S01_Calculating.lean

import LeanInVienna.Common
import Mathlib.Data.Real.Basic

/- The following exercises with `calc` blocks from *The Mechanics of Proof* by Heather Macbeth. -/

example {a b : ℚ} (h1 : a - b = 4) (h2 : a * b = 1) : (a + b) ^ 2 = 20 :=
  calc
    (a + b) ^ 2 = (a - b) ^ 2 + 4 * (a * b) := by ring
    _ = 4 ^ 2 + 4 * 1 := by rw [h1, h2]
    _ = 20 := by ring

-- Exercise: replace the words "sorry" with the correct Lean justification.
example {r s : ℝ} (h1 : s = 3) (h2 : r + 2 * s = -1) : r = -7 :=
  calc
    r = r + 2 * s - 2 * s := by sorry
    _ = -1 - 2 * s := by sorry
    _ = -1 - 2 * 3 := by sorry
    _ = -7 := by sorry

-- Exercise: replace the words "sorry" with the correct Lean justification.
example {a b m n : ℤ} (h1 : a * m + b * n = 1) (h2 : b ^ 2 = 2 * a ^ 2) :
    (2 * a * n + b * m) ^ 2 = 2 :=
  calc
    (2 * a * n + b * m) ^ 2
      = 2 * (a * m + b * n) ^ 2 + (m ^ 2 - 2 * n ^ 2) * (b ^ 2 - 2 * a ^ 2) := by sorry
    _ = 2 * 1 ^ 2 + (m ^ 2 - 2 * n ^ 2) * (2 * a ^ 2 - 2 * a ^ 2) := by sorry
    _ = 2 := by sorry

example {a b c d e f : ℤ} (h1 : a * d = b * c) (h2 : c * f = d * e) :
    d * (a * f - b * e) = 0 :=
  sorry

example {a b : ℤ} (h1 : a = 2 * b + 5) (h2 : b = 3) : a = 11 :=
  sorry

example {x : ℤ} (h1 : x + 4 = 2) : x = -2 :=
  sorry

example {a b : ℝ} (h1 : a - 5 * b = 4) (h2 : b + 2 = 3) : a = 9 :=
  sorry

example {w : ℚ} (h1 : 3 * w + 1 = 4) : w = 1 :=
  sorry

example {x : ℤ} (h1 : 2 * x + 3 = x) : x = -3 :=
  sorry

example {x y : ℤ} (h1 : 2 * x - y = 4) (h2 : y - x + 1 = 2) : x = 5 :=
  sorry

example {u v : ℚ} (h1 : u + 2 * v = 4) (h2 : u - 2 * v = 6) : u = 5 :=
  sorry

example {x y : ℝ} (h1 : x + y = 4) (h2 : 5 * x - 3 * y = 4) : x = 2 :=
  sorry

example {a b : ℚ} (h1 : a - 3 = 2 * b) : a ^ 2 - a + 3 = 4 * b ^ 2 + 10 * b + 9 :=
  sorry

example {z : ℝ} (h1 : z ^ 2 - 2 = 0) : z ^ 4 - z ^ 3 - z ^ 2 + 2 * z + 1 = 3 :=
  sorry

example {x y : ℝ} (h1 : x = 3) (h2 : y = 4 * x - 3) : y = 9 :=
  sorry

example {a b : ℤ} (h : a - b = 0) : a = b :=
  sorry

example {x y : ℤ} (h1 : x - 3 * y = 5) (h2 : y = 3) : x = 14 :=
  sorry

example {p q : ℚ} (h1 : p - 2 * q = 1) (h2 : q = -1) : p = -1 :=
  sorry

example {x y : ℚ} (h1 : y + 1 = 3) (h2 : x + 2 * y = 3) : x = -1 :=
  sorry

example {p q : ℤ} (h1 : p + 4 * q = 1) (h2 : q - 1 = 2) : p = -11 :=
  sorry

example {a b c : ℝ} (h1 : a + 2 * b + 3 * c = 7) (h2 : b + 2 * c = 3) (h3 : c = 1) : a = 2 :=
  sorry

