url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
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/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rw [modPartitions] | α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
⊢ (modPartitions s d hd h).parts = image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d) | α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
⊢ { parts := image (fun i => filter (fun j => j % d = i) (range s)) (range d), supIndep := ⋯, sup_parts := ⋯,
not_bot_mem := ⋯ }.parts =
image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | ext x | α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
⊢ { parts := image (fun i => filter (fun j => j % d = i) (range s)) (range d), supIndep := ⋯, sup_parts := ⋯,
not_bot_mem := ⋯ }.parts =
image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d) | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
⊢ x ∈
{ parts := image (fun i => filter (fun j => j % d = i) (range s)) (range d), supIndep := ⋯, sup_parts := ⋯,
not_bot_mem := ⋯ }.parts ↔
x ∈ image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | simp only [mem_image, mem_range] | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
⊢ x ∈
{ parts := image (fun i => filter (fun j => j % d = i) (range s)) (range d), supIndep := ⋯, sup_parts := ⋯,
not_bot_mem := ⋯ }.parts ↔
x ∈ image (fun i => image (fun x => i + d * x) (range ((s - i - 1) / d + 1))) (range d) | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
⊢ (∃ a < d, filter (fun j => j % d = a) (range s) = x) ↔
∃ a < d, image (fun x => a + d * x) (range ((s - a - 1) / d + 1)) = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | refine' exists_congr fun i ↦ and_congr_right fun hi ↦ _ | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
⊢ (∃ a < d, filter (fun j => j % d = a) (range s) = x) ↔
∃ a < d, image (fun x => a + d * x) (range ((s - a - 1) / d + 1)) = x | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
⊢ filter (fun j => j % d = i) (range s) = x ↔ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | suffices
((range ((s - i - 1) / d + 1)).image fun x ↦ i + d * x) = (range s).filter fun j ↦ j % d = i
by rw [this] | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
⊢ filter (fun j => j % d = i) (range s) = x ↔ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = x | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
⊢ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | clear x | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
⊢ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s) | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
⊢ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | ext j | case a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
⊢ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s) | case a.a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ j ∈ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) ↔ j ∈ filter (fun j => j % d = i) (range s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | simp only [mem_image, mem_filter, mem_range, Nat.lt_add_one_iff] | case a.a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ j ∈ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) ↔ j ∈ filter (fun j => j % d = i) (range s) | case a.a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ (∃ a ≤ (s - i - 1) / d, i + d * a = j) ↔ j < s ∧ j % d = i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | constructor | case a.a
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ (∃ a ≤ (s - i - 1) / d, i + d * a = j) ↔ j < s ∧ j % d = i | case a.a.mp
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ (∃ a ≤ (s - i - 1) / d, i + d * a = j) → j < s ∧ j % d = i
case a.a.mpr
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ j < s ∧ j % d = i → ∃ a ≤ (s - i - 1) / d, i + d * a = j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rw [this] | α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
x : Finset ℕ
i : ℕ
hi : i < d
this : image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = filter (fun j => j % d = i) (range s)
⊢ filter (fun j => j % d = i) (range s) = x ↔ image (fun x => i + d * x) (range ((s - i - 1) / d + 1)) = x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rintro ⟨j, hj, rfl⟩ | case a.a.mp
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ (∃ a ≤ (s - i - 1) / d, i + d * a = j) → j < s ∧ j % d = i | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j ≤ (s - i - 1) / d
