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This commit is contained in:
Thomas Baruchel 2023-12-03 22:54:57 +01:00
parent 4961ba9f19
commit 1e27de7157
1 changed files with 70 additions and 1 deletions
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71
src/mapping.v
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@ -308,7 +308,7 @@ Proof.
assert (forall (b: list X),
map (fun e : nat => nth e b x) (seq 0 (length b)) = b).
intro b. induction b. reflexivity. (* rewrite <- IHb at 2. *)
intro b. induction b. reflexivity.
replace (seq 0 (length (a :: b))) with (0:: map S (seq 0 (length b))).
rewrite map_cons. rewrite map_map.
@ -360,8 +360,77 @@ Proof.
intro s. rewrite map_map. induction s; intro K. reflexivity.
simpl. rewrite IHs.
rewrite nth_error_nth' with (d := x).
replace (nth_error (map (fun e : nat => nth (g e) base x)
(seq 0 (length base))) (f a))
with (Some ((fun e => nth (g e) base x)
(nth (f a) (seq 0 (length base)) 0))).
assert (forall m n f' g',
FinFun.bFun m f' -> FinFun.bFun m g'
-> (forall x : nat, x < m -> g' (f' x) = x /\ f' (g' x) = x)
-> n < m -> g' (nth (f' n) (seq 0 m) 0) = n).
intros m n f' g'. intros J J1 J2 J3.
assert (forall q p, p < q -> nth p (seq 0 q) 0 = p).
intro q. induction q; intro p'; intro J4.
apply Nat.nlt_0_r in J4. contradiction.
rewrite Nat.lt_succ_r in J4. rewrite Nat.le_lteq in J4. destruct J4.
rewrite seq_S. rewrite app_nth1. apply IHq. assumption.
rewrite seq_length. assumption. rewrite H6. rewrite seq_nth.
reflexivity. apply Nat.lt_succ_diag_r. rewrite H6.
apply J2 in J3. destruct J3. apply H7. apply J. assumption.
rewrite H6. reflexivity. assumption. assumption. assumption.
apply K. apply in_eq.
rewrite nth_error_nth' with (d := x).
replace (nth (f a) (ma
rewrite <- map_nth at 1.
assert (forall p q, p < q -> nth p (seq 0 q) 0 = p).
intro p'. induction p'; intro q; intro J4.
destruct q. apply Nat.nlt_0_r in J4. contradiction.
reflexivity.
intro m. induction m; intros n f' g'; intros J J1 J2 J3.
apply Nat.nlt_0_r in J3. contradiction. simpl in J3.
rewrite Nat.lt_succ_r in J3. rewrite Nat.le_lteq in J3. destruct J3.
rewrite seq_S. rewrite app_nth1. apply IHm. assumption.
rewrite seq_length. apply H2.
replace (seq 0 (length (a0::b))) with (0 :: (seq 1 (length b))).
rewrite cons_seq.
prouver avec map_nth
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).