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 | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_attachFin, mem_range] | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ : Set.Sized r ↑𝒞
this : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
⊢ t ∈ range'k | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ : Set.Sized r ↑𝒞
this : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
⊢ ↑t < k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ : Set.Sized r ↑𝒞
this : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
⊢ t ∈ range'k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have : toColex (image Fin.val B) < toColex (image Fin.val A) := by
rwa [toColex_image_lt_toColex_image Fin.val_strictMono] | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ : Set.Sized r ↑𝒞
this : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
⊢ ↑t < k | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⊢ ↑t < k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ : Set.Sized r ↑𝒞
this : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
⊢ ↑t < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | apply Colex.forall_lt_mono this.le _ t (mem_image.2 ⟨t, ht, rfl⟩) | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⊢ ↑t < k | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⊢ ∀ b ∈ image Fin.val A, b < k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⊢ ↑t < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | simp_rw [mem_image] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⊢ ∀ b ∈ image Fin.val A, b < k | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⊢ ∀ (b : ℕ), (∃ a ∈ A, ↑a = b) → b < k | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⊢ ∀ b ∈ image Fin.val A, b < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rintro _ ⟨a, ha, q⟩ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⊢ ∀ (b : ℕ), (∃ a ∈ A, ↑a = b) → b < k | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
⊢ b✝ < k | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⊢ ∀ (b : ℕ), (∃ a ∈ A, ↑a = b) → b < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_powersetCard] at hA | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
⊢ b✝ < k | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ⊆ range'k ∧ A.card = r
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
⊢ b✝ < k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
⊢ b✝ < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [← q, ← mem_range] | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ⊆ range'k ∧ A.card = r
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
⊢ b✝ < k | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ⊆ range'k ∧ A.card = r
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
⊢ ↑a ∈ range k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ⊆ range'k ∧ A.card = r
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
⊢ b✝ < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have := hA.1 ha | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ⊆ range'k ∧ A.card = r
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
⊢ ↑a ∈ range k | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝² : Set.Sized r ↑𝒞
this✝¹ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ⊆ range'k ∧ A.card = r
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this✝ : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
this : a ∈ range'k
⊢ ↑a ∈ range k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝¹ : Set.Sized r ↑𝒞
this✝ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ⊆ range'k ∧ A.card = r
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
⊢ ↑a ∈ range k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rwa [mem_attachFin] at this | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝² : Set.Sized r ↑𝒞
this✝¹ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ⊆ range'k ∧ A.card = r
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this✝ : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
this : a ∈ range'k
⊢ ↑a ∈ range k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝² : Set.Sized r ↑𝒞
this✝¹ : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ⊆ range'k ∧ A.card = r
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
this✝ : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
b✝ : ℕ
a : Fin n
ha : a ∈ A
q : ↑a = b✝
this : a ∈ range'k
⊢ ↑a ∈ range k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rwa [toColex_image_lt_toColex_image Fin.val_strictMono] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ : Set.Sized r ↑𝒞
this : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
⊢ { ofColex := image Fin.val B } < { ofColex := image Fin.val A } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ : Set.Sized r ↑𝒞
this : (∂ ^[i] 𝒞).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A ∈ 𝒞
HB₁ : { ofColex := B } < { ofColex := A }
HB₂ : B.card = r
t : Fin n
ht : t ∈ B
⊢ { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [this, card_powersetCard, card_attachFin, card_range] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ : Set.Sized r ↑𝒞
this : ∂ ^[i] 𝒞 = powersetCard (r - i) range'k
⊢ (∂ ^[i] 𝒞).card = k.choose (r - i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ : Set.Sized r ↑𝒞
this : ∂ ^[i] 𝒞 = powersetCard (r - i) range'k
