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import { Op, Rewrite, ChainRewrites, Sym, Equation, VerifyEquation, assert, SubstEq } from './utils/theorem';
export namespace equation {
type F<X extends Op> = { op: 'F', a: X };
export type Reflexivity<A extends Op> = Rewrite<Equation<A, A>, 'true'>;
export type Commutativity<A extends Op, B extends Op> = Rewrite<Equation<A, B>, Equation<B, A>>;
export type Congruence<A extends Op, B extends Op> = Rewrite<Equation<A, B>, Equation<F<A>, F<B>>>;
export namespace proof {
export interface SubstituteEq_Proof<A extends Op, B extends Op> {
type: 'rewrite',
known: {
'a = b': Equation<A, B>,
},
left: Equation<F<A>, F<B>>; // f(a) = f(b)
right: ChainRewrites<[
Sym<Congruence<A, B>>, // a = b
SubstEq<this['known']['a = b']>, // b = b
Reflexivity<B>,
], this['left']>;
};
export type substitution = [
assert<VerifyEquation<SubstituteEq_Proof<'A', 'B'>>>,
]
}
export namespace proof {
export interface Transitivity_Proof<A extends Op, B extends Op, C extends Op> {
type: 'rewrite',
known: {
'a = b': Equation<A, B>,
'b = c': Equation<B, C>,
},
left: Equation<A, C>; // a = c
right: ChainRewrites<[
SubstEq<this['known']['a = b']>, // b = c
Rewrite<this['known']['b = c'], 'true'>, // true
], this['left']>;
};
export type transitivity = [
assert<VerifyEquation<Transitivity_Proof<'A', 'B', 'C'>>>,
]
}
}
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