diff --git a/Katydid/Conal/Calculus.lean b/Katydid/Conal/Calculus.lean
index 67def88..d6bca24 100644
--- a/Katydid/Conal/Calculus.lean
+++ b/Katydid/Conal/Calculus.lean
@@ -5,6 +5,9 @@ import Katydid.Conal.Function
 import Katydid.Conal.Language
 import Mathlib.Logic.Equiv.Defs -- ≃
 import Katydid.Std.Tipe
+
+namespace Calculus
+
 open dLang
 open List
 open Char
@@ -55,21 +58,36 @@ def example_of_proof_relevant_parse2 : (concat (char 'a') (char 'b' ⋃ char 'c'
 
 -- ν⇃ : Lang → Set ℓ      -- “nullable”
 -- ν⇃ P = P []
-def ν' (P : dLang α) : Type u := -- backslash nu
+def null' (P : dLang α) : Type u := -- backslash nu
   P []
 
 -- δ⇃ : Lang → A → Lang   -- “derivative”
 -- δ⇃ P a w = P (a ∷ w)
-def δ' (P : dLang α) (a : α) : dLang α := -- backslash delta
+def derive' (P : dLang α) (a : α) : dLang α := -- backslash delta
   fun (w : List α) => P (a :: w)
 
-attribute [simp] ν' δ'
+-- ν : (A ✶ → B) → B                -- “nullable”
+-- ν f = f []
+def null (f: List α -> β): β :=
+  f []
+
+-- 𝒟 : (A ✶ → B) → A ✶ → (A ✶ → B)  -- “derivative”
+-- 𝒟 f u = λ v → f (u ⊙ v)
+def derives (f: List α -> β) (u: List α): (List α -> β) :=
+  λ v => f (u ++ v)
+
+-- δ : (A ✶ → B) → A → (A ✶ → B)
+-- δ f a = 𝒟 f [ a ]
+def derive (f: List α -> β) (a: α): (List α -> β) :=
+  derives f [a]
+
+attribute [simp] null' derive'
 
 -- ν∅  : ν ∅ ≡ ⊥
 -- ν∅ = refl
 def nullable_emptySet:
   ∀ (α: Type),
-    @ν' α ∅ ≡ PEmpty := by
+    @null' α ∅ ≡ PEmpty := by
   intro α
   constructor
   rfl
@@ -78,7 +96,7 @@ def nullable_emptySet:
 -- ν𝒰 = refl
 def nullable_universal:
   ∀ (α: Type),
-    @ν' α 𝒰 ≡ PUnit := by
+    @null' α 𝒰 ≡ PUnit := by
   intro α
   constructor
   rfl
@@ -89,9 +107,9 @@ def nullable_universal:
 --   (λ { tt → refl })
 --   (λ { tt → refl })
 --   (λ { refl → refl })
-def nullable_emptyStr:
+def nullable_emptystr:
   ∀ (α: Type),
-    @ν' α ε ≃ PUnit := by
+    @null' α ε ≃ PUnit := by
   intro α
   refine Equiv.mk ?a ?b ?c ?d
   intro _
@@ -107,7 +125,7 @@ def nullable_emptyStr:
 
 def nullable_emptyStr':
   ∀ (α: Type),
-    @ν' α ε ≃ PUnit :=
+    @null' α ε ≃ PUnit :=
     fun _ => Equiv.mk
       (fun _ => PUnit.unit)
       (fun _ => by constructor; rfl)
@@ -118,7 +136,7 @@ def nullable_emptyStr':
 -- ν` = mk↔′ (λ ()) (λ ()) (λ ()) (λ ())
 def nullable_char:
   ∀ (c: α),
-    ν' (char c) ≃ PEmpty := by
+    null' (char c) ≃ PEmpty := by
   intro α
   simp
   apply Equiv.mk
@@ -133,7 +151,7 @@ def nullable_char:
 
