l3lab/ntp-mathlib huggingface dataset:
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    

Proof state:
case neg.mp
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
⊢ 0 ≤ inner x y ∧ ((areaForm o) x) y = 0 → SameRay ℝ x y

Next tactic:
suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
this : 0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • (rightAngleRotation o) x)
⊢ 0 ≤ inner x y ∧ ((areaForm o) x) y = 0 → SameRay ℝ x y

Next tactic:
rw [← (o.basisRightAngleRotation x hx).sum_repr y]
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
this : 0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • (rightAngleRotation o) x)
⊢ 0 ≤
        inner x
          (Finset.sum Finset.univ fun i =>
            ((basisRightAngleRotation o x hx).repr y) i • (basisRightAngleRotation o x hx) i) ∧
      ((areaForm o) x)
          (Finset.sum Finset.univ fun i =>
            ((basisRightAngleRotation o x hx).repr y) i • (basisRightAngleRotation o x hx) i) =
        0 →
    SameRay ℝ x
      (Finset.sum Finset.univ fun i => ((basisRightAngleRotation o x hx).repr y) i • (basisRightAngleRotation o x hx) i)

Next tactic:
simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
this : 0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • (rightAngleRotation o) x)
⊢ 0 ≤ ((basisRightAngleRotation o x hx).repr y) 0 * ‖x‖ ^ 2 ∧
      ((basisRightAngleRotation o x hx).repr y) 1 * ‖x‖ ^ 2 = 0 →
    SameRay ℝ x
      (((basisRightAngleRotation o x hx).repr y) 0 • x +
        ((basisRightAngleRotation o x hx).repr y) 1 • (rightAngleRotation o) x)

Next tactic:
exact this
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    

Proof state:
case neg.mp
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
⊢ 0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • (rightAngleRotation o) x)

Next tactic:
rintro ⟨ha, hb⟩
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    rintro ⟨ha, hb⟩
    

Proof state:
case neg.mp.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
ha : 0 ≤ a * ‖x‖ ^ 2
hb : b * ‖x‖ ^ 2 = 0
⊢ SameRay ℝ x (a • x + b • (rightAngleRotation o) x)

Next tactic:
have hx' : 0 < ‖x‖ := by simpa using hx
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    rintro ⟨ha, hb⟩
    have hx' : 0 < ‖x‖ := by 

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
ha : 0 ≤ a * ‖x‖ ^ 2
hb : b * ‖x‖ ^ 2 = 0
⊢ 0 < ‖x‖

Next tactic:
simpa using hx
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    rintro ⟨ha, hb⟩
    have hx' : 0 < ‖x‖ := by simpa using hx
    

Proof state:
case neg.mp.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
ha : 0 ≤ a * ‖x‖ ^ 2
hb : b * ‖x‖ ^ 2 = 0
hx' : 0 < ‖x‖
⊢ SameRay ℝ x (a • x + b • (rightAngleRotation o) x)

Next tactic:
have ha' : 0 ≤ a := nonneg_of_mul_nonneg_left ha (by positivity)
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    rintro ⟨ha, hb⟩
    have hx' : 0 < ‖x‖ := by simpa using hx
    have ha' : 0 ≤ a := nonneg_of_mul_nonneg_left ha (by 

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
ha : 0 ≤ a * ‖x‖ ^ 2
hb : b * ‖x‖ ^ 2 = 0
hx' : 0 < ‖x‖
⊢ 0 < ‖x‖ ^ 2

Next tactic:
positivity
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    rintro ⟨ha, hb⟩
    have hx' : 0 < ‖x‖ := by simpa using hx
    have ha' : 0 ≤ a := nonneg_of_mul_nonneg_left ha (by positivity)
    

Proof state:
case neg.mp.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
ha : 0 ≤ a * ‖x‖ ^ 2
hb : b * ‖x‖ ^ 2 = 0
hx' : 0 < ‖x‖
ha' : 0 ≤ a
⊢ SameRay ℝ x (a • x + b • (rightAngleRotation o) x)

Next tactic:
have hb' : b = 0 := eq_zero_of_ne_zero_of_mul_right_eq_zero (pow_ne_zero 2 hx'.ne') hb
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    rintro ⟨ha, hb⟩
    have hx' : 0 < ‖x‖ := by simpa using hx
    have ha' : 0 ≤ a := nonneg_of_mul_nonneg_left ha (by positivity)
    have hb' : b = 0 := eq_zero_of_ne_zero_of_mul_right_eq_zero (pow_ne_zero 2 hx'.ne') hb
    

Proof state:
case neg.mp.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
a : ℝ := ((basisRightAngleRotation o x hx).repr y) 0
b : ℝ := ((basisRightAngleRotation o x hx).repr y) 1
ha : 0 ≤ a * ‖x‖ ^ 2
hb : b * ‖x‖ ^ 2 = 0
hx' : 0 < ‖x‖
ha' : 0 ≤ a
hb' : b = 0
⊢ SameRay ℝ x (a • x + b • (rightAngleRotation o) x)

Next tactic:
simpa [hb'] using SameRay.sameRay_nonneg_smul_right x ha'
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    rintro ⟨ha, hb⟩
    have hx' : 0 < ‖x‖ := by simpa using hx
    have ha' : 0 ≤ a := nonneg_of_mul_nonneg_left ha (by positivity)
    have hb' : b = 0 := eq_zero_of_ne_zero_of_mul_right_eq_zero (pow_ne_zero 2 hx'.ne') hb
    simpa [hb'] using SameRay.sameRay_nonneg_smul_right x ha'
  · 

Proof state:
case neg.mpr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
⊢ SameRay ℝ x y → 0 ≤ inner x y ∧ ((areaForm o) x) y = 0

