Update
This commit is contained in:
Thomas Baruchel
parent
bf9517f266
commit
55bb6b865b
495
src/mapping.v
495
src/mapping.v
@ -69,6 +69,142 @@ Proof.
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Qed.
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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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@ -270,362 +406,3 @@ Proof.
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*)
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*)
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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) (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 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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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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rewrite H6. apply J2 in J3. destruct J3. apply H7. apply J. assumption.
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rewrite H6. reflexivity. assumption. assumption. assumption.
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apply K. apply in_eq.
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rewrite nth_error_nth' with (d := x).
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(* aboutit à
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Some (nth (g (nth (f a) (seq 0 (length base)) 0)) base x) =
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Some
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(nth (f a) (map (fun e : nat => nth (g e) base x) (seq 0 (length base))) x)
|
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*)
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replace (nth (f a) (ma
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rewrite <- map_nth at 1.
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||||||
assert (forall p q, p < q -> nth p (seq 0 q) 0 = p).
|
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||||||
intro p'. induction p'; intro q; intro J4.
|
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||||||
destruct q. apply Nat.nlt_0_r in J4. contradiction.
|
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||||||
reflexivity.
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||||||
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||||||
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||||||
intro m. induction m; intros n f' g'; intros J J1 J2 J3.
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||||||
apply Nat.nlt_0_r in J3. contradiction. simpl in J3.
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||||||
rewrite Nat.lt_succ_r in J3. rewrite Nat.le_lteq in J3. destruct J3.
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||||||
rewrite seq_S. rewrite app_nth1. apply IHm. assumption.
|
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||||||
rewrite seq_length. apply H2.
|
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||||||
|
|
||||||
replace (seq 0 (length (a0::b))) with (0 :: (seq 1 (length b))).
|
|
||||||
rewrite cons_seq.
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||||||
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||||||
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|
||||||
prouver avec map_nth
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
|
|
||||||
rewrite nth_error_nth' with (d := nth (g 0) base x).
|
|
||||||
rewrite nth_error_nth' with (d := nth (g 0) base x).
|
|
||||||
r
|
|
||||||
replace (nth (g 0) base x) with ((fun e => nth (g e) base x) 0).
|
|
||||||
rewrite nth_error_nth' with (d := (fun e => nth (g e) base x) 0).
|
|
||||||
rewrite map_nth.
|
|
||||||
rewrite nth_error_nth' with (d := x).
|
|
||||||
|
|
||||||
rewrite nth_error_nth' with (d := (fun e => nth (g e) base x) 0).
|
|
||||||
rewrite nth_error_nth' with (d := x).
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
rewrite nth_error_nth' with (d := x).
|
|
||||||
rewrite nth_error_nth' with (d := x).
|
|
||||||
rewrite <- H4. reflexivity. apply K.
|
|
||||||
apply in_eq. apply H2. apply K.
|
|
||||||
apply in_eq. rewrite H. apply K.
|
|
||||||
apply in_eq. intro y. intro L. apply K. apply in_cons. assumption.
|
|
||||||
apply H6. intro y. intro L.
|
|
||||||
|
|
||||||
assert (forall s z, In z s -> nth_error base z <> None -> z < length base).
|
|
||||||
intros s z. intros M1 M2. apply nth_error_Some. assumption.
|
|
||||||
apply H7 with (s := l). assumption.
|
|
||||||
|
|
||||||
assert (forall s (t: list X),
|
|
||||||
map Some t = map (nth_error base) s
|
|
||||||
-> In y s -> nth_error base y <> None).
|
|
||||||
intro s. induction s; intros t; intros M1 M2.
|
|
||||||
apply in_nil in M2. contradiction.
|
|
||||||
apply in_inv in M2. destruct M2. rewrite H8 in M1.
|
|
||||||
destruct t. inversion M1. inversion M1. easy.
|
|
||||||
destruct t. inversion M1. inversion M1.
|
|
||||||
apply IHs with (t := t). assumption. assumption.
|
|
||||||
generalize L. generalize H0. apply H8.
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
assert (forall n, n < length base ->
|
|
||||||
nth_error base n = nth_error (map (fun e => nth (g e) base x) (seq 0 (length base))) (f n)).
|
|
||||||
intro n. induction n; intro M.
|
|
||||||
destruct base. destruct (f 0); reflexivity.
|
|
||||||
|
|
||||||
rewrite nth_error_nth' with (d := x).
|
|
||||||
rewrite nth_error_nth' with (d := x).
|
|
||||||
simpl.
|
|
||||||
|
|
||||||
assert (forall b,
|
|
||||||
nth_error (map (fun e => nth (g e) b x) (seq 0 (length b))) (f 0)
|
|
||||||
= nth_error b 0).
|
|
||||||
intro b. induction b. destruct (f 0); reflexivity.
|
|
||||||
replace (seq 0 (length (a0::b))) with ((seq 0 (length b)) ++ [ length b ]).
|
|
||||||
rewrite map_app.
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
induction a. destruct base. destruct (f 0); reflexivity.
|
|
||||||
destruct base. simpl.
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
intro s. induction s; intro K. reflexivity.
|
|
||||||
simpl. rewrite IHs.
|
|
||||||
|
|
||||||
rewrite map_map.
|
|
||||||
rewrite nth_error_nth' with (d := x).
|
|
||||||
rewrite nth_error_nth' with (d := x).
|
|
||||||
|
|
||||||
(*
|
|
||||||
assert (a < length base). apply K. apply in_eq.
|
|
||||||
assert (N := H6). apply H5 in N. destruct N.
|
|
||||||
*)
|
|
||||||
assert (a < length base). apply K. apply in_eq.
|
|
||||||
replace a with (f (g a)) at 1. rewrite H4.
|
|
||||||
|
|
||||||
|
|
||||||
assert (exists b, b < length base /\ a = g b).
|
|
||||||
exists (f a). split. apply H2.
|
|
||||||
assumption. apply H5 in H6. destruct H6. rewrite H6. reflexivity.
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
base = [0,1,2,3]
|
|
||||||
|
|
||||||
u = [0,1,2,3,0,1,2,3] --> l = [0,1,2,3,0,1,2,3]
|
|
||||||
p = [2,3,1,0]
|
|
||||||
v = [2,3,1,0,2,3,1,0]
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
intro u. induction u; intros v base; intro H;
|
|
||||||
destruct H; destruct H; destruct H; destruct H0.
|
|
||||||
symmetry in H0. apply map_eq_nil in H0. rewrite H0 in H1.
|
|
||||||
apply map_eq_nil in H1. rewrite H1. apply permutation_mapping_self. easy.
|
|
||||||
rewrite Permutation_nth in H. destruct H. destruct H2 as [f].
|
|
||||||
destruct H2. destruct H3.
|
|
||||||
apply FinFun.bInjective_bSurjective in H3.
|
|
||||||
apply FinFun.bSurjective_bBijective in H3. destruct H3 as [g].
|
|
||||||
destruct H3.
|
|
||||||
destruct v.
|
|
||||||
symmetry in H1. apply map_eq_nil in H1. rewrite H1 in H0.
|
|
||||||
apply map_eq_nil in H0. symmetry in H0. apply nil_cons in H0.
|
|
||||||
contradiction.
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user