prop 0.42.0

//! # Avatar Modal Logic
//!
//! This avatar modal logic builds upon the `hooo` module using Exponential Propositions.
//!
//! What makes Avatar Modal Logic different from normal Modal Logic,
//! is that there is a difference between "safe" and "unsafe" proofs:
//!
//! - `false^(false^p) => ◇p` is safe (classical in normal Modal Logic)
//! - `◇p => false^(false^p)` is unsafe (constructive in normal Modal Logic)
//!
//! The idea is to reverse decidability, but this reversed decidability should not be mixed
//! with the normal distinction between classical and constructive proofs.
//! Instead, one uses safe vs unsafe proofs.
//!
//! ### 1-Avatar and Unsafe code
//!
//! Without distinction of safe vs unsafe,
//! the usual semantics of "necessary" and "possibly" is collapsed,
//! where is it possible to prove `¬◇p == ◇¬p` and `¬□p == □¬p`.
//!
//! According [Avatar Semantics](https://advancedresearch.github.io/avatar-extensions/summary.html#avatar-semantics),
//! this is not a problem, because one can reconstruct hypercube topologies in Avatar Graphs
//! from the smallest Möbius topologies possible, by identifying the diagonals using
//! highest N-avatars. This is done by introducing a 1-avatar that covers products in
//! Avatar Algebra.
//!
//! This 1-avatar is a new-type `Pos` (possibly) that protects the content from being readable.
//! The content is accessed under special circumstances where integration of information
//! is checked by a higher N-avatar. This means, safe invariants using unsafe code.
//!
//! By constructing the semantics of "necessary" and "possibly" on top of this 1-avatar,
//! one can still prove theorems using HOOO Exponential Propositions safely.
//!
//! Safe proofs are preferrable, but one can use unsafe code in edge cases by being careful.

use crate::*;
use hooo::*;

pub use protection::*;

mod protection {
    use super::*;

    /// `◇a`.
    #[derive(Clone)]
    pub struct Pos<A>(Para<Para<A>>);
    /// `□a`.
    pub type Nec<A> = Not<Pos<Not<A>>>;

    impl<A> Pos<A> {
        /// Create a new possibly proposition.
        pub fn new(x: Para<Para<A>>) -> Self {Pos(x)}
    }

    /// `◇a => false^(false^a)`.
    ///
    /// # Safety
    ///
    /// This function is declared unsafe because it uses the "taboo" knowledge in Avatar Modal Logic.
    pub unsafe fn pos_to_para_para<A: Prop>(Pos(x): Pos<A>) -> Para<Para<A>> {x}
}

impl<A: DProp> Decidable for Pos<A> {
    fn decide() -> ExcM<Pos<A>> {
        match para_decide() {
            Left(x) => Left(Pos::new(x)),
            Right(y) => Right(imply::in_left(y, |x| unsafe {pos_to_para_para(x)})),
        }
    }
}

/// `□a => a^true`.
pub fn nec_to_tauto<A: DProp>(nec_a: Nec<A>) -> Tauto<A> {
    match tauto_decide() {
        Left(tauto_a) => tauto_a,
        Right(ntauto_a) => {
            let para_a = tauto_not_to_para(hooo_rev_not(ntauto_a));
            let x: Para<Not<A>> = npos_to_para(nec_a);
            imply::absurd()(para_rev_not(x)(para_a))
        }
    }
}

/// `a^true => □a`.
///
/// # Safety
///
/// This function is declared unsafe because it uses the "taboo" knowledge in Avatar Modal Logic.
pub unsafe fn tauto_to_nec<A: DProp>(tauto_a: Tauto<A>) -> Nec<A> {
    Rc::new(move |pos_na| {
        match para_decide() {
            Left(para_na) => pos_to_para_para(pos_na)(para_na),
            Right(npara_na) => {
                let para_a = para_not_rev_double(pow_not(npara_na));
                para_a(tauto_a(True))
            }
        }
    })
}

