generalize lemma-insert-same to arbitrary Setoids

The general idea is to enable bff to use arbitrary DecSetoids.
 * assoc needs to learn about DecSetoid
 * checkInsert needs to learn about DecSetoid
 * InsertionResult needs to learn about Setoid
 * Parameters to InsertionResult.same become weaker
 * lemma-checkInsert-restrict should work with weaker same
 * lemma-insert-same needs to learn about Setoid

1 files changed, 25 insertions, 3 deletions

diff --git a/Generic.agda b/Generic.agda
index 11b9594..33cd1ac 100644
--- a/Generic.agda
+++ b/Generic.agda
@@ -3,14 +3,16 @@ module Generic where
 import Category.Functor
 import Category.Monad
 open import Data.List using (List ; length ; replicate) renaming ([] to []L ; _∷_ to _∷L_)
-open import Data.Maybe using (Maybe ; just ; nothing)
+open import Data.Maybe using (Maybe ; just ; nothing) renaming (setoid to MaybeEq)
 open import Data.Nat using (ℕ ; zero ; suc)
 open import Data.Product using (_×_ ; _,_)
 open import Data.Vec using (Vec ; toList ; fromList ; map) renaming ([] to []V ; _∷_ to _∷V_)
+open import Data.Vec.Equality using () renaming (module Equality to VecEq)
 open import Function using (_∘_)
-import Level
+open import Level using () renaming (zero to ℓ₀)
+open import Relation.Binary using (Setoid ; module Setoid)
 open import Relation.Binary.Core using (_≡_ ; refl)
-open import Relation.Binary.PropositionalEquality using (_≗_ ; cong ; subst ; trans)
+open import Relation.Binary.PropositionalEquality using (_≗_ ; cong ; subst ; trans) renaming (setoid to PropEq)
 
 open Category.Functor.RawFunctor {Level.zero} Data.Maybe.functor using (_<$>_)
 open Category.Monad.RawMonad {Level.zero} Data.Maybe.monad using (_>>=_)
@@ -46,6 +48,14 @@ mapMV-purity : {A B : Set} {n : ℕ} → (f : A → B) → (v : Vec A n) → map
 mapMV-purity f []V = refl
 mapMV-purity f (x ∷V xs) rewrite mapMV-purity f xs = refl
 
+maybeEq-from-≡ : {A : Set} {a b : Maybe A} → Setoid._≈_ (PropEq (Maybe A)) a b → Setoid._≈_ (MaybeEq (PropEq A)) a b
+maybeEq-from-≡ {a = just x}  {b = .(just x)} refl = just refl
+maybeEq-from-≡ {a = nothing} {b = .nothing}  refl = nothing
+
+maybeEq-to-≡ : {A : Set} {a b : Maybe A} → Setoid._≈_ (MaybeEq (PropEq A)) a b → Setoid._≈_ (PropEq (Maybe A)) a b
+maybeEq-to-≡ (just refl) = refl
+maybeEq-to-≡ nothing     = refl
+
 subst-cong : {A : Set} → (T : A → Set) → {g : A → A} → {a b : A} → (f : {c : A} → T c → T (g c)) → (p : a ≡ b) →
              f ∘ subst T p ≗ subst T (cong g p) ∘ f
 subst-cong T f refl _ = refl
@@ -65,3 +75,15 @@ toList-fromList (x ∷L xs) = cong (_∷L_ x) (toList-fromList xs)
 toList-subst : {A : Set} → {n m : ℕ} (v : Vec A n) → (p : n ≡ m) →
                toList (subst (Vec A) p v) ≡ toList v
 toList-subst v refl = refl
+
+vecIsSetoid : Setoid ℓ₀ ℓ₀ → ℕ → Setoid ℓ₀ ℓ₀
+vecIsSetoid S n = record
+  { Carrier       = Vec S.Carrier n
+  ; _≈_           = λ x y → VecEq._≈_ S {n} x {n} y
+  ; isEquivalence = record
+    { refl = VecEq.refl S _
+    ; sym = VecEq.sym S
+    ; trans = VecEq.trans S }
+  }
+  where
+  module S = Setoid S