module _ where
open import bool
open import eq
open import nat
open import nat-thms


{-

0. set up agda.

  Visit the installation section of the agda website:

    https://agda.readthedocs.io/en/latest/getting-started/installation.html

  Also, get a copy of the text book:

    https://dl.acm.org/citation.cfm?id=2841316

  The book points to the IAL agda library but it points to an old
  version. You want the latest, which you can get here:

    https://github.com/cedille/ial

  If that causes you trouble, there is a virtual machine that has
  everything set up that's linked from the course webpage.

  To solve the rest of these exercises, I recommend you create a file
  in the same directory as the IAL library you've checked out (icky, I
  know). I will grade your homework by putting it there to try it out.
  It is also possible to use a `.agda-lib` file but don't worry if
  you don't get that sorted out.

Keystroke quick reference (but see the text!)

c-c c-l: load the buffer & typecheck
c-c c-,: tell me what's in the hole
c-c c-.: tell me the type of the thing
         that's in the whole (the "Have")
c-c c-c: case split on the variable
         that's in the hole
c-c c-a: do an easy proof
c-c c-r: replace the hole with its
         current contents
         (ie finish this case/proof)

meta-. : go to the definition of whatever
         thing the cursor is on top of;
         can also be typed with two
         keystrokes: esc and period

-}


{- 1. Do exercises 1.1 and 1.2 (page 14) in the text.
   Just write what they evaluate to here. You don't need
   to open bool.agda as this file already imports it, but
   you're welcome to. NB: opening bool.agda will be useful
   to understand `imp` in the optional part of problem 3.
   -}

{- 2. Do exercise 2.3 (page 41) in the text -}

{- 3. prove the following identities: -}


curry : ∀ (b1 b2 b3 : 𝔹) -> b1 imp (b2 imp b3) ≡ (b1 && b2) imp b3
curry b1 b2 b3 = {!!}

demorgan : ∀ b1 b2 -> ~ (b1 && b2) ≡ ~ b1 || ~ b2
demorgan b1 b2 = {!!}

imp-not-or : ∀ b1 b2 -> ~ b1 || b2 ≡ b1 imp b2
imp-not-or b1 b2 = {!!}


{-

4. We did right distributive in class; state and prove that
   multiplication also distributes on the left, eg, for any three
   numbers, x, y, and z:

  z * (x + y) ≡ z * x + z * y

-}

{-

5. Prove commutivity of multiplication. Note that the proof is in the
   IAL; feel free to look, but give it a try on your own first. The
   idea of the proof is similar to that of commutivity of
   addition. You case split and then, in each case need a helper
   lemma. One that says what happens when you multiply by zero on the
   right and one that says how you call "peel off one succ", roughly
   speaking.

-}