-- And

module TotalParserCombinators.And where

open import Category.Monad
open import Data.List.Any.BagAndSetEquality
open import Data.List.Any.Membership.Propositional.Properties
import Data.List.Categorical
open import Data.Product
open import Function
open import Function.Inverse using (_↔_)
import Function.Related as Related
open import Function.Related.TypeIsomorphisms
open import Level
open import Relation.Binary.Product.Pointwise

open RawMonadPlus {f = zero} Data.List.Categorical.monadPlus
  using (_⊗_)

open import TotalParserCombinators.Congruence using (_∼[_]P_; _≅P_)
open import TotalParserCombinators.Derivative using (D)
open import TotalParserCombinators.Parser
import TotalParserCombinators.Pointwise as Pointwise
open import TotalParserCombinators.Semantics using (_∈_·_)

-- _&_ is defined as a pointwise lifting of _⊗_.

  module And {R₁ R₂ : Set} = Pointwise R₁ R₂ id _⊗_ ⊗-cong

-- p₁ & p₂ returns a result if both p₁ and p₂ do.

infixr 6 _&_ _&-cong_

_&_ :  {Tok R₁ R₂ xs₁ xs₂} 
      Parser Tok R₁ xs₁  Parser Tok R₂ xs₂ 
      Parser Tok (R₁ × R₂) (xs₁  xs₂)
_&_ = And.lift

-- D distributes over _&_.

D-& :  {Tok R₁ R₂ xs₁ xs₂ t}
      (p₁ : Parser Tok R₁ xs₁) (p₂ : Parser Tok R₂ xs₂) 
      D t (p₁ & p₂) ≅P D t p₁ & D t p₂
D-& = And.D-lift

-- _&_ preserves equality.

_&-cong_ :  {k Tok R xs₁ xs₁′ xs₂ xs₂′}
             {p₁  : Parser Tok R xs₁} {p₁′ : Parser Tok R xs₁′}
             {p₂  : Parser Tok R xs₂} {p₂′ : Parser Tok R xs₂′} 
           p₁ ∼[ k ]P p₁′  p₂ ∼[ k ]P p₂′  p₁ & p₂ ∼[ k ]P p₁′ & p₂′
_&-cong_ = And.lift-cong

-- _&_ is correct.

correct :  {Tok R₁ R₂ xs₁ xs₂ x₁ x₂ s}
          (p₁ : Parser Tok R₁ xs₁) (p₂ : Parser Tok R₂ xs₂) 
          (x₁ , x₂)  p₁ & p₂ · s  (x₁  p₁ · s × x₂  p₂ · s)
correct {x₁ = x₁} {x₂} =
     F G H  H (x₁ , x₂)  (F x₁ × G x₂))
     F↔F′ G↔G′ H↔H′ 
       Related-cong (H↔H′ (x₁ , x₂)) (F↔F′ x₁ ×-↔ G↔G′ x₂))
    (sym ⊗-∈↔)
  where open Related.EquationalReasoning