409 lines
13 KiB
Coq
409 lines
13 KiB
Coq
Require Import subsequences.
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Require Import Sorting.Permutation.
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Require Import Nat.
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Require Import PeanoNat.
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Require Import List.
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Import ListNotations.
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Definition permutation_mapping {X: Type} (u v base : list X) :=
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exists p l, Permutation base p
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/\ map Some u = map (nth_error base) l
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/\ map Some v = map (nth_error p) l.
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(*
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Definition incl_without_eq {X: Type} (u v : list X) :=
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exists l, map Some u = map (nth_error v) l.
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Lemma incl_without_eq_incl {X: Type}:
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forall (u v : list X), incl_without_eq u v -> incl u v.
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Proof.
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intro u. induction u; intro v; intro H. easy. destruct H.
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destruct x. symmetry in H. apply nil_cons in H. contradiction.
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inversion H. apply incl_cons.
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symmetry in H1. apply nth_error_In in H1. assumption.
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apply IHu. exists x. assumption.
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Qed.
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Lemma incl_incl_without_eq {X: Type}:
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forall (u v : list X), incl u v -> incl_without_eq u v.
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Proof.
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intro u. induction u; intro v; intro H. exists nil. easy.
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apply incl_cons_inv in H. destruct H. apply IHu in H0. destruct H0.
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apply In_nth_error in H. destruct H. exists (x0 :: x).
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simpl. rewrite H. rewrite H0. reflexivity.
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Qed.
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*)
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Lemma permutation_mapping_self {X: Type} :
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forall (u base : list X), incl u base -> permutation_mapping u u base.
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Proof.
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intro u. induction u; intro base; intro H; exists base.
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exists nil. split; easy. apply incl_cons_inv in H. destruct H.
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apply IHu in H0. destruct H0. destruct H0. destruct H0. destruct H1.
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apply In_nth_error in H. destruct H. exists (x1 :: x0).
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split. easy. split; simpl; rewrite H; rewrite H1; reflexivity.
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Qed.
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Lemma permutation_mapping_length {X: Type} :
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forall (u v base : list X),
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permutation_mapping u v base -> length u = length v.
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Proof.
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intros u v base. intro H. destruct H. destruct H. destruct H. destruct H0.
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replace (length u) with (length (map Some u)). rewrite H0.
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replace (length v) with (length (map Some v)). rewrite H1.
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rewrite map_length. rewrite map_length. reflexivity.
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apply map_length. apply map_length.
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Qed.
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Lemma permutation_mapping_swap {X: Type} :
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forall (u v base : list X),
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permutation_mapping u v base -> permutation_mapping v u base.
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Proof.
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intros u v base.
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(* destruct u in order to get a value of type X when needed *)
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destruct u. intro H. destruct H. destruct H. destruct H. destruct H0.
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symmetry in H0. apply map_eq_nil in H0. rewrite H0 in H1.
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exists base. exists nil. split. easy. rewrite H1. split; easy.
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intro H. destruct H as [p]. destruct H as [l].
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destruct H. destruct H0. assert (I := H). rewrite Permutation_nth in H.
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destruct H. destruct H2 as [f]. destruct H2. destruct H3.
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apply FinFun.bInjective_bSurjective in H3.
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apply FinFun.bSurjective_bBijective in H3.
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destruct H3 as [g]. destruct H3.
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exists (map (fun e => nth e base x) (map g (seq 0 (length base)))).
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exists (map f l). split.
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replace base with (map (fun e => nth e base x) (seq 0 (length base))) at 1.
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apply Permutation_map. apply NoDup_Permutation_bis.
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apply seq_NoDup. rewrite map_length. rewrite seq_length. easy.
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intro a. intro J.
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assert (a < 0 + length base). apply in_seq. assumption.
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assert (forall x n h h', FinFun.bFun n h -> FinFun.bFun n h'
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-> (forall y, y < n -> h (h' y) = y /\ h' (h y) = y)
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-> x < n -> In x (map h (seq 0 n))).
