------------------------------------------------------------------------
-- The Agda standard library
--
-- Terms used in the reflection machinery
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Reflection.Term where

open import Data.List.Base hiding (_++_)
import Data.List.Properties as Lₚ
open import Data.Nat as ℕ using (ℕ; zero; suc)
open import Data.Product
import Data.Product.Properties as Product
open import Data.Maybe.Base using (Maybe; just; nothing)
open import Data.String as String using (String)
open import Reflection.Abstraction
open import Reflection.Argument
open import Reflection.Argument.Information using (visibility)
import Reflection.Argument.Visibility as Visibility; open Visibility.Visibility
import Reflection.Literal as Literal
import Reflection.Meta as Meta
open import Reflection.Name as Name using (Name)
open import Relation.Nullary
open import Relation.Nullary.Product using (_×-dec_)
open import Relation.Nullary.Decidable as Dec
open import Relation.Binary
open import Relation.Binary.PropositionalEquality

------------------------------------------------------------------------
-- Re-exporting the builtin type and constructors

open import Agda.Builtin.Reflection as Builtin public
  using (Sort; Type; Term; Clause; Pattern)
open Sort public
open Term public renaming (agda-sort to sort)
open Clause public
open Pattern public

------------------------------------------------------------------------
-- Handy synonyms

Clauses : Set
Clauses = List Clause

Telescope : Set
Telescope = List (String × Arg Type)

-- Pattern synonyms for more compact presentation

pattern vLam s t           = lam visible   (abs s t)
pattern hLam s t           = lam hidden    (abs s t)
pattern iLam s t           = lam instance′ (abs s t)
pattern Π[_∶_]_ s a ty     = pi a (abs s ty)
pattern vΠ[_∶_]_ s a ty    = Π[ s ∶ (vArg a) ] ty
pattern hΠ[_∶_]_ s a ty    = Π[ s ∶ (hArg a) ] ty
pattern iΠ[_∶_]_ s a ty    = Π[ s ∶ (iArg a) ] ty

----------------------------------------------------------------------
-- Utility functions

getName : Term → Maybe Name
getName (con c args) = just c
getName (def f args) = just f
getName _            = nothing

-- "n ⋯⟅∷⟆ xs" prepends "n" visible unknown arguments to the list of
-- arguments. Useful when constructing the list of arguments for a
-- function with initial inferable arguments.
infixr  _⋯⟨∷⟩_
_⋯⟨∷⟩_ : ℕ → Args Term → Args Term
zero  ⋯⟨∷⟩ xs = xs
suc i ⋯⟨∷⟩ xs = unknown ⟨∷⟩ (i ⋯⟨∷⟩ xs)
{-# INLINE _⋯⟨∷⟩_ #-}

-- "n ⋯⟅∷⟆ xs" prepends "n" hidden unknown arguments to the list of
-- arguments. Useful when constructing the list of arguments for a
-- function with initial implicit arguments.
infixr  _⋯⟅∷⟆_
_⋯⟅∷⟆_ : ℕ → Args Term → Args Term
zero  ⋯⟅∷⟆ xs = xs
suc i ⋯⟅∷⟆ xs = unknown ⟅∷⟆ (i ⋯⟅∷⟆ xs)
{-# INLINE _⋯⟅∷⟆_ #-}

------------------------------------------------------------------------
-- Decidable equality

clause-injective₁ : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → tel ≡ tel′
clause-injective₁ refl = refl

clause-injective₂ : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → ps ≡ ps′
clause-injective₂ refl = refl

clause-injective₃ : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → b ≡ b′
clause-injective₃ refl = refl

clause-injective : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → tel ≡ tel′ × ps ≡ ps′ × b ≡ b′
clause-injective = < clause-injective₁ , < clause-injective₂ , clause-injective₃ > >

absurd-clause-injective₁ : ∀ {tel tel′ ps ps′} → absurd-clause tel ps ≡ absurd-clause tel′ ps′ → tel ≡ tel′
absurd-clause-injective₁ refl = refl

absurd-clause-injective₂ : ∀ {tel tel′ ps ps′} → absurd-clause tel ps ≡ absurd-clause tel′ ps′ → ps ≡ ps′
absurd-clause-injective₂ refl = refl

absurd-clause-injective : ∀ {tel tel′ ps ps′} → absurd-clause tel ps ≡ absurd-clause tel′ ps′ → tel ≡ tel′ × ps ≡ ps′
absurd-clause-injective = < absurd-clause-injective₁ , absurd-clause-injective₂ >

infix  _≟-AbsTerm_ _≟-AbsType_ _≟-ArgTerm_ _≟-ArgType_ _≟-Args_
        _≟-Clause_ _≟-Clauses_ _≟_
        _≟-Sort_ _≟-Pattern_ _≟-Patterns_

