lean2/tests/lean/run/unfold_test.lean

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import data.list data.examples.vector
variables {A B : Type}
section
open list
theorem last_concat {x : A} : ∀ {l : list A} (h : concat x l ≠ []), last (concat x l) h = x
| [] h := rfl
| [a] h := rfl
| (a₁::a₂::l) h :=
by xrewrite [↑concat, ↑concat, last_cons_cons, ↓concat x (a₂::l), last_concat]
theorem reverse_append : ∀ (s t : list A), reverse (s ++ t) = (reverse t) ++ (reverse s)
| [] t2 :=
by esimp [append, reverse]; rewrite append_nil_right
| (a2 :: s2) t2 :=
by rewrite [↑append, ↑reverse, reverse_append, concat_eq_append, append.assoc, -concat_eq_append]
end
section
open vector nat prod
theorem unzip_zip : ∀ {n : nat} (v₁ : vector A n) (v₂ : vector B n), unzip (zip v₁ v₂) = (v₁, v₂)
| 0 [] [] := rfl
| (n+1) (a::va) (b::vb) := by rewrite [↑zip, ↑unzip, unzip_zip]
theorem zip_unzip : ∀ {n : nat} (v : vector (A × B) n), zip (pr₁ (unzip v)) (pr₂ (unzip v)) = v
| 0 [] := rfl
| (n+1) ((a, b) :: v) := by rewrite [↑unzip,↑zip,zip_unzip]
theorem reverse_concat : Π {n : nat} (xs : vector A n) (a : A), reverse (concat xs a) = a :: reverse xs
| 0 [] a := rfl
| (n+1) (x :: xs) a := by xrewrite [↑concat,↑reverse,reverse_concat]
theorem reverse_reverse : Π {n : nat} (xs : vector A n), reverse (reverse xs) = xs
| 0 [] := rfl
| (succ n) (x :: xs) := by rewrite [↑reverse at {1}, reverse_concat, reverse_reverse]
end