Theorem: Affine.Simplex.mongePoint_eq_affineCombination_of_pointsWithCircumcenter
File: Mathlib/Geometry/Euclidean/MongePoint.lean

tactic: rw [mongePoint_eq_smul_vsub_vadd_circumcenter,
  centroid_eq_affineCombination_of_pointsWithCircumcenter,
  circumcenter_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub,
  ← LinearMap.map_smul, weightedVSub_vadd_affineCombination]
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ s.mongePoint = (affineCombination ℝ univ s.pointsWithCircumcenter) (mongePointWeightsWithCircumcenter n)
state_after: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ (affineCombination ℝ univ s.pointsWithCircumcenter)
      ((↑(n + 2 + 1) / ↑(n + 2 - 1)) •
          (centroidWeightsWithCircumcenter univ - circumcenterWeightsWithCircumcenter (n + 2)) +
        circumcenterWeightsWithCircumcenter (n + 2)) =
    (affineCombination ℝ univ s.pointsWithCircumcenter) (mongePointWeightsWithCircumcenter n)

tactic: congr with i
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ (affineCombination ℝ univ s.pointsWithCircumcenter)
      ((↑(n + 2 + 1) / ↑(n + 2 - 1)) •
          (centroidWeightsWithCircumcenter univ - circumcenterWeightsWithCircumcenter (n + 2)) +
        circumcenterWeightsWithCircumcenter (n + 2)) =
    (affineCombination ℝ univ s.pointsWithCircumcenter) (mongePointWeightsWithCircumcenter n)
state_after: case h.e_6.h.h
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i : PointsWithCircumcenterIndex (n + 2)
⊢ ((↑(n + 2 + 1) / ↑(n + 2 - 1)) •
          (centroidWeightsWithCircumcenter univ - circumcenterWeightsWithCircumcenter (n + 2)) +
        circumcenterWeightsWithCircumcenter (n + 2))
      i =
    mongePointWeightsWithCircumcenter n i

tactic: rw [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Pi.sub_apply]
state_before: case h.e_6.h.h
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i : PointsWithCircumcenterIndex (n + 2)
⊢ ((↑(n + 2 + 1) / ↑(n + 2 - 1)) •
          (centroidWeightsWithCircumcenter univ - circumcenterWeightsWithCircumcenter (n + 2)) +
        circumcenterWeightsWithCircumcenter (n + 2))
      i =
    mongePointWeightsWithCircumcenter n i
state_after: case h.e_6.h.h
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i : PointsWithCircumcenterIndex (n + 2)
⊢ ↑(n + 2 + 1) / ↑(n + 2 - 1) *
        (centroidWeightsWithCircumcenter univ i - circumcenterWeightsWithCircumcenter (n + 2) i) +
      circumcenterWeightsWithCircumcenter (n + 2) i =
    mongePointWeightsWithCircumcenter n i

tactic: have hn1 : (n + 1 : ℝ) ≠ 0 := n.cast_add_one_ne_zero
state_before: case h.e_6.h.h
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i : PointsWithCircumcenterIndex (n + 2)
⊢ ↑(n + 2 + 1) / ↑(n + 2 - 1) *
        (centroidWeightsWithCircumcenter univ i - circumcenterWeightsWithCircumcenter (n + 2) i) +
      circumcenterWeightsWithCircumcenter (n + 2) i =
    mongePointWeightsWithCircumcenter n i
state_after: case h.e_6.h.h
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i : PointsWithCircumcenterIndex (n + 2)
hn1 : ↑n + 1 ≠ 0
⊢ ↑(n + 2 + 1) / ↑(n + 2 - 1) *
        (centroidWeightsWithCircumcenter univ i - circumcenterWeightsWithCircumcenter (n + 2) i) +
      circumcenterWeightsWithCircumcenter (n + 2) i =
    mongePointWeightsWithCircumcenter n i

