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import { Op, Rewrite, ChainRewrites, Flip } from './util';
export type Add<A extends Op, B extends Op> = { op: '+'; a: A; b: B };
export type Succ<A extends Op> = Add<'1', A>;
export type Identity<A extends Op> = Rewrite<Add<A, '0'>, A>;
export type Commutativity<A extends Op, B extends Op> = Rewrite<Add<A, B>, Add<B, A>>;
export interface Assoc0<A extends Op, B extends Op> {
type: 'rewrite',
left: Add<Add<'0', A>, B>; // (0 + a) + b
right: ChainRewrites<[
Commutativity<'0', A>, // = (a + 0) + b
Identity<A>, // = a + b
Flip<Identity<Add<A, B>>>, // = (a + b) + 0
Commutativity<Add<A, B>, '0'>, // = 0 + (a + b)
], this['left']>;
};
export type Assoc1<A extends Op, B extends Op> = Rewrite<Add<Succ<A>, B>, Succ<Add<A, B>>>;
export interface Assoc<A extends Op, B extends Op, C extends Op> {
type: 'rewrite',
left: Add<Add<A, B>, C>; // (a + b) + c
right: ChainRewrites<[
Rewrite<A, Succ<A>>, // = CHEATING
Assoc1<A, B>, // = succ(a + b) + c
Assoc1<Add<A, B>, C>, // = succ(a + b + c)
Rewrite<A, '0'>, // CHEATING. TODO: figure out recursion
Assoc0<B, C>, // Induction base case
Rewrite<'0', A>, // CHEATING
Flip<Assoc1<A, Add<B, C>>>, // = succ(a) + (b + c)
Rewrite<Succ<A>, A>, // = CHEATING
], this['left']>;
};
// type _x = ApplyRewrite<Add<Add<'A', 'B'>, 'C'>, AssocN<'A', 'B', 'C'>>
/*
Assoc: (a + b) + c = a + (b + c)
For a = 0,
(0 + b) + c
= (b + 0) + c -- comm
= b + c -- id
= (b + c) + 0 -- flip id
= 0 + (b + c) -- comm
For a' = succ(a),
(succ(a) + b) + c
= succ(a + b) + c
= succ((a + b) + c)
----- rec
= succ(a + (b + c))
= succ(a) + (b + c)
*/
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