module _ where

open import list
open import eq
open import bool

{-

Prove that if there is a `ff` in a list, i.e., if you know this:

   list-member _iff_ ff l ≡ tt

for an arbitrary (l : 𝕃 𝔹), then list-and of that list is ff, i.e.:

   list-and l ≡ ff

The definition of list-member is in list.agda.

-}

falsemeansfalse : ∀ (l : 𝕃 𝔹) -> list-member _iff_ ff l ≡ tt  -> list-and l ≡ ff
falsemeansfalse = {!!}

{-

Prove a similar result for list-or, namely that if there is a tt in
the input list, then list-or returns tt.

-}

truemeanstrue : ∀ (l : 𝕃 𝔹) -> list-member _iff_ tt l ≡ tt  -> list-or l ≡ tt
truemeanstrue = {!!}

{- prove that reversing a list twice returns the same list, i.e.

rev2x : ∀{ℓ}{A : Set ℓ} → (l : 𝕃 A) -> reverse (reverse l) ≡ l

HINT: prove two lemmas, one that says what happens when
reverse-helper's first argument end with the list element `x` and one
that says what happens when the second argument ends with the list
element `x`. Here is the statement for the first one of those: 

rh-firstarg : ∀{ℓ}{A : Set ℓ} → (x : A) -> (l m : 𝕃 A) -> 
  reverse-helper (l ++ (x :: [])) m ≡ (reverse-helper l m) ++ (x :: [])

These will be useful for finishing the inductive case in rev2x.

-}

rev2x : ∀{ℓ}{A : Set ℓ} → (l : 𝕃 A) -> reverse (reverse l) ≡ l
rev2x l = {!!}