Library mathcomp.ssreflect.tuple

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  
 Distributed under the terms of CeCILL-B.                                  *)

From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat.
From mathcomp Require Import seq choice fintype.

Set Implicit Arguments.

Tuples, i.e., sequences with a fixed (known) length. We define: n.-tuple T == the type of n-tuples of elements of type T. [tuple of s] == the tuple whose underlying sequence (value) is s. The size of s must be known: specifically, Coq must be able to infer a Canonical tuple projecting on s. in_tuple s == the (size s).-tuple with value s. [tuple] == the empty tuple. [tuple x1; ..; xn] == the explicit n.-tuple <x1; ..; xn>. [tuple E | i < n] == the n.-tuple with general term E (i : 'I_n is bound in E). tcast Emn t == the m.-tuple t cast as an n.-tuple using Emn : m = n. As n.-tuple T coerces to seq t, all seq operations (size, nth, ...) can be applied to t : n.-tuple T; we provide a few specialized instances when avoids the need for a default value. tsize t == the size of t (the n in n.-tuple T) tnth t i == the i'th component of t, where i : 'I_n. [tnth t i] == the i'th component of t, where i : nat and i < n is convertible to true. thead t == the first element of t, when n is m.+1 for some m. Most seq constructors (cons, behead, cat, rcons, belast, take, drop, rot, map, ...) can be used to build tuples via the [tuple of s] construct. Tuples are actually a subType of seq, and inherit all combinatorial structures, including the finType structure. Some useful lemmas and definitions: tuple0 : [tuple] is the only 0.-tuple tupleP : elimination view for n.+1.-tuple ord_tuple n : the n.-tuple of all i : 'I_n

Section Def.

Variables (n : nat) (T : Type).

Structure tuple_of : Type := Tuple {tval :> seq T; _ : size tval == n}.

Canonical tuple_subType := Eval hnf in [subType for tval].

Implicit Type t : tuple_of.

Definition tsize of tuple_of := n.

Lemma size_tuple t : size t = n.

Lemma tnth_default t : 'I_n T.

Definition tnth t i := nth (tnth_default t i) t i.

Lemma tnth_nth x t i : tnth t i = nth x t i.

Lemma map_tnth_enum t : map (tnth t) (enum 'I_n) = t.

Lemma eq_from_tnth t1 t2 : tnth t1 =1 tnth t2 t1 = t2.

Definition tuple t mkT : tuple_of :=
  mkT (let: Tuple _ tP := t return size t == n in tP).

Lemma tupleE t : tuple (fun sP ⇒ @Tuple t sP) = t.

End Def.

Notation "n .-tuple" := (tuple_of n)
  (at level 2, format "n .-tuple") : type_scope.

Notation "{ 'tuple' n 'of' T }" := (n.-tuple T : predArgType)
  (at level 0, only parsing) : form_scope.

Notation "[ 'tuple' 'of' s ]" := (tuple (fun sP ⇒ @Tuple _ _ s sP))
  (at level 0, format "[ 'tuple' 'of' s ]") : form_scope.

Notation "[ 'tnth' t i ]" := (tnth t (@Ordinal (tsize t) i (erefl true)))
  (at level 0, t, i at level 8, format "[ 'tnth' t i ]") : form_scope.

Canonical nil_tuple T := Tuple (isT : @size T [::] == 0).
Canonical cons_tuple n T x (t : n.-tuple T) :=
  Tuple (valP t : size (x :: t) == n.+1).

Notation "[ 'tuple' x1 ; .. ; xn ]" := [tuple of x1 :: .. [:: xn] ..]
  (at level 0, format "[ 'tuple' '[' x1 ; '/' .. ; '/' xn ']' ]")
  : form_scope.

Notation "[ 'tuple' ]" := [tuple of [::]]
(at level 0, format "[ 'tuple' ]") : form_scope.

Section CastTuple.

Variable T : Type.

Definition in_tuple (s : seq T) := Tuple (eqxx (size s)).

Definition tcast m n (eq_mn : m = n) t :=
  let: erefl in _ = n := eq_mn return n.-tuple T in t.

Lemma tcastE m n (eq_mn : m = n) t i :
  tnth (tcast eq_mn t) i = tnth t (cast_ord (esym eq_mn) i).

Lemma tcast_id n (eq_nn : n = n) t : tcast eq_nn t = t.

Lemma tcastK m n (eq_mn : m = n) : cancel (tcast eq_mn) (tcast (esym eq_mn)).

