bookshelf/TheoremProvingInLean/Chapter7.lean

/-
Chapter 7

Inductive Types
-/

-- ========================================
-- Exercise 1
--
-- Try defining other operations on the natural numbers, such as multiplication,
-- the predecessor function (with `pred 0 = 0`), truncated subtraction (with
-- `n - m = 0` when `m` is greater than or equal to `n`), and exponentiation.
-- Then try proving some of their basic properties, building on the theorems we
-- have already proved.
--
-- Since many of these are already defined in Lean’s core library, you should
-- work within a namespace named hide, or something like that, in order to avoid
-- name clashes.
-- ========================================

namespace ex1

-- As defined in the book.
inductive Nat where
  | zero : Nat
  | succ : Nat → Nat

namespace Nat

def add (m n : Nat) : Nat :=
  match n with
  | Nat.zero => m
  | Nat.succ n => Nat.succ (add m n)

instance : Add Nat where
  add := add

theorem add_zero (m : Nat) : m + Nat.zero = m :=
  rfl

theorem add_succ (m n : Nat) : m + n.succ = (m + n).succ :=
  rfl

theorem zero_add (n : Nat) : Nat.zero + n = n :=
Nat.recOn (motive := fun x => Nat.zero + x = x)
  n
  (show Nat.zero + Nat.zero = Nat.zero from rfl)
  (fun (n : Nat) (ih : Nat.zero + n = n) =>
    show Nat.zero + n.succ = n.succ from
    calc
      Nat.zero + n.succ
        = (Nat.zero + n).succ := add_succ Nat.zero n
      _ = n.succ := by rw [ih])

-- Additional definitions.
def mul (m n : Nat) : Nat :=
  match n with
  | Nat.zero => Nat.zero
  | Nat.succ n => m + mul m n

def pred (n : Nat) : Nat :=
  match n with
  | Nat.zero => Nat.zero
  | Nat.succ n => n

def sub (m n : Nat) : Nat :=
  match n with
  | Nat.zero => m
  | Nat.succ n => sub (pred m) n

def exp (m n : Nat) : Nat :=
  match n with
  | Nat.zero => Nat.zero.succ
  | Nat.succ n => mul m (exp m n)

end Nat

end ex1

-- ========================================
-- Exercise 2
--
-- Define some operations on lists, like a `length` function or the `reverse`
-- function. Prove some properties, such as the following:
--
-- a. `length (s ++ t) = length s + length t`
-- b. `length (reverse t) = length t`
-- c. `reverse (reverse t) = t`
-- ========================================

namespace ex2

variable {α : Type _}

theorem length_sum (s t : List α)
        : List.length (s ++ t) = List.length s + List.length t :=
  List.recOn s
    (by rw [List.nil_append, List.length, Nat.zero_add])
    (fun hd tl ih => by rw [
      List.length,
      List.cons_append,
      List.length,
      ih,
      Nat.add_assoc,
      Nat.add_comm t.length,
      Nat.add_assoc
    ])

theorem length_inject_anywhere (x : α) (as bs : List α)
        : List.length (as ++ [x] ++ bs) = List.length as + List.length bs + 1 := by
  induction as with
  | nil => simp
  | cons head tail ih => calc
      List.length (head :: tail ++ [x] ++ bs)
        = List.length (tail ++ [x] ++ bs) + 1 := rfl
      _ = List.length tail + List.length bs + 1 + 1 := by rw [ih]
      _ = List.length tail + (List.length bs + 1) + 1 := by rw [Nat.add_assoc (List.length tail)]
      _ = List.length tail + (1 + List.length bs) + 1 := by rw [Nat.add_comm (List.length bs)]
      _ = List.length tail + 1 + List.length bs + 1 := by rw [Nat.add_assoc (List.length tail) 1]
      _ = List.length (head :: tail) + List.length bs + 1 := rfl

