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https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have hbn : 0 < (1 - schnirelmannDensity B) := by
rw [hbeta] at hbo
rw [lt_sub_iff_add_lt, zero_add, lt_iff_le_and_ne]
exact ⟨schnirelmannDensity_le_one, hbo⟩ | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
⊢ schnirelmannDensity A * ↑n * (1 - β) + β * ↑n ≤ ↑(countelements A n) * (1 - β) + β * ↑n | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ schnirelmannDensity A * ↑n * (1 - β) + β * ↑n ≤ ↑(countelements A n) * (1 - β) + β * ↑n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | simp only [add_le_add_iff_right, sub_pos, sub_neg] | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ schnirelmannDensity A * ↑n * (1 - β) + β * ↑n ≤ ↑(countelements A n) * (1 - β) + β * ↑n | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ schnirelmannDensity A * ↑n * (1 - β) ≤ ↑(countelements A n) * (1 - β) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [← le_div_iff (hbn)] | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ schnirelmannDensity A * ↑n * (1 - β) ≤ ↑(countelements A n) * (1 - β) | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ schnirelmannDensity A * ↑n ≤ ↑(countelements A n) * (1 - β) / (1 - schnirelmannDensity B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [mul_div_assoc] | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ schnirelmannDensity A * ↑n ≤ ↑(countelements A n) * (1 - β) / (1 - schnirelmannDensity B) | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ schnirelmannDensity A * ↑n ≤ ↑(countelements A n) * ((1 - β) / (1 - schnirelmannDensity B)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1 := by
rw [div_self]
positivity | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ schnirelmannDensity A * ↑n ≤ ↑(countelements A n) * ((1 - β) / (1 - schnirelmannDensity B)) | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
⊢ schnirelmannDensity A * ↑n ≤ ↑(countelements A n) * ((1 - β) / (1 - schnirelmannDensity B)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [hun, mul_one] | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
⊢ schnirelmannDensity A * ↑n ≤ ↑(countelements A n) * ((1 - β) / (1 - schnirelmannDensity B)) | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
⊢ schnirelmannDensity A * ↑n ≤ ↑(countelements A n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact schnirelmannDensity_mul_le_card_filter | case neg
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
hun : (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1
⊢ schnirelmannDensity A * ↑n ≤ ↑(countelements A n) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [hbeta] at hbo | A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
⊢ 0 < 1 - schnirelmannDensity B | A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬schnirelmannDensity B = 1
⊢ 0 < 1 - schnirelmannDensity B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [lt_sub_iff_add_lt, zero_add, lt_iff_le_and_ne] | A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬schnirelmannDensity B = 1
⊢ 0 < 1 - schnirelmannDensity B | A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬schnirelmannDensity B = 1
⊢ schnirelmannDensity B ≤ 1 ∧ schnirelmannDensity B ≠ 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact ⟨schnirelmannDensity_le_one, hbo⟩ | A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬schnirelmannDensity B = 1
⊢ schnirelmannDensity B ≤ 1 ∧ schnirelmannDensity B ≠ 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [div_self] | A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ (1 - schnirelmannDensity B) / (1 - schnirelmannDensity B) = 1 | A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ 1 - schnirelmannDensity B ≠ 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | positivity | A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hbo : ¬β = 1
hbn : 0 < 1 - schnirelmannDensity B
⊢ 1 - schnirelmannDensity B ≠ 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | ring_nf | A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
claim :
↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤
↑(countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n)
ht : ↑(countelements A n) + β * (↑n - ↑(countelements A n)) ≤ ↑(countelements (A + B) n)
