From e7557267586b4c197663919ee83fa3e5c40e28f9 Mon Sep 17 00:00:00 2001
From: Chloe Brown <chloe.brown.00@outlook.com>
Date: Tue, 23 Jan 2024 14:54:53 +0000
Subject: Prove derivations are structurally unique.

---
 src/Cfe/Derivation/Properties.agda | 92 ++++++++++++++++++++++++++++++--------
 1 file changed, 73 insertions(+), 19 deletions(-)

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

diff --git a/src/Cfe/Derivation/Properties.agda b/src/Cfe/Derivation/Properties.agda
index 99be370..7167465 100644
--- a/src/Cfe/Derivation/Properties.agda
+++ b/src/Cfe/Derivation/Properties.agda
@@ -10,24 +10,24 @@ open Setoid over using () renaming (Carrier to C)
 
 open import Cfe.Context over using (_⊐_; Γ,Δ; ∙,∙; remove₁) renaming (wkn₂ to wkn₂ᶜ)
 open import Cfe.Derivation.Base over
-open import Cfe.Expression over
+open import Cfe.Expression over hiding (_≈_)
 open import Cfe.Fin using (zero; inj; raise!>; cast>)
 open import Cfe.Judgement over
 open import Cfe.Language over hiding (_∙_)
   renaming (_≈_ to _≈ˡ_; ≈-refl to ≈ˡ-refl; ≈-reflexive to ≈ˡ-reflexive; ≈-sym to ≈ˡ-sym)
-open import Cfe.Type over using (_⊛_; _⊨_)
+open import Cfe.Type over using (_⊛_; _⊨_; #⇒selective; ⊛⇒uniqueₗ; ⊛⇒uniqueᵣ)
 open import Cfe.Vec.Relation.Binary.Pointwise.Inductive using (Pointwise-insert)
 open import Data.Empty using (⊥-elim)
 open import Data.Fin using (Fin; zero; suc; _≟_; punchOut; punchIn)
 open import Data.Fin.Properties using (punchIn-punchOut)
 open import Data.List using (List; []; length; _++_)
 open import Data.List.Properties using (length-++)
-open import Data.List.Relation.Binary.Equality.Setoid over using (_≋_; []; _∷_)
+open import Data.List.Relation.Binary.Equality.Setoid over using (_≋_; []; _∷_; ≋-refl)
 open import Data.List.Relation.Binary.Pointwise using (Pointwise-length)
 open import Data.Nat using (ℕ; zero; suc; z≤n; s≤s; _+_) renaming (_≤_ to _≤ⁿ_)
 open import Data.Nat.Properties using (n<1+n; m≤m+n; m≤n+m; m≤n⇒m<n∨m≡n; module ≤-Reasoning)