_≟-AbsTerm_  : DecidableEquality (Abs Term)
_≟-AbsType_  : DecidableEquality (Abs Type)
_≟-ArgTerm_  : DecidableEquality (Arg Term)
_≟-ArgType_  : DecidableEquality (Arg Type)
_≟-Args_     : DecidableEquality (Args Term)
_≟-Clause_   : DecidableEquality Clause
_≟-Clauses_  : DecidableEquality Clauses
_≟_          : DecidableEquality Term
_≟-Sort_     : DecidableEquality Sort
_≟-Patterns_ : Decidable (_≡_ {A = Args Pattern})
_≟-Pattern_  : Decidable (_≡_ {A = Pattern})

-- Decidable equality 'transformers'
-- We need to inline these because the terms are not sized so termination
-- would not obvious if we were to use higher-order functions such as
-- Data.List.Properties' ≡-dec

abs s a ≟-AbsTerm abs s′ a′ = unAbs-dec (a ≟ a′)
abs s a ≟-AbsType abs s′ a′ = unAbs-dec (a ≟ a′)
arg i a ≟-ArgTerm arg i′ a′ = unArg-dec (a ≟ a′)
arg i a ≟-ArgType arg i′ a′ = unArg-dec (a ≟ a′)

[]       ≟-Args []       = yes refl
(x ∷ xs) ≟-Args (y ∷ ys) = Lₚ.∷-dec (x ≟-ArgTerm y) (xs ≟-Args ys)
[]       ≟-Args (_ ∷ _)  = no λ()
(_ ∷ _)  ≟-Args []       = no λ()

[]       ≟-Clauses []       = yes refl
(x ∷ xs) ≟-Clauses (y ∷ ys) = Lₚ.∷-dec (x ≟-Clause y) (xs ≟-Clauses ys)
[]       ≟-Clauses (_ ∷ _)  = no λ()
(_ ∷ _)  ≟-Clauses []       = no λ()

_≟-Telescope_ : DecidableEquality Telescope
[] ≟-Telescope [] = yes refl
((x , t) ∷ tel) ≟-Telescope ((x′ , t′) ∷ tel′) = Lₚ.∷-dec
  (map′ (uncurry (cong₂ _,_)) Product.,-injective ((x String.≟ x′) ×-dec (t ≟-ArgTerm t′)))
  (tel ≟-Telescope tel′)
[] ≟-Telescope (_ ∷ _) = no λ ()
(_ ∷ _) ≟-Telescope [] = no λ ()

clause tel ps b ≟-Clause clause tel′ ps′ b′ =
  Dec.map′ (λ (tel≡tel′ , ps≡ps′ , b≡b′) → cong₂ (uncurry clause) (cong₂ _,_ tel≡tel′ ps≡ps′) b≡b′)
           clause-injective
           (tel ≟-Telescope tel′ ×-dec ps ≟-Patterns ps′ ×-dec b ≟ b′)
absurd-clause tel ps ≟-Clause absurd-clause tel′ ps′ =
  Dec.map′ (uncurry (cong₂ absurd-clause))
           absurd-clause-injective
           (tel ≟-Telescope tel′ ×-dec ps ≟-Patterns ps′)
clause _ _ _      ≟-Clause absurd-clause _ _ = no λ()
absurd-clause _ _ ≟-Clause clause _ _ _      = no λ()

var-injective₁ : ∀ {x x′ args args′} → Term.var x args ≡ var x′ args′ → x ≡ x′
var-injective₁ refl = refl

var-injective₂ : ∀ {x x′ args args′} → Term.var x args ≡ var x′ args′ → args ≡ args′
var-injective₂ refl = refl

var-injective :  ∀ {x x′ args args′} → var x args ≡ var x′ args′ → x ≡ x′ × args ≡ args′
var-injective = < var-injective₁ , var-injective₂ >

