feat(geometry/euclidean): angles and some basic lemmas (#2865)

Define angles (undirected, between 0 and π, in terms of inner
product), and prove some basic lemmas involving angles, for real inner
product spaces and Euclidean affine spaces.

From the 100-theorems list, this provides versions of
* 04 Pythagorean Theorem,
* 65 Isosceles Triangle Theorem and
* 94 The Law of Cosines, with various existing definitions implicitly providing
* 91 The Triangle Inequality.

src/analysis/normed_space/real_inner_product.lean

@@ -5,6 +5,7 @@ Authors: Zhouhang Zhou
 -/
 import algebra.quadratic_discriminant
 import analysis.special_functions.pow
+import tactic.apply_fun
 import tactic.monotonicity
 
 
@@ -193,6 +194,59 @@ lemma parallelogram_law_with_norm {x y : α} :
   ∥x + y∥ * ∥x + y∥ + ∥x - y∥ * ∥x - y∥ = 2 * (∥x∥ * ∥x∥ + ∥y∥ * ∥y∥) :=
 by { simp only [(inner_self_eq_norm_square _).symm], exact parallelogram_law }
 
+/-- The inner product, in terms of the norm. -/
+lemma inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two (x y : α) :
+  inner x y = (∥x + y∥ * ∥x + y∥ - ∥x∥ * ∥x∥ - ∥y∥ * ∥y∥) / 2 :=
+begin
+  rw norm_add_mul_self,
+  ring
+end
+
+/-- The inner product, in terms of the norm. -/
+lemma inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two (x y : α) :
+  inner x y = (∥x∥ * ∥x∥ + ∥y∥ * ∥y∥ - ∥x - y∥ * ∥x - y∥) / 2 :=
+begin
+  rw norm_sub_mul_self,
+  ring
+end
+
+/-- The inner product, in terms of the norm. -/
+lemma inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four (x y : α) :
+  inner x y = (∥x + y∥ * ∥x + y∥ - ∥x - y∥ * ∥x - y∥) / 4 :=
+begin
+  rw [norm_add_mul_self, norm_sub_mul_self],
+  ring
+end
+
+/-- Pythagorean theorem, if-and-only-if vector inner product form. -/
+lemma norm_add_square_eq_norm_square_add_norm_square_iff_inner_eq_zero (x y : α) :
+  ∥x + y∥ * ∥x + y∥ = ∥x∥ * ∥x∥ + ∥y∥ * ∥y∥ ↔ inner x y = 0 :=
+begin
+  rw [norm_add_mul_self, add_right_cancel_iff, add_right_eq_self, mul_eq_zero],
+  norm_num
+end
+
+/-- Pythagorean theorem, vector inner product form. -/
+lemma norm_add_square_eq_norm_square_add_norm_square {x y : α} (h : inner x y = 0) :
+  ∥x + y∥ * ∥x + y∥ = ∥x∥ * ∥x∥ + ∥y∥ * ∥y∥ :=
+(norm_add_square_eq_norm_square_add_norm_square_iff_inner_eq_zero x y).2 h
+
+/-- Pythagorean theorem, subtracting vectors, if-and-only-if vector
+inner product form. -/
+lemma norm_sub_square_eq_norm_square_add_norm_square_iff_inner_eq_zero (x y : α) :
+  ∥x - y∥ * ∥x - y∥ = ∥x∥ * ∥x∥ + ∥y∥ * ∥y∥ ↔ inner x y = 0 :=
+begin
+  rw [norm_sub_mul_self, add_right_cancel_iff, sub_eq_add_neg, add_right_eq_self, neg_eq_zero,
+      mul_eq_zero],
+  norm_num
+end
+
+/-- Pythagorean theorem, subtracting vectors, vector inner product
+form. -/
+lemma norm_sub_square_eq_norm_square_add_norm_square {x y : α} (h : inner x y = 0) :
+  ∥x - y∥ * ∥x - y∥ = ∥x∥ * ∥x∥ + ∥y∥ * ∥y∥ :=
+(norm_sub_square_eq_norm_square_add_norm_square_iff_inner_eq_zero x y).2 h
+
 /-- An inner product space forms a normed group w.r.t. its associated norm. -/
 @[priority 100] -- see Note [lower instance priority]
 instance inner_product_space_is_normed_group : normed_group α :=
@@ -225,6 +279,165 @@ instance inner_product_space_is_normed_space : normed_space ℝ α :=
     exact mul_nonneg (abs_nonneg _) (sqrt_nonneg _)
   end }
 