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intros x' n h h'. intros J1 J2 J3 J4. replace x' with (h (h' x')) at 1.
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apply in_map. rewrite in_seq. split. apply le_0_n. apply J2. assumption.
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apply J3 in J4. destruct J4. assumption. apply H7 with (h' := f); assumption.
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assert (forall (b: list X),
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map (fun e : nat => nth e b x) (seq 0 (length b)) = b).
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intro b. induction b. reflexivity.
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replace (seq 0 (length (a :: b))) with (0:: map S (seq 0 (length b))).
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rewrite map_cons. rewrite map_map.
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assert (map (fun e : nat => nth e b x) (seq 0 (length b))
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= map (fun x0 : nat => nth (S x0) (a :: b) x) (seq 0 (length b))).
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destruct b. reflexivity. reflexivity.
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rewrite <- H6. rewrite IHb. reflexivity.
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rewrite seq_shift. rewrite cons_seq. reflexivity. apply H6.
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split.
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(* first case in split *)
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rewrite H1.
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assert (forall s, (forall y, In y s -> y < length base)
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-> map (nth_error p) s = map (nth_error base) (map f s)).
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intro s. induction s; intro K. reflexivity. simpl. rewrite IHs.
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rewrite nth_error_nth' with (d := x). rewrite nth_error_nth' with (d := x).
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rewrite <- H4. reflexivity. apply K. apply in_eq. apply H2. apply K.
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apply in_eq. rewrite H. apply K. apply in_eq. intro y. intro L. apply K.
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apply in_cons. assumption. apply H6. intro y. intro L.
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assert (forall s z, In z s -> nth_error base z <> None -> z < length base).
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intros s z. intros M1 M2. apply nth_error_Some. assumption.
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apply H7 with (s := l). assumption.
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assert (forall s (t: list X),
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map Some t = map (nth_error base) s -> In y s -> nth_error base y <> None).
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intro s. induction s; intros t; intros M1 M2.
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apply in_nil in M2. contradiction.
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apply in_inv in M2. destruct M2. rewrite H8 in M1.
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destruct t; inversion M1; easy.
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destruct t; inversion M1; apply IHs with (t := t); assumption.
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generalize L. generalize H0. apply H8.
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(* second case in split *)
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rewrite H0.
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assert (forall s, (forall y, In y s -> y < length base)
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-> map (nth_error base) s
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= map (nth_error (map (fun e => nth (g e) base x)
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(seq 0 (length base)))) (map f s)).
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intro s. rewrite map_map. induction s; intro K. reflexivity.
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simpl. rewrite IHs.
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assert (forall q p, p < q -> nth p (seq 0 q) 0 = p).
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intro q. induction q; intro p'; intro J4.
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apply Nat.nlt_0_r in J4. contradiction.
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rewrite Nat.lt_succ_r in J4. rewrite Nat.le_lteq in J4. destruct J4.
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rewrite seq_S. rewrite app_nth1. apply IHq. assumption.
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rewrite seq_length. assumption. rewrite H6. rewrite seq_nth.
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reflexivity. apply Nat.lt_succ_diag_r.
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assert (N: In a (a::s)). apply in_eq. apply K in N. assert (M := N).
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apply H5 in N. destruct N.
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rewrite nth_error_nth' with (d := x).
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replace (nth_error (map (fun e : nat => nth (g e) base x)
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(seq 0 (length base))) (f a))
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with (Some ((fun e => nth (g e) base x)
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(nth (f a) (seq 0 (length base)) 0))).
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assert (forall m n f' g',
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FinFun.bFun m f' -> FinFun.bFun m g'
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-> (forall x : nat, x < m -> g' (f' x) = x /\ f' (g' x) = x)
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-> n < m -> g' (nth (f' n) (seq 0 m) 0) = n).
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intros m n f' g'. intros J J1 J2 J3.
