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{-# OPTIONS --without-K --safe #-}
open import Relation.Binary using (Setoid)
module Cfe.Derivation.Properties
{c ℓ} (over : Setoid c ℓ)
where
open Setoid over renaming (Carrier to C; _≈_ to _∼_)
open import Cfe.Context over hiding (_≋_)
open import Cfe.Expression over hiding (_≋_)
open import Cfe.Language over hiding (≤-refl; _≈_; _<_)
open import Cfe.Language.Construct.Concatenate over using (Concat)
open import Cfe.Language.Indexed.Construct.Iterate over
open import Cfe.Judgement over
open import Cfe.Derivation.Base over
open import Cfe.Type over using (_⊛_; _⊨_)
open import Data.Bool using (T; not; true; false)
open import Data.Empty using (⊥-elim)
open import Data.Fin as F hiding (_<_)
open import Data.List hiding (null)
open import Data.List.Relation.Binary.Equality.Setoid over
open import Data.Nat as ℕ hiding (_⊔_; _^_; _<_)
open import Data.Nat.Properties using (≤-step; m≤m+n; m≤n+m; ≤-refl; n<1+n; module ≤-Reasoning)
open import Data.Nat.Induction using () renaming (<-wellFounded to <ⁿ-wellFounded)
open import Data.Product as Product
open import Data.Product.Relation.Binary.Lex.Strict
open import Data.Sum as Sum
open import Data.Vec hiding (length; _++_)
open import Data.Vec.Relation.Binary.Pointwise.Inductive
open import Data.Vec.Relation.Binary.Pointwise.Extensional
open import Function
open import Induction.WellFounded
open import Level
open import Relation.Binary
import Relation.Binary.Construct.On as On
open import Relation.Binary.PropositionalEquality as ≡ hiding (subst₂; setoid)
private
infix 4 _<_
_<_ : Rel (List C × Expression 0) _
_<_ = ×-Lex _≡_ ℕ._<_ _<ᵣₐₙₖ_ on (Product.map₁ length)
<-wellFounded : WellFounded _<_