⊢ (∂ ^[i] 𝒞).card = k.choose (r - i)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rintro ⟨A, Ah, BsubA, card_sdiff_i⟩ | case a.mp
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B : Finset (Fin n)
⊢ (∃ s ∈ 𝒞, B ⊆ s ∧ (s \ B).card = i) → B ⊆ range'k ∧ B.card = r - i | case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ∈ 𝒞
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ B ⊆ range'k ∧ B.card = r - i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B : Finset (Fin n)
⊢ (∃ s ∈ 𝒞, B ⊆ s ∧ (s \ B).card = i) → B ⊆ range'k ∧ B.card = r - i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_powersetCard] at Ah | case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ∈ 𝒞
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ B ⊆ range'k ∧ B.card = r - i | case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ⊆ range'k ∧ A.card = r
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ B ⊆ range'k ∧ B.card = r - i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ∈ 𝒞
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ B ⊆ range'k ∧ B.card = r - i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | refine' ⟨BsubA.trans Ah.1, _⟩ | case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ⊆ range'k ∧ A.card = r
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ B ⊆ range'k ∧ B.card = r - i | case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ⊆ range'k ∧ A.card = r
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ B.card = r - i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ⊆ range'k ∧ A.card = r
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ B ⊆ range'k ∧ B.card = r - i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | symm | case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ⊆ range'k ∧ A.card = r
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ B.card = r - i | case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ⊆ range'k ∧ A.card = r
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ r - i = B.card | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ⊆ range'k ∧ A.card = r
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ B.card = r - i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [Nat.sub_eq_iff_eq_add hir, ← Ah.2, ← card_sdiff_i, ← card_union_of_disjoint disjoint_sdiff,
union_sdiff_of_subset BsubA] | case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ⊆ range'k ∧ A.card = r
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ r - i = B.card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B A : Finset (Fin n)
Ah : A ⊆ range'k ∧ A.card = r
BsubA : B ⊆ A
card_sdiff_i : (A \ B).card = i
⊢ r - i = B.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [card_sdiff BsubC, hcard, hB, Nat.sub_sub_self hir] | case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B : Finset (Fin n)
hBk : B ⊆ range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B ⊆ C
hCrange : C ⊆ range'k
hcard : C.card = r
⊢ (C \ B).card = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B : Finset (Fin n)
hBk : B ⊆ range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B ⊆ C
hCrange : C ⊆ range'k
hcard : C.card = r
⊢ (C \ B).card = i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rwa [hB, card_attachFin, card_range, ← Nat.add_sub_assoc hir, Nat.add_sub_cancel_left] | case a.mpr.intro.refine_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B : Finset (Fin n)
hBk : B ⊆ range'k
hB : B.card = r - i
⊢ i + B.card ≤ range'k.card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.refine_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this : Set.Sized r ↑𝒞
B : Finset (Fin n)
hBk : B ⊆ range'k
hB : B.card = r - i
⊢ i + B.card ≤ range'k.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | cases' Nat.eq_zero_or_pos r with b h1r | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
⊢ 𝒜.card ≤ (n - 1).choose (r - 1) | case inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
⊢ 𝒜.card ≤ (n - 1).choose (r - 1)
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
⊢ 𝒜.card ≤ (n - 1).choose (r - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
⊢ 𝒜.card ≤ (n - 1).choose (r - 1)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | refine' le_of_not_lt fun size ↦ _ | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
⊢ 𝒜.card ≤ (n - 1).choose (r - 1) | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
⊢ 𝒜.card ≤ (n - 1).choose (r - 1)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | have : Disjoint 𝒜 (∂^[n - 2 * r] 𝒜ᶜˢ) := disjoint_right.2 fun A hAbar hA ↦ by
simp [mem_shadow_iterate_iff_exists_sdiff, mem_compls] at hAbar
obtain ⟨C, hC, hAC, _⟩ := hAbar
exact h𝒜 hA hC (disjoint_of_subset_left hAC disjoint_compl_right) | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | have : r ≤ n := h₃.trans (Nat.div_le_self n 2) | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this : r ≤ n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | have : 1 ≤ n := ‹1 ≤ r›.trans ‹r ≤ n› | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this : r ≤ n
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this : r ≤ n
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | have z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card := by
