mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite

Goal state before:
pp: case neg
V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
⊢ Nonempty (G.CompleteEquipartiteSubgraph (r + 1) t) ↔ ∃ K s, #s = t ∧ ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s

Tactic:
]

Goal state after:
case neg
V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
⊢ Nonempty (G.CompleteEquipartiteSubgraph (r + 1) t) ↔ ∃ K s, #s = t ∧ ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
Module: Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
x✝ : Nonempty (G.CompleteEquipartiteSubgraph (r + 1) t)
K' : G.CompleteEquipartiteSubgraph (r + 1) t
parts : Finset (Finset V)
hparts_sub : parts ⊆ K'.parts
hparts_card : #parts = (#K'.parts).pred
⊢ (r + 1).pred = r ∨ t = 0

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
x✝ : Nonempty (G.CompleteEquipartiteSubgraph (r + 1) t)
K' : G.CompleteEquipartiteSubgraph (r + 1) t
parts : Finset (Finset V)
hparts_sub : parts ⊆ K'.parts
hparts_card : #parts = (#K'.parts).pred
⊢ (r + 1).pred = r ∨ t = 0
Module: Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
x✝ : Nonempty (G.CompleteEquipartiteSubgraph (r + 1) t)
K' : G.CompleteEquipartiteSubgraph (r + 1) t
parts : Finset (Finset V)
hparts_sub : parts ⊆ K'.parts
hparts_card : #parts = (#K'.parts).pred
K : G.CompleteEquipartiteSubgraph r t :=
  { parts := parts, card_parts := ⋯, card_mem_parts := ⋯, isCompleteBetween := ⋯ }
⊢ r + 1 = r + 1

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case inr.inr
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
⊢ n' * (r - 1) + r.choose 2 + (n' % r * (n' / r) * (r - 1) + r * (r - 1) * (n' / r) ^ 2 / 2) + (n' % r).choose 2 =
    n' % r * (n' / r + 1) * (r - 1) + r * (r - 1) * (n' / r + 1) ^ 2 / 2 + (n' % r).choose 2

Tactic:
]

Goal state after:
case inr.inr
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
⊢ n' * (r - 1) + r.choose 2 + (n' % r * (n' / r) * (r - 1) + r * (r - 1) * (n' / r) ^ 2 / 2) + (n' % r).choose 2 =
    n' % r * (n' / r + 1) * (r - 1) + r * (r - 1) * (n' / r + 1) ^ 2 / 2 + (n' % r).choose 2
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case inr.inr.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
⊢ n' * (r - 1) + r.choose 2 + (n' % r * (n' / r) * (r - 1) + r.choose 2 * (n' / r) ^ 2) =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2

Tactic:
]

Goal state after:
case inr.inr.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
⊢ n' * (r - 1) + r.choose 2 + (n' % r * (n' / r) * (r - 1) + r.choose 2 * (n' / r) ^ 2) =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: V : Type u
K : Set V
G : SimpleGraph V
v w : V
vK : v ∉ K
wK : w ∉ K
a : G.Adj v w
⊢ (induce Kᶜ G).Reachable ⟨v, vK⟩ ⟨w, wK⟩

Tactic:
]

Goal state after:
V : Type u
K : Set V
G : SimpleGraph V
v w : V
vK : v ∉ K
wK : w ∉ K
a : G.Adj v w
⊢ (induce Kᶜ G).Reachable ⟨v, vK⟩ ⟨w, wK⟩
Module: Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
x✝ : ∃ K s, #s = t ∧ ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
K : G.CompleteEquipartiteSubgraph r t
s : Finset V
hs : #s = t
hadj : ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
hs_mem : s ∈ K.parts
⊢ 0 < t

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
x✝ : ∃ K s, #s = t ∧ ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
K : G.CompleteEquipartiteSubgraph r t
s : Finset V
hs : #s = t
hadj : ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
hs_mem : s ∈ K.parts
⊢ 0 < t
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case inr.inr.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' * (r - 1) + r.choose 2 + n' % r * (n' / r) * (r - 1) + r.choose 2 * (n' / r) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2