<-wellFounded = On.wellFounded (Product.map₁ length) (×-wellFounded <ⁿ-wellFounded <ᵣₐₙₖ-wellFounded)
l∈⟦e⟧⇒e⤇l : ∀ {e τ} → ∙,∙ ⊢ e ∶ τ → ∀ {l} → l ∈ ⟦ e ⟧ [] → e ⤇ l
l∈⟦e⟧⇒e⤇l {e} {τ} ∙,∙⊢e∶τ {l} l∈⟦e⟧ = All.wfRec <-wellFounded _ Pred go (l , e) ∙,∙⊢e∶τ l∈⟦e⟧
where
Pred : List C × Expression 0 → Set _
Pred (l , e) = ∀ {τ} → ∙,∙ ⊢ e ∶ τ → l ∈ ⟦ e ⟧ [] → e ⤇ l
e[μe/0]<μe : ∀ {e τ} l → ∙,∙ ⊢ μ e ∶ τ → (l , e [ μ e / F.zero ]) < (l , μ e)
e[μe/0]<μe {e} l (Fix ∙,τ⊢e∶τ)= inj₂ (≡.refl , (begin-strict
rank (e [ μ e / F.zero ]) ≡⟨ subst-preserves-rank z≤n ∙,τ⊢e∶τ (Fix ∙,τ⊢e∶τ) ⟩
rank e <⟨ n<1+n (rank e) ⟩
ℕ.suc (rank e) ≡⟨⟩
rank (μ e) ∎))
where
open ≤-Reasoning
l₁++l₂≋l⇒∣l₁∣≤∣l∣ : ∀ {l₂ l} l₁ → l₁ ++ l₂ ≋ l → (length l₁ ℕ.< length l) ⊎ (length l₁ ≡ length l)
l₁++l₂≋l⇒∣l₁∣≤∣l∣ [] [] = inj₂ ≡.refl
l₁++l₂≋l⇒∣l₁∣≤∣l∣ [] (_ ∷ _) = inj₁ (s≤s z≤n)
l₁++l₂≋l⇒∣l₁∣≤∣l∣ (_ ∷ l₁) (_ ∷ eq) = Sum.map s≤s (cong ℕ.suc) (l₁++l₂≋l⇒∣l₁∣≤∣l∣ l₁ eq)
l₁++l₂≋l⇒∣l₂∣≤∣l∣ : ∀ {l₂ l} l₁ → l₁ ++ l₂ ≋ l → (length l₂ ℕ.< length l) ⊎ (length l₁ ≡ 0)
l₁++l₂≋l⇒∣l₂∣≤∣l∣ [] _ = inj₂ ≡.refl
l₁++l₂≋l⇒∣l₂∣≤∣l∣ (_ ∷ []) (_ ∷ []) = inj₁ (s≤s z≤n)
l₁++l₂≋l⇒∣l₂∣≤∣l∣ (x ∷ []) (x∼y ∷ _ ∷ eq) = inj₁ ([ s≤s , (λ ()) ]′ (l₁++l₂≋l⇒∣l₂∣≤∣l∣ (x ∷ []) (x∼y ∷ eq)))
l₁++l₂≋l⇒∣l₂∣≤∣l∣ (_ ∷ x ∷ l₁) (_ ∷ eq) = inj₁ ([ ≤-step , (λ ()) ]′ (l₁++l₂≋l⇒∣l₂∣≤∣l∣ (x ∷ l₁) eq))
e₁<e₁∙e₂ : ∀ {l e₁} e₂ → (l∈⟦e₁∙e₂⟧ : l ∈ ⟦ e₁ ∙ e₂ ⟧ []) → (Concat.l₁ l∈⟦e₁∙e₂⟧ , e₁) < (l , e₁ ∙ e₂)
e₁<e₁∙e₂ _ l∈⟦e₁∙e₂⟧ with l₁++l₂≋l⇒∣l₁∣≤∣l∣ (Concat.l₁ l∈⟦e₁∙e₂⟧) (Concat.eq l∈⟦e₁∙e₂⟧)
... | inj₁ ∣l₁∣<∣l∣ = inj₁ ∣l₁∣<∣l∣
... | inj₂ ∣l₁∣≡∣l∣ = inj₂ (∣l₁∣≡∣l∣ , ≤-refl)
e₂<e₁∙e₂ : ∀ {l e₁ e₂ τ} → ∙,∙ ⊢ e₁ ∙ e₂ ∶ τ → (l∈⟦e₁∙e₂⟧ : l ∈ ⟦ e₁ ∙ e₂ ⟧ []) → (Concat.l₂ l∈⟦e₁∙e₂⟧ , e₂) < (l , e₁ ∙ e₂)
