Write You a Haskell Stephen Diehl

• T-Var Variables are simply pulled from the context.
• T-Lam lambdas introduce a typed variable into the environment when inferring the body.
• T-App Applications of a lambda with type t1 -> t2 to a value of type t1 yields a value of type
t2.
x : σ ∈ Γ
Γ ⊢ x : σ
Γ, x : τ 1 ⊢ e : τ 2
Γ ⊢ λx : τ 1 .e : τ 1 → τ 2
Γ ⊢ e 1 : τ 1 → τ 2 Γ ⊢ e 2 : τ 1
Γ ⊢ e 1 e 2 : τ 2
Γ ⊢ c : Bool Γ ⊢ e 1 : τ Γ ⊢ e 2 : τ
Γ ⊢ if c then e 1 else e 2 : τ
Γ ⊢ n : Int
Γ ⊢ True : Bool
Γ ⊢ False : Bool
(T-Var)
(T-Lam)
(T-App)
(T-If)
(T-Int)
(T-True)
(T-False)
e evaluation rules describe the nature by which values transition between other values and determine
the runtime behavior of the program.
e 1 → e ′ 1
e 1 e 2 → e ′ 1 e 2
e 2 → e ′ 2
v 1 e 2 → v 1 e ′ 2
(λx : τ.e 1 )v 2 → [x/v 2 ]e 1
if True then e 2 else e 3 → e 2
if False then e 2 else e 3 → e 3
e 1 → e ′ 1
if e 1 then e 2 else e 3 → if e ′ 1 then e 2 else e 3
(E-App1)
(E-App2)
(E-AppLam)
(E-IfTrue)
(E-IfFalse)
(E-If)
Since we now have the notion of scoped variables for lambda, we will implement a typing environment
Env as manifest as Γ in our typing rules.
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