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rewrite H6. apply J2 in J3. destruct J3. apply H9. apply J. assumption.
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rewrite H6. rewrite H7. reflexivity. apply H2. assumption. rewrite H6.
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symmetry. rewrite map_nth_error with (d := f a). reflexivity.
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rewrite nth_error_nth' with (d := 0). rewrite H6. reflexivity.
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apply H2. assumption. rewrite seq_length. apply H2. assumption.
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apply H2. assumption. assumption. intro y. intro G. apply K.
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apply in_cons. assumption.
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replace (map (fun e : nat => nth e base x) (map g (seq 0 (length base))))
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with (map (fun z => (fun e => nth e base x) (g z)) (seq 0 (length base))).
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apply H6.
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assert (forall w (u v: list X),
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map Some u = map (nth_error v) w -> (forall y, In y w -> y < length v)).
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intro w; induction w; intros u0 v0; intro J; intro y; intro J1.
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contradiction. destruct u0. inversion J.
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apply in_inv in J1. destruct J1. rewrite <- H7.
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inversion J. apply nth_error_Some. rewrite <- H9. easy.
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apply IHw with (u := u0). inversion J. reflexivity. assumption.
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apply H7 with (u := x::u). assumption.
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rewrite map_map. reflexivity. assumption. assumption.
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Qed.
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(*
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Lemma permutation_mapping_base {X: Type} :
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forall (u v base p: list X),
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Permutation base p -> permutation_mapping u v base
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-> permutation_mapping u v p.
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Proof.
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intros u v base p.
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(* destruct u in order to get a value of type X when needed *)
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destruct u. intros I H. destruct H. destruct H. destruct H. destruct H0.
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symmetry in H0. apply map_eq_nil in H0. rewrite H0 in H1.
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exists base. exists nil. split. easy. rewrite H1. split; easy.
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intros H I. assert (K := H).
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destruct I as [p']. destruct H0 as [l]. destruct H0.
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destruct H1.
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rewrite Permutation_nth in H. destruct H.
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destruct H3 as [f]. destruct H3. destruct H4.
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apply FinFun.bInjective_bSurjective in H4.
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apply FinFun.bSurjective_bBijective in H4. destruct H4 as [g].
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destruct H4.
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exists base. exists (map g l). split.
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apply Permutation_trans with (l' := base). apply Permutation_sym.
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assumption. apply Permutation_refl. split.
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(* first case in split *)
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rewrite H1.
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assert (forall s, (forall y, In y s -> y < length base)
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-> map (nth_error base) s = map (nth_error p) (map g s)).
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intro s. induction s; intro L. reflexivity.
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assert (M: a < length base). apply L. apply in_eq.
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simpl. rewrite IHs.
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rewrite nth_error_nth' with (d := x).
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rewrite nth_error_nth' with (d := x).
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replace a with (f (g a)) at 1.
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rewrite <- H5. reflexivity.
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apply H4. assumption.
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apply H6 in M. destruct M. assumption.
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rewrite H. apply H4. assumption. assumption.
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intro y. intro N. apply L. apply in_cons. assumption.
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apply H7. intro y. intro N.
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assert (forall s z, In z s -> nth_error base z <> None -> z < length base).
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intros s z. intros M1 M2. apply nth_error_Some. assumption.
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apply H8 with (s := l). assumption.
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assert (forall s (t: list X),
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map Some t = map (nth_error base) s
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-> In y s -> nth_error base y <> None).
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intro s. induction s; intros t; intros M1 M2.
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apply in_nil in M2. contradiction.
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apply in_inv in M2. destruct M2. rewrite H9 in M1.
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destruct t. inversion M1. inversion M1. easy.
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destruct t. inversion M1. inversion M1.
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apply IHs with (t := t). assumption. assumption.
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generalize N. generalize H1. apply H9.
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(* second case in split *)
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rewrite H2.
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assert (forall s, (forall y, In y s -> y < length base)
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-> map (nth_error p') s = map (nth_error base) (map g s)).