/// `¬◇a => false^a`.
pub fn npos_to_para<A: DProp>(npos: Not<Pos<A>>) -> Para<A> {
    match para_decide() {
        Left(para_a) => para_a,
        Right(npara_a) => imply::absurd()(pos_not(npos)(npara_a)),
    }
}

/// `false^a => ¬◇a`.
///
/// # Safety
///
/// This function is declared unsafe because it uses the "taboo" knowledge in Avatar Modal Logic.
pub unsafe fn para_to_npos<A: Prop>(para_a: Para<A>) -> Not<Pos<A>> {
    Rc::new(move |pos_a| pos_to_para_para(pos_a)(para_a))
}

/// `¬¬a => ◇a`.
pub fn not_not_to_pos<A: DProp>(nna: Not<Not<A>>) -> Pos<A> {
    Pos::new(hooo::not_not_to_para_para(nna))
}

/// `◇a => ¬¬a`.
///
/// # Safety
///
/// This function is declared unsafe because it uses the "taboo" knowledge in Avatar Modal Logic.
pub unsafe fn pos_to_not_not<A: DProp>(pos: Pos<A>) -> Not<Not<A>> {
    hooo::para_para_to_not_not(pos_to_para_para(pos))
}

/// `¬□¬a <=> ◇a`.
pub fn eq_nnecn_pos<A: DProp>() -> Eq<Not<Nec<Not<A>>>, Pos<A>> {
    fn f<A: DProp>(_: True) -> Eq<Not<Para<A>>, Not<Para<Not<Not<A>>>>> {
        (
            Rc::new(move |x| para_rev_not(para_not_triple(pow_not(x)))),
            Rc::new(move |x| para_rev_not(para_not_rev_triple(pow_not(x))))
        )
    }
    fn g<A: DProp>(_: True) -> Eq<Not<Para<A>>, Not<Para<Not<Not<A>>>>> {
        (
            Rc::new(move |x| para_rev_not(para_not_triple(pow_not(x)))),
            Rc::new(move |x| para_rev_not(para_not_rev_double(pow_not(x))))
        )
    }
    (
        Rc::new(move |nnec_na| {
            match Pos::<A>::decide() {
                Left(pos_a) => pos_a,
                Right(npos_a) => {
                    let x = pos_not(npos_a);
                    let x = para_in_arg(x, f);
                    let x = para_rev_not(x);
                    let x = imply::in_left(x, |y| unsafe {pos_to_para_para(y)});
                    imply::absurd()(nnec_na(x))
                }
            }
        }),
        Rc::new(move |x| {
            let x = unsafe {pos_to_para_para(x)};
            let x = not::double(x);
            let x = imply::in_left(x, |y| para_rev_not(y));
            let x = imply::in_left(x, |y| para_in_arg(y, tauto_eq_symmetry(g)));
            let x: Not<Not<Para<Para<Not<Not<A>>>>>> = imply::in_left(x, pow_not);
            nnpara_para_to_nnpos(x)
        })
    )
}

/// `¬◇¬a <=> □a`.
pub fn eq_nposn_nec<A: Prop>() -> Eq<Not<Pos<Not<A>>>, Nec<A>> {
    eq::refl()
}

