Add SFun module

Author: WhatisRT

formal-spec/CategoricalCrypto/SFun.agda

@@ -0,0 +1,67 @@
+{-# OPTIONS --safe #-}
+module CategoricalCrypto.SFun where
+
+open import Level renaming (zero to ℓ0)
+
+open import abstract-set-theory.Prelude hiding (id; _∘_; _⊗_; lookup; Dec; ⊤; ⊥; Functor; Bifunctor; [_]; head; tail; _++_; take)
+import abstract-set-theory.Prelude as P
+open import Data.Vec hiding (init)
+open import Data.Nat using (_+_)
+
+-- M = id, Maybe, Powerset (relation), Giry (probability)
+-- SFunType A B S = S × A → M (S × B)
+SFunType : Type → Type → Type → Type
+SFunType A B S = S × A → S × B
+
+-- explicit state
+record SFunᵉ (A B : Type) : Type₁ where
+  field
+    State : Type
+    init  : State
+    fun   : SFunType A B State
+
+private variable A B C D State : Type
+
+trace : ∀ {n} → SFunType A B State → State → Vec A n → Vec B n
+trace f s [] = []
+trace f s (a ∷ as) = let (s , b) = f (s , a) in b ∷ trace f s as
+
+take-trace : ∀ {f : SFunType A B State} {s m n} {as : Vec A (n + m)}
+           → take n (trace f s as) ≡ trace f s (take n as)
+take-trace {n = zero} = refl
+take-trace {n = suc _} {_ ∷ _} = cong (_ ∷_) take-trace
+
+-- implicit state
+record SFunⁱ (A B : Type) : Type where
+  field
+    fun : ∀ {n} → Vec A n → Vec B n
+    take-fun : ∀ {m n} {as : Vec A (m + n)} → take m (fun as) ≡ fun (take m as)
+
+  fun₁ : A → B
+  fun₁ a = head (fun [ a ])
+
+  -- the function on traces after making one fixed step
+  apply₁ : A → SFunⁱ A B
+  apply₁ a = record { fun = λ as → tail (fun (a ∷ as)) ; take-fun = {!!} }
+
+eval : SFunᵉ A B → SFunⁱ A B
+eval f = let open SFunᵉ f in record { fun = trace fun init ; take-fun = take-trace }
+
+resume : SFunⁱ A B → SFunᵉ A B
+resume f = record
+  { init = f
+  ; fun = λ where (g , a) → SFunⁱ.apply₁ g a , SFunⁱ.fun₁ g a
+  }
+
+_≈ⁱ_ : SFunⁱ A B → SFunⁱ A B → Type
+f ≈ⁱ g = let open SFunⁱ in ∀ {n} → fun f {n} ≗ fun g {n}
+
+open SFunⁱ
+≈ⁱ-ind : ∀ {f g : SFunⁱ A B} a → fun₁ f a ≡ fun₁ g a → apply₁ f a ≈ⁱ apply₁ g a → f ≈ⁱ g
+≈ⁱ-ind = {!!}
+
+_≈ᵉ_ : SFunᵉ A B → SFunᵉ A B → Type
+f ≈ᵉ g = eval f ≈ⁱ eval g
+
+resume∘eval≡id : ∀ {f : SFunᵉ A B} → resume (eval f) ≈ᵉ f
+resume∘eval≡id {f = f} {n} as = {!!}