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The similarity relation of type expressions is governed by the the following inference rules.

• Type constructors are only similar to themselves.

  ———————————
   Γ ⊢ T ≈ T

• A free type variable α is similar to itself.

   α ∈ free(Γ)
  —————————————
   Γ ⊢ α ≈ α

• Two type variables α and β are similar under a renaming Γ if they are mapped to each other by Γ.

   α ↦ β ∈ Γ
  ———————————
   Γ ⊢ α ≈ β

• Two type applications are similar if the type constructor and type argument are similar.

   Γ ⊢ τ₁ ≈ τ'₁ , Γ ⊢ τ₂ ≈ τ'₂
  —————————————————————————————
       Γ ⊢ τ₁ τ₂ ≈ τ'₁ τ'₂

• Two function types are similar if their argument and return types are similar.

   Γ ⊢ τ₁ ≈ τ'₁ , Γ ⊢ τ₂ ≈ τ'₂
  —————————————————————————————
    Γ ⊢ τ₁ -> τ₂ ≈ τ'₁ -> τ'₂

Two type schemes are similar if their abstracted types are similar under an extend Renaming that maps the corresponding type arguments to each other.

    Γ ∪ { α₁ ↦ β₁, …, αₙ ↦ βₙ } ⊢ τ ≈ τ'
————————————————————————————————————————————
 Γ ⊢ forall α₁ … αₙ. τ ≈ forall β₁ … βₙ. τ'

Two constructor declarations are similar if their field types are similar.

 Γ ⊢ τ₁ ≈ τ'₁, …, Γ ⊢ τₙ ≈ τ'ₙ
———————————————————————————————
  Γ ⊢ C τ₁ … τₙ ≈ C τ'₁ … τ'ₙ

Two type synonym declarations are similar if their right-hand sides are similar under an extended Renaming that maps the corresponding type arguments to each other.

     Γ ∪ { α₁ ↦ β₁, …, αₙ ↦ βₙ } ⊢ τ ≈ τ'
——————————————————————————————————————————————
 Γ ⊢ type T α₁ … αₙ = τ ≈ type T β₁ … βₙ = τ'

Two data type declarations are similar if their constructors are pairwise similar under an extended Renaming that maps the corresponding type arguments to each other. Note that the order of the constructors in important.

 Γ ∪ { α₁ ↦ β₁, …, αₙ ↦ βₙ } ⊢ con₁ ≈ con'₁, …,
 Γ ∪ { α₁ ↦ β₁, …, αₙ ↦ βₙ } ⊢ conₘ ≈ con'ₘ
————————————————————————————————————————————————————————————————————————
 Γ ⊢ data D α₁ … αₙ = con₁ | … | conₘ
   ≈ data D β₁ … βₙ = con'₁ | … | con'ₘ

Two pragmas are similar if they are equal except for the source span.

Two let bindings are similar if their variable pattern and expression are are similar.

 Γ' ⊢ p ≈ q  , Γ' ⊢ e ≈ f
——————————————————————————
    Γ ⊢ p = e ≈ q = f

Two variable patterns are similar if they either both don't have a type annotation or are annotated with similar types.

                     Γ ⊢ τ ≈ τ'
———————————  ——————————————————————————
 Γ ⊢ x ≈ y    Γ ⊢ (x :: τ) ≈ (y :: τ')

If one of the patterns has a bang pattern, the other one needs one as well.

                       Γ ⊢ τ ≈ τ'
—————————————  ————————————————————————————
 Γ ⊢ !x ≈ !y    Γ ⊢ !(x :: τ) ≈ !(y :: τ')

Two alternatives that match the same constructor C are similar if their right-hand sides are similar under an extended Renaming that maps the variable patterns from the first alternative to the corresponding variable patterns from the second alternative. Additionally, the type annotations of their variable patterns must be similar (if there are any).

 Γ ∪ { x₁ ↦ y₁, …, xₙ ↦ yₙ } ⊢ e ≈ f, Γ ⊢ p₁ ≈ q₁, …, Γ ⊢ pₙ ≈ qₙ
——————————————————————————————————————————————————————————————————
            Γ ⊢ C p₁ … pₙ -> e ≈ C q₁ … qₙ -> f

where xᵢ and yᵢ denote the names of the variables bound by the patterns pᵢ and qᵢ respectively.

The similarity relation of expressions is governed by the the following inference rules

• Expressions with type annotations are similar if the annotated expressions are similar and their annotations are similar.

   Γ ⊢ τ ≈ τ', Γ ⊢ e ≈ f
  ———————————————————————
   Γ ⊢ e :: τ ≈ f :: τ'

• For all variables x and y, x and y are similar under a renaming Γ if they are mapped to each other by Γ.

   x ↦ y ∈ Γ
  ———————————
   Γ ⊢ x ≈ y

• A free variable x is similar to itself.

   x ∈ free(Γ)
  —————————————
   Γ ⊢ x ≈ x

• Two lambda abstractions are similar if their right-hand sides are similar under an extended Renaming that maps the variable patterns from the first lambda abstraction to the arguments of the second one. Furthermore, the types the lambda abstraction's arguments have been annotated with must be similar (if there are any).

   Γ ∪ { x₁ ↦ y₁, …, xₙ ↦ yₙ } ⊢ e ≈ f, Γ ⊢ p₁ ≈ q₁, …, Γ ⊢ pₙ ≈ qₙ
  ——————————————————————————————————————————————————————————————————
                  Γ ⊢ \p₁ … pₙ -> e ≈ \q₁ … qₙ -> f

  where xᵢ and yᵢ denote the names of the variables bound by the patterns pᵢ and qᵢ respectively.

