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import { addition } from './index';
import { Add } from './nat';
import { ApplyRewrite, ChainRewrites, Flip, Eq, VerifyEquation } from './util';
type assert<T extends true> = T;
export type specIdentity = [
assert<Eq<
ApplyRewrite<Add<'A', Add<'B', '0'>>, addition.Identity<'B'>>,
Add<'A', 'B'>
>>,
assert<Eq<
ApplyRewrite<Add<Add<'A', 'B'>, '0'>, addition.Identity<Add<'A', 'B'>>>,
Add<'A', 'B'>
>>,
assert<Eq<
ApplyRewrite<Add<'A', 'B'>, Flip<addition.Identity<Add<'A', 'B'>>>>,
Add<Add<'A', 'B'>, '0'>
>>,
assert<Eq<
ApplyRewrite<Add<'A', Add<'B', Add<'C', '0'>>>, addition.Identity<'C'>>,
Add<'A', Add<'B', 'C'>>
>>,
]
export type specCommutativity = [
assert<Eq<
ApplyRewrite<Add<'A', 'B'>, addition.Commutativity<'A', 'B'>>,
Add<'B', 'A'>
>>,
assert<Eq<
ApplyRewrite<Add<Add<'A', 'B'>, 'C'>, addition.Commutativity<'A', 'B'>>,
Add<Add<'B', 'A'>, 'C'>
>>,
assert<Eq<
ApplyRewrite<Add<'A', Add<'B', 'C'>>, addition.Commutativity<'B', 'C'>>,
Add<'A', Add<'C', 'B'>>
>>,
]
export type specCommutativityInduction = [
assert<VerifyEquation<addition.Comm_Base<'B'>>>,
assert<VerifyEquation<addition.Comm_Inductive<'A', 'B'>>>,
]
export type specChainRewrites = [
assert<Eq<
ChainRewrites<
[
addition.Commutativity<'0', 'B'>,
addition.Identity<'B'>,
],
Add<Add<'0', 'B'>, 'C'>
>,
Add<'B', 'C'>
>>,
assert<Eq<
ChainRewrites<
[
addition.Commutativity<'0', 'B'>,
addition.Identity<'B'>,
Flip<addition.Identity<Add<'B', 'C'>>>,
addition.Commutativity<Add<'B', 'C'>, '0'>,
],
Add<Add<'0', 'B'>, 'C'>
>,
Add<'0', Add<'B', 'C'>>
>>,
]
export type specAssociativityInduction = [
assert<VerifyEquation<addition.Assoc_Base<'B', 'C'>>>,
assert<VerifyEquation<addition.Assoc_Inductive<'A', 'B', 'C'>>>,
]
|