Tactic:
]

Goal state after:
case inr.inr.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' * (r - 1) + r.choose 2 + n' % r * (n' / r) * (r - 1) + r.choose 2 * (n' / r) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: V : Type u
G : SimpleGraph V
K : Set V
C : G.ComponentCompl K
⊢ K ∩ ↑C ⊆ ∅

Tactic:
]

Goal state after:
V : Type u
G : SimpleGraph V
K : Set V
C : G.ComponentCompl K
⊢ K ∩ ↑C ⊆ ∅
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: V : Type u
G : SimpleGraph V
K : Set V
C D : G.ComponentCompl K
ne : C ≠ D
⊢ ↑C ∩ ↑D ⊆ ∅

Tactic:
]

Goal state after:
V : Type u
G : SimpleGraph V
K : Set V
C D : G.ComponentCompl K
ne : C ≠ D
⊢ ↑C ∩ ↑D ⊆ ∅
Module: Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite

Goal state before:
pp: case neg.refine_2.refine_2
V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
x✝ : ∃ K s, #s = t ∧ ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
K : G.CompleteEquipartiteSubgraph r t
s : Finset V
hs : #s = t
hadj : ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
⊢ r + 1 = r + 1 ∨ t = 0

Tactic:
]

Goal state after:
case neg.refine_2.refine_2
V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
x✝ : ∃ K s, #s = t ∧ ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
K : G.CompleteEquipartiteSubgraph r t
s : Finset V
hs : #s = t
hadj : ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
⊢ r + 1 = r + 1 ∨ t = 0
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: V : Type u
G : SimpleGraph V
K : Set V
C : G.ComponentCompl K
c d : V
x✝ : c ∈ C
dnK : d ∉ K
cd : G.Adj c d
cnK : c ∉ K
h : G.componentComplMk cnK = C
⊢ (induce Kᶜ G).Reachable ⟨d, dnK⟩ ⟨c, cnK⟩

Tactic:
]

Goal state after:
V : Type u
G : SimpleGraph V
K : Set V
C : G.ComponentCompl K
c d : V
x✝ : c ∈ C
dnK : d ∉ K
cd : G.Adj c d
cnK : c ∉ K
h : G.componentComplMk cnK = C
⊢ (induce Kᶜ G).Reachable ⟨d, dnK⟩ ⟨c, cnK⟩
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case inr.inr.e_a.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' % r * (n' / r) * (r - 1) + n' * (r - 1) + r.choose 2 =
    r.choose 2 * 2 * (n' / r) + n' % r * (n' / r + 1) * (r - 1) + r.choose 2

Tactic:
]

Goal state after:
case inr.inr.e_a.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' % r * (n' / r) * (r - 1) + n' * (r - 1) + r.choose 2 =
    r.choose 2 * 2 * (n' / r) + n' % r * (n' / r + 1) * (r - 1) + r.choose 2
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: V : Type u
G : SimpleGraph V
K : Set V
Gc : G.Preconnected
hK : K.Nonempty
v : V
vnK : v ∉ K
C : G.ComponentCompl K := G.componentComplMk vnK
dis : K ∩ ↑C ⊆ ∅ := Set.disjoint_iff.mp (ComponentCompl.disjoint_right C)
h : ∀ (ck : V × V), ck.1 ∈ G.componentComplMk vnK → ck.2 ∈ K → ¬G.Adj ck.1 ck.2
⊢ ∀ (x : V), x ∈ ↑C

Tactic:
]

Goal state after:
V : Type u
G : SimpleGraph V
K : Set V
Gc : G.Preconnected
hK : K.Nonempty
v : V
vnK : v ∉ K
C : G.ComponentCompl K := G.componentComplMk vnK
dis : K ∩ ↑C ⊆ ∅ := Set.disjoint_iff.mp (ComponentCompl.disjoint_right C)
h : ∀ (ck : V × V), ck.1 ∈ G.componentComplMk vnK → ck.2 ∈ K → ¬G.Adj ck.1 ck.2
⊢ ∀ (x : V), x ∈ ↑C
Module: Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite

