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import { addition, multiplication } from '../natural-numbers';
import { Add, Multiply, _0 } from '../utils/nat';
import { ApplyRewrite, Sym, Eq, VerifyEquation, Equation, assert, Evaluate } from '../utils/theorem';
export namespace spec_addition {
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'>, Sym<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 namespace multiplication_spec {
export type specCommutativity = [
assert<Eq<
ApplyRewrite<Multiply<Multiply<'A', 'B'>, 'C'>, multiplication.Commutativity<'A', 'B'>>,
Multiply<Multiply<'B', 'A'>, 'C'>
>>,
]
export type specDistributivity = [
assert<Eq<
Evaluate<multiplication.Distributivity<'A', 'B', 'C'>>,
Add<Multiply<'A', 'C'>, Multiply<'B', 'C'>>
>>,
]
export type specAssociativity = [
assert<Eq<
Evaluate<multiplication.Associativity<'A', 'B', 'C'>>,
Multiply<'A', Multiply<'B', 'C'>>
>>,
]
}
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