rwa [card_compls, choose_symm_of_eq_add (tsub_add_tsub_cancel ‹r ≤ n› ‹1 ≤ r›).symm] | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | have h𝒜bar : (𝒜ᶜˢ : Set (Finset (Fin n))).Sized (n - r) := by simpa using h₂.compls | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | have : n - 2 * r ≤ n - r := by
rw [tsub_le_tsub_iff_left ‹r ≤ n›]
exact Nat.le_mul_of_pos_left _ zero_lt_two | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | have kk :=
lovasz_form ‹n - 2 * r ≤ n - r› ((tsub_le_tsub_iff_left ‹1 ≤ n›).2 h1r) tsub_le_self h𝒜bar z.le | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | have q : n - r - (n - 2 * r) = r := by
rw [tsub_right_comm, Nat.sub_sub_self, two_mul]
apply Nat.add_sub_cancel
rw [mul_comm, ← Nat.le_div_iff_mul_le' zero_lt_two]
exact h₃ | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rw [q] at kk | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | have : n.choose r < (𝒜 ∪ ∂^[n - 2 * r] 𝒜ᶜˢ).card := by
rw [card_union_of_disjoint ‹_›]
convert lt_of_le_of_lt (add_le_add_left kk _) (add_lt_add_right size _) using 1
convert Nat.choose_succ_succ _ _ using 3
all_goals rwa [Nat.sub_one, Nat.succ_pred_eq_of_pos] | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | apply this.not_le | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ False | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ≤ n.choose r | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | convert Set.Sized.card_le _ | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ≤ n.choose r | case h.e'_4.h.e'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ n = Fintype.card (Fin n)
case inr.convert_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ Fintype (Fin n)
case inr.convert_5
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ Set.Sized r ↑(𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card ≤ n.choose r
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rw [coe_union, Set.sized_union] | case inr.convert_5
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ Set.Sized r ↑(𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ) | case inr.convert_5
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ Set.Sized r ↑𝒜 ∧ Set.Sized r ↑(∂ ^[n - 2 * r] 𝒜ᶜˢ) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.convert_5
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ Set.Sized r ↑(𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | refine' ⟨‹_›, _⟩ | case inr.convert_5
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ Set.Sized r ↑𝒜 ∧ Set.Sized r ↑(∂ ^[n - 2 * r] 𝒜ᶜˢ) | case inr.convert_5
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ Set.Sized r ↑(∂ ^[n - 2 * r] 𝒜ᶜˢ) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.convert_5
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ Set.Sized r ↑𝒜 ∧ Set.Sized r ↑(∂ ^[n - 2 * r] 𝒜ᶜˢ)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | convert h𝒜bar.shadow_iterate | case inr.convert_5
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ Set.Sized r ↑(∂ ^[n - 2 * r] 𝒜ᶜˢ) | case h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ r = n - r - (n - 2 * r) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.convert_5
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ Set.Sized r ↑(∂ ^[n - 2 * r] 𝒜ᶜˢ)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rw [q] | case h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ r = n - r - (n - 2 * r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ r = n - r - (n - 2 * r)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | convert Nat.zero_le _ | case inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
⊢ 𝒜.card ≤ (n - 1).choose (r - 1) | case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
⊢ 𝒜.card = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
⊢ 𝒜.card ≤ (n - 1).choose (r - 1)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rw [Finset.card_eq_zero, eq_empty_iff_forall_not_mem] | case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
⊢ 𝒜.card = 0 | case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
⊢ ∀ (x : Finset (Fin n)), x ∉ 𝒜 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
⊢ 𝒜.card = 0
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | refine' fun A HA ↦ h𝒜 HA HA _ | case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
⊢ ∀ (x : Finset (Fin n)), x ∉ 𝒜 | case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
A : Finset (Fin n)
HA : A ∈ 𝒜
⊢ Disjoint A A | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
⊢ ∀ (x : Finset (Fin n)), x ∉ 𝒜
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rw [disjoint_self_iff_empty, ← Finset.card_eq_zero, ← b] | case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
A : Finset (Fin n)
HA : A ∈ 𝒜
⊢ Disjoint A A | case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
A : Finset (Fin n)
HA : A ∈ 𝒜
⊢ A.card = r | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
A : Finset (Fin n)
HA : A ∈ 𝒜
⊢ Disjoint A A
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | exact h₂ HA | case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
A : Finset (Fin n)
HA : A ∈ 𝒜
⊢ A.card = r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
b : r = 0
A : Finset (Fin n)
HA : A ∈ 𝒜
⊢ A.card = r
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | simp [mem_shadow_iterate_iff_exists_sdiff, mem_compls] at hAbar | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
A : Finset (Fin n)