hc1 : ↑(countelements A n) * (1 - β) + β * ↑n = ↑(countelements A n) + β * (↑n - ↑(countelements A n))
hc2 : α * ↑n * (1 - β) + β * ↑n ≤ ↑(countelements A n) * (1 - β) + β * ↑n
⊢ α * ↑n * (1 - β) + β * ↑n = (α * (1 - β) + β) * ↑n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | let α := schnirelmannDensity A | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | have halpha : α = schnirelmannDensity A := rfl | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | let β := schnirelmannDensity B | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | have hbeta : β = schnirelmannDensity B := rfl | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | let γ := schnirelmannDensity (A + B) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | have hgamma : γ = schnirelmannDensity (A + B) := rfl | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | rw [← halpha, ← hbeta, ← hgamma] | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 1 - schnirelmannDensity (A + B) ≤ (1 - schnirelmannDensity A) * (1 - schnirelmannDensity B) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 1 - γ ≤ (1 - α) * (1 - β) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | linarith | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
h : 1 - γ ≤ 1 - (α + β - α * β)
⊢ 1 - γ ≤ (1 - α) * (1 - β) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | rw [sub_le_iff_le_add, add_comm_sub] | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 1 - γ ≤ 1 - (α + β - α * β) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 1 ≤ 1 + (γ - (α + β - α * β)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | nth_rewrite 1 [← add_zero 1] | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 1 ≤ 1 + (γ - (α + β - α * β)) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 1 + 0 ≤ 1 + (γ - (α + β - α * β)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | rw [add_le_add_iff_left, le_sub_comm, sub_zero] | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 1 + 0 ≤ 1 + (γ - (α + β - α * β)) | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ α + β - α * β ≤ γ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | rw [sub_eq_add_neg] | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ α + β - α * β ≤ γ | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ α + β + -(α * β) ≤ γ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | exact h0 | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
h0 : α + β - α * β ≤ γ
⊢ α + β + -(α * β) ≤ γ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | rw [halpha, hbeta, hgamma] | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ α + β - α * β ≤ γ | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ schnirelmannDensity A + schnirelmannDensity B - schnirelmannDensity A * schnirelmannDensity B ≤
schnirelmannDensity (A + B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | apply le_schnirelmannDensity_add A B | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ schnirelmannDensity A + schnirelmannDensity B - schnirelmannDensity A * schnirelmannDensity B ≤
schnirelmannDensity (A + B) | case hA
A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 0 ∈ A
case hB
A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 0 ∈ B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | exact hA | case hA
A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 0 ∈ A | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | schnirelmannDensity_for_two | [218, 1] | [239, 11] | exact hB | case hB
A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
γ : ℝ := schnirelmannDensity (A + B)
hgamma : γ = schnirelmannDensity (A + B)
⊢ 0 ∈ B | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | mannTheorem | [243, 1] | [245, 8] | sorry | A✝ B✝ : Set ℕ
n : ℕ
A B : Set ℕ
⊢ min 1 (schnirelmannDensity A + schnirelmannDensity B) ≤ schnirelmannDensity (A + B) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.prod_top | [46, 1] | [47, 53] | simp [mem_prod, LatticeHom.coe_fst] | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Lattice α
inst✝¹ : Lattice β
inst✝ : Lattice γ
L✝ M : Sublattice α
f : LatticeHom α β
s t : Set α
a✝ : α
L : Sublattice α
a : α × β
⊢ a ∈ L.prod ⊤ ↔ a ∈ comap (LatticeHom.fst α β) L | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.top_prod | [49, 1] | [50, 53] | simp [mem_prod, LatticeHom.coe_snd] | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Lattice α
