define bff on a partial getlen

The representation chosen is to give both an injection gl₁ and a
function gl₂ (formerly getlen), such that by choosing a non-identity for
gl₁ partiality of getlen can be expressed. An alternative would have
been to allow getlen to return a Maybe ℕ and have get return
  maybe (Vec A) ⊤ (getlen n)
thus sending all inputs for which getlen yields nothing to tt. It seems
that while there is no way to obtain a such a getlen predicate from an
arbitrary index Setoid I, it should be possible to manufacture a Setoid
from a predicate. Thanks to Stefan Mehner for the insightful discussion.

1 files changed, 25 insertions, 2 deletions

diff --git a/FreeTheorems.agda b/FreeTheorems.agda
index f37cada..aacb95a 100644
--- a/FreeTheorems.agda
+++ b/FreeTheorems.agda
@@ -1,10 +1,15 @@
 module FreeTheorems where
 
+open import Level using () renaming (zero to ℓ₀)
 open import Data.Nat using (ℕ)
 open import Data.List using (List ; map)
 open import Data.Vec using (Vec) renaming (map to mapV)
 open import Function using (_∘_)
-open import Relation.Binary.PropositionalEquality using (_≗_)
+open import Function.Equality using (_⟶_ ; _⟨$⟩_)
+open import Function.Injection using (module Injection) renaming (Injection to _↪_)
+open import Relation.Binary.PropositionalEquality using (_≗_ ; cong) renaming (setoid to EqSetoid)
+open import Relation.Binary using (Setoid)
+open Injection using (to)
 
 module ListList where
   get-type : Set₁
@@ -17,5 +22,23 @@ module VecVec where
   get-type : (ℕ → ℕ) → Set₁
   get-type getlen = {A : Set} {n : ℕ} → Vec A n → Vec A (getlen n)
 
+  free-theorem-type : Set₁
+  free-theorem-type = {getlen : ℕ → ℕ} → (get : get-type getlen) → {α β : Set} → (f : α → β) → {n : ℕ} → get {_} {n} ∘ mapV f ≗ mapV f ∘ get
+
+  postulate
+    free-theorem : free-theorem-type
+
+module PartialVecVec where
+  get-type : {I : Setoid ℓ₀ ℓ₀} → (I ↪ (EqSetoid ℕ)) → (I ⟶ (EqSetoid ℕ)) → Set₁
+  get-type {I} gl₁ gl₂ = {A : Set} {i : Setoid.Carrier I} → Vec A (to gl₁ ⟨$⟩ i) → Vec A (gl₂ ⟨$⟩ i)
+
   postulate
-    free-theorem : {getlen : ℕ → ℕ} → (get : get-type getlen) → {α β : Set} → (f : α → β) → {n : ℕ} → get {_} {n} ∘ mapV f ≗ mapV f ∘ get
+    free-theorem : {I : Setoid ℓ₀ ℓ₀} → (gl₁ : I ↪ (EqSetoid ℕ)) → (gl₂ : I ⟶ (EqSetoid ℕ)) (get : get-type gl₁ gl₂)  → {α β : Set} → (f : α → β) → {i : Setoid.Carrier I} → get {_} {i} ∘ mapV f ≗ mapV f ∘ get
+
+  open VecVec using () renaming (free-theorem-type to VecVec-free-theorem-type)
+
+  ≡-to-Π : {A B : Set} → (A → B) → EqSetoid A ⟶ EqSetoid B
+  ≡-to-Π f = record { _⟨$⟩_ = f; cong = cong f }
+
+  VecVec-free-theorem : VecVec-free-theorem-type
+  VecVec-free-theorem {getlen} get = free-theorem Function.Injection.id (≡-to-Π getlen) get