hAbar : A ∈ ∂ ^[n - 2 * r] 𝒜ᶜˢ
hA : A ∈ 𝒜
⊢ False | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
A : Finset (Fin n)
hA : A ∈ 𝒜
hAbar : ∃ s, sᶜ ∈ 𝒜 ∧ A ⊆ s ∧ (s \ A).card = n - 2 * r
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
A : Finset (Fin n)
hAbar : A ∈ ∂ ^[n - 2 * r] 𝒜ᶜˢ
hA : A ∈ 𝒜
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | obtain ⟨C, hC, hAC, _⟩ := hAbar | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
A : Finset (Fin n)
hA : A ∈ 𝒜
hAbar : ∃ s, sᶜ ∈ 𝒜 ∧ A ⊆ s ∧ (s \ A).card = n - 2 * r
⊢ False | case intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
A : Finset (Fin n)
hA : A ∈ 𝒜
C : Finset (Fin n)
hC : Cᶜ ∈ 𝒜
hAC : A ⊆ C
right✝ : (C \ A).card = n - 2 * r
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
A : Finset (Fin n)
hA : A ∈ 𝒜
hAbar : ∃ s, sᶜ ∈ 𝒜 ∧ A ⊆ s ∧ (s \ A).card = n - 2 * r
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | exact h𝒜 hA hC (disjoint_of_subset_left hAC disjoint_compl_right) | case intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
A : Finset (Fin n)
hA : A ∈ 𝒜
C : Finset (Fin n)
hC : Cᶜ ∈ 𝒜
hAC : A ⊆ C
right✝ : (C \ A).card = n - 2 * r
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
A : Finset (Fin n)
hA : A ∈ 𝒜
C : Finset (Fin n)
hC : Cᶜ ∈ 𝒜
hAC : A ⊆ C
right✝ : (C \ A).card = n - 2 * r
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rwa [card_compls, choose_symm_of_eq_add (tsub_add_tsub_cancel ‹r ≤ n› ‹1 ≤ r›).symm] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
⊢ (n - 1).choose (n - r) < 𝒜ᶜˢ.card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
⊢ (n - 1).choose (n - r) < 𝒜ᶜˢ.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | simpa using h₂.compls | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
⊢ Set.Sized (n - r) ↑𝒜ᶜˢ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
⊢ Set.Sized (n - r) ↑𝒜ᶜˢ
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rw [tsub_le_tsub_iff_left ‹r ≤ n›] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
⊢ n - 2 * r ≤ n - r | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
⊢ r ≤ 2 * r | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
⊢ n - 2 * r ≤ n - r
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | exact Nat.le_mul_of_pos_left _ zero_lt_two | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
⊢ r ≤ 2 * r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝¹ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝ : r ≤ n
this : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
⊢ r ≤ 2 * r
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rw [tsub_right_comm, Nat.sub_sub_self, two_mul] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ n - r - (n - 2 * r) = r | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ r + r - r = r
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ 2 * r ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ n - r - (n - 2 * r) = r
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | apply Nat.add_sub_cancel | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ r + r - r = r
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ 2 * r ≤ n | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ 2 * r ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ r + r - r = r
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ 2 * r ≤ n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rw [mul_comm, ← Nat.le_div_iff_mul_le' zero_lt_two] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ 2 * r ≤ n | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ r ≤ n / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ 2 * r ≤ n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | exact h₃ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ r ≤ n / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose (n - r - (n - 2 * r)) ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ r ≤ n / 2
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rw [card_union_of_disjoint ‹_›] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n.choose r < 𝒜.card + (∂ ^[n - 2 * r] 𝒜ᶜˢ).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | convert lt_of_le_of_lt (add_le_add_left kk _) (add_lt_add_right size _) using 1 | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n.choose r < 𝒜.card + (∂ ^[n - 2 * r] 𝒜ᶜˢ).card | case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n.choose r = (n - 1).choose (r - 1) + (n - 1).choose r | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n.choose r < 𝒜.card + (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | convert Nat.choose_succ_succ _ _ using 3 | case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n.choose r = (n - 1).choose (r - 1) + (n - 1).choose r | case h.e'_2.h.e'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n = (n - 1).succ
case h.e'_2.h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ r = (r - 1).succ
case h.e'_3.h.e'_6.h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ r = (r - 1).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n.choose r = (n - 1).choose (r - 1) + (n - 1).choose r
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | all_goals rwa [Nat.sub_one, Nat.succ_pred_eq_of_pos] | case h.e'_2.h.e'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n = (n - 1).succ
case h.e'_2.h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ r = (r - 1).succ
