

End Sequences.

Infix "++" := cat : seq_scope.

Notation count_mem x := (count (pred_of_simpl (pred1 x))).

Section EqSeq.

Variables (n0 : nat) (T : eqType) (x0 : T).
Local Notation nth := (nth x0).
Implicit Type s : seq T.
Implicit Types x y z : T.

Fixpoint eqseq s1 s2 {struct s2} :=
  match s1, s2 with
  | [::], [::] => true
  | x1 :: s1', x2 :: s2' => (x1 == x2) && eqseq s1' s2'
  | _, _ => false
  end.

Lemma eqseqP : Equality.axiom eqseq.
Proof.
move; elim=> [|x1 s1 IHs] [|x2 s2]; do [by constructor | simpl].
case: (x1 =P x2) => [<-|neqx]; last by right; case.
by apply: (iffP (IHs s2)) => [<-|[]].
Qed.

Canonical seq_eqMixin := EqMixin eqseqP.
Canonical seq_eqType := Eval hnf in EqType (seq T) seq_eqMixin.

Fixpoint mem_seq (s : seq T) :=
  if s is y :: s' then xpredU1 y (mem_seq s') else xpred0.

Definition eqseq_class := seq T.
Identity Coercion seq_of_eqseq : eqseq_class >-> seq.
Coercion pred_of_eq_seq (s : eqseq_class) : {pred T} := [eta mem_seq s].

Canonical seq_predType := @PredType T (seq T) pred_of_eq_seq.

Fixpoint uniq s := if s is x :: s' then (x \notin s') && uniq s' else true.

End EqSeq.

Definition bitseq := seq bool.
Canonical bitseq_eqType := Eval hnf in [eqType of bitseq].
Canonical bitseq_predType := Eval hnf in [predType of bitseq].

Section Pmap.

Variables (aT rT : Type) (f : aT -> option rT) (g : rT -> aT).

Fixpoint pmap s :=
  if s is x :: s' then let r := pmap s' in oapp (cons^~ r) r (f x) else [::].

End Pmap.

Fixpoint iota m n := if n is n'.+1 then m :: iota m.+1 n' else [::].

Section FoldRight.

Variables (T : Type) (R : Type) (f : T -> R -> R) (z0 : R).

Fixpoint foldr s := if s is x :: s' then f x (foldr s') else z0.

End FoldRight.

Lemma mem_iota m n i : (i \in iota m n) = (m <= i) && (i < m + n).
Admitted.

(* choice ------------------------------------------------------------- *)

Module Choice.

Section ClassDef.

Record mixin_of T := Mixin {
  find : pred T -> nat -> option T;
  _ : forall P n x, find P n = Some x -> P x;
  _ : forall P : pred T, (exists x, P x) -> exists n, find P n;
  _ : forall P Q : pred T, P =1 Q -> find P =1 find Q
}.

Record class_of T := Class {base : Equality.class_of T; mixin : mixin_of T}.
Local Coercion base : class_of >->  Equality.class_of.

Structure type := Pack {sort; _ : class_of sort; _ : Type}.
Local Coercion sort : type >-> Sortclass.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c _ as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack T c T.
Let xT := let: Pack T _ _ := cT in T.
Notation xclass := (class : class_of xT).

Definition pack m :=
  fun b bT & phant_id (Equality.class bT) b => Pack (@Class T b m) T.

(* Inheritance *)
Definition eqType := @Equality.Pack cT xclass xT.

End ClassDef.

Module Import Exports.
Coercion base : class_of >-> Equality.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Notation choiceType := type.
Notation choiceMixin := mixin_of.
Notation ChoiceType T m := (@pack T m _ _ id).
Notation "[ 'choiceType' 'of' T 'for' cT ]" :=  (@clone T cT _ idfun)
  (at level 0, format "[ 'choiceType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'choiceType' 'of' T ]" := (@clone T _ _ id)
  (at level 0, format "[ 'choiceType' 'of' T ]") : form_scope.

End Exports.

End Choice.
Export Choice.Exports.

Section ChoiceTheory.

Variable T : choiceType.

Section CanChoice.

Variables (sT : Type) (f : sT -> T).

Lemma PcanChoiceMixin f' : pcancel f f' -> choiceMixin sT.
Admitted.

Definition CanChoiceMixin f' (fK : cancel f f') :=
  PcanChoiceMixin (can_pcan fK).

End CanChoice.

Section SubChoice.

Variables (P : pred T) (sT : subType P).