+/-- The inner product of two vectors, divided by the product of their
+norms, has absolute value at most 1. -/
+lemma abs_inner_div_norm_mul_norm_le_one (x y : α) : abs (inner x y / (∥x∥ * ∥y∥)) ≤ 1 :=
+begin
+  rw abs_div,
+  by_cases h : 0 = abs (∥x∥ * ∥y∥),
+  { rw [←h, div_zero],
+    norm_num },
+  { apply div_le_of_le_mul (lt_of_le_of_ne (ge_iff_le.mp (abs_nonneg (∥x∥ * ∥y∥))) h),
+    convert abs_inner_le_norm x y using 1,
+    rw [abs_mul, abs_of_nonneg (norm_nonneg x), abs_of_nonneg (norm_nonneg y), mul_one] }
+end
+
+/-- The inner product of a vector with a multiple of itself. -/
+lemma inner_smul_self_left (x : α) (r : ℝ) : inner (r • x) x = r * (∥x∥ * ∥x∥) :=
+by rw [inner_smul_left, ←inner_self_eq_norm_square]
+
+/-- The inner product of a vector with a multiple of itself. -/
+lemma inner_smul_self_right (x : α) (r : ℝ) : inner x (r • x) = r * (∥x∥ * ∥x∥) :=
+by rw [inner_smul_right, ←inner_self_eq_norm_square]
+
+/-- The inner product of a nonzero vector with a nonzero multiple of
+itself, divided by the product of their norms, has absolute value
+1. -/
+lemma abs_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul
+  {x : α} {r : ℝ} (hx : x ≠ 0) (hr : r ≠ 0) : abs (inner x (r • x) / (∥x∥ * ∥r • x∥)) = 1 :=
+begin
+  rw [inner_smul_self_right, norm_smul, real.norm_eq_abs, ←mul_assoc ∥x∥, mul_comm _ (abs r),
+      mul_assoc, abs_div, abs_mul r, abs_mul (abs r), abs_abs, div_self],
+  exact mul_ne_zero (λ h, hr (eq_zero_of_abs_eq_zero h))
+    (λ h, hx (norm_eq_zero.1 (eq_zero_of_mul_self_eq_zero (eq_zero_of_abs_eq_zero h))))
+end
+
+/-- The inner product of a nonzero vector with a positive multiple of
+itself, divided by the product of their norms, has value 1. -/
+lemma inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul
+  {x : α} {r : ℝ} (hx : x ≠ 0) (hr : 0 < r) : inner x (r • x) / (∥x∥ * ∥r • x∥) = 1 :=
+begin
+  rw [inner_smul_self_right, norm_smul, real.norm_eq_abs, ←mul_assoc ∥x∥, mul_comm _ (abs r),
+      mul_assoc, abs_of_nonneg (le_of_lt hr), div_self],
+  exact mul_ne_zero (ne_of_gt hr)
+    (λ h, hx (norm_eq_zero.1 (eq_zero_of_mul_self_eq_zero h)))
+end
+
+/-- The inner product of a nonzero vector with a negative multiple of
+itself, divided by the product of their norms, has value -1. -/
+lemma inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul
+  {x : α} {r : ℝ} (hx : x ≠ 0) (hr : r < 0) : inner x (r • x) / (∥x∥ * ∥r • x∥) = -1 :=
+begin
+  rw [inner_smul_self_right, norm_smul, real.norm_eq_abs, ←mul_assoc ∥x∥, mul_comm _ (abs r),
+      mul_assoc, abs_of_neg hr, ←neg_mul_eq_neg_mul, div_neg_eq_neg_div, div_self],
+  exact mul_ne_zero (ne_of_lt hr)
+    (λ h, hx (norm_eq_zero.1 (eq_zero_of_mul_self_eq_zero h)))
+end
+
+/-- The inner product of two vectors, divided by the product of their
+norms, has absolute value 1 if and only if they are nonzero and one is
+a multiple of the other. One form of equality case for Cauchy-Schwarz. -/
+lemma abs_inner_div_norm_mul_norm_eq_one_iff (x y : α) :
+  abs (inner x y / (∥x∥ * ∥y∥)) = 1 ↔ (x ≠ 0 ∧ ∃ (r : ℝ), r ≠ 0 ∧ y = r • x) :=
+begin
+  split,
+  { intro h,
+    have hx0 : x ≠ 0,
+    { intro hx0,
+      rw [hx0, inner_zero_left, zero_div] at h,
+      norm_num at h,
+      exact h },
+    refine and.intro hx0 _,
+    set r := inner x y / (∥x∥ * ∥x∥) with hr,