⊢ i + d * j < s ∧ (i + d * j) % d = i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rw [Nat.add_mul_mod_self_left, Nat.mod_eq_of_lt hi, eq_self_iff_true, and_true_iff, ←
Nat.lt_sub_iff_add_lt', mul_comm] | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j ≤ (s - i - 1) / d
⊢ i + d * j < s ∧ (i + d * j) % d = i | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j ≤ (s - i - 1) / d
⊢ j * d < s - i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rwa [Nat.le_div_iff_mul_le hd.bot_lt, Nat.le_sub_iff_add_le, Nat.succ_le_iff] at hj | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j ≤ (s - i - 1) / d
⊢ j * d < s - i | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j * d ≤ s - i - 1
⊢ 1 ≤ s - i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rw [Nat.succ_le_iff] | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j * d ≤ s - i - 1
⊢ 1 ≤ s - i | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j * d ≤ s - i - 1
⊢ 0 < s - i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | exact Nat.sub_pos_of_lt (hi.trans_le h) | case a.a.mp.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
hj : j * d ≤ s - i - 1
⊢ 0 < s - i | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rintro ⟨hj, rfl⟩ | case a.a.mpr
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
i : ℕ
hi : i < d
j : ℕ
⊢ j < s ∧ j % d = i → ∃ a ≤ (s - i - 1) / d, i + d * a = j | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ ∃ a ≤ (s - j % d - 1) / d, j % d + d * a = j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | refine' ⟨j / d, _, Nat.mod_add_div _ _⟩ | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ ∃ a ≤ (s - j % d - 1) / d, j % d + d * a = j | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ j / d ≤ (s - j % d - 1) / d |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rwa [Nat.le_div_iff_mul_le' hd.bot_lt, Nat.le_sub_iff_add_le, Nat.le_sub_iff_add_le',
← add_assoc, mul_comm, Nat.mod_add_div, Nat.add_one_le_iff] | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ j / d ≤ (s - j % d - 1) / d | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ j % d ≤ s
case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ 1 ≤ s - j % d |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | rw [Nat.succ_le_iff] | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ 1 ≤ s - j % d | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ 0 < s - j % d |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | exact Nat.sub_pos_of_lt (hi.trans_le h) | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ 0 < s - j % d | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Order/Partition/Finpartition.lean | Finpartition.modPartitions_parts_eq | [68, 1] | [94, 44] | exact hi.le.trans h | case a.a.mpr.intro
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s d : ℕ
hd : d ≠ 0
h : d ≤ s
j : ℕ
hj : j < s
hi : j % d < d
⊢ j % d ≤ s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.coe_eq_one | [20, 1] | [21, 85] | rfl | α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
⊢ ↑s = ↑⊥ ↔ ↑s = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.smul_coe | [23, 1] | [25, 97] | ext | α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
ha : a ∈ s
⊢ a • ↑s = ↑s | case h
α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
ha : a ∈ s
x✝ : α
⊢ x✝ ∈ a • ↑s ↔ x✝ ∈ ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.smul_coe | [23, 1] | [25, 97] | rw [Set.mem_smul_set_iff_inv_smul_mem] | case h
α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
ha : a ∈ s
x✝ : α
⊢ x✝ ∈ a • ↑s ↔ x✝ ∈ ↑s | case h
α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
ha : a ∈ s
x✝ : α
⊢ a⁻¹ • x✝ ∈ ↑s ↔ x✝ ∈ ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.smul_coe | [23, 1] | [25, 97] | exact Subgroup.mul_mem_cancel_left _ (inv_mem ha) | case h
α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
ha : a ∈ s
x✝ : α