Definition sub_choiceMixin := PcanChoiceMixin (@valK T P sT).
Definition sub_choiceClass := @Choice.Class sT (sub_eqMixin sT) sub_choiceMixin.
Canonical sub_choiceType := Choice.Pack sub_choiceClass sT.

End SubChoice.

Fact seq_choiceMixin : choiceMixin (seq T).
Admitted.
Canonical seq_choiceType := Eval hnf in ChoiceType (seq T) seq_choiceMixin.
End ChoiceTheory.

Fact nat_choiceMixin : choiceMixin nat.
Proof.
pose f := [fun (P : pred nat) n => if P n then Some n else None].
exists f => [P n m | P [n Pn] | P Q eqPQ n] /=; last by rewrite eqPQ.
  by case: ifP => // Pn [<-].
by exists n; rewrite Pn.
Qed.
Canonical nat_choiceType := Eval hnf in ChoiceType nat nat_choiceMixin.

Definition bool_choiceMixin := CanChoiceMixin oddb.
Canonical bool_choiceType := Eval hnf in ChoiceType bool bool_choiceMixin.
Canonical bitseq_choiceType := Eval hnf in [choiceType of bitseq].

Notation "[ 'choiceMixin' 'of' T 'by' <: ]" :=
  (sub_choiceMixin _ : choiceMixin T)
  (at level 0, format "[ 'choiceMixin' 'of' T 'by' <: ]") : form_scope.

Module Countable.

Record mixin_of (T : Type) : Type := Mixin {
  pickle : T -> nat;
  unpickle : nat -> option T;
  pickleK : pcancel pickle unpickle
}.

Definition EqMixin T m := PcanEqMixin (@pickleK T m).
Definition ChoiceMixin T m := PcanChoiceMixin (@pickleK T m).

Section ClassDef.

Record class_of T := Class { base : Choice.class_of T; mixin : mixin_of T }.
Local Coercion base : class_of >-> Choice.class_of.

Structure type : Type := Pack {sort : Type; _ : class_of sort; _ : Type}.
Local Coercion sort : type >-> Sortclass.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c _ as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack T c T.
Let xT := let: Pack T _ _ := cT in T.
Notation xclass := (class : class_of xT).

Definition pack m :=
  fun bT b & phant_id (Choice.class bT) b => Pack (@Class T b m) T.

Definition eqType := @Equality.Pack cT xclass xT.
Definition choiceType := @Choice.Pack cT xclass xT.

End ClassDef.

Module Exports.
Coercion base : class_of >-> Choice.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Notation countType := type.
Notation CountType T m := (@pack T m _ _ id).
Notation CountMixin := Mixin.
Notation CountChoiceMixin := ChoiceMixin.
Notation "[ 'countType' 'of' T 'for' cT ]" := (@clone T cT _ idfun)
(at level 0, format "[ 'countType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'countType' 'of' T ]" := (@clone T _ _ id)
  (at level 0, format "[ 'countType' 'of' T ]") : form_scope.

End Exports.

End Countable.
Export Countable.Exports.

Definition unpickle T := Countable.unpickle (Countable.class T).
Definition pickle T := Countable.pickle (Countable.class T).
Arguments unpickle [T].
Prenex Implicits pickle unpickle.

Section CountableTheory.

Variable T : countType.

Lemma pickleK : @pcancel nat T pickle unpickle.
Proof. exact: Countable.pickleK. Qed.

Definition pickle_inv n :=
  obind (fun x : T => if pickle x == n then Some x else None) (unpickle n).

Lemma pickle_invK : ocancel pickle_inv pickle.
Proof.
by rewrite /pickle_inv => n; case def_x: (unpickle n) => //= [x]; case: eqP.
Qed.

Lemma pickleK_inv : pcancel pickle pickle_inv.
Proof. by rewrite /pickle_inv => x; rewrite pickleK /= eqxx. Qed.

Lemma pcan_pickleK sT f f' :
  @pcancel T sT f f' -> pcancel (pickle \o f) (pcomp f' unpickle).
Proof. by move=> fK x; rewrite /pcomp pickleK /= fK. Qed.

Definition PcanCountMixin sT f f' (fK : pcancel f f') :=
  @CountMixin sT _ _ (pcan_pickleK fK).

Definition CanCountMixin sT f f' (fK : cancel f f') :=
  @PcanCountMixin sT _ _ (can_pcan fK).

Definition sub_countMixin P sT := PcanCountMixin (@valK T P sT).

