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(************************************************************************)
(* * 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) *)
(************************************************************************)
Require Arith_base.
(** [fin n] is a convenient way to represent \[1 .. n\]
[fin n] can be seen as a n-uplet of unit. [F1] is the first element of
the n-uplet. If [f] is the k-th element of the (n-1)-uplet, [FS f] is the
(k+1)-th element of the n-uplet.
Author: Pierre Boutillier
Institution: PPS, INRIA 12/2010-01/2012-07/2012
*)
Inductive t : nat -> Set :=
|F1 : forall {n}, t (S n)
|FS : forall {n}, t n -> t (S n).
Section SCHEMES.
Definition case0 P (p: t 0): P p :=
match p with | F1 | FS _ => fun devil => False_rect (@IDProp) devil (* subterm !!! *) end.
Definition caseS' {n : nat} (p : t (S n)) : forall (P : t (S n) -> Type)
(P1 : P F1) (PS : forall (p : t n), P (FS p)), P p :=
match p with
| @F1 k => fun P P1 PS => P1
| FS pp => fun P P1 PS => PS pp
end.
Definition caseS (P: forall {n}, t (S n) -> Type)
(P1: forall n, @P n F1) (PS : forall {n} (p: t n), P (FS p))
{n} (p: t (S n)) : P p := caseS' p P (P1 n) PS.
Definition rectS (P: forall {n}, t (S n) -> Type)
(P1: forall n, @P n F1) (PS : forall {n} (p: t (S n)), P p -> P (FS p)):
forall {n} (p: t (S n)), P p :=
fix rectS_fix {n} (p: t (S n)): P p:=
match p with
| @F1 k => P1 k
| @FS 0 pp => case0 (fun f => P (FS f)) pp
| @FS (S k) pp => PS pp (rectS_fix pp)
end.
Definition rect2 (P : forall {n} (a b : t n), Type)
(H0 : forall n, @P (S n) F1 F1)
(H1 : forall {n} (f : t n), P F1 (FS f))
(H2 : forall {n} (f : t n), P (FS f) F1)
(HS : forall {n} (f g : t n), P f g -> P (FS f) (FS g)) :
forall {n} (a b : t n), P a b :=
fix rect2_fix {n} (a : t n) {struct a} : forall (b : t n), P a b :=
match a with
| @F1 m => fun (b : t (S m)) => caseS' b (P F1) (H0 _) H1
| @FS m a' => fun (b : t (S m)) =>
caseS' b (fun b => P (@FS m a') b) (H2 a') (fun b' => HS _ _ (rect2_fix a' b'))
end.
End SCHEMES.
Definition FS_inj {n} (x y: t n) (eq: FS x = FS y): x = y :=
match eq in _ = a return
match a as a' in t m return match m with |0 => Prop |S n' => t n' -> Prop end
with F1 => fun _ => True |FS y => fun x' => x' = y end x with
eq_refl => eq_refl
end.
(** [to_nat f] = p iff [f] is the p{^ th} element of [fin m]. *)
Fixpoint to_nat {m} (n : t m) : {i | i < m} :=
match n with
|@F1 j => exist _ 0 (Lt.lt_0_Sn j)
|FS p => match to_nat p with |exist _ i P => exist _ (S i) (Lt.lt_n_S _ _ P) end
end.
(** [of_nat p n] answers the p{^ th} element of [fin n] if p < n or a proof of
p >= n else *)
Fixpoint of_nat (p n : nat) : (t n) + { exists m, p = n + m } :=
match n with
|0 => inright _ (ex_intro _ p eq_refl)
|S n' => match p with
|0 => inleft _ (F1)
|S p' => match of_nat p' n' with
|inleft f => inleft _ (FS f)
|inright arg => inright _ (match arg with |ex_intro _ m e =>
ex_intro (fun x => S p' = S n' + x) m (f_equal S e) end)
end
end
end.
(** [of_nat_lt p n H] answers the p{^ th} element of [fin n]
it behaves much better than [of_nat p n] on open term *)
Fixpoint of_nat_lt {p n : nat} : p < n -> t n :=
match n with
|0 => fun H : p < 0 => False_rect _ (Lt.lt_n_O p H)
|S n' => match p with
|0 => fun _ => @F1 n'
|S p' => fun H => FS (of_nat_lt (Lt.lt_S_n _ _ H))
end
end.
Lemma of_nat_ext {p}{n} (h h' : p < n) : of_nat_lt h = of_nat_lt h'.
Proof.
now rewrite (Peano_dec.le_unique _ _ h h').
Qed.
Lemma of_nat_to_nat_inv {m} (p : t m) : of_nat_lt (proj2_sig (to_nat p)) = p.
Proof.
induction p; simpl.
- reflexivity.
- destruct (to_nat p); simpl in *. f_equal. subst p. apply of_nat_ext.
Qed.
Lemma to_nat_of_nat {p}{n} (h : p < n) : to_nat (of_nat_lt h) = exist _ p h.
Proof.
revert n h.
induction p; (destruct n ; intros h; [ destruct (Lt.lt_n_O _ h) | cbn]);
[ | rewrite (IHp _ (Lt.lt_S_n p n h))]; f_equal; apply Peano_dec.le_unique.
