Type inference for The Simply Typed Lambda Calculus

%% Type inference for The Simply Typed Lambda Calculus

:- op(150, xfx, ⊢).
:- op(140, xfx, :).
:- op(100, xfy, ->).
:- op(100, yfx, $).

Γ ⊢    Term : Type          :- atom(Term), member(Term : Type, Γ).
Γ ⊢ λ(A, B) : Alpha -> Beta :- [A : Alpha|Γ] ⊢ B : Beta.
Γ ⊢   A $ B : Beta          :- Γ ⊢ A : Alpha -> Beta, Γ ⊢ B : Alpha.

/** Examples:

% Typing fix
?- Γ ⊢ y $ f : Y, Γ ⊢ f $ (y $ f) : Y.
Γ = [y: (Y->Y)->Y, f:Y->Y|_G330]

% Typing the Y combinator
?- Γ ⊢ λ(f, (λ(f,f $ f)) $ (λ(g,f $ (g $ g)))) : Y.
Y = (_G309 -> _G309) -> _G309 

% Typing flip id
?- Id = λ(i, i), Flip = λ(f, λ(y, λ(x, f $ x $ y))),
   Γ ⊢ Flip $ Id : T.
T = _G395 -> (_G395 -> _G411) -> _G411

**/