End CountableTheory.
Notation "[ 'countMixin' 'of' T 'by' <: ]" :=
    (sub_countMixin _ : Countable.mixin_of T)
  (at level 0, format "[ 'countMixin' 'of' T 'by' <: ]") : form_scope.

Section SubCountType.

Variables (T : choiceType) (P : pred T).
Import Countable.

Structure subCountType : Type :=
  SubCountType {subCount_sort :> subType P; _ : mixin_of subCount_sort}.

Coercion sub_countType (sT : subCountType) :=
  Eval hnf in pack (let: SubCountType _ m := sT return mixin_of sT in m) id.
Canonical sub_countType.

Definition pack_subCountType U :=
  fun sT cT & sub_sort sT * sort cT -> U * U =>
  fun b m   & phant_id (Class b m) (class cT) => @SubCountType sT m.

End SubCountType.

(* This assumes that T has both countType and subType structures. *)
Notation "[ 'subCountType' 'of' T ]" :=
    (@pack_subCountType _ _ T _ _ id _ _ id)
  (at level 0, format "[ 'subCountType' 'of' T ]") : form_scope.

Lemma nat_pickleK : pcancel id (@Some nat). Proof. by []. Qed.
Definition nat_countMixin := CountMixin nat_pickleK.
Canonical nat_countType := Eval hnf in CountType nat nat_countMixin.

(* fintype --------------------------------------------------------- *)

Module Finite.

Section RawMixin.

Variable T : eqType.

Definition axiom e := forall x : T, count_mem x e = 1.

Lemma uniq_enumP e : uniq e -> e =i T -> axiom e.
Admitted.

Record mixin_of := Mixin {
  mixin_base : Countable.mixin_of T;
  mixin_enum : seq T;
  _ : axiom mixin_enum
}.

End RawMixin.

Section Mixins.

Variable T : countType.

Definition EnumMixin :=
  let: Countable.Pack _ (Countable.Class _ m) _ as cT := T
    return forall e : seq cT, axiom e -> mixin_of cT in
  @Mixin (EqType _ _) m.

Definition UniqMixin e Ue eT := @EnumMixin e (uniq_enumP Ue eT).

Variable n : nat.

End Mixins.

Section ClassDef.

Record class_of T := Class {
  base : Choice.class_of T;
  mixin : mixin_of (Equality.Pack base T)
}.
Definition base2 T c := Countable.Class (@base T c) (mixin_base (mixin c)).
Local Coercion base : class_of >-> Choice.class_of.

Structure type : Type := Pack {sort; _ : class_of sort; _ : Type}.
Local Coercion sort : type >-> Sortclass.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c _ as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack T c T.
Let xT := let: Pack T _ _ := cT in T.
Notation xclass := (class : class_of xT).

Definition pack b0 (m0 : mixin_of (EqType T b0)) :=
  fun bT b & phant_id (Choice.class bT) b =>
  fun m & phant_id m0 m => Pack (@Class T b m) T.

Definition eqType := @Equality.Pack cT xclass xT.
Definition choiceType := @Choice.Pack cT xclass xT.
Definition countType := @Countable.Pack cT (base2 xclass) xT.

End ClassDef.

Module Import Exports.
Coercion mixin_base : mixin_of >-> Countable.mixin_of.
Coercion base : class_of >-> Choice.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion base2 : class_of >-> Countable.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Coercion countType : type >-> Countable.type.
Canonical countType.
Notation finType := type.
Notation FinType T m := (@pack T _ m _ _ id _ id).
Notation FinMixin := EnumMixin.
Notation UniqFinMixin := UniqMixin.
Notation "[ 'finType' 'of' T 'for' cT ]" := (@clone T cT _ idfun)
  (at level 0, format "[ 'finType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'finType' 'of' T ]" := (@clone T _ _ id)
  (at level 0, format "[ 'finType' 'of' T ]") : form_scope.
End Exports.

Module Type EnumSig.
Parameter enum : forall cT : type, seq cT.
Axiom enumDef : enum = fun cT => mixin_enum (class cT).
End EnumSig.

Module EnumDef : EnumSig.
Definition enum cT := mixin_enum (class cT).
Definition enumDef := erefl enum.
End EnumDef.

Notation enum := EnumDef.enum.

End Finite.
Export Finite.Exports.

Section SubFinType.

Variables (T : choiceType) (P : pred T).
Import Finite.

Structure subFinType := SubFinType {
  subFin_sort :> subType P;
  _ : mixin_of (sub_eqType subFin_sort)
}.