⊢ a⁻¹ • x✝ ∈ ↑s ↔ x✝ ∈ ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | rw [← Nat.card_prod, Nat.card_congr] | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ Nat.card ↥s * Nat.card ↑(QuotientGroup.mk '' t) = Nat.card ↑(t * ↑s) | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↥s × ↑(QuotientGroup.mk '' t) ≃ ↑(t * ↑s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | apply Equiv.trans (QuotientGroup.preimageMkEquivSubgroupProdSet _ _).symm | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↥s × ↑(QuotientGroup.mk '' t) ≃ ↑(t * ↑s) | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' t)) ≃ ↑(t * ↑s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | rw [QuotientGroup.preimage_image_mk] | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' t)) ≃ ↑(t * ↑s) | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↑(⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t) ≃ ↑(t * ↑s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | apply Set.BijOn.equiv id | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↑(⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t) ≃ ↑(t * ↑s) | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ Set.BijOn id (⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t) (t * ↑s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | convert Set.bijOn_id _ | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ Set.BijOn id (⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t) (t * ↑s) | case h.e'_5
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ t * ↑s = ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | ext x | case h.e'_5
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ t * ↑s = ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t | case h.e'_5.h
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ t * ↑s ↔ x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | constructor | case h.e'_5.h
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ t * ↑s ↔ x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t | case h.e'_5.h.mp
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ t * ↑s → x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t
case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t → x ∈ t * ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | simp only [Set.mem_iUnion, Set.mem_preimage, Subtype.exists, exists_prop, Set.mem_mul] | case h.e'_5.h.mp
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ t * ↑s → x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t | case h.e'_5.h.mp
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ (∃ x_1 ∈ t, ∃ y ∈ ↑s, x_1 * y = x) → ∃ a ∈ s, x * a ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | aesop | case h.e'_5.h.mp
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ (∃ x_1 ∈ t, ∃ y ∈ ↑s, x_1 * y = x) → ∃ a ∈ s, x * a ∈ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | simp only [Set.mem_iUnion, Set.mem_preimage, Subtype.exists, exists_prop, forall_exists_index,
and_imp] | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t → x ∈ t * ↑s | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ ∀ x_1 ∈ s, x * x_1 ∈ t → x ∈ t * ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | intro y hys hxy | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ ∀ x_1 ∈ s, x * x_1 ∈ t → x ∈ t * ↑s | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x y : α
hys : y ∈ s
hxy : x * y ∈ t
⊢ x ∈ t * ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | rw [← mul_inv_cancel_right x y] | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x y : α
hys : y ∈ s
hxy : x * y ∈ t
⊢ x ∈ t * ↑s | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x y : α
hys : y ∈ s
hxy : x * y ∈ t
⊢ x * y * y⁻¹ ∈ t * ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | apply Set.mul_mem_mul hxy (by simpa) | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x y : α
hys : y ∈ s
hxy : x * y ∈ t
⊢ x * y * y⁻¹ ∈ t * ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | simpa | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x y : α
hys : y ∈ s
hxy : x * y ∈ t
⊢ y⁻¹ ∈ ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.pairwiseDisjoint_smul | [53, 1] | [60, 80] | rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ hab | α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
⊢ (range fun a => a • ↑s).PairwiseDisjoint id | case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : (fun a => a • ↑s) a ≠ (fun a => a • ↑s) b
⊢ (Disjoint on id) ((fun a => a • ↑s) a) ((fun a => a • ↑s) b) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.pairwiseDisjoint_smul | [53, 1] | [60, 80] | simp only [Function.onFun, id_eq, disjoint_left] at hab ⊢ | case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : (fun a => a • ↑s) a ≠ (fun a => a • ↑s) b
⊢ (Disjoint on id) ((fun a => a • ↑s) a) ((fun a => a • ↑s) b) | case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
⊢ ∀ ⦃a_1 : β⦄, a_1 ∈ a • ↑s → a_1 ∉ b • ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.pairwiseDisjoint_smul | [53, 1] | [60, 80] | rintro _ ⟨c, hc, rfl⟩ ⟨d, hd, (hcd : b • d = a • c)⟩ | case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