con-injective₁ : ∀ {c c′ args args′} → Term.con c args ≡ con c′ args′ → c ≡ c′
con-injective₁ refl = refl

con-injective₂ : ∀ {c c′ args args′} → Term.con c args ≡ con c′ args′ → args ≡ args′
con-injective₂ refl = refl

con-injective : ∀ {c c′ args args′} → con c args ≡ con c′ args′ → c ≡ c′ × args ≡ args′
con-injective = < con-injective₁ , con-injective₂ >

def-injective₁ : ∀ {f f′ args args′} → def f args ≡ def f′ args′ → f ≡ f′
def-injective₁ refl = refl

def-injective₂ : ∀ {f f′ args args′} → def f args ≡ def f′ args′ → args ≡ args′
def-injective₂ refl = refl

def-injective : ∀ {f f′ args args′} → def f args ≡ def f′ args′ → f ≡ f′ × args ≡ args′
def-injective = < def-injective₁ , def-injective₂ >

meta-injective₁ : ∀ {x x′ args args′} → meta x args ≡ meta x′ args′ → x ≡ x′
meta-injective₁ refl = refl

meta-injective₂ : ∀ {x x′ args args′} → meta x args ≡ meta x′ args′ → args ≡ args′
meta-injective₂ refl = refl

meta-injective : ∀ {x x′ args args′} → meta x args ≡ meta x′ args′ → x ≡ x′ × args ≡ args′
meta-injective = < meta-injective₁ , meta-injective₂ >

lam-injective₁ : ∀ {v v′ t t′} → lam v t ≡ lam v′ t′ → v ≡ v′
lam-injective₁ refl = refl

lam-injective₂ : ∀ {v v′ t t′} → lam v t ≡ lam v′ t′ → t ≡ t′
lam-injective₂ refl = refl

lam-injective : ∀ {v v′ t t′} → lam v t ≡ lam v′ t′ → v ≡ v′ × t ≡ t′
lam-injective = < lam-injective₁ , lam-injective₂ >

pat-lam-injective₁ : ∀ {cs cs′ args args′} → pat-lam cs args ≡ pat-lam cs′ args′ → cs ≡ cs′
pat-lam-injective₁ refl = refl

pat-lam-injective₂ : ∀ {cs cs′ args args′} → pat-lam cs args ≡ pat-lam cs′ args′ → args ≡ args′
pat-lam-injective₂ refl = refl

pat-lam-injective : ∀ {cs cs′ args args′} → pat-lam cs args ≡ pat-lam cs′ args′ → cs ≡ cs′ × args ≡ args′
pat-lam-injective = < pat-lam-injective₁ , pat-lam-injective₂ >

pi-injective₁ : ∀ {t₁ t₁′ t₂ t₂′} → pi t₁ t₂ ≡ pi t₁′ t₂′ → t₁ ≡ t₁′
pi-injective₁ refl = refl

pi-injective₂ : ∀ {t₁ t₁′ t₂ t₂′} → pi t₁ t₂ ≡ pi t₁′ t₂′ → t₂ ≡ t₂′
pi-injective₂ refl = refl

pi-injective : ∀ {t₁ t₁′ t₂ t₂′} → pi t₁ t₂ ≡ pi t₁′ t₂′ → t₁ ≡ t₁′ × t₂ ≡ t₂′
pi-injective = < pi-injective₁ , pi-injective₂ >

sort-injective : ∀ {x y} → sort x ≡ sort y → x ≡ y
sort-injective refl = refl

lit-injective : ∀ {x y} → lit x ≡ lit y → x ≡ y
lit-injective refl = refl

set-injective : ∀ {x y} → set x ≡ set y → x ≡ y
set-injective refl = refl

slit-injective : ∀ {x y} → Sort.lit x ≡ lit y → x ≡ y
slit-injective refl = refl

prop-injective : ∀ {x y} → prop x ≡ prop y → x ≡ y
prop-injective refl = refl

propLit-injective : ∀ {x y} → propLit x ≡ propLit y → x ≡ y
propLit-injective refl = refl