 open import Data.Product using (_×_; _,_; -,_; ∃-syntax; map₂; proj₁; proj₂)
-open import Data.Sum using (inj₁; inj₂; [_,_]′)
+open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Vec using ([]; _∷_; [_]; lookup; insert)
 open import Data.Vec.Properties using (insert-lookup; insert-punchIn)
 open import Data.Vec.Relation.Binary.Pointwise.Inductive as Pw using ([]; _∷_)
@@ -39,6 +39,14 @@ open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong)
 open import Relation.Nullary
 open import Relation.Nullary.Decidable using (fromWitness)
 
+-- Lemma
+w,e[μe/0]<ₗₑₓw,μe : ∀ {e τ} → [ τ ] ⊐ suc zero ⊢ e ∶ τ → ∀ w → (w , e [ μ e / zero ]) <ₗₑₓ (w , μ e)
+w,e[μe/0]<ₗₑₓw,μe {e} ctx⊢e∶τ w = inj₂ (refl , (begin-strict
+  rank (e [ μ e / zero ]) ≡⟨ subst₂-pres-rank ctx⊢e∶τ zero (Fix ctx⊢e∶τ) ⟩
+  rank e                  <⟨ rank-μ e ⟩
+  rank (μ e)              ∎))
+  where open ≤-Reasoning
+
 parse : ∀ {e τ} → ∙,∙ ⊢ e ∶ τ → ∀ {w} → w ∈ ⟦ e ⟧ [] → e ⤇ w
 parse {e = e} ctx⊢e∶τ {w} w∈⟦e⟧ = All.wfRec <ₗₑₓ-wellFounded _ Pred go (w , e) ctx⊢e∶τ w∈⟦e⟧
   where
@@ -51,17 +59,10 @@ parse {e = e} ctx⊢e∶τ {w} w∈⟦e⟧ = All.wfRec <ₗₑₓ-wellFounded _
   go (w  , μ e)     rec (Fix ctx⊢e∶τ) w∈⟦e⟧ =
     Fix (rec
       (w , e [ μ e / zero ])
-      w,e[μe/0]<ₗₑₓw,μe
+      (w,e[μe/0]<ₗₑₓw,μe ctx⊢e∶τ w)
       (subst₂ ctx⊢e∶τ zero (Fix ctx⊢e∶τ))
       (∈-resp-⊆ ⟦μe⟧⊆⟦e[μe/0]⟧ w∈⟦e⟧))
     where
-    w,e[μe/0]<ₗₑₓw,μe : (w , e [ μ e / zero ]) <ₗₑₓ (w , μ e)
-    w,e[μe/0]<ₗₑₓw,μe = inj₂ (refl , (begin-strict
-      rank (e [ μ e / zero ]) ≡⟨ subst₂-pres-rank ctx⊢e∶τ zero (Fix ctx⊢e∶τ) ⟩
-      rank e                  <⟨ rank-μ e ⟩
-      rank (μ e)              ∎))
-      where open ≤-Reasoning
-
     ⟦μe⟧⊆⟦e[μe/0]⟧ : ⟦ μ e ⟧ [] ⊆ ⟦ e [ μ e / zero ] ⟧ []
     ⟦μe⟧⊆⟦e[μe/0]⟧ = begin
       ⟦ μ e ⟧ []              ⊆⟨  ⋃-unroll (⟦⟧-mono-env e ∘ (_∷ [])) ⟩
@@ -93,15 +94,9 @@ generate {e = e} ctx⊢e∶τ {w} e⤇w = All.wfRec <ₗₑₓ-wellFounded _ Pre
