From da35892c463cd6b9a492c6aee09726a41031ca93 Mon Sep 17 00:00:00 2001
From: Chloe Brown <chloe.brown.00@outlook.com>
Date: Mon, 29 Mar 2021 15:56:46 +0100
Subject: Rename parse to derivation.

---
 src/Cfe/Parse/Properties.agda | 132 ------------------------------------------
 1 file changed, 132 deletions(-)
 delete mode 100644 src/Cfe/Parse/Properties.agda

(limited to 'src/Cfe/Parse/Properties.agda')

diff --git a/src/Cfe/Parse/Properties.agda b/src/Cfe/Parse/Properties.agda
deleted file mode 100644
index 1b1d247..0000000
--- a/src/Cfe/Parse/Properties.agda
+++ /dev/null
@@ -1,132 +0,0 @@
-{-# OPTIONS --without-K --safe #-}
-
-open import Relation.Binary using (Setoid)
-
-module Cfe.Parse.Properties
-  {c ℓ} (over : Setoid c ℓ)
-  where
-
-open Setoid over renaming (Carrier to C)
-
-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.Parse.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._<_ ℕ._<_ on (Product.map length rank)
-
-  <-wellFounded : WellFounded _<_
-  <-wellFounded = On.wellFounded (Product.map length rank) (×-wellFounded <ⁿ-wellFounded <ⁿ-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)
-
-  e₁<e₁∨e₂ : ∀ l e₁ e₂ → (l , e₁) < (l , e₁ ∨ e₂)
-  e₁<e₁∨e₂ _ e₁ e₂ = inj₂ (≡.refl , (begin-strict
-    rank e₁                     ≤⟨ m≤m+n (rank e₁) (rank e₂) ⟩
-    rank e₁ ℕ.+ rank e₂         <⟨ n<1+n (rank e₁ ℕ.+ rank e₂ ) ⟩
-    ℕ.suc (rank e₁ ℕ.+ rank e₂) ≡⟨⟩
-    rank (e₁ ∨ e₂)              ∎))
-    where
-    open ≤-Reasoning
-
-  e₂<e₁∨e₂ : ∀ l e₁ e₂ → (l , e₂) < (l , e₁ ∨ e₂)
-  e₂<e₁∨e₂ _ e₁ e₂ = inj₂ (≡.refl , (begin-strict
-    rank e₂                     ≤⟨ m≤n+m (rank e₂) (rank e₁) ⟩
-    rank e₁ ℕ.+ rank e₂         <⟨ n<1+n (rank e₁ ℕ.+ rank e₂ ) ⟩
-    ℕ.suc (rank e₁ ℕ.+ rank e₂) ≡⟨⟩
-    rank (e₁ ∨ e₂)              ∎))
-    where
-    open ≤-Reasoning
-
-  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) (e₁<e₁∨e₂ l e₁ e₂) ∙,∙⊢e₁∶τ₁ l∈⟦e₁⟧)
-  go (l , e₁ ∨ e₂) rec (Vee _ ∙,∙⊢e₂∶τ₂ _) (inj₂ l∈⟦e₂⟧) = Veeʳ (rec (l , e₂) (e₂<e₁∨e₂ l e₁ e₂) ∙,∙⊢e₂∶τ₂ l∈⟦e₂⟧)
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