 def nullable_char':
   ∀ (c: α),
-    ν' (char c) -> PEmpty := by
+    null' (char c) -> PEmpty := by
   intro
   refine (fun x => ?c)
   simp at x
@@ -148,7 +166,7 @@ def nullable_char':
 -- ν∪ = refl
 def nullable_or:
   ∀ (P Q: dLang α),
-    ν' (P ⋃ Q) ≡ (Sum (ν' P) (ν' Q)) := by
+    null' (P ⋃ Q) ≡ (Sum (null' P) (null' Q)) := by
   intro P Q
   constructor
   rfl
@@ -157,7 +175,7 @@ def nullable_or:
 -- ν∩ = refl
 def nullable_and:
   ∀ (P Q: dLang α),
-    ν' (P ⋂ Q) ≡ (Prod (ν' P) (ν' Q)) := by
+    null' (P ⋂ Q) ≡ (Prod (null' P) (null' Q)) := by
   intro P Q
   constructor
   rfl
@@ -166,7 +184,7 @@ def nullable_and:
 -- ν· = refl
 def nullable_scalar:
   ∀ (s: Type) (P: dLang α),
-    ν' (dLang.scalar s P) ≡ (Prod s (ν' P)) := by
+    null' (dLang.scalar s P) ≡ (Prod s (null' P)) := by
   intro P Q
   constructor
   rfl
@@ -179,7 +197,7 @@ def nullable_scalar:
 --   (λ { (([] , []) , refl , νP , νQ) → refl})
 def nullable_concat:
   ∀ (P Q: dLang α),
-    ν' (concat P Q) ≃ (Prod (ν' Q) (ν' P)) := by
+    null' (concat P Q) ≃ (Prod (null' Q) (null' P)) := by
   -- TODO
   sorry
 
@@ -213,7 +231,7 @@ def nullable_concat:
 --   ∎ where open ↔R
 def nullable_star:
   ∀ (P: dLang α),
-    ν' (P *) ≃ List (ν' P) := by
+    null' (P *) ≃ List (null' P) := by
   -- TODO
   sorry
 
@@ -221,7 +239,7 @@ def nullable_star:
 -- δ∅ = refl
 def derivative_emptySet:
   ∀ (a: α),
-    (δ' ∅ a) ≡ ∅ := by
+    (derive' ∅ a) ≡ ∅ := by
   intro a
   constructor
   rfl
@@ -230,14 +248,14 @@ def derivative_emptySet:
 -- δ𝒰 = refl
 def derivative_universal:
   ∀ (a: α),
-    (δ' 𝒰 a) ≡ 𝒰 := by
+    (derive' 𝒰 a) ≡ 𝒰 := by
   intro a
   constructor
   rfl
 
 -- δ𝟏  : δ 𝟏 a ⟷ ∅
 -- δ𝟏 = mk↔′ (λ ()) (λ ()) (λ ()) (λ ())
-def derivative_emptyStr: ∀ (w: List α), (δ' ε a) w <=> ∅ w := by
+def derivative_emptyStr: ∀ (w: List α), (derive' ε a) w <=> ∅ w := by
   intro w
   constructor
   · intro D
@@ -257,9 +275,9 @@ def derivative_emptyStr: ∀ (w: List α), (δ' ε a) w <=> ∅ w := by
 -- TODO: Redo this definition to do extensional isomorphism: `⟷` properly
 def derivative_char:
   ∀ (a: α) (c: α),
-    (δ' (char c) a) ≡ dLang.scalar (a ≡ c) ε := by
+    (derive' (char c) a) ≡ dLang.scalar (a ≡ c) ε := by
     intros a c
-    unfold δ'
+    unfold derive'
     unfold char
     unfold emptystr
     unfold scalar
@@ -269,7 +287,7 @@ def derivative_char:
 -- δ∪ = refl
 def derivative_or:
   ∀ (a: α) (P Q: dLang α),
-    (δ' (P ⋃ Q) a) ≡ ((δ' P a) ⋃ (δ' Q a)) := by
+    (derive' (P ⋃ Q) a) ≡ ((derive' P a) ⋃ (derive' Q a)) := by
   intro a P Q
   constructor
   rfl
@@ -278,7 +296,7 @@ def derivative_or:
 -- δ∩ = refl
 def derivative_and:
   ∀ (a: α) (P Q: dLang α),
-    (δ' (P ⋂ Q) a) ≡ ((δ' P a) ⋂ (δ' Q a)) := by
+    (derive' (P ⋂ Q) a) ≡ ((derive' P a) ⋂ (derive' Q a)) := by
   intro a P Q
   constructor
   rfl
@@ -287,7 +305,7 @@ def derivative_and:
 -- δ· = refl
 def derivative_scalar:
   ∀ (a: α) (s: Type) (P: dLang α),
-    (δ (dLang.scalar s P) a) ≡ (dLang.scalar s (δ' P a)) := by
+    (δ (dLang.scalar s P) a) ≡ (dLang.scalar s (derive' P a)) := by
   intro a s P
   constructor
   rfl
@@ -306,7 +324,7 @@ def derivative_scalar:
 def derivative_concat:
   ∀ (a: α) (P Q: dLang α),
   -- TODO: Redo this definition to do extensional isomorphism: `⟷` properly
-    (δ' (concat P Q) a) ≡ dLang.scalar (ν' P) ((δ' Q a) ⋃ (concat (δ' P a) Q)) := by
+    (derive' (concat P Q) a) ≡ dLang.scalar (null' P) ((derive' Q a) ⋃ (concat (derive' P a) Q)) := by
   -- TODO
   sorry
 
