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mathlib-archive / miu_language.basic

# An MIU Decision Procedure in Lean #

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The MIU formal system was introduced by Douglas Hofstadter in the first chapter of his 1979 book, Gödel, Escher, Bach. The system is defined by four rules of inference, one axiom, and an alphabet of three symbols: M, I, and U.

Hofstadter's central question is: can the string "MU" be derived?

It transpires that there is a simple decision procedure for this system. A string is derivable if and only if it starts with M, contains no other Ms, and the number of Is in the string is congruent to 1 or 2 modulo 3.

The principal aim of this project is to give a Lean proof that the derivability of a string is a decidable predicate.

## The MIU System #

In Hofstadter's description, an atom is any one of M, I or U. A string is a finite sequence of zero or more symbols. To simplify notation, we write a sequence [I,U,U,M], for example, as IUUM.

The four rules of inference are:

1. xI → xIU,
2. Mx → Mxx,
3. xIIIy → xUy,
4. xUUy → xy,

where the notation α → β is to be interpreted as 'if α is derivable, then β is derivable'.

• MI is derivable.

In Lean, it is natural to treat the rules of inference and the axiom on an equal footing via an inductive data type derivable designed so that derivable x represents the notion that the string x can be derived from the axiom by the rules of inference. The axiom is represented as a nonrecursive constructor for derivable. This mirrors the translation of Peano's axiom '0 is a natural number' into the nonrecursive constructor zero of the inductive type nat.

## Tags #

miu, derivable strings

### Declarations and instance derivations for miu_atom and miustr#

inductive miu.miu_atom  :

The atoms of MIU can be represented as an enumerated type in Lean.

Instances for miu.miu_atom
@[protected, instance]

The annotation @[derive decidable_eq] above assigns the attribute derive to miu_atom, through which Lean automatically derives that miu_atom is an instance of decidable_eq. The use of derive is crucial in this project and will lead to the automatic derivation of decidability.

@[protected, instance]

We show that the type miu_atom is inhabited, giving M (for no particular reason) as the default element.

Equations

miu_atom.repr is the 'natural' function from miu_atom to string.

Equations
@[protected, instance]

Using miu_atom.repr, we prove that miu_atomis an instance ofhas_repr.

Equations
@[protected, instance]
@[protected, instance]
def miu.miustr  :

For simplicity, an miustr is just a list of elements of type miu_atom.

Equations
Instances for miu.miustr

For display purposes, an miustr can be represented as a string.

Equations
@[protected, instance]
Equations

In the other direction, we set up a coercion from string to miustr.

Equations
• miu.lchar_to_miustr (c :: cs) = let ms : miu.miustr := in miu.lchar_to_miustr._match_1 ms c
• miu.lchar_to_miustr._match_1 ms _x = list.nil
• miu.lchar_to_miustr._match_1 ms 'U' =
• miu.lchar_to_miustr._match_1 ms 'I' =
• miu.lchar_to_miustr._match_1 ms 'M' =
@[protected, instance]
Equations

### Derivability #

inductive miu.derivable  :

The inductive type derivable has five constructors. The nonrecursive constructor mk corresponds to Hofstadter's axiom that "MI" is derivable. Each of the constructors r1, r2, r3, r4 corresponds to the one of Hofstadter's rules of inference.

Instances for miu.derivable`