Prove `hooo::tauto_from_para_transitivity` (challenge)

[Sven Nilsen]

I will give 200 USD to the first person that can prove (⊥^(a == b) ∧ ⊥^(b == c)) => (a == c)^⊤.

It is currently believed that proving the contradiction is impossible, but if somebody does it, then the reward will be equal.

The proof, if found, will be added to hooo::tauto_from_para_transitivity.

Currently, there are only 4 axioms for HOOO Exponential Propositions, see #502. Finding this proof will reduce to a set of 3 axioms.

[Pâris DOUADY]

Could be fun to integrate exponential types to https://incredible.pm/ and try to make the visual proof

[Sven Nilsen]

I found it:

/// `(false^(a == b) ∧ false^(b == c)) => (a == c)^true`.
pub fn tauto_from_para_transitivity<A: Prop, B: Prop, C: Prop>(
    para_eq_ab: Para<Eq<A, B>>,
    para_eq_bc: Para<Eq<B, C>>,
) -> Tauto<Eq<A, C>> {
    let x: Not<Eq<Para<A>, Para<B>>> = hooo_dual_eq(para_eq_ab);
    let y: Not<Eq<Para<B>, Para<C>>> = hooo_dual_eq(para_eq_bc);
    match (para_decide::<A>(), para_decide::<B>(), para_decide::<C>()) {
        (Left(para_a), Left(para_b), _) => imply::absurd()(x((para_b.map_any(), para_a.map_any()))),
        (_, Left(para_b), Left(para_c)) => imply::absurd()(y((para_c.map_any(), para_b.map_any()))),
        (Left(para_a), _, Left(para_c)) => {
            let z: Eq<Tauto<A>, Tauto<C>> = (
                Rc::new(move |tauto_a| imply::absurd()(para_a(tauto_a(True)))),
                Rc::new(move |tauto_c| imply::absurd()(para_c(tauto_c(True))))
            );
            hooo_rev_eq(z)
        }
        (_, Right(npara_b), Right(npara_c)) => {
            imply::absurd()(y(eq_not_para_to_eq_para((npara_c.map_any(), npara_b.map_any()))))
        }
        (Right(npara_a), Right(npara_b), _) => {
            imply::absurd()(x(eq_not_para_to_eq_para((npara_b.map_any(), npara_a.map_any()))))
        }
        (Right(npara_a), _, Right(npara_c)) => {
            let y: Eq<Para<A>, Para<C>> = eq_not_para_to_eq_para((npara_c.map_any(), npara_a.map_any()));
            let y: Para<Not<Eq<A, C>>> = hooo_dual_rev_neq(y);
            match program::<Eq<A, C>>() {
                Left(Left(tauto_eq_ac)) => tauto_eq_ac,
                Left(Right(para_eq_ac)) => imply::absurd()(pow_rev_not(y)(para_eq_ac)),
                Right(para_uni_eq_ac) => {
                    let nuni_eq_ac = Rc::new(move |x| para_uni_eq_ac(x));
                    let (x, _): And<Not<Tauto<Eq<A, C>>>, Not<Para<Eq<A, C>>>> = and::from_de_morgan(nuni_eq_ac);
                    let x: Tauto<Not<Eq<A, C>>> = hooo_rev_not(x);
                    imply::absurd()(y(x(True)))
                }
            }
        }
    }
}