   go (w , Char c)  rec (Char c) (Char c∼y) = c∼y ∷ []
   go (w , μ e)     rec (Fix ctx⊢e∶τ) (Fix e[μe/0]⤇w) = ∈-resp-≈ (μ-roll e []) w∈⟦e[μe/0]⟧
     where
-    w,e[μe/0]<ₗₑₓw,μe : (w , e [ μ e / zero ]) <ₗₑₓ (w , μ e)
-    w,e[μe/0]<ₗₑₓw,μe = inj₂ (refl , (begin-strict
-      rank (e [ μ e / zero ]) ≡⟨ subst₂-pres-rank ctx⊢e∶τ zero (Fix ctx⊢e∶τ) ⟩
-      rank e                  <⟨ rank-μ e ⟩
-      rank (μ e)              ∎))
-      where open ≤-Reasoning
 
     w∈⟦e[μe/0]⟧ : w ∈ ⟦ e [ μ e / zero ] ⟧ []
-    w∈⟦e[μe/0]⟧ = rec (w , e [ μ e / zero ]) w,e[μe/0]<ₗₑₓw,μe (subst₂ ctx⊢e∶τ zero (Fix ctx⊢e∶τ)) e[μe/0]⤇w
+    w∈⟦e[μe/0]⟧ = rec (w , e [ μ e / zero ]) (w,e[μe/0]<ₗₑₓw,μe ctx⊢e∶τ w) (subst₂ ctx⊢e∶τ zero (Fix ctx⊢e∶τ)) e[μe/0]⤇w
   go (w , e₁ ∙ e₂) rec (Cat ctx⊢e₁∶τ₁ ctx⊢e₂∶τ₂ τ₁⊛τ₂) (Cat {w₁ = w₁} {w₂} e₁⤇w₁ e₂⤇w₂ eq) =
     w₁ , w₂ , w₁∈⟦e₁⟧ , w₂∈⟦e₂⟧ , eq
     where
@@ -115,3 +110,62 @@ generate {e = e} ctx⊢e∶τ {w} e⤇w = All.wfRec <ₗₑₓ-wellFounded _ Pre
     inj₁ (rec (w , e₁) (inj₂ (refl , rank-∨ˡ e₁ e₂)) ctx⊢e₁∶τ₁ e₁⤇w)
   go (w , e₁ ∨ e₂) rec (Vee ctx⊢e₁∶τ₁ ctx⊢e₂∶τ₂ τ₁#τ₂) (Veeʳ e₂⤇w) =
     inj₂ (rec (w , e₂) (inj₂ (refl , rank-∨ʳ e₁ e₂)) ctx⊢e₂∶τ₂ e₂⤇w)
+
+parse-unique : ∀ {e τ} → ∙,∙ ⊢ e ∶ τ → ∀ {w} (d₁ d₂ : e ⤇ w) → d₁ ≈ d₂
+parse-unique {e = e} ctx⊢e∶τ {w} d₁ d₂ =
+  All.wfRec <ₗₑₓ-wellFounded _ Pred go (w , e) ctx⊢e∶τ ≋-refl d₁ d₂
+  where
+  Pred : (List C × Expression 0) → Set _
+  Pred (w , e) = ∀ {τ} → ∙,∙ ⊢ e ∶ τ → ∀ {w′} → w ≋ w′ → (d₁ : e ⤇ w) → (d₂ : e ⤇ w′) → d₁ ≈ d₂
+
+  go : ∀ w,e → WfRec _<ₗₑₓ_ Pred w,e → Pred w,e
+  go (w , ε)       rec Eps           eq Eps        Eps         = Eps
+  go (w , Char c)  rec (Char c)      eq (Char c∼y) (Char c∼y′) = Char c∼y c∼y′
+  go (w , μ e)     rec (Fix ctx⊢e∶τ) eq (Fix d₁)   (Fix d₂)    =
+    Fix (rec
+      (w , e [ μ e / zero ])
+      (w,e[μe/0]<ₗₑₓw,μe ctx⊢e∶τ w)
+      (subst₂ ctx⊢e∶τ zero (Fix ctx⊢e∶τ))
+      eq d₁ d₂)
+  go (w , e₁ ∨ e₂) rec (Vee ctx⊢e₁∶τ₁ ctx⊢e₂∶τ₂ τ₁#τ₂) eq (Veeˡ d₁) (Veeˡ d₂) =