⊢ ∀ ⦃a_1 : β⦄, a_1 ∈ a • ↑s → a_1 ∉ b • ↑s | case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
c : β
hc : c ∈ ↑s
d : β
hd : d ∈ ↑s
hcd : b • d = a • c
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.pairwiseDisjoint_smul | [53, 1] | [60, 80] | refine' hab _ | case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
c : β
hc : c ∈ ↑s
d : β
hd : d ∈ ↑s
hcd : b • d = a • c
⊢ False | case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
c : β
hc : c ∈ ↑s
d : β
hd : d ∈ ↑s
hcd : b • d = a • c
⊢ a • ↑s = b • ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.pairwiseDisjoint_smul | [53, 1] | [60, 80] | rw [← smul_coe hc, ← smul_assoc, ← hcd, smul_assoc, smul_coe hc, smul_coe hd] | case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
c : β
hc : c ∈ ↑s
d : β
hd : d ∈ ↑s
hcd : b • d = a • c
⊢ a • ↑s = b • ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/BigOperators/LocallyFinite.lean | Finset.mul_prod_Ico | [12, 1] | [14, 36] | rw [Icc_eq_cons_Ico h, prod_cons] | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : CommMonoid β
f : α → β
a b : α
inst✝ : LocallyFiniteOrder α
h : a ≤ b
⊢ f b * ∏ x ∈ Ico a b, f x = ∏ x ∈ Icc a b, f x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/BigOperators/LocallyFinite.lean | Finset.mul_prod_Ioc | [16, 1] | [18, 36] | rw [Icc_eq_cons_Ioc h, prod_cons] | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : CommMonoid β
f : α → β
a b : α
inst✝ : LocallyFiniteOrder α
h : a ≤ b
⊢ f a * ∏ x ∈ Ioc a b, f x = ∏ x ∈ Icc a b, f x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/BigOperators/LocallyFinite.lean | Finset.mul_prod_Ioi | [25, 1] | [27, 34] | rw [Ici_eq_cons_Ioi, prod_cons] | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : CommMonoid β
f : α → β
a✝ b : α
inst✝ : LocallyFiniteOrderTop α
a : α
⊢ f a * ∏ x ∈ Ioi a, f x = ∏ x ∈ Ici a, f x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/BigOperators/LocallyFinite.lean | Finset.mul_prod_Iio | [34, 1] | [36, 34] | rw [Iic_eq_cons_Iio, prod_cons] | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : CommMonoid β
f : α → β
a✝ b : α
inst✝ : LocallyFiniteOrderBot α
a : α
⊢ f a * ∏ x ∈ Iio a, f x = ∏ x ∈ Iic a, f x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.Pure.exists_facet | [36, 1] | [36, 89] | sorry | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.Pure n
hs : s ∈ K
⊢ ∃ t ∈ K.facets, s ⊆ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.Pure.exists_face_of_card_le | [38, 1] | [44, 10] | by_cases H : s ∈ K.facets | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.Pure n
h : k ≤ n + 1
hs : s ∈ K
hcard : s.card ≤ k
⊢ ∃ t ∈ K, s ⊆ t ∧ t.card = k | case pos
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.Pure n
h : k ≤ n + 1
hs : s ∈ K
hcard : s.card ≤ k
H : s ∈ K.facets
⊢ ∃ t ∈ K, s ⊆ t ∧ t.card = k
case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.Pure n
h : k ≤ n + 1
hs : s ∈ K
hcard : s.card ≤ k
H : s ∉ K.facets
⊢ ∃ t ∈ K, s ⊆ t ∧ t.card = k |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.Pure.exists_face_of_card_le | [38, 1] | [44, 10] | exact ⟨s, hs, Subset.rfl, hcard.antisymm <| h.trans (hK.2 H).ge⟩ | case pos
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.Pure n
h : k ≤ n + 1
hs : s ∈ K
hcard : s.card ≤ k
H : s ∈ K.facets
⊢ ∃ t ∈ K, s ⊆ t ∧ t.card = k | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.Pure.exists_face_of_card_le | [38, 1] | [44, 10] | unfold facets at H | case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.Pure n
h : k ≤ n + 1
hs : s ∈ K
hcard : s.card ≤ k
H : s ∉ K.facets
⊢ ∃ t ∈ K, s ⊆ t ∧ t.card = k | case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.Pure n
h : k ≤ n + 1
hs : s ∈ K
hcard : s.card ≤ k
H : s ∉ {s | s ∈ K.faces ∧ ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t}
⊢ ∃ t ∈ K, s ⊆ t ∧ t.card = k |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.Pure.exists_face_of_card_le | [38, 1] | [44, 10] | simp at H | case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.Pure n
h : k ≤ n + 1
hs : s ∈ K
hcard : s.card ≤ k
H : s ∉ {s | s ∈ K.faces ∧ ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t}
⊢ ∃ t ∈ K, s ⊆ t ∧ t.card = k | case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.Pure n
h : k ≤ n + 1
hs : s ∈ K
hcard : s.card ≤ k
H : s ∈ K → ∃ x ∈ K, s ⊆ x ∧ ¬s = x