Goal state before:
pp: case neg.refine_2.refine_4
V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
x✝ : ∃ K s, #s = t ∧ ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
K : G.CompleteEquipartiteSubgraph r t
s : Finset V
hs : #s = t
hadj : ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
⊢ (insert s ↑K.parts).Pairwise fun x1 x2 ↦ G.IsCompleteBetween ↑x1 ↑x2

Tactic:
]

Goal state after:
case neg.refine_2.refine_4
V : Type u_1
G : SimpleGraph V
r t : ℕ
ht : ¬t = 0
x✝ : ∃ K s, #s = t ∧ ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
K : G.CompleteEquipartiteSubgraph r t
s : Finset V
hs : #s = t
hadj : ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s
⊢ (insert s ↑K.parts).Pairwise fun x1 x2 ↦ G.IsCompleteBetween ↑x1 ↑x2
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: V : Type u
G : SimpleGraph V
K L : Set V
C : G.ComponentCompl L
h : K ⊆ L
D : G.ComponentCompl K
⊢ hom h C = D ↔ ∃ x ∈ ↑C, x ∈ ↑D

Tactic:
]

Goal state after:
V : Type u
G : SimpleGraph V
K L : Set V
C : G.ComponentCompl L
h : K ⊆ L
D : G.ComponentCompl K
⊢ hom h C = D ↔ ∃ x ∈ ↑C, x ∈ ↑D
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: case mpr
V : Type u
G : SimpleGraph V
K L : Set V
C : G.ComponentCompl L
h : K ⊆ L
x✝ : V
xnL : x✝ ∉ L
x : V
w✝¹ : x ∉ L
e₁ : G.componentComplMk w✝¹ = G.componentComplMk xnL
w✝ : x ∉ K
⊢ hom h (G.componentComplMk w✝¹) = G.componentComplMk w✝

Tactic:
]

Goal state after:
case mpr
V : Type u
G : SimpleGraph V
K L : Set V
C : G.ComponentCompl L
h : K ⊆ L
x✝ : V
xnL : x✝ ∉ L
x : V
w✝¹ : x ∉ L
e₁ : G.componentComplMk w✝¹ = G.componentComplMk xnL
w✝ : x ∉ K
⊢ hom h (G.componentComplMk w✝¹) = G.componentComplMk w✝
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case inr.inr.e_a.e_a.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' * (r - 1) + n' % r * (n' / r) * (r - 1) = n' % r * (r - 1) + r * (r - 1) * (n' / r) + n' % r * (n' / r) * (r - 1)

Tactic:
]

Goal state after:
case inr.inr.e_a.e_a.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' * (r - 1) + n' % r * (n' / r) * (r - 1) = n' % r * (r - 1) + r * (r - 1) * (n' / r) + n' % r * (n' / r) * (r - 1)
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: V : Type u
G : SimpleGraph V
L : Set V
C : G.ComponentCompl L
⊢ C = C

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: V : Type u
G : SimpleGraph V
K L M : Set V
C : G.ComponentCompl L
h : K ⊆ L
h' : M ⊆ K
⊢ ConnectedComponent.map (induceHom Hom.id ⋯) C = ConnectedComponent.map (induceHom (Hom.id.comp Hom.id) ⋯) C

Tactic:
]

Goal state after:
V : Type u
G : SimpleGraph V
K L M : Set V
C : G.ComponentCompl L
h : K ⊆ L
h' : M ⊆ K
⊢ ConnectedComponent.map (induceHom Hom.id ⋯) C = ConnectedComponent.map (induceHom (Hom.id.comp Hom.id) ⋯) C
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case inr.inr.e_a.e_a.e_a.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' * (r - 1) = n' * (r - 1)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: case inl
V : Type u
G : SimpleGraph V
K L M : Set V
inst✝ : G.LocallyFinite
Gpc : Fact G.Preconnected
⊢ Finite (induce Set.univ G).ConnectedComponent

Tactic:
]