inst✝¹ : Lattice β
inst✝ : Lattice γ
L✝ M : Sublattice α
f : LatticeHom α β
s t : Set α
a✝ : α
L : Sublattice β
a : α × β
⊢ a ∈ ⊤.prod L ↔ a ∈ comap (LatticeHom.snd α β) L | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.le_prod_iff | [61, 1] | [63, 36] | simp [SetLike.le_def, forall_and] | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Lattice α
inst✝¹ : Lattice β
inst✝ : Lattice γ
L M✝ : Sublattice α
f : LatticeHom α β
s t : Set α
a : α
M : Sublattice β
N : Sublattice (α × β)
⊢ N ≤ L.prod M ↔ N ≤ comap (LatticeHom.fst α β) L ∧ N ≤ comap (LatticeHom.snd α β) M | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.prod_eq_bot | [65, 1] | [66, 53] | simpa only [← coe_inj] using Set.prod_eq_empty_iff | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Lattice α
inst✝¹ : Lattice β
inst✝ : Lattice γ
L M✝ : Sublattice α
f : LatticeHom α β
s t : Set α
a : α
M : Sublattice β
⊢ L.prod M = ⊥ ↔ L = ⊥ ∨ M = ⊥ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.prod_eq_top | [68, 1] | [69, 85] | simpa only [← coe_inj] using Set.prod_eq_univ | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : Lattice α
inst✝³ : Lattice β
inst✝² : Lattice γ
L M✝ : Sublattice α
f : LatticeHom α β
s t : Set α
a : α
inst✝¹ : Nonempty α
inst✝ : Nonempty β
M : Sublattice β
⊢ L.prod M = ⊤ ↔ L = ⊤ ∧ M = ⊤ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.pi_empty | [93, 1] | [93, 96] | simp [mem_pi] | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : Lattice α
inst✝² : Lattice β
inst✝¹ : Lattice γ
L✝ M : Sublattice α
f : LatticeHom α β
s t : Set α
a✝ : α
κ : Type u_5
π : κ → Type u_6
inst✝ : (i : κ) → Lattice (π i)
L : (i : κ) → Sublattice (π i)
a : (i : κ) → π i
⊢ a ∈ pi ∅ L ↔ a ∈ ⊤ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.pi_top | [95, 1] | [96, 31] | simp [mem_pi] | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : Lattice α
inst✝² : Lattice β
inst✝¹ : Lattice γ
L M : Sublattice α
f : LatticeHom α β
s✝ t : Set α
a✝ : α
κ : Type u_5
π : κ → Type u_6
inst✝ : (i : κ) → Lattice (π i)
s : Set κ
a : (i : κ) → π i
⊢ (a ∈ pi s fun i => ⊤) ↔ a ∈ ⊤ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.pi_bot | [98, 1] | [99, 31] | simp [mem_pi] | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : Lattice α
inst✝³ : Lattice β
inst✝² : Lattice γ
L M : Sublattice α
f : LatticeHom α β
s t : Set α
a✝ : α
κ : Type u_5
π : κ → Type u_6
inst✝¹ : (i : κ) → Lattice (π i)
inst✝ : Nonempty κ
a : (i : κ) → π i
⊢ (a ∈ pi univ fun i => ⊥) ↔ a ∈ ⊥ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.le_pi | [101, 1] | [102, 101] | simp [SetLike.le_def] | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : Lattice α
inst✝² : Lattice β
inst✝¹ : Lattice γ
L✝ M✝ : Sublattice α
f : LatticeHom α β
s✝ t : Set α
a : α
κ : Type u_5
π : κ → Type u_6
inst✝ : (i : κ) → Lattice (π i)
s : Set κ
L : (i : κ) → Sublattice (π i)
M : Sublattice ((i : κ) → π i)
⊢ M ≤ pi s L ↔ ∀ i ∈ s, M ≤ comap (Pi.evalLatticeHom π i) (L i) | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : Lattice α
inst✝² : Lattice β
inst✝¹ : Lattice γ
L✝ M✝ : Sublattice α
f : LatticeHom α β
s✝ t : Set α
a : α
κ : Type u_5
π : κ → Type u_6
inst✝ : (i : κ) → Lattice (π i)
s : Set κ
L : (i : κ) → Sublattice (π i)
M : Sublattice ((i : κ) → π i)
⊢ (∀ ⦃x : (i : κ) → π i⦄, x ∈ M → ∀ i ∈ s, x i ∈ L i) ↔ ∀ i ∈ s, ∀ ⦃x : (i : κ) → π i⦄, x ∈ M → x i ∈ L i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.le_pi | [101, 1] | [102, 101] | aesop | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : Lattice α
inst✝² : Lattice β
inst✝¹ : Lattice γ
L✝ M✝ : Sublattice α
f : LatticeHom α β
s✝ t : Set α
a : α
κ : Type u_5
π : κ → Type u_6
inst✝ : (i : κ) → Lattice (π i)
s : Set κ
L : (i : κ) → Sublattice (π i)
M : Sublattice ((i : κ) → π i)
⊢ (∀ ⦃x : (i : κ) → π i⦄, x ∈ M → ∀ i ∈ s, x i ∈ L i) ↔ ∀ i ∈ s, ∀ ⦃x : (i : κ) → π i⦄, x ∈ M → x i ∈ L i | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.pi_univ_eq_bot | [104, 1] | [105, 28] | simp_rw [← coe_inj] | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : Lattice α
inst✝² : Lattice β
inst✝¹ : Lattice γ
L✝ M : Sublattice α
f : LatticeHom α β
s t : Set α
a : α
κ : Type u_5
π : κ → Type u_6
inst✝ : (i : κ) → Lattice (π i)