case h.e'_3.h.e'_6.h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ r = (r - 1).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ n = (n - 1).succ
case h.e'_2.h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ r = (r - 1).succ
case h.e'_3.h.e'_6.h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ r = (r - 1).succ
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rwa [Nat.sub_one, Nat.succ_pred_eq_of_pos] | case h.e'_3.h.e'_6.h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ r = (r - 1).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_6.h.e'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝² : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝¹ : r ≤ n
this✝ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
⊢ r = (r - 1).succ
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.EKR | [386, 1] | [436, 9] | rw [Fintype.card_fin] | case h.e'_4.h.e'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ n = Fintype.card (Fin n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset (Fin n))
r : ℕ
h𝒜 : (↑𝒜).Intersecting
h₂ : Set.Sized r ↑𝒜
h₃ : r ≤ n / 2
h1r : r > 0
size : (n - 1).choose (r - 1) < 𝒜.card
this✝³ : Disjoint 𝒜 (∂ ^[n - 2 * r] 𝒜ᶜˢ)
this✝² : r ≤ n
this✝¹ : 1 ≤ n
z : (n - 1).choose (n - r) < 𝒜ᶜˢ.card
h𝒜bar : Set.Sized (n - r) ↑𝒜ᶜˢ
this✝ : n - 2 * r ≤ n - r
kk : (n - 1).choose r ≤ (∂ ^[n - 2 * r] 𝒜ᶜˢ).card
q : n - r - (n - 2 * r) = r
this : n.choose r < (𝒜 ∪ ∂ ^[n - 2 * r] 𝒜ᶜˢ).card
⊢ n = Fintype.card (Fin n)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_eq | [8, 1] | [14, 47] | simp [List.length_eq_zero] | case zero
α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length = 0}.Finite | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length = 0}.Finite
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_eq | [8, 1] | [14, 47] | ext (_ | _) <;> simp [n.succ_ne_zero.symm] | case succ
α : Type u_1
inst✝ : Finite α
n✝ n : ℕ
ih : {l | l.length = n}.Finite
⊢ {l | l.length = n + 1} = image2 (fun x x_1 => x :: x_1) univ {l | l.length = n} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type u_1
inst✝ : Finite α
n✝ n : ℕ
ih : {l | l.length = n}.Finite
⊢ {l | l.length = n + 1} = image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_eq | [8, 1] | [14, 47] | rw [this] | α : Type u_1
inst✝ : Finite α
n✝ n : ℕ
ih : {l | l.length = n}.Finite
this : {l | l.length = n + 1} = image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}
⊢ {l | l.length = n + 1}.Finite | α : Type u_1
inst✝ : Finite α
n✝ n : ℕ
ih : {l | l.length = n}.Finite
this : {l | l.length = n + 1} = image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}
⊢ (image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Finite α
n✝ n : ℕ
ih : {l | l.length = n}.Finite
this : {l | l.length = n + 1} = image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}
⊢ {l | l.length = n + 1}.Finite
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_eq | [8, 1] | [14, 47] | exact Set.finite_univ.image2 _ ih | α : Type u_1
inst✝ : Finite α
n✝ n : ℕ
ih : {l | l.length = n}.Finite
this : {l | l.length = n + 1} = image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}
⊢ (image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}).Finite | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Finite α
n✝ n : ℕ
ih : {l | l.length = n}.Finite
this : {l | l.length = n + 1} = image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}
⊢ (image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}).Finite
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_lt | [16, 1] | [17, 90] | convert (Finset.range n).finite_toSet.biUnion fun i _ ↦ finite_length_eq α i | α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length < n}.Finite | case h.e'_2
α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length < n} = ⋃ i ∈ ↑(Finset.range n), {l | l.length = i} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length < n}.Finite
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_lt | [16, 1] | [17, 90] | ext | case h.e'_2
α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length < n} = ⋃ i ∈ ↑(Finset.range n), {l | l.length = i} | case h.e'_2.h
α : Type u_1
inst✝ : Finite α
n : ℕ
x✝ : List α
⊢ x✝ ∈ {l | l.length < n} ↔ x✝ ∈ ⋃ i ∈ ↑(Finset.range n), {l | l.length = i} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length < n} = ⋃ i ∈ ↑(Finset.range n), {l | l.length = i}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_lt | [16, 1] | [17, 90] | simp | case h.e'_2.h
α : Type u_1
inst✝ : Finite α
n : ℕ
x✝ : List α
⊢ x✝ ∈ {l | l.length < n} ↔ x✝ ∈ ⋃ i ∈ ↑(Finset.range n), {l | l.length = i} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h
α : Type u_1
inst✝ : Finite α
n : ℕ
x✝ : List α
⊢ x✝ ∈ {l | l.length < n} ↔ x✝ ∈ ⋃ i ∈ ↑(Finset.range n), {l | l.length = i}
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_le | [19, 1] | [20, 59] | simpa [Nat.lt_succ_iff] using finite_length_lt α (n + 1) | α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length ≤ n}.Finite | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length ≤ n}.Finite
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.getLast_filter | [6, 1] | [18, 42] | rw [List.getLast_singleton'] at h' | α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a : α
h : filter p [a] ≠ []