inf-injective : ∀ {x y} → inf x ≡ inf y → x ≡ y
inf-injective refl = refl

var x args ≟ var x′ args′ =
  Dec.map′ (uncurry (cong₂ var)) var-injective (x ℕ.≟ x′ ×-dec args ≟-Args args′)
con c args ≟ con c′ args′ =
  Dec.map′ (uncurry (cong₂ con)) con-injective (c Name.≟ c′ ×-dec args ≟-Args args′)
def f args ≟ def f′ args′ =
  Dec.map′ (uncurry (cong₂ def)) def-injective (f Name.≟ f′ ×-dec args ≟-Args args′)
meta x args ≟ meta x′ args′ =
  Dec.map′ (uncurry (cong₂ meta)) meta-injective (x Meta.≟ x′   ×-dec args ≟-Args args′)
lam v t    ≟ lam v′ t′    =
  Dec.map′ (uncurry (cong₂ lam)) lam-injective (v Visibility.≟ v′ ×-dec t ≟-AbsTerm t′)
pat-lam cs args ≟ pat-lam cs′ args′ =
  Dec.map′ (uncurry (cong₂ pat-lam)) pat-lam-injective (cs ≟-Clauses cs′ ×-dec args ≟-Args args′)
pi t₁ t₂   ≟ pi t₁′ t₂′   =
  Dec.map′ (uncurry (cong₂ pi))  pi-injective (t₁ ≟-ArgType t₁′  ×-dec t₂ ≟-AbsType t₂′)
sort s     ≟ sort s′      = Dec.map′ (cong sort)  sort-injective (s ≟-Sort s′)
lit l      ≟ lit l′       = Dec.map′ (cong lit)   lit-injective (l Literal.≟ l′)
unknown    ≟ unknown      = yes refl

var x args ≟ con c args′ = no λ()
var x args ≟ def f args′ = no λ()
var x args ≟ lam v t     = no λ()
var x args ≟ pi t₁ t₂    = no λ()
var x args ≟ sort _      = no λ()
var x args ≟ lit _      = no λ()
var x args ≟ meta _ _    = no λ()
var x args ≟ unknown     = no λ()
con c args ≟ var x args′ = no λ()
con c args ≟ def f args′ = no λ()
con c args ≟ lam v t     = no λ()
con c args ≟ pi t₁ t₂    = no λ()
con c args ≟ sort _      = no λ()
con c args ≟ lit _      = no λ()
con c args ≟ meta _ _    = no λ()
con c args ≟ unknown     = no λ()
def f args ≟ var x args′ = no λ()
def f args ≟ con c args′ = no λ()
def f args ≟ lam v t     = no λ()
def f args ≟ pi t₁ t₂    = no λ()
def f args ≟ sort _      = no λ()
def f args ≟ lit _      = no λ()
def f args ≟ meta _ _    = no λ()
def f args ≟ unknown     = no λ()
lam v t    ≟ var x args  = no λ()
lam v t    ≟ con c args  = no λ()
lam v t    ≟ def f args  = no λ()
lam v t    ≟ pi t₁ t₂    = no λ()
lam v t    ≟ sort _      = no λ()
lam v t    ≟ lit _      = no λ()
lam v t    ≟ meta _ _    = no λ()
lam v t    ≟ unknown     = no λ()
pi t₁ t₂   ≟ var x args  = no λ()
pi t₁ t₂   ≟ con c args  = no λ()
pi t₁ t₂   ≟ def f args  = no λ()
pi t₁ t₂   ≟ lam v t     = no λ()
pi t₁ t₂   ≟ sort _      = no λ()
pi t₁ t₂   ≟ lit _      = no λ()
pi t₁ t₂   ≟ meta _ _    = no λ()
pi t₁ t₂   ≟ unknown     = no λ()
sort _     ≟ var x args  = no λ()
sort _     ≟ con c args  = no λ()
sort _     ≟ def f args  = no λ()
sort _     ≟ lam v t     = no λ()
sort _     ≟ pi t₁ t₂    = no λ()
sort _     ≟ lit _       = no λ()
sort _     ≟ meta _ _    = no λ()
sort _     ≟ unknown     = no λ()
lit _     ≟ var x args  = no λ()
lit _     ≟ con c args  = no λ()
lit _     ≟ def f args  = no λ()
lit _     ≟ lam v t     = no λ()
lit _     ≟ pi t₁ t₂    = no λ()
lit _     ≟ sort _      = no λ()
lit _     ≟ meta _ _    = no λ()
lit _     ≟ unknown     = no λ()
meta _ _   ≟ var x args  = no λ()
meta _ _   ≟ con c args  = no λ()
meta _ _   ≟ def f args  = no λ()
meta _ _   ≟ lam v t     = no λ()
meta _ _   ≟ pi t₁ t₂    = no λ()
meta _ _   ≟ sort _      = no λ()
meta _ _   ≟ lit _       = no λ()
meta _ _   ≟ unknown     = no λ()
unknown    ≟ var x args  = no λ()
unknown    ≟ con c args  = no λ()
unknown    ≟ def f args  = no λ()
unknown    ≟ lam v t     = no λ()
unknown    ≟ pi t₁ t₂    = no λ()
unknown    ≟ sort _      = no λ()
unknown    ≟ lit _       = no λ()
unknown    ≟ meta _ _    = no λ()
pat-lam _ _ ≟ var x args  = no λ()
pat-lam _ _ ≟ con c args  = no λ()
pat-lam _ _ ≟ def f args  = no λ()
pat-lam _ _ ≟ lam v t     = no λ()
pat-lam _ _ ≟ pi t₁ t₂    = no λ()
pat-lam _ _ ≟ sort _      = no λ()
pat-lam _ _ ≟ lit _       = no λ()
pat-lam _ _ ≟ meta _ _    = no λ()
pat-lam _ _ ≟ unknown     = no λ()
var x args  ≟ pat-lam _ _ = no λ()
con c args  ≟ pat-lam _ _ = no λ()
def f args  ≟ pat-lam _ _ = no λ()
lam v t     ≟ pat-lam _ _ = no λ()
pi t₁ t₂    ≟ pat-lam _ _ = no λ()
sort _      ≟ pat-lam _ _ = no λ()
lit _       ≟ pat-lam _ _ = no λ()
meta _ _    ≟ pat-lam _ _ = no λ()
unknown     ≟ pat-lam _ _ = no λ()