@@ -346,6 +364,8 @@ def derivative_concat:
 def derivative_star:
   ∀ (a: α) (P: dLang α),
   -- TODO: Redo this definition to do extensional isomorphism: `⟷` properly
-    (δ' (P *) a) ≡ dLang.scalar (List (ν' P)) (concat (δ' P a) (P *)) := by
+    (derive' (P *) a) ≡ dLang.scalar (List (null' P)) (concat (derive' P a) (P *)) := by
   -- TODO
   sorry
+
+end Calculus
diff --git a/Katydid/Conal/Decidability.lean b/Katydid/Conal/Decidability.lean
index c81afce..a79e334 100644
--- a/Katydid/Conal/Decidability.lean
+++ b/Katydid/Conal/Decidability.lean
@@ -1,6 +1,7 @@
 -- A translation to Lean from Agda
 -- https://github.com/conal/paper-2021-language-derivatives/blob/main/Decidability.lagda
 
+import Katydid.Conal.Function
 
 -- data Dec (A: Set l):Set l where
 --   yes: A → Dec A
@@ -10,13 +11,19 @@ inductive Dec (α: Type u): Type u where
   | no: (α -> PEmpty.{u}) -> Dec α
 
 -- ⊥? : Dec ⊥
--- ⊥? =no(𝜆())
+-- ⊥? = no(𝜆())
 def empty? : Dec PEmpty :=
   Dec.no (by intro; contradiction)
 
+-- ⊤‽  : Dec ⊤
+-- ⊤‽  = yes tt
 def unit? : Dec PUnit :=
   Dec.yes PUnit.unit
 
+-- _⊎‽_  : Dec A → Dec B → Dec (A ⊎ B)
+-- no ¬a  ⊎‽ no ¬b  = no [ ¬a , ¬b ]
+-- yes a  ⊎‽ no ¬b  = yes (inj₁ a)
+-- _      ⊎‽ yes b  = yes (inj₂ b)
 def sum? (a: Dec α) (b: Dec β): Dec (α ⊕ β) :=
   match (a, b) with
   | (Dec.no a, Dec.no b) =>
@@ -30,10 +37,44 @@ def sum? (a: Dec α) (b: Dec β): Dec (α ⊕ β) :=
   | (_, Dec.yes b) =>
     Dec.yes (Sum.inr b)
 
+-- _×‽_  : Dec A → Dec B → Dec (A × B)
+-- yes a  ×‽ yes b  = yes (a , b)
+-- no ¬a  ×‽ yes b  = no (¬a ∘ proj₁)
+-- _      ×‽ no ¬b  = no (¬b ∘ proj₂)
 def prod? (a: Dec α) (b: Dec β): Dec (α × β) :=
   match (a, b) with
   | (Dec.yes a, Dec.yes b) => Dec.yes (Prod.mk a b)
   | (Dec.no a, Dec.yes _) => Dec.no (fun ⟨a', _⟩ => a a')
   | (_, Dec.no b) => Dec.no (fun ⟨_, b'⟩ => b b')
 