Goal state after:
case inl
V : Type u
G : SimpleGraph V
K L M : Set V
inst✝ : G.LocallyFinite
Gpc : Fact G.Preconnected
⊢ Finite (induce Set.univ G).ConnectedComponent
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: n r : ℕ
⊢ (n ^ 2 - (n % r) ^ 2) * (r - 1) + 2 * r * (n % r).choose 2 ≤ (r - 1) * n ^ 2

Tactic:
]

Goal state after:
n r : ℕ
⊢ (n ^ 2 - (n % r) ^ 2) * (r - 1) + 2 * r * (n % r).choose 2 ≤ (r - 1) * n ^ 2
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: n r : ℕ
⊢ n ^ 2 * (r - 1) + 2 * r * (n % r).choose 2 - (n % r) ^ 2 * (r - 1) ≤ (r - 1) * n ^ 2

pp: n r : ℕ
⊢ (n % r) ^ 2 * (r - 1) ≤ n ^ 2 * (r - 1)

Tactic:
]

Goal state after:
n r : ℕ
⊢ n ^ 2 * (r - 1) + 2 * r * (n % r).choose 2 - (n % r) ^ 2 * (r - 1) ≤ (r - 1) * n ^ 2

n r : ℕ
⊢ (n % r) ^ 2 * (r - 1) ≤ n ^ 2 * (r - 1)
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: n r : ℕ
⊢ n ^ 2 * (r - 1) ≤ n ^ 2 * (r - 1)

pp: case ha
n r : ℕ
⊢ 0 ≤ n % r

Tactic:
]

Goal state after:
case ha
n r : ℕ
⊢ 0 ≤ n % r
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: n r : ℕ
⊢ (n % r - 1) * r * (n % r) ≤ (r - 1) * (n % r) * (n % r)

Tactic:
]

Goal state after:
n r : ℕ
⊢ (n % r - 1) * r * (n % r) ≤ (r - 1) * (n % r) * (n % r)
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: V : Type u
G : SimpleGraph V
s : (j : (Finset V)ᵒᵖ) → G.componentComplFunctor.obj j
sec : s ∈ G.end
K L : (Finset V)ᵒᵖ
h : L ⟶ K
v : V
vnL : v ∉ Opposite.unop L
hs : s L = G.componentComplMk vnL
⊢ (ConcreteCategory.hom (G.componentComplFunctor.map h)) (G.componentComplMk vnL) = G.componentComplMk ⋯

Tactic:
]

Goal state after:
V : Type u
G : SimpleGraph V
s : (j : (Finset V)ᵒᵖ) → G.componentComplFunctor.obj j
sec : s ∈ G.end
K L : (Finset V)ᵒᵖ
h : L ⟶ K
v : V
vnL : v ∉ Opposite.unop L
hs : s L = G.componentComplMk vnL
⊢ (ConcreteCategory.hom (G.componentComplFunctor.map h)) (G.componentComplMk vnL) = G.componentComplMk ⋯
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity

Goal state before:
pp: W : Type u_1
H : SimpleGraph W
n : ℕ
hn : n ≥ 2
⊢ ∀ ⦃G : SimpleGraph (Fin (n + 1))⦄ [inst : DecidableRel G.Adj],
    H.Free G → ↑(#G.edgeFinset) ≤ ↑(extremalNumber n H) / ↑(n.choose 2) * ↑((n + 1).choose 2)

Tactic:
]

Goal state after:
W : Type u_1
H : SimpleGraph W
n : ℕ
hn : n ≥ 2
⊢ ∀ ⦃G : SimpleGraph (Fin (n + 1))⦄ [inst : DecidableRel G.Adj],
    H.Free G → ↑(#G.edgeFinset) ≤ ↑(extremalNumber n H) / ↑(n.choose 2) * ↑((n + 1).choose 2)
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case h₁.inr
n r : ℕ
hr : r > 0
⊢ r * (n % r) - r ≤ r * (n % r) - n % r

Tactic:
]

Goal state after:
case h₁.inr
n r : ℕ
hr : r > 0
⊢ r * (n % r) - r ≤ r * (n % r) - n % r
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Zarankiewicz