L : (i : κ) → Sublattice (π i)
⊢ pi univ L = ⊥ ↔ ∃ i, L i = ⊥ | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : Lattice α
inst✝² : Lattice β
inst✝¹ : Lattice γ
L✝ M : Sublattice α
f : LatticeHom α β
s t : Set α
a : α
κ : Type u_5
π : κ → Type u_6
inst✝ : (i : κ) → Lattice (π i)
L : (i : κ) → Sublattice (π i)
⊢ ↑(pi univ L) = ↑⊥ ↔ ∃ i, ↑(L i) = ↑⊥ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Sublattice.lean | Sublattice.pi_univ_eq_bot | [104, 1] | [105, 28] | simp | ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : Lattice α
inst✝² : Lattice β
inst✝¹ : Lattice γ
L✝ M : Sublattice α
f : LatticeHom α β
s t : Set α
a : α
κ : Type u_5
π : κ → Type u_6
inst✝ : (i : κ) → Lattice (π i)
L : (i : κ) → Sublattice (π i)
⊢ ↑(pi univ L) = ↑⊥ ↔ ∃ i, ↑(L i) = ↑⊥ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | mem_combiFrontier_iff | [36, 1] | [37, 97] | simp [combiFrontier] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ x ∈ combiFrontier 𝕜 s ↔ ∃ t ⊂ s, x ∈ (convexHull 𝕜) ↑t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_empty | [44, 1] | [47, 42] | apply Set.eq_empty_of_subset_empty | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ combiFrontier 𝕜 ∅ = ∅ | case a
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ combiFrontier 𝕜 ∅ ⊆ ∅ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_empty | [44, 1] | [47, 42] | convert combiFrontier_subset_convexHull | case a
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ combiFrontier 𝕜 ∅ ⊆ ∅ | case h.e'_4
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ ∅ = (convexHull 𝕜) ↑∅ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_empty | [44, 1] | [47, 42] | rw [Finset.coe_empty, convexHull_empty] | case h.e'_4
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ ∅ = (convexHull 𝕜) ↑∅ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_empty | [49, 1] | [52, 42] | apply Set.eq_empty_of_subset_empty | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ combiInterior 𝕜 ∅ = ∅ | case a
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ combiInterior 𝕜 ∅ ⊆ ∅ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_empty | [49, 1] | [52, 42] | convert combiInterior_subset_convexHull | case a
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ combiInterior 𝕜 ∅ ⊆ ∅ | case h.e'_4
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ ∅ = (convexHull 𝕜) ↑∅ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_empty | [49, 1] | [52, 42] | rw [Finset.coe_empty, convexHull_empty] | case h.e'_4
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ ∅ = (convexHull 𝕜) ↑∅ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_singleton | [54, 1] | [59, 14] | refine eq_empty_of_subset_empty fun y hy ↦ ?_ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ combiFrontier 𝕜 {x} = ∅ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x y : E
hy : y ∈ combiFrontier 𝕜 {x}
⊢ y ∈ ∅ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_singleton | [54, 1] | [59, 14] | rw [mem_combiFrontier_iff] at hy | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x y : E
hy : y ∈ combiFrontier 𝕜 {x}
⊢ y ∈ ∅ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x y : E
hy : ∃ t ⊂ {x}, y ∈ (convexHull 𝕜) ↑t
⊢ y ∈ ∅ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_singleton | [54, 1] | [59, 14] | obtain ⟨s, hs, hys⟩ := hy | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x y : E
hy : ∃ t ⊂ {x}, y ∈ (convexHull 𝕜) ↑t
⊢ y ∈ ∅ | case intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x y : E
s : Finset E
hs : s ⊂ {x}
hys : y ∈ (convexHull 𝕜) ↑s
⊢ y ∈ ∅ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_singleton | [54, 1] | [59, 14] | rw [Finset.eq_empty_of_ssubset_singleton hs] at hys | case intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x y : E
s : Finset E
hs : s ⊂ {x}
hys : y ∈ (convexHull 𝕜) ↑s
⊢ y ∈ ∅ | case intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x y : E
s : Finset E
hs : s ⊂ {x}
hys : y ∈ (convexHull 𝕜) ↑∅
⊢ y ∈ ∅ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_singleton | [54, 1] | [59, 14] | simp at hys | case intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x y : E
s : Finset E
hs : s ⊂ {x}
hys : y ∈ (convexHull 𝕜) ↑∅