e₂<e₁∙e₂ (Cat ∙,∙⊢e₁∶τ₁ _ τ₁⊛τ₂) l∈⟦e₁∙e₂⟧ with l₁++l₂≋l⇒∣l₂∣≤∣l∣ (Concat.l₁ l∈⟦e₁∙e₂⟧) (Concat.eq l∈⟦e₁∙e₂⟧)
... | inj₁ ∣l₂∣<∣l∣ = inj₁ ∣l₂∣<∣l∣
... | inj₂ ∣l₁∣≡0 with Concat.l₁ l∈⟦e₁∙e₂⟧ | Concat.l₁∈A l∈⟦e₁∙e₂⟧ | (_⊛_.τ₁.Null τ₁⊛τ₂) | _⊛_.¬n₁ τ₁⊛τ₂ | _⊨_.n⇒n (soundness ∙,∙⊢e₁∶τ₁ [] (ext (λ ()))) | ∣l₁∣≡0
... | [] | ε∈A | false | _ | n⇒n | refl = ⊥-elim (n⇒n ε∈A)
l∈⟦e⟧ⁿ⇒l∈⟦e[μe/0]⟧ : ∀ {l} e n → l ∈ ((λ X → ⟦ e ⟧ (X ∷ [])) ^ n) (⟦ ⊥ ⟧ []) → l ∈ ⟦ e [ μ e / F.zero ] ⟧ []
l∈⟦e⟧ⁿ⇒l∈⟦e[μe/0]⟧ e (suc n) l∈⟦e⟧ⁿ = _≤_.f
(begin
((λ X → ⟦ e ⟧ (X ∷ [])) ^ (ℕ.suc n)) (⟦ ⊥ ⟧ []) ≡⟨⟩
⟦ e ⟧ (((λ X → ⟦ e ⟧ (X ∷ [])) ^ n) (⟦ ⊥ ⟧ []) ∷ []) ≤⟨ mono e (fⁿ≤⋃f (λ X → ⟦ e ⟧ (X ∷ [])) n ∷ []) ⟩
⟦ e ⟧ (⋃ (λ X → ⟦ e ⟧ (X ∷ [])) ∷ []) ≡⟨⟩
⟦ e ⟧ (⟦ μ e ⟧ [] ∷ []) ≈˘⟨ subst-fun e (μ e) F.zero [] ⟩
⟦ e [ μ e / F.zero ] ⟧ [] ∎)
l∈⟦e⟧ⁿ
where
open import Relation.Binary.Reasoning.PartialOrder (poset _)
go : ∀ l,e → WfRec _<_ Pred l,e → Pred l,e
go (l , e) rec Eps (lift refl) = Eps
go (l , e) rec (Char c) (lift (c∼y ∷ [])) = Char c∼y
go (l , μ e) rec (Fix ∙,τ⊢e∶τ) (n , l∈⟦e⟧ⁿ) =
Fix (rec
(l , e [ μ e / F.zero ])
(e[μe/0]<μe l (Fix ∙,τ⊢e∶τ))
(subst₂ z≤n ∙,τ⊢e∶τ (Fix ∙,τ⊢e∶τ))
(l∈⟦e⟧ⁿ⇒l∈⟦e[μe/0]⟧ e n l∈⟦e⟧ⁿ))
go (l , e₁ ∙ e₂) rec (∙,∙⊢e₁∙e₂∶τ @ (Cat ∙,∙⊢e₁∶τ₁ ∙,∙⊢e₂∶τ₂ _)) l∈⟦e⟧ =
Cat
(rec (l∈⟦e⟧.l₁ , e₁) (e₁<e₁∙e₂ e₂ l∈⟦e⟧) ∙,∙⊢e₁∶τ₁ l∈⟦e⟧.l₁∈A)
(rec (l∈⟦e⟧.l₂ , e₂) (e₂<e₁∙e₂ ∙,∙⊢e₁∙e₂∶τ l∈⟦e⟧) ∙,∙⊢e₂∶τ₂ l∈⟦e⟧.l₂∈B)
l∈⟦e⟧.eq
where
module l∈⟦e⟧ = Concat l∈⟦e⟧
go (l , e₁ ∨ e₂) rec (Vee ∙,∙⊢e₁∶τ₁ _ _) (inj₁ l∈⟦e₁⟧) =
Veeˡ (rec (l , e₁) (inj₂ (≡.refl , e₁<ᵣₐₙₖe₁∨e₂ e₁ e₂)) ∙,∙⊢e₁∶τ₁ l∈⟦e₁⟧)
go (l , e₁ ∨ e₂) rec (Vee _ ∙,∙⊢e₂∶τ₂ _) (inj₂ l∈⟦e₂⟧) =
Veeʳ (rec (l , e₂) (inj₂ (≡.refl , e₂<ᵣₐₙₖe₁∨e₂ e₁ e₂)) ∙,∙⊢e₂∶τ₂ l∈⟦e₂⟧)
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