theorem list_reverse_aux_append (as bs : List α)
        : List.reverseAux as bs = List.reverse as ++ bs := by
  induction as generalizing bs with
  | nil => rw [List.reverseAux, List.reverse, List.reverseAux, List.nil_append]
  | cons head tail ih => calc
      List.reverseAux (head :: tail) bs
        = List.reverseAux tail (head :: bs) := rfl
      _ = List.reverse tail ++ (head :: bs) := by rw [ih]
      _ = List.reverse tail ++ ([head] ++ bs) := rfl
      _ = List.reverse tail ++ [head] ++ bs := by rw [List.append_assoc]
      _ = List.reverseAux tail [head] ++ bs := by rw [ih]
      _ = List.reverseAux (head :: tail) [] ++ bs := rfl

theorem length_reverse (t : List α)
        : List.length (List.reverse t) = List.length t := by
  induction t with
  | nil => simp
  | cons head tail ih => calc
      List.length (List.reverse (head :: tail))
        = List.length (List.reverseAux tail [head]) := rfl
      _ = List.length (List.reverse tail ++ [head]) := by rw [list_reverse_aux_append]
      _ = List.length (List.reverse tail) + List.length [head] := by simp
      _ = List.length tail + List.length [head] := by rw [ih]
      _ = List.length tail + 1 := rfl
      _ = List.length [] + List.length tail + 1 := by simp
      _ = List.length ([] ++ [head] ++ tail) := by rw [length_inject_anywhere]
      _ = List.length (head :: tail) := rfl

theorem reverse_reverse_aux (as bs : List α)
        : List.reverse (as ++ bs) = List.reverse bs ++ List.reverse as := by
  induction as generalizing bs with
  | nil => simp
  | cons head tail ih => calc
      List.reverse (head :: tail ++ bs)
        = List.reverseAux (head :: tail ++ bs) [] := rfl
      _ = List.reverseAux (tail ++ bs) [head] := rfl
      _ = List.reverse (tail ++ bs) ++ [head] := by rw [list_reverse_aux_append]
      _ = List.reverse bs ++ List.reverse tail ++ [head] := by rw [ih]
      _ = List.reverse bs ++ (List.reverse tail ++ [head]) := by rw [List.append_assoc]
      _ = List.reverse bs ++ (List.reverseAux tail [head]) := by rw [list_reverse_aux_append]
      _ = List.reverse bs ++ (List.reverseAux (head :: tail) []) := rfl
      _ = List.reverse bs ++ List.reverse (head :: tail) := rfl

theorem reverse_reverse (t : List α)
        : List.reverse (List.reverse t) = t := by
  induction t with
  | nil => simp
  | cons head tail ih => calc
      List.reverse (List.reverse (head :: tail))
        = List.reverse (List.reverseAux tail [head]) := rfl
      _ = List.reverse (List.reverse tail ++ [head]) := by rw [list_reverse_aux_append]
      _ = List.reverse [head] ++ List.reverse (List.reverse tail) := by rw [reverse_reverse_aux]
      _ = List.reverse [head] ++ tail := by rw [ih]
      _ = [head] ++ tail := by simp
      _ = head :: tail := rfl

end ex2

-- ========================================
-- Exercise 3
--
-- Define an inductive data type consisting of terms built up from the following
-- constructors:
--
-- • `const n`, a constant denoting the natural number `n`
-- • `var n`, a variable, numbered `n`
-- • `plus s t`, denoting the sum of `s` and `t`
-- • `times s t`, denoting the product of `s` and `t`
--
-- Recursively define a function that evaluates any such term with respect to an
-- assignment of values to the variables.
-- ========================================

namespace ex3

inductive Foo : Type _
  | const : Nat → Foo
  | var : Nat → Foo
  | plus : Foo → Foo → Foo
  | times : Foo → Foo → Foo

def value_at : Nat → List Nat → Nat
| _,       [] => default
| 0,       vs => List.head! vs
| (i + 1), vs => value_at i (List.tail! vs)

-- The provided "variables" are supplied in a 0-indexed list.
def eval_foo : Foo → List Nat → Nat
  | (Foo.const n)  , _  => n
  | (Foo.var n)    , vs => value_at n vs
  | (Foo.plus m n) , vs => eval_foo m vs + eval_foo n vs
  | (Foo.times m n), vs => eval_foo m vs * eval_foo n vs

end ex3