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intro s. induction s; intro R. reflexivity.
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simpl. rewrite IHs.
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rewrite nth_error_nth' with (d := x).
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rewrite nth_error_nth' with (d := x).
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replace a with (g (f a)) at 1.
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rewrite <- H5. reflexivity.
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apply H4. assumption.
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apply H6 in M. destruct M. assumption.
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rewrite H. apply H4. assumption. assumption.
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intro y. intro N. apply L. apply in_cons. assumption.
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apply H7. intro y. intro N.
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assert (forall s z, In z s -> nth_error base z <> None -> z < length base).
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intros s z. intros M1 M2. apply nth_error_Some. assumption.
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apply H8 with (s := l). assumption.
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assert (forall s (t: list X),
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map Some t = map (nth_error base) s
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-> In y s -> nth_error base y <> None).
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intro s. induction s; intros t; intros M1 M2.
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apply in_nil in M2. contradiction.
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apply in_inv in M2. destruct M2. rewrite H9 in M1.
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destruct t. inversion M1. inversion M1. easy.
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destruct t. inversion M1. inversion M1.
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apply IHs with (t := t). assumption. assumption.
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generalize N. generalize H1. apply H9.
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exists p'. exists (map g l). split.
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apply Permutation_trans with (l' := base). apply Permutation_sym.
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assumption. assumption. split.
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(* first case in split *)
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rewrite H1.
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assert (forall s, (forall y, In y s -> y < length base)
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-> map (nth_error base) s = map (nth_error p) (map g s)).
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intro s. induction s; intro L. reflexivity.
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assert (M: a < length base). apply L. apply in_eq.
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simpl. rewrite IHs.
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rewrite nth_error_nth' with (d := x).
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rewrite nth_error_nth' with (d := x).
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replace a with (f (g a)) at 1.
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rewrite <- H5. reflexivity.
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apply H4. assumption.
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apply H6 in M. destruct M. assumption.
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rewrite H. apply H4. assumption. assumption.
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intro y. intro N. apply L. apply in_cons. assumption.
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apply H7. intro y. intro N.
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assert (forall s z, In z s -> nth_error base z <> None -> z < length base).
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intros s z. intros M1 M2. apply nth_error_Some. assumption.
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apply H8 with (s := l). assumption.
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assert (forall s (t: list X),
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map Some t = map (nth_error base) s
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-> In y s -> nth_error base y <> None).
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intro s. induction s; intros t; intros M1 M2.
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apply in_nil in M2. contradiction.
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apply in_inv in M2. destruct M2. rewrite H9 in M1.
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destruct t. inversion M1. inversion M1. easy.
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destruct t. inversion M1. inversion M1.
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apply IHs with (t := t). assumption. assumption.
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generalize N. generalize H1. apply H9.
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(* second case in split *)
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rewrite H2.
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apply FinFun.bInjective_bSurjective in H3.
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apply FinFun.bSurjective_bBijective in H3. destruct H3 as [g].
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destruct H3.
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assert (forall s, (forall y, In y s -> y < length base)
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-> map (nth_error base) s = map (nth_error p) (map f s)).
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intro s. induction s; intro K. reflexivity.
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simpl. rewrite IHs.
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rewrite map_map.
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rewrite nth_error_nth' with (d := x).
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rewrite nth_error_nth' with (d := x).
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(*
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assert (a < length base). apply K. apply in_eq.
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assert (N := H6). apply H5 in N. destruct N.
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*)
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replace a with (f (g a)) at 1. rewrite H4.
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assert (exists b, b < length base /\ a = g b).
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exists (f a).
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rewrite <- H4. reflexivity. apply K.
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apply in_eq. apply H2. apply K.
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apply in_eq. rewrite H. apply K.
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apply in_eq. intro y. intro L. apply K. apply in_cons. assumption.
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apply H5. intro y. intro L.
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*)
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