Qed.
Lemma to_nat_inj {n} (p q : t n) :
proj1_sig (to_nat p) = proj1_sig (to_nat q) -> p = q.
Proof.
intro H.
rewrite <- (of_nat_to_nat_inv p), <- (of_nat_to_nat_inv q).
destruct (to_nat p) as (np,hp), (to_nat q) as (nq,hq); simpl in *.
revert hp hq. rewrite H. apply of_nat_ext.
Qed.
(** [weak p f] answers a function witch is the identity for the p{^ th} first
element of [fin (p + m)] and [FS (FS .. (FS (f k)))] for [FS (FS .. (FS k))]
with p FSs *)
Fixpoint weak {m}{n} p (f : t m -> t n) :
t (p + m) -> t (p + n) :=
match p as p' return t (p' + m) -> t (p' + n) with
|0 => f
|S p' => fun x => match x with
|@F1 n' => fun eq : n' = p' + m => F1
|@FS n' y => fun eq : n' = p' + m => FS (weak p' f (eq_rect _ t y _ eq))
end (eq_refl _)
end.
(** The p{^ th} element of [fin m] viewed as the p{^ th} element of
[fin (m + n)] *)
Fixpoint L {m} n (p : t m) : t (m + n) :=
match p with |F1 => F1 |FS p' => FS (L n p') end.
Lemma L_sanity {m} n (p : t m) : proj1_sig (to_nat (L n p)) = proj1_sig (to_nat p).
Proof.
induction p.
- reflexivity.
- simpl; destruct (to_nat (L n p)); simpl in *; rewrite IHp. now destruct (to_nat p).
Qed.
(** The p{^ th} element of [fin m] viewed as the p{^ th} element of
[fin (n + m)]
Really really inefficient !!! *)
Definition L_R {m} n (p : t m) : t (n + m).
Proof.
induction n.
- exact p.
- exact ((fix LS k (p: t k) :=
match p with
|@F1 k' => @F1 (S k')
|FS p' => FS (LS _ p')
end) _ IHn).
Defined.
(** The p{^ th} element of [fin m] viewed as the (n + p){^ th} element of
[fin (n + m)] *)
Fixpoint R {m} n (p : t m) : t (n + m) :=
match n with |0 => p |S n' => FS (R n' p) end.
Lemma R_sanity {m} n (p : t m) : proj1_sig (to_nat (R n p)) = n + proj1_sig (to_nat p).
Proof.
induction n.
- reflexivity.
- simpl; destruct (to_nat (R n p)); simpl in *; rewrite IHn. now destruct (to_nat p).
Qed.
Fixpoint depair {m n} (o : t m) (p : t n) : t (m * n) :=
match o with
|@F1 m' => L (m' * n) p
|FS o' => R n (depair o' p)
end.
Lemma depair_sanity {m n} (o : t m) (p : t n) :
proj1_sig (to_nat (depair o p)) = n * (proj1_sig (to_nat o)) + (proj1_sig (to_nat p)).
Proof.
induction o ; simpl.
- rewrite L_sanity. now rewrite Mult.mult_0_r.
- rewrite R_sanity. rewrite IHo.
rewrite Plus.plus_assoc. destruct (to_nat o); simpl; rewrite Mult.mult_succ_r.
now rewrite (Plus.plus_comm n).
Qed.
Fixpoint eqb {m n} (p : t m) (q : t n) :=
match p, q with
| @F1 m', @F1 n' => EqNat.beq_nat m' n'
| FS _, F1 => false
| F1, FS _ => false
| FS p', FS q' => eqb p' q'
end.
Lemma eqb_nat_eq : forall m n (p : t m) (q : t n), eqb p q = true -> m = n.
Proof.
intros m n p; revert n; induction p; destruct q; simpl; intros; f_equal.
- now apply EqNat.beq_nat_true.
- easy.
- easy.
- eapply IHp. eassumption.
Qed.
Lemma eqb_eq : forall n (p q : t n), eqb p q = true <-> p = q.
Proof.
apply rect2; simpl; intros.
- split; intros ; [ reflexivity | now apply EqNat.beq_nat_true_iff ].
- now split.
- now split.
- eapply iff_trans.
+ eassumption.
+ split.
* intros; now f_equal.
* apply FS_inj.
Qed.
Lemma eq_dec {n} (x y : t n): {x = y} + {x <> y}.
Proof.
case_eq (eqb x y); intros.
- left; now apply eqb_eq.
- right. intros Heq. apply <- eqb_eq in Heq. congruence.
Defined.
Definition cast: forall {m} (v: t m) {n}, m = n -> t n.
Proof.
refine (fix cast {m} (v: t m) {struct v} :=
match v in t m' return forall n, m' = n -> t n with
|F1 => fun n => match n with
| 0 => fun H => False_rect _ _
| S n' => fun H => F1
end
|FS f => fun n => match n with
| 0 => fun H => False_rect _ _
| S n' => fun H => FS (cast f n' (f_equal pred H))
end
end); discriminate.
Defined.
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