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//! # Prop
//! Propositional logic with types in Rust.
//!
//! A library in Rust for theorem proving with [Intuitionistic Propositional Logic](https://en.wikipedia.org/wiki/Intuitionistic_logic).
//! Supports theorem proving in [Classical Propositional Logic](https://en.wikipedia.org/wiki/Propositional_calculus).
//!
//! - Used in research on [Path Semantics](https://github.com/advancedresearch/path_semantics)
//! - Brought to you by the [AdvancedResearch Community](https://advancedresearch.github.io/)
//! - [Join us on Discord!](https://discord.gg/JkrhJJRBR2)
//!
//! Abbreviations:
//!
//! - IPL: Intuitionistic/Constructive Propositional Logic
//! - EL: Existential Logic (Excluded Middle of Non-Existence)
//! - PL: Classical Propositional Logic
//! - PSI: Path Semantical Intuitionistic/Constructive Propositional Logic
//! - PSEL: Path Semantical Existential Logic
//! - PSL: Path Semantical Classical Propositional Logic
//! - PSQ: Path Semantical Quantum Propositional Logic
//! - HOOO EP: Higher Order Operator Overloading Exponential Propositions
//! - MEL: Middle Exponential Logic
//!
//! ### Motivation
//!
//! [Path Semantics](https://github.com/advancedresearch/path_semantics)
//! extends dependent types with normal paths and is also used to extend
//! Classical Propositional Logic with multiple levels of propositions.
//! It is also used to explore higher dimensional mathematics.
//! A popular research subject in Path Semantics is [Avatar Extensions](https://advancedresearch.github.io/avatar-extensions/summary.html).
//!
//! When researching, in some cases it is useful to figure out whether a proof is
//! provable in classical logic, but not in constructive logic.
//! This requires comparing proofs easily.
//!
//! This library uses a lifting mechanism for making it easier
//! to produce proofs in classical logic and compare them to
//! proofs in constructive logic.
//!
//! ### Design
//!
//! This library contains:
//!
//! - `Prop`: Propositions that might or might not be decidable (constructive logic)
//! - `EProp`: Existential propositions (existential logic)
//! - `DProp`: Decidable propositions (classical logic)
//! - `LProp`: Like `Prop`, but with path semantics (path semantical constructive logic)
//! - `ELProp`: Like `EProp`, but with path semantics (path semantical existential logic)
//! - `DLProp`: Like `DProp`, but with path semantics (path semantical classical logic)
//! - Automatic lifting of Excluded Middle of Non-Existence to existential propositions
//! - Automatic lifting of Excluded Middle to decidable propositions
//! - Double Negation for proofs of `Prop`
//! - A model of Path Semantical Quality/Aquality in IPL (see "quality" module)
//! - A model of Path Semantical Qubit in IPL (see "qubit" module)
//! - A model of Path Semantical Con-Quality in IPL (see "con_qubit" module)
//! - A model of Seshatic Queenity (see "queenity" module)
//! - Formalization of the core axiom of Path Semantics
//! - Exponential Propositions (HOOO) for tautological/paradoxical theorem proving
//! - A model of S5 Modal Logic derived from HOOO EP
//! - A model of Avatar Modal Logic derived from HOOO EP and Theory of Avatar Extensions
//! - A model of Middle Exponential Logic using EL and HOOO EP
//! - Tactics organized in modules by constructs (e.g. `and` or `imply`)
//!
//! ### Examples
//!
//! ```rust
//! use prop::*;
//!
//! fn proof<A: Prop, B: Prop>(f: Imply<A, B>, a: A) -> B {
//! imply::modus_ponens(f, a)
//! }
//! ```
//!
//! Notice that there is no `DProp` used here,
//! which means that it is a constructive proof.
//!
//! ```rust
//! use prop::*;
//!
//! fn proof<A: DProp, B: DProp>(f: Imply<Not<A>, Not<B>>) -> Imply<B, A> {
//! imply::rev_modus_tollens(f)
//! }
//! ```
//!
//! Here, `DProp` is needed because `rev_modus_tollens` needs Excluded Middle.
//! This limits the proof to decidable propositions.
//!
//! ### Path Semantics
//!
//! Path Semantics is an extremely expressive language for mathematical programming.
//! It uses a single core axiom, which models semantics of symbols.
//!
//! Basically, mathematical languages contain a hidden symmetry due to use of symbols.
//! Counter-intuitively, symbols are not "inherently" in logic.
//!
//! One way to put it, is that the symbols "themselves" encode laws of mathematics.
//! The hidden symmetry can be exploited to prove semantics and sometimes
//! improve performance of automated theorem provers.
//!
//! For more information, see the [Path Semantics Project](https://github.com/advancedresearch/path_semantics).
use Rc;
use *;
/// Logical true.
;
/// Logical false.
/// Sum type of left and right case.
/// Logical AND.
pub type And<T, U> = ;
/// Double negation.
pub type Dneg<T> = ;
/// Logical EQ.
pub type Eq<T, U> = ;
/// Alternative to Logical EQ.
pub type Iff<T, U> = ;
/// Logical IMPLY.
pub type Imply<T, U> = ;
/// Excluded middle.
pub type ExcM<T> = ;
/// Logical NOT.
pub type Not<T> = ;
/// Logical OR.
pub type Or<T, U> = ;
/// A proposition that might be decidable or undecidable.
/// Implemented by decidable types.
/// Shorthand for decidable proposition.