Definition pack_subFinType U :=
  fun cT b m & phant_id (class cT) (@Class U b m) =>
  fun sT m'  & phant_id m' m => @SubFinType sT m'.

Implicit Type sT : subFinType.

Definition subFin_mixin sT :=
  let: SubFinType _ m := sT return mixin_of (sub_eqType sT) in m.

Coercion subFinType_subCountType sT := @SubCountType _ _ sT (subFin_mixin sT).
Canonical subFinType_subCountType.

Coercion subFinType_finType sT :=
  Pack (@Class sT (sub_choiceClass sT) (subFin_mixin sT)) sT.
Canonical subFinType_finType.

Definition enum_mem T (mA : mem_pred _) := filter mA (Finite.enum T).
Definition image_mem T T' f mA : seq T' := map f (@enum_mem T mA).
Definition codom T T' f := @image_mem T T' f (mem T).

Lemma codom_val sT x : (x \in codom (val : sT -> T)) = P x.
Admitted.

End SubFinType.

(* This assumes that T has both finType and subCountType structures. *)
Notation "[ 'subFinType' 'of' T ]" := (@pack_subFinType _ _ T _ _ _ id _ _ id)
  (at level 0, format "[ 'subFinType' 'of' T ]") : form_scope.

Section OrdinalSub.

Variable n : nat.

Inductive ordinal : predArgType := Ordinal m of m < n.

Coercion nat_of_ord i := let: Ordinal m _ := i in m.

Canonical ordinal_subType := [subType for nat_of_ord].
Definition ordinal_eqMixin := Eval hnf in [eqMixin of ordinal by <:].
Canonical ordinal_eqType := Eval hnf in EqType ordinal ordinal_eqMixin.
Definition ordinal_choiceMixin := [choiceMixin of ordinal by <:].
Canonical ordinal_choiceType :=
  Eval hnf in ChoiceType ordinal ordinal_choiceMixin.
Definition ordinal_countMixin := [countMixin of ordinal by <:].
Canonical ordinal_countType := Eval hnf in CountType ordinal ordinal_countMixin.
Canonical ordinal_subCountType := [subCountType of ordinal].

Lemma ltn_ord (i : ordinal) : i < n. Proof. exact: valP i. Qed.

Lemma ord_inj : injective nat_of_ord. Proof. exact: val_inj. Qed.

Definition ord_enum : seq ordinal := pmap insub (iota 0 n).

Lemma val_ord_enum : map val ord_enum = iota 0 n.
Admitted.

Lemma ord_enum_uniq : uniq ord_enum.
Admitted.

Lemma mem_ord_enum i : i \in ord_enum.
Admitted.

Definition ordinal_finMixin :=
  Eval hnf in UniqFinMixin ord_enum_uniq mem_ord_enum.
Canonical ordinal_finType := Eval hnf in FinType ordinal ordinal_finMixin.
Canonical ordinal_subFinType := Eval hnf in [subFinType of ordinal].

End OrdinalSub.

Notation "''I_' n" := (ordinal n)
  (at level 8, n at level 2, format "''I_' n").

(* bigop ----------------------------------------------------------------- *)

Reserved Notation "\big [ op / idx ]_ i F"
  (at level 36, F at level 36, op, idx at level 10, i at level 0,
     right associativity,
           format "'[' \big [ op / idx ]_ i '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i <- r | P ) F"
  (at level 36, F at level 36, op, idx at level 10, i, r at level 50,
           format "'[' \big [ op / idx ]_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i <- r ) F"
  (at level 36, F at level 36, op, idx at level 10, i, r at level 50,
           format "'[' \big [ op / idx ]_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( m <= i < n | P ) F"
  (at level 36, F at level 36, op, idx at level 10, m, i, n at level 50,
           format "'[' \big [ op / idx ]_ ( m <= i < n | P ) F ']'").
Reserved Notation "\big [ op / idx ]_ ( m <= i < n ) F"
  (at level 36, F at level 36, op, idx at level 10, i, m, n at level 50,
           format "'[' \big [ op / idx ]_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i | P ) F"
  (at level 36, F at level 36, op, idx at level 10, i at level 50,
           format "'[' \big [ op / idx ]_ ( i | P ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i : t | P ) F"
  (at level 36, F at level 36, op, idx at level 10, i at level 50,
           format "'[' \big [ op / idx ]_ ( i : t | P ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i : t ) F"
  (at level 36, F at level 36, op, idx at level 10, i at level 50,
           format "'[' \big [ op / idx ]_ ( i : t ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i < n | P ) F"
  (at level 36, F at level 36, op, idx at level 10, i, n at level 50,
           format "'[' \big [ op / idx ]_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i < n ) F"
  (at level 36, F at level 36, op, idx at level 10, i, n at level 50,
           format "'[' \big [ op / idx ]_ ( i < n ) F ']'").
Reserved Notation "\big [ op / idx ]_ ( i 'in' A | P ) F"
  (at level 36, F at level 36, op, idx at level 10, i, A at level 50,
           format "'[' \big [ op / idx ]_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i 'in' A ) F"
  (at level 36, F at level 36, op, idx at level 10, i, A at level 50,
           format "'[' \big [ op / idx ]_ ( i 'in' A ) '/ ' F ']'").