-def list?: Dec α -> Dec (List α) := fun _ => Dec.yes []
+-- _✶‽ : Dec A → Dec (A ✶)
+-- _ ✶‽ = yes []
+def list?: Dec α -> Dec (List α) :=
+  fun _ => Dec.yes []
+
+-- map′ : (A → B) → (B → A) → Dec A → Dec B
+-- map′ A→B B→A (yes a) = yes (A→B a)
+-- map′ A→B B→A (no ¬a) = no (¬a ∘ B→A)
+def map' (ab: A -> B) (ba: B -> A) (deca: Dec A): Dec B :=
+  match deca with
+  | Dec.yes a =>
+    Dec.yes (ab a)
+  | Dec.no nota =>
+    Dec.no (nota ∘ ba)
+
+-- The following defintions are only so simple because of our approximation of <=> in Function.lean
+
+-- map‽⇔ : A ⇔ B → Dec A → Dec B
+-- map‽⇔ A⇔B = map′ (to ⟨$⟩_) (from ⟨$⟩_) where open Equivalence A⇔B
+def map? (ab: A <=> B) (deca: Dec A): Dec B :=
+  map' ab.mp ab.mpr deca
+
+-- _▹_ : A ↔ B → Dec A → Dec B
+-- f ▹ a? = map‽⇔ (↔→⇔ f) a?
+def apply (f: A <=> B) (deca: Dec A): Dec B :=
+  map? f deca
+
+-- _◃_ : B ↔ A → Dec A → Dec B
+-- g ◃ a? = ↔Eq.sym g ▹ a?
+def apply' (f: B <=> A) (deca: Dec A): Dec B :=
+  map? f.sym deca
diff --git a/Katydid/Conal/Function.lean b/Katydid/Conal/Function.lean
index ee7d4da..6ce9721 100644
--- a/Katydid/Conal/Function.lean
+++ b/Katydid/Conal/Function.lean
@@ -58,7 +58,7 @@ inductive Inverses (f: A -> B) (g: B -> A): Type (u + 1) where
   -- Modus ponens for if and only if, reversed. If `a ↔ b` and `b`, then `a`. -/
   -- mpr : b → a
 
--- We use this weaker form of Inverses, but redefine it work Type
+-- We use this weaker form of Inverses, but redefined Iff to work Type instead of Prop
 
 structure TIff (a b: Type u): Type (u + 1) where
   intro ::
@@ -67,6 +67,11 @@ structure TIff (a b: Type u): Type (u + 1) where
 
 infixr:100 " <=> " => TIff
 
+-- ↔Eq.sym
+def TIff.sym (tiff: A <=> B): B <=> A :=
+  match tiff with
+  | TIff.intro mp mpr => TIff.intro mpr mp
+
 -- Extensional (or “pointwise”) isomorphism relates predicates isomorphic on every argument: P ←→ Q = ∀ {w} → P w ↔ Q w
 
 def EIff {w: List α} (a b: List α -> Type u) := (a w) <=> (b w)
diff --git a/Katydid/Conal/Symbolic.lean b/Katydid/Conal/Symbolic.lean
index 2dcb2b1..7db055b 100644
--- a/Katydid/Conal/Symbolic.lean
+++ b/Katydid/Conal/Symbolic.lean
@@ -4,17 +4,15 @@
 import Katydid.Conal.Decidability
 import Katydid.Conal.Function
 import Katydid.Conal.Language
+import Katydid.Conal.Calculus
 
-variable {P Q : dLang α}
-variable {s : Type u}
-
-inductive Lang : (List α -> Type u) -> Type (u + 1) where
+inductive Lang {P Q : dLang α}: (List α -> Type u) -> Type (u + 1) where
   -- ∅ : Lang ◇.∅
-  | emptySet : Lang dLang.emptyset
+  | emptyset : Lang dLang.emptyset
   -- 𝒰 : Lang ◇.𝒰
   | universal : Lang dLang.universal
   -- 𝟏 : Lang ◇.𝟏
-  | emptyStr : Lang dLang.emptystr
+  | emptystr : Lang dLang.emptystr
   -- ` : (a : A) → Lang (◇.` a)
   | char {a: Type u}: (a: α) -> Lang (dLang.char a)
   -- _∪_ : Lang P → Lang Q → Lang (P ◇.∪ Q)
@@ -28,4 +26,4 @@ inductive Lang : (List α -> Type u) -> Type (u + 1) where
   -- _☆  : Lang P → Lang (P ◇.☆)
   | star : Lang P -> Lang (dLang.star P)
   -- _◂_  : (Q ⟷ P) → Lang P → Lang Q
-  | iso: (Q ⟷ P) -> Lang P -> Lang Q
+  | iso : (Q ⟷ P) -> Lang P -> Lang Q