Lemma tcastKV m n (eq_mn : m = n) : cancel (tcast (esym eq_mn)) (tcast eq_mn).

Lemma tcast_trans m n p (eq_mn : m = n) (eq_np : n = p) t:
  tcast (etrans eq_mn eq_np) t = tcast eq_np (tcast eq_mn t).

Lemma tvalK n (t : n.-tuple T) : in_tuple t = tcast (esym (size_tuple t)) t.

Lemma in_tupleE s : in_tuple s = s :> seq T.

End CastTuple.

Section SeqTuple.

Variables (n m : nat) (T U rT : Type).
Implicit Type t : n.-tuple T.

Lemma rcons_tupleP t x : size (rcons t x) == n.+1.
Canonical rcons_tuple t x := Tuple (rcons_tupleP t x).

Lemma nseq_tupleP x : @size T (nseq n x) == n.
Canonical nseq_tuple x := Tuple (nseq_tupleP x).

Lemma iota_tupleP : size (iota m n) == n.
Canonical iota_tuple := Tuple iota_tupleP.

Lemma behead_tupleP t : size (behead t) == n.-1.
Canonical behead_tuple t := Tuple (behead_tupleP t).

Lemma belast_tupleP x t : size (belast x t) == n.
Canonical belast_tuple x t := Tuple (belast_tupleP x t).

Lemma cat_tupleP t (u : m.-tuple T) : size (t ++ u) == n + m.
Canonical cat_tuple t u := Tuple (cat_tupleP t u).

Lemma take_tupleP t : size (take m t) == minn m n.
Canonical take_tuple t := Tuple (take_tupleP t).

Lemma drop_tupleP t : size (drop m t) == n - m.
Canonical drop_tuple t := Tuple (drop_tupleP t).

Lemma rev_tupleP t : size (rev t) == n.
Canonical rev_tuple t := Tuple (rev_tupleP t).

Lemma rot_tupleP t : size (rot m t) == n.
Canonical rot_tuple t := Tuple (rot_tupleP t).

Lemma rotr_tupleP t : size (rotr m t) == n.
Canonical rotr_tuple t := Tuple (rotr_tupleP t).

Lemma map_tupleP f t : @size rT (map f t) == n.
Canonical map_tuple f t := Tuple (map_tupleP f t).

Lemma scanl_tupleP f x t : @size rT (scanl f x t) == n.
Canonical scanl_tuple f x t := Tuple (scanl_tupleP f x t).

Lemma pairmap_tupleP f x t : @size rT (pairmap f x t) == n.
Canonical pairmap_tuple f x t := Tuple (pairmap_tupleP f x t).

Lemma zip_tupleP t (u : n.-tuple U) : size (zip t u) == n.
Canonical zip_tuple t u := Tuple (zip_tupleP t u).

Lemma allpairs_tupleP f t (u : m.-tuple U) : @size rT (allpairs f t u) == n × m.
Canonical allpairs_tuple f t u := Tuple (allpairs_tupleP f t u).

Definition thead (u : n.+1.-tuple T) := tnth u ord0.

Lemma tnth0 x t : tnth [tuple of x :: t] ord0 = x.

Lemma tnthS x t i : tnth [tuple of x :: t] (lift ord0 i) = tnth t i.

Lemma theadE x t : thead [tuple of x :: t] = x.

Lemma tuple0 : all_equal_to ([tuple] : 0.-tuple T).

Variant tuple1_spec : n.+1.-tuple T Type :=
  Tuple1spec x t : tuple1_spec [tuple of x :: t].

Lemma tupleP u : tuple1_spec u.

Lemma tnth_map f t i : tnth [tuple of map f t] i = f (tnth t i) :> rT.

Lemma tnth_nseq x i : tnth [tuple of nseq n x] i = x.

End SeqTuple.

Lemma tnth_behead n T (t : n.+1.-tuple T) i :
  tnth [tuple of behead t] i = tnth t (inord i.+1).

Lemma tuple_eta n T (t : n.+1.-tuple T) : t = [tuple of thead t :: behead t].

Section TupleQuantifiers.

Variables (n : nat) (T : Type).
Implicit Types (a : pred T) (t : n.-tuple T).

Lemma forallb_tnth a t : [ i, a (tnth t i)] = all a t.

Lemma existsb_tnth a t : [ i, a (tnth t i)] = has a t.

Lemma all_tnthP a t : reflect ( i, a (tnth t i)) (all a t).

Lemma has_tnthP a t : reflect ( i, a (tnth t i)) (has a t).

End TupleQuantifiers.

Section EqTuple.