⊢ ∃ t ∈ K, s ⊆ t ∧ t.card = k |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.Pure.exists_face_of_card_le | [38, 1] | [44, 10] | sorry | case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.Pure n
h : k ≤ n + 1
hs : s ∈ K
hcard : s.card ≤ k
H : s ∈ K → ∃ x ∈ K, s ⊆ x ∧ ¬s = x
⊢ ∃ t ∈ K, s ⊆ t ∧ t.card = k | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.pure_unique | [46, 1] | [49, 59] | obtain ⟨s, hs⟩ := hK | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
hK : K.faces.Nonempty
ha : K.Pure a
hb : K.Pure b
⊢ a = b | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
ha : K.Pure a
hb : K.Pure b
s : Finset E
hs : s ∈ K.faces
⊢ a = b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.pure_unique | [46, 1] | [49, 59] | obtain ⟨t, ht, -⟩ := ha.exists_facet hs | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
ha : K.Pure a
hb : K.Pure b
s : Finset E
hs : s ∈ K.faces
⊢ a = b | case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
ha : K.Pure a
hb : K.Pure b
s : Finset E
hs : s ∈ K.faces
t : Finset E
ht : t ∈ K.facets
⊢ a = b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.pure_unique | [46, 1] | [49, 59] | exact add_right_cancel ((ha.2 ht).symm.trans <| hb.2 ht) | case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
ha : K.Pure a
hb : K.Pure b
s : Finset E
hs : s ∈ K.faces
t : Finset E
ht : t ∈ K.facets
⊢ a = b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.pure_iff | [55, 1] | [64, 22] | refine' ⟨fun hK s hs => _, fun hK => ⟨fun s hs => _, fun s hs => _⟩⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
⊢ K.Pure n ↔ ∀ ⦃s : Finset E⦄, s ∈ K → ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : K.Pure n
s : Finset E
hs : s ∈ K
⊢ ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : ∀ ⦃s : Finset E⦄, s ∈ K → ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t
s : Finset E
hs : s ∈ K
⊢ s.card ≤ n + 1
case refine'_3
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : ∀ ⦃s : Finset E⦄, s ∈ K → ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t
s : Finset E
hs : s ∈ K.facets
⊢ s.card = n + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.pure_iff | [55, 1] | [64, 22] | obtain ⟨t, ht, hst⟩ := hK.exists_facet hs | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : K.Pure n
s : Finset E
hs : s ∈ K
⊢ ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t | case refine'_1.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : K.Pure n
s : Finset E
hs : s ∈ K
t : Finset E
ht : t ∈ K.facets
hst : s ⊆ t
⊢ ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.pure_iff | [55, 1] | [64, 22] | exact ⟨t, ht.1, hK.2 ht, hst⟩ | case refine'_1.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : K.Pure n
s : Finset E
hs : s ∈ K
t : Finset E
ht : t ∈ K.facets
hst : s ⊆ t
⊢ ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.pure_iff | [55, 1] | [64, 22] | obtain ⟨t, _, htcard, hst⟩ := hK hs | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : ∀ ⦃s : Finset E⦄, s ∈ K → ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t
s : Finset E
hs : s ∈ K
⊢ s.card ≤ n + 1 | case refine'_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : ∀ ⦃s : Finset E⦄, s ∈ K → ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t
s : Finset E
hs : s ∈ K
t : Finset E
left✝ : t ∈ K
htcard : t.card = n + 1
hst : s ⊆ t
⊢ s.card ≤ n + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.pure_iff | [55, 1] | [64, 22] | exact (Finset.card_le_card hst).trans_eq htcard | case refine'_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : ∀ ⦃s : Finset E⦄, s ∈ K → ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t
s : Finset E
hs : s ∈ K
t : Finset E
left✝ : t ∈ K
htcard : t.card = n + 1
hst : s ⊆ t
⊢ s.card ≤ n + 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.pure_iff | [55, 1] | [64, 22] | obtain ⟨t, ht, htcard, hst⟩ := hK hs.1 | case refine'_3
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : ∀ ⦃s : Finset E⦄, s ∈ K → ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t
s : Finset E
hs : s ∈ K.facets
⊢ s.card = n + 1 | case refine'_3.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : ∀ ⦃s : Finset E⦄, s ∈ K → ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t
s : Finset E
hs : s ∈ K.facets
t : Finset E
ht : t ∈ K
htcard : t.card = n + 1
hst : s ⊆ t
⊢ s.card = n + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.pure_iff | [55, 1] | [64, 22] | rwa [hs.2 ht hst] | case refine'_3.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