• Two application expressions are similar if the callees and the arguments are similar.

   Γ ⊢ e₁ ≈ f₁, Γ ⊢ e₂ ≈ f₂
  ——————————————————————————
      Γ ⊢ e₁ e₂ ≈ f₁ f₂

• Two visible type application expressions are similar if the visibly applied expressions and the type arguments are similar.

   Γ ⊢ e ≈ f, Γ ⊢ τ ≈ τ'
  ———————————————————————
      Γ ⊢ e @τ ≈ f @τ'

• Two case expressions are similar if the expressions they match are similar and their alternatives are pairwise similar. Note that the order of the alternatives is important.

   Γ ⊢ e ≈ f,      Γ ⊢ alt₁ ≈ alt'₁,      …,      Γ ⊢ altₙ ≈ alt'ₙ
  —————————————————————————————————————————————————————————————————
   Γ ⊢ case e of { alt₁; …; altₙ } ≈ case f of { alt'₁; …; alt'ₙ }

• Two if expressions are similar if their conditions and branches are similar.

   Γ ⊢ e₁ ≈ f₁,       Γ ⊢ e₂ ≈ f₂,       Γ ⊢ e₃ ≈ f₃
  ———————————————————————————————————————————————————
   Γ ⊢ if e₁ then e₂ else e₃ ≈ if f₁ then f₂ else f₃

• Two let expressions are similar if their bindings and the expression are similar under an extended Renaming Γ' that maps the variables bound by the first let expression to the variables bound by the second let expression.

   Γ' ⊢ e ≈ f,    Γ' ⊢ b₁ ≈ d₁,    …,    Γ' ⊢ bₙ ≈ dₙ
  ——————————————————————————————————————————————————————
   Γ ⊢ let { b₁; …; bₙ } in e ≈ let { d₁; …; dₙ } in f

  where Γ' = Γ ∪ { x₁ ↦ y₁, …, xₙ ↦ yₙ } and x₁, …, xₙ and y₁, …, yₙ denote the names of variables bound by b₁, …, bₙ and d₁, …, dₙ, respectively.

• All AST nodes which do not contain any variables are similar to themselves under every Renaming Γ.

  ———————————  ———————————————————————————  ———————————————————————
   Γ ⊢ C ≈ C    Γ ⊢ undefined ≈ undefined    Γ ⊢ error s ≈ error s

  Where C is a constructor or integer literal and s is an arbitrary error message.

Two function declarations are similar if their right-hand sides are similar under an extended Renaming that maps the corresponding type arguments and arguments to each other. If the arguments are annotated, their type annotations must be similar under an extended Renaming that maps the corresponding type arguments to each other.

 Γ ∪ { α₁ ↦ β₁, …, αₘ ↦ βₘ, x₁ ↦ y₁, …, xₙ ↦ yₙ } ⊢ e ≈ f
 Γ ∪ { α₁ ↦ β₁, …, αₘ ↦ βₘ } ⊢ p₁ ≈ q₁, …,
 Γ ∪ { α₁ ↦ β₁, …, αₘ ↦ βₘ } ⊢ pₙ ≈ qₙ
——————————————————————————————————————————————————————————
 Γ ⊢ g @α₁ … @αₘ p₁ … pₙ = e ≈ g @β₁ … @βₘ q₁ … qₙ = f

where xᵢ and yᵢ denote the names of the variables bound by the patterns pᵢ and qᵢ respectively.

If the function has an explicit return type annotation, the annotated return typed must be similar as well.

 Γ ∪ { α₁ ↦ β₁, …, αₘ ↦ βₘ, x₁ ↦ y₁, …, xₙ ↦ yₙ } ⊢ e ≈ f
 Γ ∪ { α₁ ↦ β₁, …, αₘ ↦ βₘ } ⊢ p₁ ≈ q₁, …,
 Γ ∪ { α₁ ↦ β₁, …, αₘ ↦ βₘ } ⊢ pₙ ≈ qₙ,
 Γ ∪ { α₁ ↦ β₁, …, αₘ ↦ βₘ } ⊢ τ ≈ τ',
——————————————————————————————————————————————————————————
 Γ ⊢ g @α₁ … @αₘ p₁ … pₙ :: τ = e ≈ g @β₁ … @βₘ q₁ … qₙ :: τ' = f

Two type signatures are similar if they annotate the type of the same functions with similar types.

 Γ ⊢ τ ≈ τ'
————————————————————————————————————————
 Γ ⊢ f₁, … , fₙ :: τ ≈ f₁, … , fₙ :: τ'

Only top-level declarations of the same type can be similar.

Two import declarations are similar if they import the same module.

Two lists of nodes are similar if the corresponding elements are similar and each element has a corresponding element (i.e., the lists are of the same length).

  Γ ⊢ x₁ ≈ y₁, …, Γ ⊢ xₙ ≈ yₙ
———————————————————————————————
 Γ ⊢ [x₁, …, xₙ] ≈ [y₁, …, yₙ]

Two optional nodes are similar when they are either both not present or both are present and similar.

                               Γ ⊢ x ≈ y
———————————————————————  —————————————————————
 Γ ⊢ Nothing ≈ Nothing    Γ ⊢ Just x ≈ Just y

Two modules are similar if their names and contents are similar.