Theorem Proving in Lean. Solutions to chapter 2 exercises.

src/theorem-proving-in-lean/exercises-2.lean

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+/- Exercises 2.10
+ -
+ - Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
+-/
+
+-- Borrowed from the book.
+def double (x : ℕ) : ℕ := x + x
+def do_twice (f : ℕ → ℕ) (x : ℕ) : ℕ := f (f x)
+
+-- Exercise 1
+--
+-- Define the function `Do_Twice`, as described in Section 2.4.
+section ex_1
+def Do_Twice (f : (ℕ → ℕ) → (ℕ → ℕ)) (x : ℕ → ℕ)
+  : (ℕ → ℕ) :=
+f (f x)
+end ex_1
+
+-- Exercise 2
+--
+-- Define the functions `curry` and `uncurry`, as described in Section 2.4.
+section ex_2
+def curry (α β γ : Type*) (f : α × β → γ)
+  : α → β → γ :=
+λ α β, f (α, β)
+
+def uncurry (α β γ : Type*) (f : α → β → γ)
+  : α × β → γ :=
+λ x, f x.1 x.2
+end ex_2
+
+-- Borrowed from the book.
+universe u
+constant vec : Type u → ℕ → Type u
+
+namespace vec
+constant empty : Π (α : Type u), vec α 0
+constant cons : Π (α : Type u) (n : ℕ), α → vec α n → vec α (n + 1)
+constant append : Π (α : Type u) (n m : ℕ), vec α m → vec α n → vec α (n + m)
+end vec
+
+-- Exercise 3
+--
+-- Above, we used the example `vec α n` for vectors of elements of type `α` of
+-- length `n`. Declare a constant `vec_add` that could represent a function that
+-- adds two vectors of natural numbers of the same length, and a constant
+-- `vec_reverse` that can represent a function that reverses its argument. Use
+-- implicit arguments for parameters that can be inferred. Declare some
+-- variables and check some expressions involving the constants that you have
+-- declared.
+section ex_3
+
+namespace vec
+constant add :
+  Π {α : Type u} {n : ℕ}, vec α n → vec α n → vec α n
+constant reverse :
+  Π {α : Type u} {n : ℕ}, vec α n → vec α n
+end vec
+
+-- Check results.
+variables a b : vec Prop 1
+variables c d : vec Prop 2
+#check vec.add a b
+-- #check vec.add a c
+#check vec.reverse a
+
+end ex_3
+
+-- Exercise 4
+--
+-- Similarly, declare a constant `matrix` so that `matrix α m n` could represent
+-- the type of `m` by `n` matrices. Declare some constants to represent
+-- functions on this type, such as matrix addition and multiplication, and
+-- (using vec) multiplication of a matrix by a vector. Once again, declare some
+-- variables and check some expressions involving the constants that you have
+-- declared.
+constant matrix : Type u → ℕ → ℕ → Type u
+
+section ex_4
+
+namespace matrix
+constant add : Π {α : Type u} {m n : ℕ}, matrix α m n → matrix α m n → matrix α m n
+constant mul : Π {α : Type u} {m n p : ℕ}, matrix α m n → matrix α n p → matrix α m p
+constant app : Π {α : Type u} {m n : ℕ}, matrix α m n → vec α n → vec α m
+end matrix
+
+variables a b : matrix Prop 5 7
+variable c : matrix Prop 7 3
+variable d : vec Prop 3
+
+#check matrix.add a b
+-- #check matrix.add a c
+#check matrix.mul a c
+#check matrix.app c d
+
+end ex_4