Module Monoid.

Section Definitions.
Variables (T : Type) (idm : T).

Structure law := Law {
  operator : T -> T -> T;
  _ : associative operator;
  _ : left_id idm operator;
  _ : right_id idm operator
}.
Local Coercion operator : law >-> Funclass.

Structure com_law := ComLaw {
   com_operator : law;
   _ : commutative com_operator
}.
Local Coercion com_operator : com_law >-> law.

Structure mul_law := MulLaw {
  mul_operator : T -> T -> T;
  _ : left_zero idm mul_operator;
  _ : right_zero idm mul_operator
}.
Local Coercion mul_operator : mul_law >-> Funclass.

Structure add_law (mul : T -> T -> T) := AddLaw {
  add_operator : com_law;
  _ : left_distributive mul add_operator;
  _ : right_distributive mul add_operator
}.
Local Coercion add_operator : add_law >-> com_law.

Let op_id (op1 op2 : T -> T -> T) := phant_id op1 op2.

Definition clone_law op :=
  fun (opL : law) & op_id opL op =>
  fun opmA op1m opm1 (opL' := @Law op opmA op1m opm1)
    & phant_id opL' opL => opL'.

Definition clone_com_law op :=
  fun (opL : law) (opC : com_law) & op_id opL op & op_id opC op =>
  fun opmC (opC' := @ComLaw opL opmC) & phant_id opC' opC => opC'.

Definition clone_mul_law op :=
  fun (opM : mul_law) & op_id opM op =>
  fun op0m opm0 (opM' := @MulLaw op op0m opm0) & phant_id opM' opM => opM'.

Definition clone_add_law mop aop :=
  fun (opC : com_law) (opA : add_law mop) & op_id opC aop & op_id opA aop =>
  fun mopDm mopmD (opA' := @AddLaw mop opC mopDm mopmD)
    & phant_id opA' opA => opA'.

End Definitions.

Module Import Exports.
Coercion operator : law >-> Funclass.
Coercion com_operator : com_law >-> law.
Coercion mul_operator : mul_law >-> Funclass.
Coercion add_operator : add_law >-> com_law.
Notation "[ 'law' 'of' f ]" := (@clone_law _ _ f _ id _ _ _ id)
  (at level 0, format"[ 'law' 'of' f ]") : form_scope.
Notation "[ 'com_law' 'of' f ]" := (@clone_com_law _ _ f _ _ id id _ id)
  (at level 0, format "[ 'com_law' 'of' f ]") : form_scope.
Notation "[ 'mul_law' 'of' f ]" := (@clone_mul_law _ _ f _ id _ _ id)
  (at level 0, format"[ 'mul_law' 'of' f ]") : form_scope.
Notation "[ 'add_law' m 'of' a ]" := (@clone_add_law _ _ m a _ _ id id _ _ id)
  (at level 0, format "[ 'add_law' m 'of' a ]") : form_scope.
End Exports.

Section CommutativeAxioms.

Variable (T : Type) (zero one : T) (mul add : T -> T -> T) (inv : T -> T).
Hypothesis mulC : commutative mul.

Lemma mulC_id : left_id one mul -> right_id one mul.
Proof. by move=> mul1x x; rewrite mulC. Qed.

Lemma mulC_zero : left_zero zero mul -> right_zero zero mul.
Proof. by move=> mul0x x; rewrite mulC. Qed.

Lemma mulC_dist : left_distributive mul add -> right_distributive mul add.
Proof. by move=> mul_addl x y z; rewrite !(mulC x). Qed.

End CommutativeAxioms.
Module Theory.

Section Theory.
Variables (T : Type) (idm : T).