tactic: cases i <;>
    simp_rw [centroidWeightsWithCircumcenter, circumcenterWeightsWithCircumcenter,
      mongePointWeightsWithCircumcenter] <;>
  rw [add_tsub_assoc_of_le (by decide : 1 ≤ 2), (by decide : 2 - 1 = 1)]
state_before: case h.e_6.h.h
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i : PointsWithCircumcenterIndex (n + 2)
hn1 : ↑n + 1 ≠ 0
⊢ ↑(n + 2 + 1) / ↑(n + 2 - 1) *
        (centroidWeightsWithCircumcenter univ i - circumcenterWeightsWithCircumcenter (n + 2) i) +
      circumcenterWeightsWithCircumcenter (n + 2) i =
    mongePointWeightsWithCircumcenter n i
state_after: case h.e_6.h.h.pointIndex
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
a✝ : Fin (n + 2 + 1)
⊢ ↑(n + 2 + 1) / ↑(n + 1) * ((if a✝ ∈ univ then (↑univ.card)⁻¹ else 0) - 0) + 0 = (↑(n + 1))⁻¹

case h.e_6.h.h.circumcenterIndex
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
⊢ ↑(n + 2 + 1) / ↑(n + 1) * (0 - 1) + 1 = -2 / ↑(n + 1)

tactic: decide
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
⊢ 1 ≤ 2
state_after: no goals

tactic: decide
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
⊢ 2 - 1 = 1
state_after: no goals

tactic: rw [if_pos (mem_univ _), sub_zero, add_zero, card_fin]
state_before: case h.e_6.h.h.pointIndex
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
a✝ : Fin (n + 2 + 1)
⊢ ↑(n + 2 + 1) / ↑(n + 1) * ((if a✝ ∈ univ then (↑univ.card)⁻¹ else 0) - 0) + 0 = (↑(n + 1))⁻¹
state_after: case h.e_6.h.h.pointIndex
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
a✝ : Fin (n + 2 + 1)
⊢ ↑(n + 2 + 1) / ↑(n + 1) * (↑(n + 2 + 1))⁻¹ = (↑(n + 1))⁻¹

tactic: have hn3 : (n + 2 + 1 : ℝ) ≠ 0 := by norm_cast
state_before: case h.e_6.h.h.pointIndex
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
a✝ : Fin (n + 2 + 1)
⊢ ↑(n + 2 + 1) / ↑(n + 1) * (↑(n + 2 + 1))⁻¹ = (↑(n + 1))⁻¹
state_after: case h.e_6.h.h.pointIndex
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
a✝ : Fin (n + 2 + 1)
hn3 : ↑n + 2 + 1 ≠ 0
⊢ ↑(n + 2 + 1) / ↑(n + 1) * (↑(n + 2 + 1))⁻¹ = (↑(n + 1))⁻¹

tactic: field_simp [hn1, hn3, mul_comm]
state_before: case h.e_6.h.h.pointIndex
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
a✝ : Fin (n + 2 + 1)
hn3 : ↑n + 2 + 1 ≠ 0
⊢ ↑(n + 2 + 1) / ↑(n + 1) * (↑(n + 2 + 1))⁻¹ = (↑(n + 1))⁻¹
state_after: no goals

tactic: norm_cast
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
a✝ : Fin (n + 2 + 1)
⊢ ↑n + 2 + 1 ≠ 0
state_after: no goals

tactic: field_simp [hn1]
state_before: case h.e_6.h.h.circumcenterIndex
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
⊢ ↑(n + 2 + 1) / ↑(n + 1) * (0 - 1) + 1 = -2 / ↑(n + 1)
state_after: case h.e_6.h.h.circumcenterIndex
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
⊢ -1 + (-2 + -↑n) + (↑n + 1) = -2

tactic: ring
state_before: case h.e_6.h.h.circumcenterIndex
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
hn1 : ↑n + 1 ≠ 0
⊢ -1 + (-2 + -↑n) + (↑n + 1) = -2
state_after: no goals