add derivative proves

These proves were originally copied from https://github.com/katydid/symbolic-automatic-derivatives/blob/main/Sadol/Calculus.lean and then rewritten in Prop instead of Type

Katydid.lean

@@ -8,5 +8,6 @@ import Katydid.Example.SimpLibrary
 
 import Katydid.Regex.Regex
 import Katydid.Regex.Language
+import Katydid.Regex.Calculus
 
 import Katydid.Expr.Desc

Katydid/Regex/Calculus.lean

@@ -0,0 +1,139 @@
+import Katydid.Regex.Language
+
+namespace Calculus
+
+open Language
+open List
+open Char
+open String
+
+def null' (P: Lang α): Prop :=
+  P []
+
+def derive' (P: Lang α) (a: α): Lang α :=
+  fun (w: List α) => P (a :: w)
+
+def null {α: Type} (f: List α -> Prop): Prop :=
+  f []
+
+def derives {α: Type} (f: List α -> Prop) (u: List α): (List α -> Prop) :=
+  λ v => f (u ++ v)
+
+def derive {α: Type} (f: List α -> Prop) (a: α): (List α -> Prop) :=
+  derives f [a]
+
+attribute [simp] null derive derives
+
+def derives_emptylist : derives f [] = f :=
+  rfl
+
+def derives_strings (f: List α -> Prop) (u v: List α): derives f (u ++ v) = derives (derives f u) v :=
+  match u with
+  | [] => rfl
+  | (a :: u') => derives_strings (derive f a) u' v
+
+def null_derives (f: List α -> Prop) (u: List α): (null ∘ derives f) u = f u := by
+  simp
+
+def derives_foldl (f: List α -> Prop) (u: List α): (derives f) u = (List.foldl derive f) u :=
+  match u with
+  | [] => rfl
+  | (a :: as) => by sorry
+
+def null_emptyset {α: Type}:
+  @null α emptyset = False :=
+  rfl
+
+def null_universal {α: Type}:
+  @null α universal = True :=
+  rfl
+
+def null_emptystr {α: Type}:
+  @null α emptystr <-> True :=
+  Iff.intro
+    (fun _ => True.intro)
+    (fun _ => rfl)
+
+def null_char {α: Type} {c: α}:
+  null (char c) <-> False :=
+  Iff.intro nofun nofun
+
+def null_or {α: Type} {P Q: Lang α}:
+  null (or P Q) = ((null P) \/ (null Q)) := by
+  rfl
+
+def null_and {α: Type} {P Q: Lang α}:
+  null (and P Q) = ((null P) /\ (null Q)) :=
+  rfl
+
+def null_concat {α: Type} {P Q: Lang α}:
+  null (concat P Q) <-> ((null P) /\ (null Q)) := by
+  refine Iff.intro ?toFun ?invFun
+  case toFun =>
+    intro ⟨x, y, hx, hy, hxy⟩
+    simp at hxy
+    simp [hxy] at hx hy
+    exact ⟨hx, hy⟩
+  case invFun =>
+    exact fun ⟨x, y⟩ => ⟨[], [], x, y, rfl⟩
+
+-- def null_star {α: Type} {P: Lang α}:
+--   null (star P) <-> (List (null P)) := by
+--   refine Iff.intro ?toFun ?invFun
+--   case toFun =>
+--     sorry
+--   case invFun =>
+--     sorry
+
+def derive_emptyset {α: Type} {a: α}:
+  (derive emptyset a) = emptyset :=
+  rfl
+
+def derive_universal {α: Type} {a: α}:
+  (derive universal a) = universal :=
+  rfl
+
+def derive_emptystr {α: Type} {a: α} {w: List α}:
+  (derive emptystr a) w <-> emptyset w :=
+  Iff.intro nofun nofun
+
+def derive_char {α: Type} {a: α} {c: α} {w: List α}:
+  (derive (char c) a) w <-> (scalar (a = c) emptystr) w := by
+  refine Iff.intro ?toFun ?invFun
+  case toFun =>
+    intro D
+    cases D with | refl =>
+    exact ⟨ rfl, rfl ⟩
+  case invFun =>
+    intro ⟨ H1 , H2  ⟩
+    cases H1 with | refl =>
+    cases H2 with | refl =>
+    exact rfl
+
+def derive_or {α: Type} {a: α} {P Q: Lang α}:
+  (derive (or P Q) a) = (or (derive P a) (derive Q a)) :=
+  rfl
+
+def derive_and {α: Type} {a: α} {P Q: Lang α}:
+  (derive (and P Q) a) = (and (derive P a) (derive Q a)) :=
+  rfl
+
+def derive_scalar {α: Type} {a: α} {s: Prop} {P: Lang α}:
+  (derive (scalar s P) a) = (scalar s (derive P a)) :=
+  rfl
+
+def derive_concat {α: Type} {a: α} {P Q: Lang α} {w: List α}:
+  (derive (concat P Q) a) w <-> (scalar (null P) (or (derive Q a) (concat (derive P a) Q))) w := by
+  refine Iff.intro ?toFun ?invFun
+  case toFun =>
+    sorry
+  case invFun =>
+    sorry
+
+-- def derive_star {α: Type} {a: α} {P: Lang α} {w: List α}:
+--   (derive (star P) a) w <-> (scalar (List (null P)) (concat (derive P a) (star P))) w := by
+--   refine Iff.intro ?toFun ?invFun
+--   case toFun =>
+--     sorry
+--   case invFun =>
+--     sorry

Katydid/Regex/Language.lean

@@ -14,6 +14,11 @@ def emptystr {α: Type} : Lang α :=
 def char {α: Type} (a : α): Lang α :=
   fun w => w = [a]
 
+-- scalar is used as an and to make derive char not require an if statement
+-- (derive (char c) a) w <-> (scalar (a = c) emptystr)
+def scalar {α: Type} (s : Prop) (P : Lang α) : Lang α :=
+  fun w => s /\ P w
+
 def or {α: Type} (P : Lang α) (Q : Lang α) : Lang α :=
   fun w => P w \/ Q w
 
@@ -30,7 +35,7 @@ inductive All {α: Type} (P : α -> Prop) : (List α -> Prop) where
 
 def star {α: Type} (P : Lang α) : Lang α :=
   fun (w : List α) =>
-    ∃ (ws : List (List α)), All P ws /\ w = (List.join ws)
+    ∃ (ws : List (List α)), All P ws /\ w = (List.flatten ws)
 
 -- attribute [simp] allows these definitions to be unfolded when using the simp tactic.
 attribute [simp] universal emptyset emptystr char or and concat star
@@ -46,6 +51,7 @@ example: Lang Char := char 'b'
 example: Lang Char := (or (char 'a') emptyset)
 example: Lang Char := (and (char 'a') (char 'b'))
 example: Lang Nat := (and (char 1) (char 2))
+example: Lang Nat := (scalar True (char 2))
 example: Lang Nat := (concat (char 1) (char 2))
 
 end Language