hK : ∀ ⦃s : Finset E⦄, s ∈ K → ∃ t ∈ K, t.card = n + 1 ∧ s ⊆ t
s : Finset E
hs : s ∈ K.facets
t : Finset E
ht : t ∈ K
htcard : t.card = n + 1
hst : s ⊆ t
⊢ s.card = n + 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.facets_mono | [66, 1] | [70, 23] | refine' fun s hs => ⟨h hs.1, fun t ht hst => Finset.eq_of_subset_of_card_le hst _⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s : Finset E
K₁ K₂ : SimplicialComplex 𝕜 E
h : K₁ ≤ K₂
hK₁ : K₁.Pure n
hK₂ : K₂.Pure n
⊢ K₁.facets ⊆ K₂.facets | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
K₁ K₂ : SimplicialComplex 𝕜 E
h : K₁ ≤ K₂
hK₁ : K₁.Pure n
hK₂ : K₂.Pure n
s : Finset E
hs : s ∈ K₁.facets
t : Finset E
ht : t ∈ K₂.faces
hst : s ⊆ t
⊢ t.card ≤ s.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.facets_mono | [66, 1] | [70, 23] | rw [hK₁.2 hs] | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
K₁ K₂ : SimplicialComplex 𝕜 E
h : K₁ ≤ K₂
hK₁ : K₁.Pure n
hK₂ : K₂.Pure n
s : Finset E
hs : s ∈ K₁.facets
t : Finset E
ht : t ∈ K₂.faces
hst : s ⊆ t
⊢ t.card ≤ s.card | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
K₁ K₂ : SimplicialComplex 𝕜 E
h : K₁ ≤ K₂
hK₁ : K₁.Pure n
hK₂ : K₂.Pure n
s : Finset E
hs : s ∈ K₁.facets
t : Finset E
ht : t ∈ K₂.faces
hst : s ⊆ t
⊢ t.card ≤ n + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Pure.lean | Geometry.SimplicialComplex.facets_mono | [66, 1] | [70, 23] | exact hK₂.card_le ht | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ : Finset E
K₁ K₂ : SimplicialComplex 𝕜 E
h : K₁ ≤ K₂
hK₁ : K₁.Pure n
hK₂ : K₂.Pure n
s : Finset E
hs : s ∈ K₁.facets
t : Finset E
ht : t ∈ K₂.faces
hst : s ⊆ t
⊢ t.card ≤ n + 1 | 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/Mathlib/Combinatorics/SimpleGraph/Finite.lean | SimpleGraph.disjoint_edgeFinset | [8, 1] | [10, 68] | simp_rw [← Finset.disjoint_coe, coe_edgeFinset, disjoint_edgeSet] | α : Type u_1
G H : SimpleGraph α
inst✝¹ : Fintype ↑G.edgeSet
inst✝ : Fintype ↑H.edgeSet
⊢ Disjoint G.edgeFinset H.edgeFinset ↔ Disjoint G H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Finite.lean | SimpleGraph.edgeFinset_eq_empty | [12, 1] | [13, 40] | rw [← edgeFinset_bot, edgeFinset_inj] | α : Type u_1
G H : SimpleGraph α
inst✝ : Fintype ↑G.edgeSet
⊢ G.edgeFinset = ∅ ↔ G = ⊥ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Finite.lean | SimpleGraph.edgeFinset_nonempty | [15, 1] | [16, 60] | rw [Finset.nonempty_iff_ne_empty, edgeFinset_eq_empty.ne] | α : Type u_1
G H : SimpleGraph α
inst✝ : Fintype ↑G.edgeSet
⊢ G.edgeFinset.Nonempty ↔ G ≠ ⊥ | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_08/Diaphontine.lean | eqn_solvable_divides | [50, 1] | [62, 32] | intro ⟨x, y, h⟩ | a b c : ℤ
⊢ (∃ x y, a * x + b * y = c) → ↑(Int.gcd a b) ∣ c | a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ c |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_08/Diaphontine.lean | eqn_solvable_divides | [50, 1] | [62, 32] | rw [← h] | a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ c | a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ a * x + b * y |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_08/Diaphontine.lean | eqn_solvable_divides | [50, 1] | [62, 32] | apply dvd_add | a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ a * x + b * y | case h₁
a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ a * x
case h₂
a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ b * y |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_08/Diaphontine.lean | eqn_solvable_divides | [50, 1] | [62, 32] | trans a | case h₁
a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ a * x | a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ a
a b c x y : ℤ
h : a * x + b * y = c
⊢ a ∣ a * x |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_08/Diaphontine.lean | eqn_solvable_divides | [50, 1] | [62, 32] | apply Int.gcd_dvd_left | a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ a | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_08/Diaphontine.lean | eqn_solvable_divides | [50, 1] | [62, 32] | apply Int.dvd_mul_right | a b c x y : ℤ
h : a * x + b * y = c
⊢ a ∣ a * x | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_08/Diaphontine.lean | eqn_solvable_divides | [50, 1] | [62, 32] | trans b | case h₂
a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ b * y | a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ b
a b c x y : ℤ
h : a * x + b * y = c
⊢ b ∣ b * y |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.