example {u v : ℚ} (h1 : 4 * u + v = 3) (h2 : v = 2) : u = 1 / 4 :=
  sorry

example {c : ℚ} (h1 : 4 * c + 1 = 3 * c - 2) : c = -3 :=
  sorry


-- An example.
example (a b c : ℝ) : a * b * c = b * (a * c) := by
  rw [mul_comm a b]
  rw [mul_assoc b a c]

-- Try these.
example (a b c : ℝ) : c * b * a = b * (a * c) := by
  rw [mul_comm c b]
  rw [mul_assoc b c a]
  rw [mul_comm a c]


example (a b c : ℝ) : a * (b * c) = b * (a * c) := by
  rw [mul_comm a c] -- ⊢ a * (b * c) = b * (c * a)
  let b' := b * c
  rw [mul_comm a b'] -- ⊢ b' * a = b * (c * a)
  rw [mul_assoc b c a]
  -- ring

-- An example.
example (a b c : ℝ) : a * b * c = b * c * a := by
  rw [mul_assoc a b c]
  let b' := b * c
  rw [mul_comm a b']

/- Try doing the first of these without providing any arguments at all,
   and the second with only one argument. -/
example (a b c : ℝ) : a * (b * c) = b * (c * a) := by
  let b' := b * c
  rw[mul_comm a b']
  rw[mul_assoc b c a]
  --rw[mul_comm]
  --rw[mul_assoc]

example (a b c : ℝ) : a * (b * c) = b * (a * c) := by
  sorry

-- Using facts from the local context.
example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by
  rw [h']
  rw [← mul_assoc]
  rw [h]
  rw [mul_assoc]

example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by
  sorry

example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by
  sorry

example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by
  rw [h', ← mul_assoc, h, mul_assoc]

section

variable (a b c d e f : ℝ)

example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by
  rw [h', ← mul_assoc, h, mul_assoc]

end

section
variable (a b c : ℝ)

#check a
#check a + b
#check (a : ℝ)
#check mul_comm a b
#check (mul_comm a b : a * b = b * a)
#check mul_assoc c a b
#check mul_comm a
#check mul_comm

end

section
variable (a b : ℝ)

example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by
  rw [mul_add, add_mul, add_mul]
  rw [← add_assoc, add_assoc (a * a)]
  rw [mul_comm b a, ← two_mul]

example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b :=
  calc
    (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by
      rw [mul_add, add_mul, add_mul]
    _ = a * a + (b * a + a * b) + b * b := by
      rw [← add_assoc, add_assoc (a * a)]
    _ = a * a + 2 * (a * b) + b * b := by
      rw [mul_comm b a, ← two_mul]

example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b :=
  calc
    (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by
      sorry
    _ = a * a + (b * a + a * b) + b * b := by
      sorry
    _ = a * a + 2 * (a * b) + b * b := by
      sorry

end

-- Try these. For the second, use the theorems listed underneath.
section
variable (a b c d : ℝ)

example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := by
  sorry

example (a b : ℝ) : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by
  sorry

#check pow_two a
#check mul_sub a b c
#check add_mul a b c
#check add_sub a b c
#check sub_sub a b c
#check add_zero a

end

-- Examples.

section
variable (a b c d : ℝ)

example (a b c d : ℝ) (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by
  rw [hyp'] at hyp
  rw [mul_comm d a] at hyp
  rw [← two_mul (a * d)] at hyp
  rw [← mul_assoc 2 a d] at hyp
  exact hyp

example : c * b * a = b * (a * c) := by
  ring

example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by
  ring

example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by
  ring

example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by
  rw [hyp, hyp']
  ring

end

example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := by
  nth_rw 2 [h]
  rw [add_mul]