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import { Add, Assoc, Assoc0, Assoc1, Commutativity, Identity, Succ } from './index';
import { ApplyRewrite, ChainRewrites, Flip, Rewrite } from './util';
type Eq<a, b> = ([a] extends [b] ? ([b] extends [a] ? true : false & { a: a; b: b }) : false & { a: a; b: b });
type assert<T extends true> = T;
export type specIdentity = [
assert<Eq<
ApplyRewrite<Add<'A', Add<'B', '0'>>, Identity<'B'>>,
Add<'A', 'B'>
>>,
assert<Eq<
ApplyRewrite<Add<Add<'A', 'B'>, '0'>, Identity<Add<'A', 'B'>>>,
Add<'A', 'B'>
>>,
assert<Eq<
ApplyRewrite<Add<'A', 'B'>, Flip<Identity<Add<'A', 'B'>>>>,
Add<Add<'A', 'B'>, '0'>
>>,
assert<Eq<
ApplyRewrite<Add<'A', Add<'B', Add<'C', '0'>>>, Identity<'C'>>,
Add<'A', Add<'B', 'C'>>
>>,
]
export type specCommutativity = [
assert<Eq<
ApplyRewrite<Add<'A', 'B'>, Commutativity<'A', 'B'>>,
Add<'B', 'A'>
>>,
assert<Eq<
ApplyRewrite<Add<Add<'A', 'B'>, 'C'>, Commutativity<'A', 'B'>>,
Add<Add<'B', 'A'>, 'C'>
>>,
assert<Eq<
ApplyRewrite<Add<'A', Add<'B', 'C'>>, Commutativity<'B', 'C'>>,
Add<'A', Add<'C', 'B'>>
>>,
]
export type specChainRewrites = [
assert<Eq<
ChainRewrites<
[
Commutativity<'0', 'B'>,
Identity<'B'>,
],
Add<Add<'0', 'B'>, 'C'>
>,
Add<'B', 'C'>
>>,
assert<Eq<
ChainRewrites<
[
Commutativity<'0', 'B'>,
Identity<'B'>,
Flip<Identity<Add<'B', 'C'>>>,
Commutativity<Add<'B', 'C'>, '0'>,
],
Add<Add<'0', 'B'>, 'C'>
>,
Add<'0', Add<'B', 'C'>>
>>,
]
export type specAssoc0 = [
assert<Eq<
ApplyRewrite<Add<Add<'0', 'B'>, 'C'>, Assoc0<'B', 'C'>>,
Add<'0', Add<'B', 'C'>>
>>,
assert<Eq<
ApplyRewrite<Add<Add<'0', Add<'A', 'B'>>, 'C'>, Assoc0<Add<'A', 'B'>, 'C'>>,
Add<'0', Add<Add<'A', 'B'>, 'C'>>
>>,
]
export type specAssocN = [
assert<Eq<
ApplyRewrite<Add<Add<'A', 'B'>, 'C'>, Assoc<'A', 'B', 'C'>>,
Add<'A', Add<'B', 'C'>>
>>,
]
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