h' : p ([a].getLast ⋯) = true
⊢ (filter p [a]).getLast h = [a].getLast ⋯ | α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a : α
h : filter p [a] ≠ []
h' : p a = true
⊢ (filter p [a]).getLast h = [a].getLast ⋯ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a : α
h : filter p [a] ≠ []
h' : p ([a].getLast ⋯) = true
⊢ (filter p [a]).getLast h = [a].getLast ⋯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.getLast_filter | [6, 1] | [18, 42] | simp [List.filter_cons, h'] | α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a : α
h : filter p [a] ≠ []
h' : p a = true
⊢ (filter p [a]).getLast h = [a].getLast ⋯ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a : α
h : filter p [a] ≠ []
h' : p a = true
⊢ (filter p [a]).getLast h = [a].getLast ⋯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.getLast_filter | [6, 1] | [18, 42] | rw [List.getLast_cons_cons] at h' ⊢ | α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h : filter p (a :: b :: as) ≠ []
h' : p ((a :: b :: as).getLast ⋯) = true
⊢ (filter p (a :: b :: as)).getLast h = (a :: b :: as).getLast ⋯ | α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
⊢ (filter p (a :: b :: as)).getLast h = (b :: as).getLast ⋯ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h : filter p (a :: b :: as) ≠ []
h' : p ((a :: b :: as).getLast ⋯) = true
⊢ (filter p (a :: b :: as)).getLast h = (a :: b :: as).getLast ⋯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.getLast_filter | [6, 1] | [18, 42] | simp only [List.filter_cons (x := a)] at h ⊢ | α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
⊢ (filter p (a :: b :: as)).getLast h = (b :: as).getLast ⋯ | α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
⊢ (if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
⊢ (filter p (a :: b :: as)).getLast h = (b :: as).getLast ⋯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.getLast_filter | [6, 1] | [18, 42] | obtain ha | ha := Bool.eq_false_or_eq_true (p a) | α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
⊢ (if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯ | case inl
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = true
⊢ (if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯
case inr
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = false
⊢ (if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
⊢ (if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.getLast_filter | [6, 1] | [18, 42] | simp only [ha, ite_true] | case inl
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = true
⊢ (if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯ | case inl
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = true
⊢ (a :: filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯ | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = true
⊢ (if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.getLast_filter | [6, 1] | [18, 42] | rw [getLast_cons, getLast_filter (b :: as) _ h'] | case inl
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = true
⊢ (a :: filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯ | α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = true
⊢ filter p (b :: as) ≠ [] | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = true
⊢ (a :: filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.getLast_filter | [6, 1] | [18, 42] | exact ne_nil_of_mem $ mem_filter.2 ⟨getLast_mem _, h'⟩ | α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = true
⊢ filter p (b :: as) ≠ [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = true
⊢ filter p (b :: as) ≠ []
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.getLast_filter | [6, 1] | [18, 42] | simp only [ha, cond_false] at h ⊢ | case inr
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = false
⊢ (if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯ | case inr
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
ha : p a = false
h : (if False then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
⊢ (if False then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
ha : p a = false
⊢ (if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.getLast_filter | [6, 1] | [18, 42] | exact getLast_filter (b :: as) h h' | case inr
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
ha : p a = false
h : (if False then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
⊢ (if False then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α✝ : Type u_1
l l' l₀ l₁ l₂ : List α✝
a✝ b✝ : α✝
m n : ℕ
α : Type u_2
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
ha : p a = false
h : (if False then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
⊢ (if False then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.cons_sublist_cons_iff' | [20, 1] | [27, 30] | constructor | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ a :: l₁ <+ b :: l₂ ↔ a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂ | case mp
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ a :: l₁ <+ b :: l₂ → a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂
case mpr
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂ → a :: l₁ <+ b :: l₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ a :: l₁ <+ b :: l₂ ↔ a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.cons_sublist_cons_iff' | [20, 1] | [27, 30] | rintro (_ | _) | case mp