⊢ y ∈ ∅ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_singleton | [61, 1] | [64, 7] | unfold combiInterior | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ combiInterior 𝕜 {x} = {x} | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ (convexHull 𝕜) ↑{x} \ combiFrontier 𝕜 {x} = {x} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_singleton | [61, 1] | [64, 7] | rw [combiFrontier_singleton] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ (convexHull 𝕜) ↑{x} \ combiFrontier 𝕜 {x} = {x} | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ (convexHull 𝕜) ↑{x} \ ∅ = {x} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiInterior_singleton | [61, 1] | [64, 7] | simp | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ (convexHull 𝕜) ↑{x} \ ∅ = {x} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | apply Subset.antisymm _ _ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ (convexHull 𝕜) ↑s = ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ (convexHull 𝕜) ↑s ⊆ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t ⊆ (convexHull 𝕜) ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | refine s.strongInductionOn ?_ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ (convexHull 𝕜) ↑s ⊆ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ ∀ (s : Finset E),
(∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1) →
(convexHull 𝕜) ↑s ⊆ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | rintro s ih x hx | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ ∀ (s : Finset E),
(∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1) →
(convexHull 𝕜) ↑s ⊆ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x✝ : E
s : Finset E
ih : ∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
x : E
hx : x ∈ (convexHull 𝕜) ↑s
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | by_cases h : x ∈ combiFrontier 𝕜 s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x✝ : E
s : Finset E
ih : ∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
x : E
hx : x ∈ (convexHull 𝕜) ↑s
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x✝ : E
s : Finset E
ih : ∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
x : E
hx : x ∈ (convexHull 𝕜) ↑s
h : x ∈ combiFrontier 𝕜 s
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t
case neg
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x✝ : E
s : Finset E
ih : ∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
x : E
hx : x ∈ (convexHull 𝕜) ↑s
h : x ∉ combiFrontier 𝕜 s
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | rw [mem_combiFrontier_iff] at h | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x✝ : E
s : Finset E
ih : ∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
x : E
hx : x ∈ (convexHull 𝕜) ↑s
h : x ∈ combiFrontier 𝕜 s
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x✝ : E
s : Finset E
ih : ∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
x : E
hx : x ∈ (convexHull 𝕜) ↑s
h : ∃ t ⊂ s, x ∈ (convexHull 𝕜) ↑t
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | obtain ⟨t, st, ht⟩ := h | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x✝ : E
s : Finset E
ih : ∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
x : E
hx : x ∈ (convexHull 𝕜) ↑s
h : ∃ t ⊂ s, x ∈ (convexHull 𝕜) ↑t
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t | case pos.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t✝ : Finset E
x✝ : E
s : Finset E
ih : ∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
x : E
hx : x ∈ (convexHull 𝕜) ↑s
t : Finset E
st : t ⊂ s
ht : x ∈ (convexHull 𝕜) ↑t
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | specialize ih _ st ht | case pos.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t✝ : Finset E
x✝ : E
s : Finset E
ih : ∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
x : E
hx : x ∈ (convexHull 𝕜) ↑s
t : Finset E
st : t ⊂ s
ht : x ∈ (convexHull 𝕜) ↑t
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t | case pos.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t✝ : Finset E
x✝ : E
s : Finset E
x : E
hx : x ∈ (convexHull 𝕜) ↑s
t : Finset E
st : t ⊂ s
ht : x ∈ (convexHull 𝕜) ↑t
ih : x ∈ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | simp only [exists_prop, Set.mem_iUnion] at ih ⊢ | case pos.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t✝ : Finset E
x✝ : E
s : Finset E
x : E
hx : x ∈ (convexHull 𝕜) ↑s
t : Finset E