set t   ≟-Sort set t′  = Dec.map′ (cong set) set-injective (t ≟ t′)
lit n   ≟-Sort lit n′  = Dec.map′ (cong lit) slit-injective (n ℕ.≟ n′)
prop t   ≟-Sort prop t′  = Dec.map′ (cong prop) prop-injective (t ≟ t′)
propLit n   ≟-Sort propLit n′  = Dec.map′ (cong propLit) propLit-injective (n ℕ.≟ n′)
inf n   ≟-Sort inf n′  = Dec.map′ (cong inf) inf-injective (n ℕ.≟ n′)
unknown ≟-Sort unknown = yes refl
set _   ≟-Sort lit _   = no λ()
set _   ≟-Sort prop _   = no λ()
set _   ≟-Sort propLit _   = no λ()
set _   ≟-Sort inf _   = no λ()
set _   ≟-Sort unknown = no λ()
lit _   ≟-Sort set _   = no λ()
lit _   ≟-Sort prop _   = no λ()
lit _   ≟-Sort propLit _   = no λ()
lit _   ≟-Sort inf _   = no λ()
lit _   ≟-Sort unknown = no λ()
prop _  ≟-Sort set _   = no λ()
prop _  ≟-Sort lit _   = no λ()
prop _  ≟-Sort propLit _   = no λ()
prop _  ≟-Sort inf _   = no λ()
prop _  ≟-Sort unknown   = no λ()
propLit _  ≟-Sort set _   = no λ()
propLit _  ≟-Sort lit _   = no λ()
propLit _  ≟-Sort prop _   = no λ()
propLit _  ≟-Sort inf _   = no λ()
propLit _  ≟-Sort unknown   = no λ()
inf _  ≟-Sort set _   = no λ()
inf _  ≟-Sort lit _   = no λ()
inf _  ≟-Sort prop _   = no λ()
inf _  ≟-Sort propLit _   = no λ()
inf _  ≟-Sort unknown   = no λ()
unknown ≟-Sort set _   = no λ()
unknown ≟-Sort lit _   = no λ()
unknown ≟-Sort prop _   = no λ()
unknown ≟-Sort propLit _   = no λ()
unknown ≟-Sort inf _   = no λ()