+    use r,
+    set t := y - r • x with ht,
+    have ht0 : inner x t = 0,
+    { rw [ht, inner_sub_right, inner_smul_right, hr, ←inner_self_eq_norm_square,
+          div_mul_cancel _ (λ h, hx0 (inner_self_eq_zero.1 h)), sub_self] },
+    rw [←sub_add_cancel y (r • x), ←ht, inner_add_right, ht0, zero_add, inner_smul_right,
+        inner_self_eq_norm_square, ←mul_assoc, mul_comm,
+        mul_div_mul_left _ _ (λ h, hx0 (norm_eq_zero.1 h)), abs_div, abs_mul,
+        abs_of_nonneg (norm_nonneg _), abs_of_nonneg (norm_nonneg _), ←real.norm_eq_abs,
+        ←norm_smul] at h,
+    have hr0 : r ≠ 0,
+    { intro hr0,
+      rw [hr0, zero_smul, norm_zero, zero_div] at h,
+      norm_num at h },
+    refine and.intro hr0 _,
+    have h2 : ∥r • x∥ ^ 2 = ∥t + r • x∥ ^ 2,
+    { congr' 1,
+      refine eq_of_div_eq_one _ _ h,
+      intro h0,
+      rw [h0, div_zero] at h,
+      norm_num at h },
+    rw [pow_two, pow_two, ←inner_self_eq_norm_square, ←inner_self_eq_norm_square,
+        inner_add_add_self] at h2,
+    conv_rhs at h2 {
+      congr,
+      congr,
+      skip,
+      rw [inner_smul_right, inner_comm, ht0, mul_zero, mul_zero]
+    },
+    symmetry' at h2,
+    rw [add_zero, add_left_eq_self, inner_self_eq_zero] at h2,
+    rw h2 at ht,
+    exact eq_of_sub_eq_zero ht.symm },
+  { intro h,
+    rcases h with ⟨hx, ⟨r, ⟨hr, hy⟩⟩⟩,
+    rw hy,
+    exact abs_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul hx hr }
+end
+
+/-- The inner product of two vectors, divided by the product of their
+norms, has value 1 if and only if they are nonzero and one is
+a positive multiple of the other. -/
+lemma inner_div_norm_mul_norm_eq_one_iff (x y : α) :
+  inner x y / (∥x∥ * ∥y∥) = 1 ↔ (x ≠ 0 ∧ ∃ (r : ℝ), 0 < r ∧ y = r • x) :=
+begin
+  split,
+  { intro h,
+    have ha := h,
+    apply_fun abs at ha,
+    norm_num at ha,
+    rcases (abs_inner_div_norm_mul_norm_eq_one_iff x y).1 ha with ⟨hx, ⟨r, ⟨hr, hy⟩⟩⟩,
+    use [hx, r],
+    refine and.intro _ hy,
+    by_contradiction hrneg,
+    rw hy at h,
+    rw inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul hx
+      (lt_of_le_of_ne' (le_of_not_lt hrneg) hr) at h,
+    norm_num at h },
+  { intro h,
+    rcases h with ⟨hx, ⟨r, ⟨hr, hy⟩⟩⟩,
+    rw hy,
+    exact inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul hx hr }
+end
+
+/-- The inner product of two vectors, divided by the product of their
+norms, has value -1 if and only if they are nonzero and one is
+a negative multiple of the other. -/
+lemma inner_div_norm_mul_norm_eq_neg_one_iff (x y : α) :
+  inner x y / (∥x∥ * ∥y∥) = -1 ↔ (x ≠ 0 ∧ ∃ (r : ℝ), r < 0 ∧ y = r • x) :=
+begin
+  split,
+  { intro h,
+    have ha := h,
+    apply_fun abs at ha,
+    norm_num at ha,
+    rcases (abs_inner_div_norm_mul_norm_eq_one_iff x y).1 ha with ⟨hx, ⟨r, ⟨hr, hy⟩⟩⟩,
+    use [hx, r],
+    refine and.intro _ hy,
+    by_contradiction hrpos,
+    rw hy at h,
+    rw inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul hx
+      (lt_of_le_of_ne' (le_of_not_lt hrpos) hr.symm) at h,
+    norm_num at h },
+  { intro h,
+    rcases h with ⟨hx, ⟨r, ⟨hr, hy⟩⟩⟩,
+    rw hy,
+    exact inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul hx hr }
+end
+
 end norm
 
 -- TODO [Lean 3.15]: drop some of these `show`s