Section Plain.
Variable mul : law idm.
Lemma mul1m : left_id idm mul. Proof. by case mul. Qed.
Lemma mulm1 : right_id idm mul. Proof. by case mul. Qed.
Lemma mulmA : associative mul. Proof. by case mul. Qed.
(*Lemma iteropE n x : iterop n mul x idm = iter n (mul x) idm.*)

End Plain.

Section Commutative.
Variable mul : com_law idm.
Lemma mulmC : commutative mul. Proof. by case mul. Qed.
Lemma mulmCA : left_commutative mul.
Proof. by move=> x y z; rewrite !mulmA (mulmC x). Qed.
Lemma mulmAC : right_commutative mul.
Proof. by move=> x y z; rewrite -!mulmA (mulmC y). Qed.
Lemma mulmACA : interchange mul mul.
Proof. by move=> x y z t; rewrite -!mulmA (mulmCA y). Qed.
End Commutative.

Section Mul.
Variable mul : mul_law idm.
Lemma mul0m : left_zero idm mul. Proof. by case mul. Qed.
Lemma mulm0 : right_zero idm mul. Proof. by case mul. Qed.
End Mul.

Section Add.
Variables (mul : T -> T -> T) (add : add_law idm mul).
Lemma addmA : associative add. Proof. exact: mulmA. Qed.
Lemma addmC : commutative add. Proof. exact: mulmC. Qed.
Lemma addmCA : left_commutative add. Proof. exact: mulmCA. Qed.
Lemma addmAC : right_commutative add. Proof. exact: mulmAC. Qed.
Lemma add0m : left_id idm add. Proof. exact: mul1m. Qed.
Lemma addm0 : right_id idm add. Proof. exact: mulm1. Qed.
Lemma mulm_addl : left_distributive mul add. Proof. by case add. Qed.
Lemma mulm_addr : right_distributive mul add. Proof. by case add. Qed.
End Add.

Definition simpm := (mulm1, mulm0, mul1m, mul0m, mulmA).

End Theory.

End Theory.
Include Theory.

End Monoid.
Export Monoid.Exports.

Section PervasiveMonoids.

Import Monoid.

Canonical andb_monoid := Law andbA andTb andbT.
Canonical andb_comoid := ComLaw andbC.

Canonical andb_muloid := MulLaw andFb andbF.
Canonical orb_monoid := Law orbA orFb orbF.
Canonical orb_comoid := ComLaw orbC.
Canonical orb_muloid := MulLaw orTb orbT.
Canonical addb_monoid := Law addbA addFb addbF.
Canonical addb_comoid := ComLaw addbC.
Canonical orb_addoid := AddLaw andb_orl andb_orr.
Canonical andb_addoid := AddLaw orb_andl orb_andr.
Canonical addb_addoid := AddLaw andb_addl andb_addr.

Canonical addn_monoid := Law addnA add0n addn0.
Canonical addn_comoid := ComLaw addnC.
Canonical muln_monoid := Law mulnA mul1n muln1.
Canonical muln_comoid := ComLaw mulnC.
Canonical muln_muloid := MulLaw mul0n muln0.
Canonical addn_addoid := AddLaw mulnDl mulnDr.

Canonical cat_monoid T := Law (@catA T) (@cat0s T) (@cats0 T).

End PervasiveMonoids.
Delimit Scope big_scope with BIG.
Open Scope big_scope.

(* The bigbody wrapper is a workaround for a quirk of the Coq pretty-printer, *)
(* which would fail to redisplay the \big notation when the <general_term> or *)
(* <condition> do not depend on the bound index. The BigBody constructor      *)
(* packages both in in a term in which i occurs; it also depends on the       *)
(* iterated <op>, as this can give more information on the expected type of   *)
(* the <general_term>, thus allowing for the insertion of coercions.          *)
CoInductive bigbody R I := BigBody of I & (R -> R -> R) & bool & R.

Definition applybig {R I} (body : bigbody R I) x :=
  let: BigBody _ op b v := body in if b then op v x else x.

Definition reducebig R I idx r (body : I -> bigbody R I) :=
  foldr (applybig \o body) idx r.

Module Type BigOpSig.
Parameter bigop : forall R I, R -> seq I -> (I -> bigbody R I) -> R.
Axiom bigopE : bigop = reducebig.
End BigOpSig.

Module BigOp : BigOpSig.
Definition bigop := reducebig.
Lemma bigopE : bigop = reducebig. Proof. by []. Qed.
End BigOp.

