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feat(library/data/list/basic): add 'firstn' definition and theorems

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Leonardo de Moura 2015-07-19 20:15:40 -07:00
parent d2f64d7744
commit 6936d71030
1 changed files with 45 additions and 0 deletions
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library/data/list/basic.lean
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@ -592,6 +592,51 @@ theorem sub_of_mem_of_sub_of_qeq {a : A} {l : list A} {u v : list A} : a ∉ l
(suppose x = a, by substvars; contradiction)
(suppose x ∈ u, this)
end qeq
section firstn
variable {A : Type}
definition firstn : nat → list A → list A
| 0 l := []
| (n+1) [] := []
| (n+1) (a::l) := a :: firstn n l
lemma firstn_zero : ∀ (l : list A), firstn 0 l = [] :=
by intros; reflexivity
lemma firstn_nil : ∀ n, firstn n [] = ([] : list A)
| 0 := rfl
| (n+1) := rfl
lemma firstn_cons : ∀ n (a : A) (l : list A), firstn (succ n) (a::l) = a :: firstn n l :=
by intros; reflexivity
lemma firstn_all : ∀ (l : list A), firstn (length l) l = l
| [] := rfl
| (a::l) := begin unfold [length, firstn], rewrite firstn_all end
lemma firstn_all_of_ge : ∀ {n} {l : list A}, n ≥ length l → firstn n l = l
| 0 [] h := rfl
| 0 (a::l) h := absurd h (not_le_of_gt !succ_pos)
| (n+1) [] h := rfl
| (n+1) (a::l) h := begin unfold firstn, rewrite [firstn_all_of_ge (le_of_succ_le_succ h)] end
lemma firstn_firstn : ∀ (n m) (l : list A), firstn n (firstn m l) = firstn (min n m) l
| n 0 l := by rewrite [min_zero, firstn_zero, firstn_nil]
| 0 m l := by rewrite [zero_min]
| (succ n) (succ m) nil := by rewrite [*firstn_nil]
| (succ n) (succ m) (a::l) := by rewrite [*firstn_cons, firstn_firstn, min_succ_succ]
lemma length_firstn_le : ∀ (n) (l : list A), length (firstn n l) ≤ n
| 0 l := by rewrite [firstn_zero]
| (succ n) (a::l) := by rewrite [firstn_cons, length_cons, add_one]; apply succ_le_succ; apply length_firstn_le
| (succ n) [] := by rewrite [firstn_nil, length_nil]; apply zero_le
lemma length_firstn_eq : ∀ (n) (l : list A), length (firstn n l) = min n (length l)
| 0 l := by rewrite [firstn_zero, zero_min]
| (succ n) (a::l) := by rewrite [firstn_cons, *length_cons, *add_one, min_succ_succ, length_firstn_eq]
| (succ n) [] := by rewrite [firstn_nil]
end firstn
end list
attribute list.has_decidable_eq [instance]