pat-con-injective₁ : ∀ {c c′ args args′} → Pattern.con c args ≡ con c′ args′ → c ≡ c′
pat-con-injective₁ refl = refl

pat-con-injective₂ : ∀ {c c′ args args′} → Pattern.con c args ≡ con c′ args′ → args ≡ args′
pat-con-injective₂ refl = refl

pat-con-injective : ∀ {c c′ args args′} → Pattern.con c args ≡ con c′ args′ → c ≡ c′ × args ≡ args′
pat-con-injective = < pat-con-injective₁ , pat-con-injective₂ >

pat-var-injective : ∀ {x y} → var x ≡ var y → x ≡ y
pat-var-injective refl = refl

pat-lit-injective : ∀ {x y} → Pattern.lit x ≡ lit y → x ≡ y
pat-lit-injective refl = refl

proj-injective : ∀ {x y} → proj x ≡ proj y → x ≡ y
proj-injective refl = refl

dot-injective : ∀ {x y} → dot x ≡ dot y → x ≡ y
dot-injective refl = refl

absurd-injective : ∀ {x y} → absurd x ≡ absurd y → x ≡ y
absurd-injective refl = refl

con c ps ≟-Pattern con c′ ps′ = Dec.map′ (uncurry (cong₂ con)) pat-con-injective (c Name.≟ c′ ×-dec ps ≟-Patterns ps′)
var x    ≟-Pattern var x′     = Dec.map′ (cong var) pat-var-injective (x ℕ.≟ x′)
lit l    ≟-Pattern lit l′     = Dec.map′ (cong lit) pat-lit-injective (l Literal.≟ l′)
proj a   ≟-Pattern proj a′    = Dec.map′ (cong proj) proj-injective (a Name.≟ a′)
dot t    ≟-Pattern dot t′     = Dec.map′ (cong dot) dot-injective (t ≟ t′)
absurd x ≟-Pattern absurd x′  = Dec.map′ (cong absurd) absurd-injective (x ℕ.≟ x′)

con x x₁ ≟-Pattern dot x₂ = no (λ ())
con x x₁ ≟-Pattern var x₂ = no (λ ())
con x x₁ ≟-Pattern lit x₂ = no (λ ())
con x x₁ ≟-Pattern proj x₂ = no (λ ())
con x x₁ ≟-Pattern absurd x₂ = no (λ ())
dot x ≟-Pattern con x₁ x₂ = no (λ ())
dot x ≟-Pattern var x₁ = no (λ ())
dot x ≟-Pattern lit x₁ = no (λ ())
dot x ≟-Pattern proj x₁ = no (λ ())
dot x ≟-Pattern absurd x₁ = no (λ ())
var s ≟-Pattern con x x₁ = no (λ ())
var s ≟-Pattern dot x = no (λ ())
var s ≟-Pattern lit x = no (λ ())
var s ≟-Pattern proj x = no (λ ())
var s ≟-Pattern absurd x = no (λ ())
lit x ≟-Pattern con x₁ x₂ = no (λ ())
lit x ≟-Pattern dot x₁ = no (λ ())
lit x ≟-Pattern var _ = no (λ ())
lit x ≟-Pattern proj x₁ = no (λ ())
lit x ≟-Pattern absurd x₁ = no (λ ())
proj x ≟-Pattern con x₁ x₂ = no (λ ())
proj x ≟-Pattern dot x₁ = no (λ ())
proj x ≟-Pattern var _ = no (λ ())
proj x ≟-Pattern lit x₁ = no (λ ())
proj x ≟-Pattern absurd x₁ = no (λ ())
absurd x ≟-Pattern con x₁ x₂ = no (λ ())
absurd x ≟-Pattern dot x₁ = no (λ ())
absurd x ≟-Pattern var _ = no (λ ())
absurd x ≟-Pattern lit x₁ = no (λ ())
absurd x ≟-Pattern proj x₁ = no (λ ())

[]             ≟-Patterns []             = yes refl
(arg i p ∷ xs) ≟-Patterns (arg j q ∷ ys) = Lₚ.∷-dec (unArg-dec (p ≟-Pattern q)) (xs ≟-Patterns ys)

[]      ≟-Patterns (_ ∷ _) = no λ()
(_ ∷ _) ≟-Patterns []      = no λ()