# Add fixity for (==) and (/=)

David Feuer david.feuer at gmail.com
Mon Sep 17 09:22:09 UTC 2018

```Looks good to me! Do you have an opinion about infixl vs infixr?

On Mon, Sep 17, 2018, 5:21 AM Dannyu NDos <ndospark320 at gmail.com> wrote:

>
>
> ---------- Forwarded message ---------
> From: Dannyu NDos <ndospark320 at gmail.com>
> Date: 2018년 9월 17일 (월) 오후 6:18
> Subject: Re: Add fixity for (==) and (/=)
> To: <david.feuer at gmail.com>
>
>
> Proof by truth table (F is False, T is True):
> p q r (p == q) (q == r) ((p == q) == r) (p == (q == r))
> F F F        T        T               F               F
> F F T        T        F               T               T
> F T F        F        F               T               T
> F T T        F        T               F               F
> T F F        F        T               T               T
> T F T        F        F               F               F
> T T F        T        F               F               F
> T T T        T        T               T               T
> That proves associativity of (==).
>
> Also note that p /= q ≡ not p == q. Proof:
> p q (p /= q) (not p) (not p == q)
> F F        F       T            F
> F T        T       T            T
> T F        T       F            T
> T T        F       F            F
> And by symmetry of (/=), p /= q ≡ p == not q.
>
> Then:
> (p /= q) /= r ≡ (not p == q) == not r ≡ not p == (q == not r) ≡ p /= (q /=
> r).
> Hence (/=) is associative.
>
> Q.E.D.
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