failed attempt to use std lib reasoning

src/plfa/part3/Compositional.lagda.md

@@ -94,10 +94,12 @@ sub-ℱ {v = v₁ ↦ v₂ ⊔ v₁ ↦ v₃} {v₁ ↦ (v₂ ⊔ v₃)} ⟨ N2
 sub-ℱ d (⊑-trans x₁ x₂) = sub-ℱ (sub-ℱ d x₂) x₁
 ```
 
+<!--
 [PLW:
   If denotations were strengthened to be downward closed,
   we could rewrite the signature replacing (ℰ N) by d : Denotation (Γ , ★)]
 [JGS: I'll look into this.]
+-->
 
 With this subsumption property in hand, we can prove the forward
 direction of the semantic equation for lambda.  The proof is by

src/plfa/part3/Denotational.lagda.md

@@ -54,16 +54,19 @@ down a denotational semantics of the lambda calculus.
 
 ## Imports
 
+<!-- JGS: for equational reasoning
+open import Relation.Binary using (Setoid)
+-->
 ```
-open import Relation.Binary.PropositionalEquality
+open import Agda.Primitive using (lzero; lsuc)
-  using (_≡_; _≢_; refl; sym; cong; cong₂; cong-app)
+open import Data.Empty using (⊥-elim)
 open import Data.Product using (_×_; Σ; Σ-syntax; ∃; ∃-syntax; proj₁; proj₂)
   renaming (_,_ to ⟨_,_⟩)
 open import Data.Sum
-open import Agda.Primitive using (lzero)
+open import Relation.Binary.PropositionalEquality
+  using (_≡_; _≢_; refl; sym; cong; cong₂; cong-app)
 open import Relation.Nullary using (¬_)
 open import Relation.Nullary.Negation using (contradiction)
-open import Data.Empty using (⊥-elim)
 open import Function using (_∘_)
 open import plfa.part2.Untyped
   using (Context; ★; _∋_; ∅; _,_; Z; S_; _⊢_; `_; _·_; ƛ_;
@@ -187,7 +190,9 @@ property.
                             (⊑-fun (⊑-conj-R2 ⊑-refl) ⊑-refl))
 ```
 
-<!-- above might read more nicely if we introduce inequational reasoning -->
+<!--
+ [PLW: above might read more nicely if we introduce inequational reasoning.]
+ -->
 
 If the join `u ⊔ v` is less than another value `w`,
 then both `u` and `v` are less than `w`.
@@ -246,8 +251,7 @@ last γ = γ Z
 
 init-last : ∀ {Γ} → (γ : Env (Γ , ★)) → γ ≡ (init γ `, last γ)
 init-last {Γ} γ = extensionality lemma
-  where
+  where lemma : ∀ (x : Γ , ★ ∋ ★) → γ x ≡ (init γ `, last γ) x
-  lemma : ∀ (x : Γ , ★ ∋ ★) → γ x ≡ (init γ `, last γ) x
         lemma Z      =  refl
         lemma (S x)  =  refl
 ```
@@ -576,7 +580,25 @@ equal, that is, `ℰ M ≃ ℰ N`.
 The following submodule introduces equational reasoning for the `≃`
 relation.
 
+<!--
+
+JGS: I couldn't get this to work. The definitions here were accepted
+by Agda, but then the uses in the Compositional chapter got rejected.
+
+denotation-setoid : Context → Setoid (lsuc lzero) lzero
+denotation-setoid Γ = record
+  { Carrier = Denotation Γ
+  ; _≈_ = _≃_
+  ; isEquivalence = record { refl = ≃-refl ; sym = ≃-sym ; trans = ≃-trans } }
+-->
+<!--
+  The following went inside the module ≃-Reasoning:
+  
+  open import Relation.Binary.Reasoning.Setoid (denotation-setoid Γ)
+    renaming (begin_ to start_; _≈⟨_⟩_ to _≃⟨_⟩_; _∎ to _☐) public
+-->
 ```
+
 module ≃-Reasoning {Γ : Context} where
 
   infix  1 start_
@@ -589,12 +611,6 @@ module ≃-Reasoning {Γ : Context} where
     → x ≃ y
   start x≃y  =  x≃y
 
-  _≃⟨⟩_ : ∀ (x : Denotation Γ) {y : Denotation Γ}
-    → x ≃ y
-      -----
-    → x ≃ y
-  x ≃⟨⟩ x≃y  =  x≃y
-
   _≃⟨_⟩_ : ∀ (x : Denotation Γ) {y z : Denotation Γ}
     → x ≃ y
     → y ≃ z
@@ -602,6 +618,12 @@ module ≃-Reasoning {Γ : Context} where
     → x ≃ z
   (x ≃⟨ x≃y ⟩ y≃z) =  ≃-trans x≃y y≃z
 
+  _≃⟨⟩_ : ∀ (x : Denotation Γ) {y : Denotation Γ}
+    → x ≃ y
+      -----
+    → x ≃ y
+  x ≃⟨⟩ x≃y  =  x≃y
+
   _☐ : ∀ (x : Denotation Γ)
       -----
     → x ≃ x