+    Veeˡ (rec (w , e₁) (inj₂ (refl , rank-∨ˡ e₁ e₂)) ctx⊢e₁∶τ₁ eq d₁ d₂)
+  go (w , e₁ ∨ e₂) rec (Vee ctx⊢e₁∶τ₁ ctx⊢e₂∶τ₂ τ₁#τ₂) eq (Veeˡ d₁) (Veeʳ d₂) =
+    let w∈⟦e₁⟧ = ⟦ e₁ ⟧ [] .∈-resp-≋ eq (generate ctx⊢e₁∶τ₁ d₁) in
+    let w∈⟦e₂⟧ = generate ctx⊢e₂∶τ₂ d₂ in
+    let ⟦e₁⟧⊨τ₁ = soundness ctx⊢e₁∶τ₁ [] [] in
+    let ⟦e₂⟧⊨τ₂ = soundness ctx⊢e₂∶τ₂ [] [] in
+    ⊥-elim (#⇒selective τ₁#τ₂ ⟦e₁⟧⊨τ₁ ⟦e₂⟧⊨τ₂ (w∈⟦e₁⟧ , w∈⟦e₂⟧))
+    where open Language
+  go (w , e₁ ∨ e₂) rec (Vee ctx⊢e₁∶τ₁ ctx⊢e₂∶τ₂ τ₁#τ₂) eq (Veeʳ d₁) (Veeˡ d₂) =
+    let w∈⟦e₁⟧ = generate ctx⊢e₁∶τ₁ d₂ in
+    let w∈⟦e₂⟧ = ⟦ e₂ ⟧ [] .∈-resp-≋ eq (generate ctx⊢e₂∶τ₂ d₁) in
+    let ⟦e₁⟧⊨τ₁ = soundness ctx⊢e₁∶τ₁ [] [] in
+    let ⟦e₂⟧⊨τ₂ = soundness ctx⊢e₂∶τ₂ [] [] in
+    ⊥-elim (#⇒selective τ₁#τ₂ ⟦e₁⟧⊨τ₁ ⟦e₂⟧⊨τ₂ (w∈⟦e₁⟧ , w∈⟦e₂⟧))
+    where open Language
+  go (w , e₁ ∨ e₂) rec (Vee ctx⊢e₁∶τ₁ ctx⊢e₂∶τ₂ τ₁#τ₂) eq (Veeʳ d₁) (Veeʳ d₂) =
+    Veeʳ (rec (w , e₂) (inj₂ (refl , rank-∨ʳ e₁ e₂)) ctx⊢e₂∶τ₂ eq d₁ d₂)
+  go (w , e₁ ∙ e₂) rec ctx⊢e₁∙e₂:τ₁∙τ₂@(Cat ctx⊢e₁∶τ₁ ctx⊢e₂∶τ₂ τ₁⊛τ₂) eq
+    d₁@(Cat {w₁ = w₁} {w₂} d₁₁ d₁₂ eq₁)
+    d₂@(Cat {w₁ = w₃} {w₄} d₂₁ d₂₂ eq₂) =
+    Cat
+      (rec (w₁ , e₁) (lex-∙ˡ e₁ e₂ w₁ eq₁) ctx⊢e₁∶τ₁ w₁≋w₃ d₁₁ d₂₁)
+      (rec (w₂ , e₂) (lex-∙ʳ e₁ e₂ [] ε∉⟦e₁⟧ w₁∈⟦e₁⟧ eq₁) ctx⊢e₂∶τ₂ w₂≋w₄ d₁₂ d₂₂)
+      eq₁
+      eq₂
+    where
+    open _⊛_ using (¬n₁); open _⊨_ using (n⇒n); open Language
+    ⟦e₁⟧⊨τ₁ = soundness ctx⊢e₁∶τ₁ [] []
+    ⟦e₂⟧⊨τ₂ = soundness ctx⊢e₂∶τ₂ [] []
+
+    ε∉⟦e₁⟧ = τ₁⊛τ₂ .¬n₁ ∘ ⟦e₁⟧⊨τ₁ .n⇒n
+
+    w₁w₂∈⟦e₁∙e₂⟧ = ⟦ e₁ ∙ e₂ ⟧ [] .∈-resp-≋ eq (generate ctx⊢e₁∙e₂:τ₁∙τ₂ d₁)
+    w₃w₄∈⟦e₁∙e₂⟧ = generate ctx⊢e₁∙e₂:τ₁∙τ₂ d₂
+    w₁∈⟦e₁⟧ = generate ctx⊢e₁∶τ₁ d₁₁
+
+    w₁≋w₃ : w₁ ≋ w₃
+    w₁≋w₃ = ⊛⇒uniqueₗ τ₁⊛τ₂ ⟦e₁⟧⊨τ₁ ⟦e₂⟧⊨τ₂ w₁w₂∈⟦e₁∙e₂⟧ w₃w₄∈⟦e₁∙e₂⟧
+
+    w₂≋w₄ : w₂ ≋ w₄
+    w₂≋w₄ = ⊛⇒uniqueᵣ τ₁⊛τ₂ ⟦e₁⟧⊨τ₁ ⟦e₂⟧⊨τ₂ w₁w₂∈⟦e₁∙e₂⟧ w₃w₄∈⟦e₁∙e₂⟧
-- 
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