Goal state before:
pp: m n s t : ℕ
V : Type u_1
W : Type u_2
α : Type u_3
β : Type u_4
inst✝³ : Fintype V
inst✝² : Fintype W
inst✝¹ : Fintype α
inst✝ : Fintype β
hm : Fintype.card V = m
hn : Fintype.card W = n
hs : Fintype.card α = s
ht : Fintype.card β = t
e₁ : completeBipartiteGraph V W ≃g completeBipartiteGraph (Fin m) (Fin n) :=
  completeBipartiteGraphCongr (Fintype.equivFinOfCardEq hm) (Fintype.equivFinOfCardEq hn)
K : SimpleGraph (Fin s ⊕ Fin t) := completeBipartiteGraph (Fin s) (Fin t)
e₂ : completeBipartiteGraph α β ≃g completeBipartiteGraph (Fin s) (Fin t) :=
  completeBipartiteGraphCongr (Fintype.equivFinOfCardEq hs) (Fintype.equivFinOfCardEq ht)
⊢ (({G | G ≤ completeBipartiteGraph (Fin m) (Fin n) ∧ (completeBipartiteGraph (Fin s) (Fin t)).Free G}.sup fun x ↦
        #x.edgeFinset) ≤
      {G | G ≤ completeBipartiteGraph V W ∧ (completeBipartiteGraph α β).Free G}.sup fun x ↦ #x.edgeFinset) ∧
    ({G | G ≤ completeBipartiteGraph V W ∧ (completeBipartiteGraph α β).Free G}.sup fun x ↦ #x.edgeFinset) ≤
      {G | G ≤ completeBipartiteGraph (Fin m) (Fin n) ∧ (completeBipartiteGraph (Fin s) (Fin t)).Free G}.sup fun x ↦
        #x.edgeFinset

Tactic:
]

Goal state after:
m n s t : ℕ
V : Type u_1
W : Type u_2
α : Type u_3
β : Type u_4
inst✝³ : Fintype V
inst✝² : Fintype W
inst✝¹ : Fintype α
inst✝ : Fintype β
hm : Fintype.card V = m
hn : Fintype.card W = n
hs : Fintype.card α = s
ht : Fintype.card β = t
e₁ : completeBipartiteGraph V W ≃g completeBipartiteGraph (Fin m) (Fin n) :=
  completeBipartiteGraphCongr (Fintype.equivFinOfCardEq hm) (Fintype.equivFinOfCardEq hn)
K : SimpleGraph (Fin s ⊕ Fin t) := completeBipartiteGraph (Fin s) (Fin t)
e₂ : completeBipartiteGraph α β ≃g completeBipartiteGraph (Fin s) (Fin t) :=
  completeBipartiteGraphCongr (Fintype.equivFinOfCardEq hs) (Fintype.equivFinOfCardEq ht)
⊢ (({G | G ≤ completeBipartiteGraph (Fin m) (Fin n) ∧ (completeBipartiteGraph (Fin s) (Fin t)).Free G}.sup fun x ↦
        #x.edgeFinset) ≤
      {G | G ≤ completeBipartiteGraph V W ∧ (completeBipartiteGraph α β).Free G}.sup fun x ↦ #x.edgeFinset) ∧
    ({G | G ≤ completeBipartiteGraph V W ∧ (completeBipartiteGraph α β).Free G}.sup fun x ↦ #x.edgeFinset) ≤
      {G | G ≤ completeBipartiteGraph (Fin m) (Fin n) ∧ (completeBipartiteGraph (Fin s) (Fin t)).Free G}.sup fun x ↦
        #x.edgeFinset
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case inl
V : Type u_1
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
cf : Fintype.card V = 0
⊢ let n := Fintype.card V;
  #G.edgeFinset ≤ (n ^ 2 - (n % 0) ^ 2) * (0 - 1) / (2 * 0) + (n % 0).choose 2

Tactic:
]