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ a :: l₁ <+ b :: l₂ → a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂ | case mp.cons
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
a✝ : a :: l₁ <+ l₂
⊢ a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂
case mp.cons₂
α : Type u_1
l l' l₀ l₁ l₂ : List α
a : α
m n : ℕ
a✝ : l₁ <+ l₂
⊢ a :: l₁ <+ l₂ ∨ a = a ∧ l₁ <+ l₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ a :: l₁ <+ b :: l₂ → a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.cons_sublist_cons_iff' | [20, 1] | [27, 30] | exact Or.inl ‹_› | case mp.cons
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
a✝ : a :: l₁ <+ l₂
⊢ a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.cons
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
a✝ : a :: l₁ <+ l₂
⊢ a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.cons_sublist_cons_iff' | [20, 1] | [27, 30] | exact Or.inr ⟨rfl, ‹_›⟩ | case mp.cons₂
α : Type u_1
l l' l₀ l₁ l₂ : List α
a : α
m n : ℕ
a✝ : l₁ <+ l₂
⊢ a :: l₁ <+ l₂ ∨ a = a ∧ l₁ <+ l₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.cons₂
α : Type u_1
l l' l₀ l₁ l₂ : List α
a : α
m n : ℕ
a✝ : l₁ <+ l₂
⊢ a :: l₁ <+ l₂ ∨ a = a ∧ l₁ <+ l₂
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.cons_sublist_cons_iff' | [20, 1] | [27, 30] | rintro (h | ⟨rfl, h⟩) | case mpr
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂ → a :: l₁ <+ b :: l₂ | case mpr.inl
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
h : a :: l₁ <+ l₂
⊢ a :: l₁ <+ b :: l₂
case mpr.inr.intro
α : Type u_1
l l' l₀ l₁ l₂ : List α
a : α
m n : ℕ
h : l₁ <+ l₂
⊢ a :: l₁ <+ a :: l₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ a :: l₁ <+ l₂ ∨ a = b ∧ l₁ <+ l₂ → a :: l₁ <+ b :: l₂
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.cons_sublist_cons_iff' | [20, 1] | [27, 30] | exact sublist_cons_of_sublist _ h | case mpr.inl
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
h : a :: l₁ <+ l₂
⊢ a :: l₁ <+ b :: l₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inl
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
h : a :: l₁ <+ l₂
⊢ a :: l₁ <+ b :: l₂
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.cons_sublist_cons_iff' | [20, 1] | [27, 30] | rwa [cons_sublist_cons] | case mpr.inr.intro
α : Type u_1
l l' l₀ l₁ l₂ : List α
a : α
m n : ℕ
h : l₁ <+ l₂
⊢ a :: l₁ <+ a :: l₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.intro
α : Type u_1
l l' l₀ l₁ l₂ : List α
a : α
m n : ℕ
h : l₁ <+ l₂
⊢ a :: l₁ <+ a :: l₂
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.subperm_nil | [33, 1] | [34, 77] | rintro rfl | α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ l = [] → l <+~ [] | α : Type u_1
l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ [] <+~ [] | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
l l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ l = [] → l <+~ []
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/Basic.lean | List.subperm_nil | [33, 1] | [34, 77] | rfl | α : Type u_1
l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ [] <+~ [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
l' l₀ l₁ l₂ : List α
a b : α
m n : ℕ
⊢ [] <+~ []
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_intermediate | [14, 1] | [19, 16] | induction' s using Quotient.inductionOn with l₀ | α : Type u_1
s t : Multiset α
n : ℕ
hst : s ≤ t
hs : card s ≤ n
ht : n ≤ card t
⊢ ∃ u, s ≤ u ∧ u ≤ t ∧ card u = n | case h
α : Type u_1
s t : Multiset α
n : ℕ
ht : n ≤ card t
l₀ : List α
hst : ⟦l₀⟧ ≤ t
hs : card ⟦l₀⟧ ≤ n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ t ∧ card u = n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s t : Multiset α
n : ℕ
hst : s ≤ t
hs : card s ≤ n
ht : n ≤ card t
⊢ ∃ u, s ≤ u ∧ u ≤ t ∧ card u = n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_intermediate | [14, 1] | [19, 16] | induction' t using Quotient.inductionOn with l₂ | case h
α : Type u_1
s t : Multiset α
n : ℕ
ht : n ≤ card t
l₀ : List α
hst : ⟦l₀⟧ ≤ t
hs : card ⟦l₀⟧ ≤ n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ t ∧ card u = n | case h.h
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
s t : Multiset α
n : ℕ
ht : n ≤ card t
l₀ : List α
hst : ⟦l₀⟧ ≤ t
hs : card ⟦l₀⟧ ≤ n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ t ∧ card u = n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_intermediate | [14, 1] | [19, 16] | obtain ⟨l₁, h⟩ := hst.exists_intermediate hs ht | case h.h
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n | case h.h.intro
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
l₁ : List α
h : l₀.Subperm l₁ ∧ l₁.Subperm l₂ ∧ l₁.length = n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_intermediate | [14, 1] | [19, 16] | exact ⟨l₁, h⟩ | case h.h.intro
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
l₁ : List α
h : l₀.Subperm l₁ ∧ l₁.Subperm l₂ ∧ l₁.length = n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.intro
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
l₁ : List α
h : l₀.Subperm l₁ ∧ l₁.Subperm l₂ ∧ l₁.length = n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_le_card_eq | [21, 1] | [22, 65] | simpa using exists_intermediate (zero_le _) (Nat.zero_le _) hn | α : Type u_1
s t : Multiset α
n : ℕ
hn : n ≤ card s
⊢ ∃ t ≤ s, card t = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s t : Multiset α