st : t ⊂ s
ht : x ∈ (convexHull 𝕜) ↑t
ih : x ∈ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t | case pos.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t✝ : Finset E
x✝ : E
s : Finset E
x : E
hx : x ∈ (convexHull 𝕜) ↑s
t : Finset E
st : t ⊂ s
ht : x ∈ (convexHull 𝕜) ↑t
ih : ∃ i ⊆ t, x ∈ combiInterior 𝕜 i
⊢ ∃ i ⊆ s, x ∈ combiInterior 𝕜 i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | obtain ⟨Z, Zt, hZ⟩ := ih | case pos.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t✝ : Finset E
x✝ : E
s : Finset E
x : E
hx : x ∈ (convexHull 𝕜) ↑s
t : Finset E
st : t ⊂ s
ht : x ∈ (convexHull 𝕜) ↑t
ih : ∃ i ⊆ t, x ∈ combiInterior 𝕜 i
⊢ ∃ i ⊆ s, x ∈ combiInterior 𝕜 i | case pos.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t✝ : Finset E
x✝ : E
s : Finset E
x : E
hx : x ∈ (convexHull 𝕜) ↑s
t : Finset E
st : t ⊂ s
ht : x ∈ (convexHull 𝕜) ↑t
Z : Finset E
Zt : Z ⊆ t
hZ : x ∈ combiInterior 𝕜 Z
⊢ ∃ i ⊆ s, x ∈ combiInterior 𝕜 i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | exact ⟨_, Zt.trans st.1, hZ⟩ | case pos.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t✝ : Finset E
x✝ : E
s : Finset E
x : E
hx : x ∈ (convexHull 𝕜) ↑s
t : Finset E
st : t ⊂ s
ht : x ∈ (convexHull 𝕜) ↑t
Z : Finset E
Zt : Z ⊆ t
hZ : x ∈ combiInterior 𝕜 Z
⊢ ∃ i ⊆ s, x ∈ combiInterior 𝕜 i | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | exact subset_iUnion₂ s Subset.rfl ⟨hx, h⟩ | case neg
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t : Finset E
x✝ : E
s : Finset E
ih : ∀ t ⊂ s, (convexHull 𝕜) ↑t ⊆ ⋃ t_1, ⋃ (_ : t_1 ⊆ t), combiInterior 𝕜 t_1
x : E
hx : x ∈ (convexHull 𝕜) ↑s
h : x ∉ combiFrontier 𝕜 s
⊢ x ∈ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | simplex_combiInteriors_cover | [70, 1] | [82, 75] | exact iUnion₂_subset fun t ht ↦ diff_subset.trans $ convexHull_mono ht | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ ⋃ t, ⋃ (_ : t ⊆ s), combiInterior 𝕜 t ⊆ (convexHull 𝕜) ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | ext x | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x : E
⊢ combiFrontier 𝕜 s =
{x | ∃ w, ∃ (_ : ∀ y ∈ s, 0 ≤ w y) (_ : ∑ y ∈ s, w y = 1) (_ : ∃ y ∈ s, w y = 0), s.centerMass w id = x} | case h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x✝ x : E
⊢ x ∈ combiFrontier 𝕜 s ↔
x ∈ {x | ∃ w, ∃ (_ : ∀ y ∈ s, 0 ≤ w y) (_ : ∑ y ∈ s, w y = 1) (_ : ∃ y ∈ s, w y = 0), s.centerMass w id = x} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | simp_rw [combiFrontier, Set.mem_iUnion, Set.mem_setOf_eq] | case h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x✝ x : E
⊢ x ∈ combiFrontier 𝕜 s ↔
x ∈ {x | ∃ w, ∃ (_ : ∀ y ∈ s, 0 ≤ w y) (_ : ∑ y ∈ s, w y = 1) (_ : ∃ y ∈ s, w y = 0), s.centerMass w id = x} | case h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x✝ x : E
⊢ (∃ i, ∃ (_ : i ⊂ s), x ∈ (convexHull 𝕜) ↑i) ↔
∃ w, ∃ (_ : ∀ y ∈ s, 0 ≤ w y) (_ : ∑ y ∈ s, w y = 1) (_ : ∃ y ∈ s, w y = 0), s.centerMass w id = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | constructor | case h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x✝ x : E
⊢ (∃ i, ∃ (_ : i ⊂ s), x ∈ (convexHull 𝕜) ↑i) ↔
∃ w, ∃ (_ : ∀ y ∈ s, 0 ≤ w y) (_ : ∑ y ∈ s, w y = 1) (_ : ∃ y ∈ s, w y = 0), s.centerMass w id = x | case h.mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x✝ x : E
⊢ (∃ i, ∃ (_ : i ⊂ s), x ∈ (convexHull 𝕜) ↑i) →
∃ w, ∃ (_ : ∀ y ∈ s, 0 ≤ w y) (_ : ∑ y ∈ s, w y = 1) (_ : ∃ y ∈ s, w y = 0), s.centerMass w id = x
case h.mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x✝ x : E
⊢ (∃ w, ∃ (_ : ∀ y ∈ s, 0 ≤ w y) (_ : ∑ y ∈ s, w y = 1) (_ : ∃ y ∈ s, w y = 0), s.centerMass w id = x) →
∃ i, ∃ (_ : i ⊂ s), x ∈ (convexHull 𝕜) ↑i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | simp only [and_imp, exists_prop, exists_imp] | case h.mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x✝ x : E
⊢ (∃ i, ∃ (_ : i ⊂ s), x ∈ (convexHull 𝕜) ↑i) →
∃ w, ∃ (_ : ∀ y ∈ s, 0 ≤ w y) (_ : ∑ y ∈ s, w y = 1) (_ : ∃ y ∈ s, w y = 0), s.centerMass w id = x | case h.mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x✝ x : E
⊢ ∀ x_1 ⊂ s,
x ∈ (convexHull 𝕜) ↑x_1 → ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | intro t ts hx | case h.mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Finset E
x✝ x : E
⊢ ∀ x_1 ⊂ s,
