Library tuple
Tuples, i.e., lists 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, which
must have a known size; specifically, Coq must be able
to infer a tuple whose Canonical Projection is s, e.g.,
[tuple] is the empty tuple, and
[tuple x1; ..; xn] the explicit n.-tuple <x1; ..; xn>.
As n.-tuple T coerces to seq t, all operations for seq (size, nth, ...)
can be applied to t : n.-tuple T; we provide a few specialized instances
when this 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.
thead t : the first element of t, when n is canonically positive.
Most seq constructors (behead, cat, 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 Structure tuple_subType :=
Eval hnf in [subType for tval by tuple_of_rect].
Implicit Type t : tuple_of.
Definition tsize of tuple_of := n.
Lemma size_tuple : forall t, size t = n.
Lemma tnth_default : forall t (i : 'I_n), T.
Definition tnth t i := nth (tnth_default t i) t i.
Lemma tnth_nth : forall x t i, tnth t i = nth x t i.
Lemma map_tnth_enum : forall t, map (tnth t) (enum 'I_n) = t.
Lemma eq_from_tnth : forall 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 : forall 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 Structure nil_tuple T :=
Tuple (erefl _ : @size T [::] == 0).
Canonical Structure cons_tuple n T x (t : n.-tuple T) :=
Tuple (valP t : size (x :: t) == n.+1).
Notation "[ 'tuple' x1 ; .. ; xn ]" := [tuple of x1 :: ..
Notation "[ 'tuple' ]" := [tuple of [::]] (at level 0, format "[ 'tuple' ]") : form_scope.
Section SeqTuple.
Variables (n : nat) (T rT : Type).
Notation tT := (n.-tuple T).
Lemma nseq_tupleP : forall x : T, size (nseq n x) == n.
Canonical Structure nseq_tuple x := Tuple (nseq_tupleP x).
Lemma iota_tupleP : forall m, size (iota m n) == n.
Canonical Structure iota_tuple m := Tuple (iota_tupleP m).
Lemma behead_tupleP : forall t : tT, size (behead t) == n.-1.
Canonical Structure behead_tuple t := Tuple (behead_tupleP t).
Lemma belast_tupleP : forall x (t : tT), size (belast x t) == n.
Canonical Structure belast_tuple x t := Tuple (belast_tupleP x t).
Lemma cat_tupleP : forall n1 n2 (t1 : n1.-tuple T) (t2 : n2.-tuple T),
size (t1 ++ t2) == n1 + n2.
Canonical Structure cat_tuple n1 n2 t1 t2 := Tuple (@cat_tupleP n1 n2 t1 t2).
Lemma take_tupleP : forall m (t : tT), size (take m t) == minn m n.
Canonical Structure take_tuple m t := Tuple (take_tupleP m t).
Lemma drop_tupleP : forall m (t : tT), size (drop m t) == n - m.
Canonical Structure drop_tuple m t := Tuple (drop_tupleP m t).
Lemma rev_tupleP : forall t : tT, size (rev t) == n.
Canonical Structure rev_tuple t := Tuple (rev_tupleP t).
Lemma rot_tupleP : forall m (t : tT), size (rot m t) == n.
Canonical Structure rot_tuple m t := Tuple (rot_tupleP m t).
Lemma rotr_tupleP : forall m (t : tT), size (rotr m t) == n.
Canonical Structure rotr_tuple m t := Tuple (rotr_tupleP m t).
Lemma map_tupleP : forall f (t : tT), @size rT (map f t) == n.
Canonical Structure map_tuple f t := Tuple (map_tupleP f t).
Lemma scanl_tupleP : forall f x (t : tT), @size rT (scanl f x t) == n.
Canonical Structure scanl_tuple f x t := Tuple (scanl_tupleP f x t).
Lemma pairmap_tupleP : forall f x (t : tT), @size rT (pairmap f x t) == n.
Canonical Structure pairmap_tuple f x t := Tuple (pairmap_tupleP f x t).
Lemma zip_tupleP : forall t1 t2 : tT, size (zip t1 t2) == n.
Canonical Structure zip_tuple t1 t2 := Tuple (zip_tupleP t1 t2).
Definition thead n (t : n.+1.-tuple T) := tnth t ord0.
Lemma tnth0 : forall x n (t : n.-tuple T), tnth [tuple of x :: t] ord0 = x.
Lemma theadE : forall x n (t : n.-tuple T), thead [tuple of x :: t] = x.
Lemma tuple0 : @all_equal_to (0.-tuple T) [tuple].
CoInductive tuple1_spec : n.+1.-tuple T -> Type :=
Tuple1spec x (t : tT) : tuple1_spec [tuple of x :: t].
Lemma tupleP : forall t, tuple1_spec t.
Lemma tnth_map : forall f (t : tT) i,
tnth [tuple of map f t] i = f (tnth t i) :> rT.
End SeqTuple.
Lemma tnth_behead : forall n T (t : n.+1.-tuple T) i,
tnth [tuple of behead t] i = tnth t (inord i.+1).
Lemma tuple_eta : forall n T (t : n.+1.-tuple T),
t = [tuple of thead t :: behead t].
Section EqTuple.
Variables (n : nat) (T : eqType).
Definition tuple_eqMixin := Eval hnf in [eqMixin of n.-tuple T by <:].
Canonical Structure tuple_eqType :=
Eval hnf in EqType (n.-tuple T) tuple_eqMixin.
Canonical Structure tuple_predType :=
Eval hnf in mkPredType (fun t : n.-tuple T => mem_seq t).
Lemma memtE : forall t : n.-tuple T, mem t = mem (tval t).
End EqTuple.
Definition tuple_choiceMixin n (T : choiceType) :=
[choiceMixin of n.-tuple T by <:].
Canonical Structure 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 Structure tuple_countType n (T : countType) :=
Eval hnf in CountType (n.-tuple T) (tuple_countMixin n T).
Canonical Structure 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 (map (fun x => map (cons x) e) (Finite.enum T)) 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).
Notation tT := (n.-tuple T).
Canonical Structure tuple_finMixin :=
Eval hnf in FinMixin (@FinTuple.enumP n T).
Canonical Structure tuple_finType := Eval hnf in FinType tT tuple_finMixin.
Canonical Structure tuple_subFinType := Eval hnf in [subFinType of tT].
Lemma card_tuple : #|{:n.-tuple T}| = #|T| ^ n.
Lemma enum_tupleP : forall A : pred T, size (enum A) == #|A|.
Canonical Structure 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 : forall T' (t : n.-tuple T'),
t = [tuple of map (tnth t) ord_tuple].
Lemma tnth_ord_tuple : forall i, tnth ord_tuple i = i.
End UseFinTuple.