Next tactic:
intro h
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    rintro ⟨ha, hb⟩
    have hx' : 0 < ‖x‖ := by simpa using hx
    have ha' : 0 ≤ a := nonneg_of_mul_nonneg_left ha (by positivity)
    have hb' : b = 0 := eq_zero_of_ne_zero_of_mul_right_eq_zero (pow_ne_zero 2 hx'.ne') hb
    simpa [hb'] using SameRay.sameRay_nonneg_smul_right x ha'
  · intro h
    

Proof state:
case neg.mpr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
hx : ¬x = 0
h : SameRay ℝ x y
⊢ 0 ≤ inner x y ∧ ((areaForm o) x) y = 0

Next tactic:
obtain ⟨r, hr, rfl⟩ := h.exists_nonneg_left hx
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    rintro ⟨ha, hb⟩
    have hx' : 0 < ‖x‖ := by simpa using hx
    have ha' : 0 ≤ a := nonneg_of_mul_nonneg_left ha (by positivity)
    have hb' : b = 0 := eq_zero_of_ne_zero_of_mul_right_eq_zero (pow_ne_zero 2 hx'.ne') hb
    simpa [hb'] using SameRay.sameRay_nonneg_smul_right x ha'
  · intro h
    obtain ⟨r, hr, rfl⟩ := h.exists_nonneg_left hx
    

Proof state:
case neg.mpr.intro.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
hx : ¬x = 0
r : ℝ
hr : 0 ≤ r
h : SameRay ℝ x (r • x)
⊢ 0 ≤ inner x (r • x) ∧ ((areaForm o) x) (r • x) = 0

Next tactic:
simp only [inner_smul_right, real_inner_self_eq_norm_sq, LinearMap.map_smulₛₗ,
      areaForm_apply_self, Algebra.id.smul_eq_mul, mul_zero, eq_self_iff_true,
      and_true_iff]
theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
    0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by
  by_cases hx : x = 0
  · simp [hx]
  constructor
  · let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
    let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
    suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
      rw [← (o.basisRightAngleRotation x hx).sum_repr y]
      simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero,
        Fin.succ_zero_eq_one', Finset.univ_eq_empty, Finset.sum_empty, areaForm_apply_self,
        map_smul, map_add, real_inner_smul_right, inner_add_right, Matrix.cons_val_one,
        Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,
        mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right,
        real_inner_self_eq_norm_sq]
      exact this
    rintro ⟨ha, hb⟩
    have hx' : 0 < ‖x‖ := by simpa using hx
    have ha' : 0 ≤ a := nonneg_of_mul_nonneg_left ha (by positivity)
    have hb' : b = 0 := eq_zero_of_ne_zero_of_mul_right_eq_zero (pow_ne_zero 2 hx'.ne') hb
    simpa [hb'] using SameRay.sameRay_nonneg_smul_right x ha'
  · intro h
    obtain ⟨r, hr, rfl⟩ := h.exists_nonneg_left hx
    simp only [inner_smul_right, real_inner_self_eq_norm_sq, LinearMap.map_smulₛₗ,
      areaForm_apply_self, Algebra.id.smul_eq_mul, mul_zero, eq_self_iff_true,
      and_true_iff]
    

Proof state:
case neg.mpr.intro.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
hx : ¬x = 0
r : ℝ
hr : 0 ≤ r
h : SameRay ℝ x (r • x)
⊢ 0 ≤ r * ‖x‖ ^ 2

Next tactic:
positivity
theorem kahler_swap (x y : E) : o.kahler x y = conj (o.kahler y x) := by
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ ((kahler o) x) y = (starRingEnd ℂ) (((kahler o) y) x)

Next tactic:
have : ∀ r : ℝ, Complex.ofReal' r = @IsROrC.ofReal ℂ _ r := fun r => rfl
theorem kahler_swap (x y : E) : o.kahler x y = conj (o.kahler y x) := by
  have : ∀ r : ℝ, Complex.ofReal' r = @IsROrC.ofReal ℂ _ r := fun r => rfl
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
this : ∀ (r : ℝ), ↑r = ↑r
⊢ ((kahler o) x) y = (starRingEnd ℂ) (((kahler o) y) x)

Next tactic:
simp only [kahler_apply_apply]
theorem kahler_swap (x y : E) : o.kahler x y = conj (o.kahler y x) := by
  have : ∀ r : ℝ, Complex.ofReal' r = @IsROrC.ofReal ℂ _ r := fun r => rfl
  simp only [kahler_apply_apply]
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
this : ∀ (r : ℝ), ↑r = ↑r
⊢ ↑(inner x y) + ((areaForm o) x) y • Complex.I = (starRingEnd ℂ) (↑(inner y x) + ((areaForm o) y) x • Complex.I)

Next tactic:
rw [real_inner_comm, areaForm_swap]
theorem kahler_swap (x y : E) : o.kahler x y = conj (o.kahler y x) := by
  have : ∀ r : ℝ, Complex.ofReal' r = @IsROrC.ofReal ℂ _ r := fun r => rfl
  simp only [kahler_apply_apply]
  rw [real_inner_comm, areaForm_swap]
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
this : ∀ (r : ℝ), ↑r = ↑r
⊢ ↑(inner y x) + -((areaForm o) y) x • Complex.I = (starRingEnd ℂ) (↑(inner y x) + ((areaForm o) y) x • Complex.I)

Next tactic:
simp [this]
@[simp]
theorem kahler_apply_self (x : E) : o.kahler x x = ‖x‖ ^ 2 := by
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
⊢ ((kahler o) x) x = ↑‖x‖ ^ 2

Next tactic:
simp [kahler_apply_apply, real_inner_self_eq_norm_sq]