bookshelf/Bookshelf/Avigad/Chapter_7.lean

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/-! # Avigad.Chapter7
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Inductive Types
-/
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namespace Avigad.Chapter7
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/-! ### Exercise 1
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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 Leans core library, you should work
within a namespace named hide, or something like that, in order to avoid name
clashes.
-/
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namespace ex1
-- As defined in the book.
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inductive Nat where
| zero : Nat
| succ : Nat → Nat
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namespace Nat
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def add (m n : Nat) : Nat :=
match n with
| Nat.zero => m
| Nat.succ n => Nat.succ (add m n)
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instance : Add Nat where
add := add
theorem add_zero (m : Nat) : m + Nat.zero = m :=
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rfl
theorem add_succ (m n : Nat) : m + n.succ = (m + n).succ :=
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rfl
theorem zero_add (n : Nat) : Nat.zero + n = n :=
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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
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def exp (m n : Nat) : Nat :=
match n with
| Nat.zero => Nat.zero.succ
| Nat.succ n => mul m (exp m n)
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end Nat
end ex1
/-! ### Exercise 2
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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`
-/
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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
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end ex2
/-! ### Exercise 3
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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.
-/
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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
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end Avigad.Chapter7