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Diffstat (limited to 'theories/Vectors/Fin.v')
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diff --git a/theories/Vectors/Fin.v b/theories/Vectors/Fin.v new file mode 100644 index 00000000..28e355fb --- /dev/null +++ b/theories/Vectors/Fin.v @@ -0,0 +1,176 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +Require Arith_base. + +(** [fin n] is a convinient way to represent \[1 .. n\] + +[fin n] can be seen as a n-uplet of unit where [F1] is the first element of +the n-uplet and [FS] set (n-1)-uplet of all the element but the first. + + Author: Pierre Boutillier + Institution: PPS, INRIA 12/2010 +*) + +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 as p' in t n return + match n as n' return t n' -> Type + with |0 => fun f0 => P f0 |S _ => fun _ => @ID end p' + with |F1 _ => @id |FS _ _ => @id 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 := + match p with + |F1 k => P1 k + |FS k pp => PS pp + end. + +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): forall (b: t n), P a b := +match a with + |F1 m => fun (b: t (S m)) => match b as b' in t n' + return match n',b' with + |0,_ => @ID + |S n0,b0 => P F1 b0 + end with + |F1 m' => H0 m' + |FS m' b' => H1 b' + end + |FS m a' => fun (b: t (S m)) => match b with + |F1 m' => fun aa: t m' => H2 aa + |FS m' b' => fun aa: t m' => HS aa b' (rect2_fix aa b') + end a' +end. +End SCHEMES. + +(** [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 in t k return {i | i< k} with + |F1 j => exist (fun i => i< S j) 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 (fun x => p = 0 + x) p (@eq_refl _ p)) + |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_to_nat_inv {m} (p : t m) : of_nat_lt (proj2_sig (to_nat p)) = p. +Proof. +induction p. + reflexivity. + simpl; destruct (to_nat p). simpl. subst p; repeat f_equal. apply Peano_dec.le_unique. +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 ineficient !!! *) +Definition L_R {m} n (p : t m) : t (n + m). +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 m' 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)). +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. |