n : ℕ
hn : n ≤ card s
⊢ ∃ t ≤ s, card t = n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/BigOperators/Ring.lean | Finset.sum_boole_mul' | [12, 1] | [13, 69] | simp | ι : Type u_1
α : Type u_2
inst✝¹ : NonAssocSemiring α
inst✝ : DecidableEq ι
s : Finset ι
f : ι → α
i : ι
⊢ ∑ j ∈ s, (if i = j then 1 else 0) * f j = if i ∈ s then f i else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
inst✝¹ : NonAssocSemiring α
inst✝ : DecidableEq ι
s : Finset ι
f : ι → α
i : ι
⊢ ∑ j ∈ s, (if i = j then 1 else 0) * f j = if i ∈ s then f i else 0
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosRenyi/Basic.lean | ErdosRenyi.meas | [51, 1] | [54, 35] | simpa using hG.meas H.edgeFinset | α : Type u_1
Ω : Type u_2
inst✝⁵ : MeasurableSpace Ω
G : Ω → SimpleGraph α
H : SimpleGraph α
inst✝⁴ : (ω : Ω) → DecidableRel (G ω).Adj
p : ℝ≥0
μ : Measure Ω
inst✝³ : IsProbabilityMeasure μ
hG : ErdosRenyi G p μ
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel H.Adj
⊢ μ {ω | G ω = H} = ↑p ^ H.edgeFinset.card * (1 - ↑p) ^ (Fintype.card (Sym2 α) - H.edgeFinset.card) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
Ω : Type u_2
inst✝⁵ : MeasurableSpace Ω
G : Ω → SimpleGraph α
H : SimpleGraph α
inst✝⁴ : (ω : Ω) → DecidableRel (G ω).Adj
p : ℝ≥0
μ : Measure Ω
inst✝³ : IsProbabilityMeasure μ
hG : ErdosRenyi G p μ
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel H.Adj
⊢ μ {ω | G ω = H} = ↑p ^ H.edgeFinset.card * (1 - ↑p) ^ (Fintype.card (Sym2 α) - H.edgeFinset.card)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ExampleSheets/Graph/ES1.lean | GraphTheory.ES1.q4 | [67, 1] | [70, 8] | cases nonempty_fintype α | ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : Finite α
inst✝ : Nontrivial α
G : SimpleGraph α
hG : G.Connected
⊢ ∃ a, (⊤.deleteVerts {a}).coe.Connected | case intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : Finite α
inst✝ : Nontrivial α
G : SimpleGraph α
hG : G.Connected
val✝ : Fintype α
⊢ ∃ a, (⊤.deleteVerts {a}).coe.Connected | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : Finite α
inst✝ : Nontrivial α
G : SimpleGraph α
hG : G.Connected
⊢ ∃ a, (⊤.deleteVerts {a}).coe.Connected
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ExampleSheets/Graph/ES1.lean | GraphTheory.ES1.q4 | [67, 1] | [70, 8] | sorry | case intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : Finite α
inst✝ : Nontrivial α
G : SimpleGraph α
hG : G.Connected
val✝ : Fintype α
⊢ ∃ a, (⊤.deleteVerts {a}).coe.Connected | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : Finite α
inst✝ : Nontrivial α
G : SimpleGraph α
hG : G.Connected
val✝ : Fintype α
⊢ ∃ a, (⊤.deleteVerts {a}).coe.Connected
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ExampleSheets/Graph/ES1.lean | GraphTheory.ES1.q5 | [79, 1] | [83, 8] | cases isEmpty_or_nonempty α | ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
hG : G.IsAcyclic
⊢ G.edgeFinset.card ≤ card α - 1 | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
hG : G.IsAcyclic
h✝ : IsEmpty α
⊢ G.edgeFinset.card ≤ card α - 1
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
hG : G.IsAcyclic
h✝ : Nonempty α
⊢ G.edgeFinset.card ≤ card α - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
hG : G.IsAcyclic
⊢ G.edgeFinset.card ≤ card α - 1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ExampleSheets/Graph/ES1.lean | GraphTheory.ES1.q5 | [79, 1] | [83, 8] | sorry | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
hG : G.IsAcyclic
h✝ : Nonempty α
⊢ G.edgeFinset.card ≤ card α - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
hG : G.IsAcyclic
h✝ : Nonempty α
⊢ G.edgeFinset.card ≤ card α - 1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ExampleSheets/Graph/ES1.lean | GraphTheory.ES1.q5 | [79, 1] | [83, 8] | simp | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
hG : G.IsAcyclic
h✝ : IsEmpty α
⊢ G.edgeFinset.card ≤ card α - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
hG : G.IsAcyclic
h✝ : IsEmpty α
⊢ G.edgeFinset.card ≤ card α - 1
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.length_rtake | [10, 1] | [11, 65] | rw [Nat.sub_sub_eq_min, min_comm] | α : Type u_1
l✝ l' l₀ l₁ l₂ : List α
a b : α
m n✝ n : ℕ
l : List α
⊢ l.length - (l.length - n) = min n l.length | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
l✝ l' l₀ l₁ l₂ : List α
a b : α
m n✝ n : ℕ
l : List α
⊢ l.length - (l.length - n) = min n l.length
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.take_reverse | [13, 1] | [14, 54] | rw [rtake_eq_reverse_take_reverse, reverse_reverse] | α : Type u_1
l✝ l' l₀ l₁ l₂ : List α
a b : α
m n✝ n : ℕ
l : List α
⊢ take n l.reverse = (l.rtake n).reverse | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
l✝ l' l₀ l₁ l₂ : List α
a b : α
m n✝ n : ℕ
l : List α
⊢ take n l.reverse = (l.rtake n).reverse
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/List/DropRight.lean | List.rtake_reverse | [16, 1] | [17, 54] | rw [rtake_eq_reverse_take_reverse, reverse_reverse] | α : Type u_1
l✝ l' l₀ l₁ l₂ : List α
a b : α
m n✝ n : ℕ
l : List α
⊢ l.reverse.rtake n = (take n l).reverse | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
l✝ l' l₀ l₁ l₂ : List α
a b : α
m n✝ n : ℕ
l : List α
⊢ l.reverse.rtake n = (take n l).reverse
TACTIC:
|
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