/// `¬□a == ◇¬a`.
pub fn eq_nnec_posn<A: DProp>() -> Eq<Not<Nec<A>>, Pos<Not<A>>> {
    fn g<A: DProp>(_: True) -> Eq<Not<Not<Para<Not<A>>>>, Para<Not<A>>> {
        fn f<A: DProp>(_: True) -> Eq<Not<Not<Not<A>>>, Not<A>> {
            (Rc::new(move |x| not::rev_triple(x)), Rc::new(move |x| not::double(x)))
        }
        (
            Rc::new(move |x| {
                let x = imply::in_left(x, |y| para_rev_not(y));
                para_in_arg(pow_not(x), f)
            }),
            Rc::new(move |x| {
                let x = para_in_arg(x, tauto_eq_symmetry(f));
                let x: Not<Para<Not<Not<A>>>> = para_rev_not(x);
                imply::in_left(x, pow_not)
            })
        )
    }
    (
        Rc::new(move |nnec_a: Not<Not<Pos<Not<A>>>>| {
            let x = unsafe {nnpos_to_nnpara_para(nnec_a)};
            let x = imply::in_left(x, |y| para_rev_not(y));
            let x = pow_not(x);
            Pos::new(para_in_arg(x, g))
        }),
        Rc::new(move |pos_na| {
            let x = unsafe {pos_to_para_para(pos_na)};
            let x: Para<Not<Not<Para<Not<A>>>>> = para_in_arg(x, tauto_eq_symmetry(g));
            let x = para_rev_not(x);
            let x: Not<Not<Para<Para<Not<A>>>>> = imply::in_left(x, pow_not);
            imply::in_left(x, |y| imply::in_left(y, Pos::new))
        })
    )
}

/// `¬◇a == □¬a`.
pub fn eq_posn_nnec<A: DProp>() -> Eq<Not<Pos<A>>, Nec<Not<A>>> {
    fn f<A: DProp>(_: True) -> Eq<Not<Para<A>>, Not<Para<Not<Not<A>>>>> {
        fn g<A: Prop>(_: True) -> Eq<Not<Not<Not<A>>>, Not<A>> {
            (Rc::new(move |x| not::rev_triple(x)), Rc::new(move |x| not::double(x)))
        }
        (
            Rc::new(move |x| {
                let x = pow_not(x);
                let x = para_in_arg(x, tauto_eq_symmetry(g));
                para_rev_not(x)
            }),
            Rc::new(move |x| {
                let x = pow_not(x);
                let x = para_in_arg(x, g);
                para_rev_not(x)
            })
        )
    }
    (
        Rc::new(move |x| {
            let x = pos_not(x);
            let x = para_in_arg(x, f);
            imply::in_left(para_rev_not(x), |y| unsafe {pos_to_para_para(y)})
        }),
        Rc::new(move |x| {
            let x = pos_not(x);
            let x = para_in_arg(x, tauto_eq_symmetry(f));
            imply::in_left(para_rev_not(x), |y| unsafe {pos_to_para_para(y)})
        })
    )
}

/// `¬¬◇a => ¬¬(false^(false^a))`.
///
/// # Safety
///
/// This function is declared unsafe because it uses the "taboo" knowledge of Avatar Modal Logic.
pub unsafe fn nnpos_to_nnpara_para<A: Prop>(x: Not<Not<Pos<A>>>) -> Not<Not<Para<Para<A>>>> {
    imply::in_left(x, |y| imply::in_left(y, |x| pos_to_para_para(x)))
}

/// `¬¬◇a => ¬¬(false^(false^a))`.
pub fn nnpara_para_to_nnpos<A: Prop>(x: Not<Not<Para<Para<A>>>>) -> Not<Not<Pos<A>>> {
    imply::in_left(x, |y| imply::in_left(y, |x| Pos::new(x)))
}

/// `(false^(false^a) => ◇a)^true) => (false^(false^a) == ◇a)^true)`.
pub fn to_pos_tauto_eq<A: Prop>(
    y: Tauto<Imply<Para<Para<A>>, Pos<A>>>
) -> Tauto<Eq<Para<Para<A>>, Pos<A>>> {
    fn f<A: Prop>(_: True) -> Imply<Pos<A>, Para<Para<A>>> {
        Rc::new(move |pos_a| unsafe {pos_to_para_para(pos_a)})
    }
    hooo_rev_and((y, f::<A>))
}

/// `¬◇a => false^(¬(false^a))`.
pub fn pos_not<A: DProp>(x: Not<Pos<A>>) -> Para<Not<Para<A>>> {
    pow_not(imply::in_left(x, |y| Pos::new(y)))
}