new-lang/proof.py

from __future__ import annotations
from typing import Any, Callable, Union

from dataclasses import dataclass

# -----------------------------------------------------------------------------
# Terms
# -----------------------------------------------------------------------------

class Term:
    pass

@dataclass(frozen=True)
class App(Term):
    f : Term
    x : Term

@dataclass(frozen=True)
class Const(Term):
    name : str

@dataclass(frozen=True)
class Var(Term):
    name : str

# --- Bool constructors ---

Bool = Union["Truth", "Contradiction"]

@dataclass(frozen=True)
class Truth(Term):
    pass

@dataclass(frozen=True)
class Contradiction(Term):
    pass

# --- Nat constructors ---

Nat = Union["Zero", "Succ"]

@dataclass(frozen=True)
class Zero(Term):
    pass

@dataclass(frozen=True)
class Succ(Term):
    n : Term

def kernel_from_int(n : int) -> Nat:
    if n == 0:
        return Zero()
    elif n < 0:
        raise ValueError("Int is not Nat")
    else:
        return Succ(n=kernel_from_int(n=n-1))

# -----------------------------------------------------------------------------
# Propositions
# -----------------------------------------------------------------------------

class Prop:
    pass

@dataclass(frozen=True)
class Eq(Prop):
    lhs : Term
    rhs : Term

# -----------------------------------------------------------------------------
# Proofs
# -----------------------------------------------------------------------------

class Proof:
    pass

@dataclass(frozen=True)
class Refl(Proof):
    pass

# -----------------------------------------------------------------------------
# Normalization function
# -----------------------------------------------------------------------------

@dataclass(frozen=True)
class F:
    arity : int
    func : Callable[..., Term | None]

def apply_rules() -> dict[str, F]:
    def __bool_not(x : Term) -> Term | None:
        match x:
            case Truth():
                return Contradiction()
            
            case Contradiction():
                return Truth()
    
    def __nat_add(x : Term, y : Term) -> Term | None:
        match x:
            case Zero():
                return y
            
            case Succ():
                return Succ(
                    n=normalize(App(
                        f=App(Const("Nat.add"), x=x.n),
                        x=y,
                    ))
                )
    
    def __nat_iszero(x : Term) -> Term | None:
        match x:
            case Zero():
                return Truth()
            
            case Succ():
                return Contradiction()
    
    return {
        "Bool.not": F(arity=1, func=__bool_not),
        "Nat.add" : F(arity=2, func=__nat_add),
        "Nat.isZero" : F(arity=1, func=__nat_iszero),
    }
                

def normalize(term : Term) -> Term:
    match term:
        case App():
            f = normalize(term.f)
            x = normalize(term.x)

            cursor = term
            items = [ term ]

            while True:
                match cursor.f:
                    case App():
                        cursor = cursor.f
                        items.append(cursor)
                    
                    case Const():
                        name = cursor.f.name
                        rule = apply_rules().get(name, None)

                        if rule is None:
                            return App(f, x) # Unknown function
                        if len(items) != rule.arity:
                            # If len(items) is too small, we don't have
                            # enough inputs yet to evaluate the function
                            # If len(items) is too large, we already evaluated
                            # the function before and couldn't optimize.
                            return App(f, x)
                        
                        out = rule.func(*reversed([i.x for i in items]))

                        return out if out is not None else App(f, x)

                    case Contradiction():
                        if len(items) != 2:
                            # Either evaluating too early or too late
                            return term
                        
                        # Return the "second" item
                        return x

                    case Truth():
                        if len(items) != 2:
                            # Either evaluating too early or too late
                            return App(f, x)
                        
                        # Return the "first" item
                        return cursor.x

        case Const():
            return term
        
        case Succ():
            return Succ(n=normalize(term.n))
        
        case Term():
            return term

        case Var():
            return term
        
        case Zero():
            return term

# -----------------------------------------------------------------------------
# Proof checker
# -----------------------------------------------------------------------------

def check(goal: Prop, proof : Proof) -> bool:
    match ( goal, proof ):
        case ( Eq(), Refl() ):
            return normalize(goal.lhs) == normalize(goal.rhs)
    
        case _:
            return False

# -----------------------------------------------------------------------------
# -----------------------------------------------------------------------------
# -----------------------------------------------------------------------------

if __name__ == "__main__":
    # 2 + 3 == 5
    print(check(
        Eq(
            lhs=App(f=App(f=Const("Nat.add"), x=kernel_from_int(2)), x=kernel_from_int(3)),
            rhs=kernel_from_int(5),
        ),
        Refl(),
    ))

    # 0 + x == x
    print(check(
        Eq(
            lhs=App(f=App(f=Const("Nat.add"), x=Zero()), x=Var("x")),
            rhs=Var("x")
        ),
        Refl(),
    ))

    # x + 0 == x
    print(check(
        Eq(
            lhs=App(f=App(f=Const("Nat.add"), x=Var("x")), x=Zero()),
            rhs=Var("x")
        ),
        Refl(),
    ))