Goal state after:
case inl
V : Type u_1
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
cf : Fintype.card V = 0
⊢ let n := Fintype.card V;
  #G.edgeFinset ≤ (n ^ 2 - (n % 0) ^ 2) * (0 - 1) / (2 * 0) + (n % 0).choose 2
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case e'_4
V : Type u_1
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
r : ℕ
cf : G.CliqueFree (r + 1)
hr : r > 0
H : SimpleGraph V
w✝ : DecidableRel H.Adj
maxH : H.IsTuranMaximal r
⊢ (Fintype.card V ^ 2 - (Fintype.card V % r) ^ 2) * (r - 1) / (2 * r) + (Fintype.card V % r).choose 2 =
    (Fintype.card V ^ 2 - (Fintype.card V % r) ^ 2) * (r - 1) / (2 * r) + (Fintype.card V % r).choose 2

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity

Goal state before:
pp: W : Type u_1
H : SimpleGraph W
n : ℕ
hn : n ≥ 2
G : SimpleGraph (Fin (n + 1))
inst✝ : DecidableRel G.Adj
h : H.Free G
⊢ #G.edgeFinset * (n - 1) ≤ (n + 1) * extremalNumber n H

Tactic:
]

Goal state after:
W : Type u_1
H : SimpleGraph W
n : ℕ
hn : n ≥ 2
G : SimpleGraph (Fin (n + 1))
inst✝ : DecidableRel G.Adj
h : H.Free G
⊢ #G.edgeFinset * (n - 1) ≤ (n + 1) * extremalNumber n H
Module: Mathlib.Combinatorics.SimpleGraph.Finsubgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
inst✝ : Finite V
ι : Sort u_1
f : ι → G.Finsubgraph
⊢ ⨆ i, ↑(f i) = ⨆ i, ↑(f i)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Finsubgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
inst✝ : Finite V
ι : Sort u_1
f : ι → G.Finsubgraph
⊢ ⨅ i, ↑(f i) = ⨅ i, ↑(f i)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity

Goal state before:
pp: W : Type u_1
H : SimpleGraph W
x : ℝ
x✝ : x ∈ {x | ∃ n ∈ Set.Ici 2, ↑(extremalNumber n H) / ↑(n.choose 2) = x}
w✝ : ℕ
left✝ : w✝ ∈ Set.Ici 2
hx : ↑(extremalNumber w✝ H) / ↑(w✝.choose 2) = x
⊢ 0 ≤ ↑(extremalNumber w✝ H) / ↑(w✝.choose 2)

Tactic:
]

Goal state after:
W : Type u_1
H : SimpleGraph W
x : ℝ
x✝ : x ∈ {x | ∃ n ∈ Set.Ici 2, ↑(extremalNumber n H) / ↑(n.choose 2) = x}
w✝ : ℕ
left✝ : w✝ ∈ Set.Ici 2
hx : ↑(extremalNumber w✝ H) / ↑(w✝.choose 2) = x
⊢ 0 ≤ ↑(extremalNumber w✝ H) / ↑(w✝.choose 2)
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity

Goal state before:
pp: W : Type u_1
H : SimpleGraph W
h_bdd : BddBelow {x | ∃ n ∈ Set.Ici 2, ↑(extremalNumber n H) / ↑(n.choose 2) = x}
h_tto :
  Tendsto (fun n ↦ ↑(extremalNumber n H) / ↑(n.choose 2)) atTop
    (𝓝 (atTop.limUnder fun n ↦ ↑(extremalNumber n H) / ↑(n.choose 2)))
⊢ Tendsto (fun n ↦ ↑(extremalNumber n H) / ↑(n.choose 2)) atTop (𝓝 H.turanDensity)

Tactic:
]

Goal state after:
W : Type u_1
H : SimpleGraph W
h_bdd : BddBelow {x | ∃ n ∈ Set.Ici 2, ↑(extremalNumber n H) / ↑(n.choose 2) = x}
h_tto :
  Tendsto (fun n ↦ ↑(extremalNumber n H) / ↑(n.choose 2)) atTop
    (𝓝 (atTop.limUnder fun n ↦ ↑(extremalNumber n H) / ↑(n.choose 2)))
⊢ Tendsto (fun n ↦ ↑(extremalNumber n H) / ↑(n.choose 2)) atTop (𝓝 H.turanDensity)