x ∈ (convexHull 𝕜) ↑x_1 → ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x | case h.mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
hx : x ∈ (convexHull 𝕜) ↑t
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [Finset.convexHull_eq, Set.mem_setOf_eq] at hx | case h.mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
hx : x ∈ (convexHull 𝕜) ↑t
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x | case h.mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
hx : ∃ w, (∀ y ∈ t, 0 ≤ w y) ∧ ∑ y ∈ t, w y = 1 ∧ t.centerMass w id = x
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rcases hx with ⟨w, hw₀, hw₁, hx⟩ | case h.mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
hx : ∃ w, (∀ y ∈ t, 0 ≤ w y) ∧ ∑ y ∈ t, w y = 1 ∧ t.centerMass w id = x
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x | case h.mp.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rcases Finset.exists_of_ssubset ts with ⟨y, hys, hyt⟩ | case h.mp.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x | case h.mp.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | let w' z := if z ∈ t then w z else 0 | case h.mp.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x | case h.mp.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | have hw'₁ : s.sum w' = 1 := by
rwa [← Finset.sum_subset ts.1, Finset.sum_extend_by_zero]
simp only [ite_eq_right_iff]
tauto | case h.mp.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x | case h.mp.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | refine' ⟨w', _, hw'₁, ⟨_, ‹y ∈ s›, _⟩, _⟩ | case h.mp.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ (∃ y ∈ s, w y = 0) ∧ s.centerMass w id = x | case h.mp.intro.intro.intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∀ y ∈ s, 0 ≤ w' y
case h.mp.intro.intro.intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ w' y = 0
case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ s.centerMass w' id = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [← Finset.centerMass_subset id ts.1] | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ s.centerMass w' id = x | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ t.centerMass w' id = x
case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∀ i ∈ s, i ∉ t → w' i = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | exact fun i _ hi => if_neg hi | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∀ i ∈ s, i ∉ t → w' i = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rwa [← Finset.sum_subset ts.1, Finset.sum_extend_by_zero] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
⊢ s.sum w' = 1 | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
⊢ ∀ x ∈ s, x ∉ t → (if x ∈ t then w x else 0) = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | simp only [ite_eq_right_iff] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
⊢ ∀ x ∈ s, x ∉ t → (if x ∈ t then w x else 0) = 0 | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
⊢ ∀ x ∈ s, x ∉ t → x ∈ t → w x = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | tauto | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
⊢ ∀ x ∈ s, x ∉ t → x ∈ t → w x = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rintro y - | case h.mp.intro.intro.intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∀ y ∈ s, 0 ≤ w' y | case h.mp.intro.intro.intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y✝ : E
hys : y✝ ∈ s
hyt : y✝ ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
y : E
⊢ 0 ≤ w' y |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | change 0 ≤ ite (y ∈ t) (w y) 0 | case h.mp.intro.intro.intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y✝ : E
hys : y✝ ∈ s
hyt : y✝ ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
y : E
⊢ 0 ≤ w' y | case h.mp.intro.intro.intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y✝ : E
hys : y✝ ∈ s
hyt : y✝ ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
y : E
⊢ 0 ≤ if y ∈ t then w y else 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | split_ifs | case h.mp.intro.intro.intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y✝ : E
hys : y✝ ∈ s
hyt : y✝ ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
y : E
⊢ 0 ≤ if y ∈ t then w y else 0 | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y✝ : E
hys : y✝ ∈ s
hyt : y✝ ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