Notation bigop := BigOp.bigop (only parsing).
Canonical bigop_unlock := Unlockable BigOp.bigopE.

Definition index_iota m n := iota m (n - m).

Definition index_enum (T : finType) := Finite.enum T.

Lemma mem_index_iota m n i : i \in index_iota m n = (m <= i < n).
Admitted.

Lemma mem_index_enum T i : i \in index_enum T.
Admitted.

Hint Resolve mem_index_enum.

(*
Lemma filter_index_enum T P : filter P (index_enum T) = enum P.
Proof. by []. Qed.
*)

Notation "\big [ op / idx ]_ ( i <- r | P ) F" :=
  (bigop idx r (fun i => BigBody i op P%B F)) : big_scope.
Notation "\big [ op / idx ]_ ( i <- r ) F" :=
  (bigop idx r (fun i => BigBody i op true F)) : big_scope.
Notation "\big [ op / idx ]_ ( m <= i < n | P ) F" :=
  (bigop idx (index_iota m n) (fun i : nat => BigBody i op P%B F))
     : big_scope.
Notation "\big [ op / idx ]_ ( m <= i < n ) F" :=
  (bigop idx (index_iota m n) (fun i : nat => BigBody i op true F))
     : big_scope.
Notation "\big [ op / idx ]_ ( i | P ) F" :=
  (bigop idx (index_enum _) (fun i => BigBody i op P%B F)) : big_scope.
Notation "\big [ op / idx ]_ i F" :=
  (bigop idx (index_enum _) (fun i => BigBody i op true F)) : big_scope.
Notation "\big [ op / idx ]_ ( i : t | P ) F" :=
  (bigop idx (index_enum _) (fun i : t => BigBody i op P%B F))
     (only parsing) : big_scope.
Notation "\big [ op / idx ]_ ( i : t ) F" :=
  (bigop idx (index_enum _) (fun i : t => BigBody i op true F))
     (only parsing) : big_scope.
Notation "\big [ op / idx ]_ ( i < n | P ) F" :=
  (\big[op/idx]_(i : ordinal n | P%B) F) : big_scope.
Notation "\big [ op / idx ]_ ( i < n ) F" :=
  (\big[op/idx]_(i : ordinal n) F) : big_scope.
Notation "\big [ op / idx ]_ ( i 'in' A | P ) F" :=
  (\big[op/idx]_(i | (i \in A) && P) F) : big_scope.
Notation "\big [ op / idx ]_ ( i 'in' A ) F" :=
  (\big[op/idx]_(i | i \in A) F) : big_scope.

Notation BIG_F := (F in \big[_/_]_(i <- _ | _) F i)%pattern.
Notation BIG_P := (P in \big[_/_]_(i <- _ | P i) _)%pattern.

(* Induction loading *)
Lemma big_load R (K K' : R -> Type) idx op I r (P : pred I) F :
  K (\big[op/idx]_(i <- r | P i) F i) * K' (\big[op/idx]_(i <- r | P i) F i)
  -> K' (\big[op/idx]_(i <- r | P i) F i).
Proof. by case. Qed.

Arguments big_load [R] K [K'] idx op [I].

Section Elim3.

Variables (R1 R2 R3 : Type) (K : R1 -> R2 -> R3 -> Type).
Variables (id1 : R1) (op1 : R1 -> R1 -> R1).
Variables (id2 : R2) (op2 : R2 -> R2 -> R2).
Variables (id3 : R3) (op3 : R3 -> R3 -> R3).

Hypothesis Kid : K id1 id2 id3.

Lemma big_rec3 I r (P : pred I) F1 F2 F3
    (K_F : forall i y1 y2 y3, P i -> K y1 y2 y3 ->
       K (op1 (F1 i) y1) (op2 (F2 i) y2) (op3 (F3 i) y3)) :
  K (\big[op1/id1]_(i <- r | P i) F1 i)
    (\big[op2/id2]_(i <- r | P i) F2 i)
    (\big[op3/id3]_(i <- r | P i) F3 i).
Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: K_F. Qed.

Hypothesis Kop : forall x1 x2 x3 y1 y2 y3,
  K x1 x2 x3 -> K y1 y2 y3-> K (op1 x1 y1) (op2 x2 y2) (op3 x3 y3).
Lemma big_ind3 I r (P : pred I) F1 F2 F3
   (K_F : forall i, P i -> K (F1 i) (F2 i) (F3 i)) :
  K (\big[op1/id1]_(i <- r | P i) F1 i)
    (\big[op2/id2]_(i <- r | P i) F2 i)
    (\big[op3/id3]_(i <- r | P i) F3 i).
Proof. by apply: big_rec3 => i x1 x2 x3 /K_F; apply: Kop. Qed.