Variables (n : nat) (T : eqType).

Definition tuple_eqMixin := Eval hnf in [eqMixin of n.-tuple T by <:].
Canonical tuple_eqType := Eval hnf in EqType (n.-tuple T) tuple_eqMixin.

Canonical tuple_predType := PredType (pred_of_seq : n.-tuple T pred T).

Lemma eqEtuple (t1 t2 : n.-tuple T) :
  (t1 == t2) = [ i, tnth t1 i == tnth t2 i].

Lemma memtE (t : n.-tuple T) : mem t = mem (tval t).

Lemma mem_tnth i (t : n.-tuple T) : tnth t i \in t.

Lemma memt_nth x0 (t : n.-tuple T) i : i < n nth x0 t i \in t.

Lemma tnthP (t : n.-tuple T) x : reflect ( i, x = tnth t i) (x \in t).

Lemma seq_tnthP (s : seq T) x : x \in s {i | x = tnth (in_tuple s) i}.

End EqTuple.

Definition tuple_choiceMixin n (T : choiceType) :=
  [choiceMixin of n.-tuple T by <:].

Canonical tuple_choiceType n (T : choiceType) :=
  Eval hnf in ChoiceType (n.-tuple T) (tuple_choiceMixin n T).

Definition tuple_countMixin n (T : countType) :=
  [countMixin of n.-tuple T by <:].

Canonical tuple_countType n (T : countType) :=
  Eval hnf in CountType (n.-tuple T) (tuple_countMixin n T).

Canonical tuple_subCountType n (T : countType) :=
  Eval hnf in [subCountType of n.-tuple T].

Module Type FinTupleSig.
Section FinTupleSig.
Variables (n : nat) (T : finType).
Parameter enum : seq (n.-tuple T).
Axiom enumP : Finite.axiom enum.
Axiom size_enum : size enum = #|T| ^ n.
End FinTupleSig.
End FinTupleSig.

Module FinTuple : FinTupleSig.
Section FinTuple.
Variables (n : nat) (T : finType).

Definition enum : seq (n.-tuple T) :=
  let extend e := flatten (codom (fun xmap (cons x) e)) in
  pmap insub (iter n extend [::[::]]).

Lemma enumP : Finite.axiom enum.

Lemma size_enum : size enum = #|T| ^ n.

End FinTuple.
End FinTuple.

Section UseFinTuple.

Variables (n : nat) (T : finType).

tuple_finMixin could, in principle, be made Canonical to allow for folding Finite.enum of a finite tuple type (see comments around eqE in eqtype.v), but in practice it will not work because the mixin_enum projector has been buried under an opaque alias, to avoid some performance issues during type inference.
Definition tuple_finMixin := Eval hnf in FinMixin (@FinTuple.enumP n T).
Canonical tuple_finType := Eval hnf in FinType (n.-tuple T) tuple_finMixin.
Canonical tuple_subFinType := Eval hnf in [subFinType of n.-tuple T].

Lemma card_tuple : #|{:n.-tuple T}| = #|T| ^ n.

Lemma enum_tupleP (A : {pred T}) : size (enum A) == #|A|.
Canonical enum_tuple A := Tuple (enum_tupleP A).

Definition ord_tuple : n.-tuple 'I_n := Tuple (introT eqP (size_enum_ord n)).
Lemma val_ord_tuple : val ord_tuple = enum 'I_n.

Lemma tuple_map_ord U (t : n.-tuple U) : t = [tuple of map (tnth t) ord_tuple].

Lemma tnth_ord_tuple i : tnth ord_tuple i = i.

Section ImageTuple.

Variables (T' : Type) (f : T T') (A : {pred T}).

Canonical image_tuple : #|A|.-tuple T' := [tuple of image f A].
Canonical codom_tuple : #|T|.-tuple T' := [tuple of codom f].

End ImageTuple.

Section MkTuple.

Variables (T' : Type) (f : 'I_n T').

Definition mktuple := map_tuple f ord_tuple.

Lemma tnth_mktuple i : tnth mktuple i = f i.

Lemma nth_mktuple x0 (i : 'I_n) : nth x0 mktuple i = f i.

End MkTuple.

Lemma eq_mktuple T' (f1 f2 : 'I_n T') :
  f1 =1 f2 mktuple f1 = mktuple f2.

End UseFinTuple.

Notation "[ 'tuple' F | i < n ]" := (mktuple (fun i : 'I_nF))
  (at level 0, i at level 0,
   format "[ '[hv' 'tuple' F '/' | i < n ] ']'") : form_scope.