y : E
h✝ : y ∈ t
⊢ 0 ≤ w y
case neg
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y✝ : E
hys : y✝ ∈ s
hyt : y✝ ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
y : E
h✝ : y ∉ t
⊢ 0 ≤ 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | apply hw₀ y ‹_› | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y✝ : E
hys : y✝ ∈ s
hyt : y✝ ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
y : E
h✝ : y ∈ t
⊢ 0 ≤ w y | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rfl | case neg
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y✝ : E
hys : y✝ ∈ s
hyt : y✝ ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
y : E
h✝ : y ∉ t
⊢ 0 ≤ 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | apply if_neg ‹y ∉ t› | case h.mp.intro.intro.intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ w' y = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [Finset.centerMass_eq_of_sum_1] | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ t.centerMass w' id = x | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∑ i ∈ t, w' i • id i = x
case h.mp.intro.intro.intro.intro.intro.refine'_3.hw
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∑ i ∈ t, w' i = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rwa [Finset.sum_extend_by_zero] | case h.mp.intro.intro.intro.intro.intro.refine'_3.hw
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∑ i ∈ t, w' i = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [Finset.centerMass_eq_of_sum_1 _ _ hw₁] at hx | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : t.centerMass w id = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∑ i ∈ t, w' i • id i = x | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : ∑ i ∈ t, w i • id i = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∑ i ∈ t, w' i • id i = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [← hx] | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : ∑ i ∈ t, w i • id i = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∑ i ∈ t, w' i • id i = x | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : ∑ i ∈ t, w i • id i = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∑ i ∈ t, w' i • id i = ∑ i ∈ t, w i • id i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | apply Finset.sum_congr rfl | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : ∑ i ∈ t, w i • id i = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∑ i ∈ t, w' i • id i = ∑ i ∈ t, w i • id i | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : ∑ i ∈ t, w i • id i = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∀ x ∈ t, w' x • id x = w x • id x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | intro x hx | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝ x : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx : ∑ i ∈ t, w i • id i = x
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
⊢ ∀ x ∈ t, w' x • id x = w x • id x | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝¹ x✝ : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx✝ : ∑ i ∈ t, w i • id i = x✝
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
x : E
hx : x ∈ t
⊢ w' x • id x = w x • id x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | change ite _ _ _ • _ = _ | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝¹ x✝ : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx✝ : ∑ i ∈ t, w i • id i = x✝
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
x : E
hx : x ∈ t
⊢ w' x • id x = w x • id x | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝¹ x✝ : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx✝ : ∑ i ∈ t, w i • id i = x✝
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
x : E
hx : x ∈ t
⊢ (if x ∈ t then w x else 0) • id x = w x • id x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | combiFrontier_eq | [90, 1] | [135, 14] | rw [if_pos hx] | case h.mp.intro.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t✝ : Finset E
x✝¹ x✝ : E
t : Finset E
ts : t ⊂ s
w : E → 𝕜
hw₀ : ∀ y ∈ t, 0 ≤ w y
hw₁ : ∑ y ∈ t, w y = 1
hx✝ : ∑ i ∈ t, w i • id i = x✝
y : E
hys : y ∈ s
hyt : y ∉ t
w' : E → 𝕜 := fun z => if z ∈ t then w z else 0
hw'₁ : s.sum w' = 1
x : E
hx : x ∈ t
⊢ (if x ∈ t then w x else 0) • id x = w x • id x | no goals |
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