End Elim3.

Arguments big_rec3 [R1 R2 R3] K [id1 op1 id2 op2 id3 op3] _ [I r P F1 F2 F3].
Arguments big_ind3 [R1 R2 R3] K [id1 op1 id2 op2 id3 op3] _ _ [I r P F1 F2 F3].

Section Elim2.

Variables (R1 R2 : Type) (K : R1 -> R2 -> Type) (f : R2 -> R1).
Variables (id1 : R1) (op1 : R1 -> R1 -> R1).
Variables (id2 : R2) (op2 : R2 -> R2 -> R2).

Hypothesis Kid : K id1 id2.

Lemma big_rec2 I r (P : pred I) F1 F2
    (K_F : forall i y1 y2, P i -> K y1 y2 ->
       K (op1 (F1 i) y1) (op2 (F2 i) y2)) :
  K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i).
Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: K_F. Qed.

Hypothesis Kop : forall x1 x2 y1 y2,
  K x1 x2 -> K y1 y2 -> K (op1 x1 y1) (op2 x2 y2).
Lemma big_ind2 I r (P : pred I) F1 F2 (K_F : forall i, P i -> K (F1 i) (F2 i)) :
  K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i).
Proof. by apply: big_rec2 => i x1 x2 /K_F; apply: Kop. Qed.

Hypotheses (f_op : {morph f : x y / op2 x y >-> op1 x y}) (f_id : f id2 = id1).
Lemma big_morph I r (P : pred I) F :
  f (\big[op2/id2]_(i <- r | P i) F i) = \big[op1/id1]_(i <- r | P i) f (F i).
Proof. by rewrite unlock; elim: r => //= i r <-; rewrite -f_op -fun_if. Qed.

End Elim2.

Arguments big_rec2 [R1 R2] K [id1 op1 id2 op2] _ [I r P F1 F2].
Arguments big_ind2 [R1 R2] K [id1 op1 id2 op2] _ _ [I r P F1 F2].
Arguments big_morph [R1 R2] f [id1 op1 id2 op2] _ _ [I].

Section Elim1.

Variables (R : Type) (K : R -> Type) (f : R -> R).
Variables (idx : R) (op op' : R -> R -> R).

Hypothesis Kid : K idx.

Lemma big_rec I r (P : pred I) F
    (Kop : forall i x, P i -> K x -> K (op (F i) x)) :
  K (\big[op/idx]_(i <- r | P i) F i).
Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: Kop. Qed.

Hypothesis Kop : forall x y, K x -> K y -> K (op x y).
Lemma big_ind I r (P : pred I) F (K_F : forall i, P i -> K (F i)) :
  K (\big[op/idx]_(i <- r | P i) F i).
Proof. by apply: big_rec => // i x /K_F /Kop; apply. Qed.

Hypothesis Kop' : forall x y, K x -> K y -> op x y = op' x y.
Lemma eq_big_op I r (P : pred I) F (K_F : forall i, P i -> K (F i)) :
  \big[op/idx]_(i <- r | P i) F i = \big[op'/idx]_(i <- r | P i) F i.
Proof.
by elim/(big_load K): _; elim/big_rec2: _ => // i _ y Pi [Ky <-]; auto.
Qed.

Hypotheses (fM : {morph f : x y / op x y}) (f_id : f idx = idx).
Lemma big_endo I r (P : pred I) F :
  f (\big[op/idx]_(i <- r | P i) F i) = \big[op/idx]_(i <- r | P i) f (F i).
Proof. exact: big_morph. Qed.

End Elim1.

Arguments big_rec [R] K [idx op] _ [I r P F].
Arguments big_ind [R] K [idx op] _ _ [I r P F].
Arguments eq_big_op [R] K [idx op] op' _ _ _ [I].
Arguments big_endo [R] f [idx op] _ _ [I].

(* zmodp -------------------------------------------------------------------- *)

Lemma ord1 : all_equal_to (@Ordinal 1 0 is_true_true : 'I_1).
Admitted.

¤ Dauer der Verarbeitung: 0.78 Sekunden (vorverarbeitet) ¤

Druckansicht unsichere Verbindung
Druckansicht sprechenden Kalenders
in der Quellcodebibliothek suchen

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.