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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
h : i ≤ p.length
h' : p.getVert (i - 1) = p.getVert (i + 1)
hl : 3 ≤ p.length
hi' : ¬i ≥ p.length - 1
⊢ i ≤ p.length - 1 + 1

Tactic:
lia

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
h : i ≤ p.length
h' : p.getVert (i - 1) = p.getVert (i + 1)
hl : 3 ≤ p.length
hi' : ¬i ≥ p.length - 1
⊢ i + 1 ≤ p.length - 1

Tactic:
lia

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

Goal state before:
pp: case neg
V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
h : i ≤ p.length
h' : p.getVert (i - 1) = p.getVert (i + 1)
hl : 3 ≤ p.length
hi' : ¬i ≥ p.length - 1
this : i - 1 = i + 1
⊢ False

Tactic:
lia

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

Goal state before:
pp: case cons
V : Type u
G : SimpleGraph V
u v✝ : V
h' : G.Adj u v✝
p : G.Walk v✝ u
x✝ : (cons h' p).tail.IsPath ∧ 3 ≤ (cons h' p).length
h₂ : 3 ≤ p.length + 1
h₁ : p.IsPath
hh : s(u, v✝) ∈ p.edges
this✝ : p.support[0] = p.support[p.length - 1]
this : ⟨0, ⋯⟩ = ⟨p.length - 1, ⋯⟩
⊢ False

Tactic:
lia

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v w : V
inst✝ : DecidableEq V
p : G.Walk u v
hp : p.IsPath
hw : w ∈ p.support
h : v ≠ w
n : ℕ
hn : p.getVert n = v
hnl : n ≤ (p.takeUntil w hw).length
this : (p.takeUntil w hw).length < p.length
⊢ n ≤ p.length

Tactic:
lia

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v w : V
inst✝ : DecidableEq V
p : G.Walk u v
hp : p.IsPath
hw : w ∈ p.support
h : v ≠ w
n : ℕ
hn : p.getVert n = v
hnl : n ≤ (p.takeUntil w hw).length
this✝ : (p.takeUntil w hw).length < p.length
this : n = p.length
⊢ False

Tactic:
lia

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝ : Fintype α
f : completeGraph α ⊑ G
⊢ ¬G.CliqueFree (Fintype.card α)

Tactic:
not_cliqueFree_iff_top_isContained

Goal state after:
α : Type u_1
G : SimpleGraph α
inst✝ : Fintype α
f : completeGraph α ⊑ G
⊢ completeGraph (Fin (Fintype.card α)) ⊑ G
Module: Mathlib.Combinatorics.SimpleGraph.Clique

Goal state before:
pp: α : Type u_1
β : Type u_2
G : SimpleGraph α
n : ℕ
H : SimpleGraph β
hle : H ⊑ G
h : ¬H.CliqueFree n
⊢ ¬G.CliqueFree n

Tactic:
not_cliqueFree_iff_top_isContained

Goal state after:
α : Type u_1
β : Type u_2
G : SimpleGraph α
n : ℕ
H : SimpleGraph β
hle : H ⊑ G
h : completeGraph (Fin n) ⊑ H
⊢ completeGraph (Fin n) ⊑ G
Module: Mathlib.Combinatorics.SimpleGraph.Clique

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
n : ℕ
inst✝ : DecidableEq α
s t : α
h : ¬(G.replaceVertex s t).CliqueFree n
φ : Fin n ↪ α
hφ : ∀ {a b : Fin n}, (G.replaceVertex s t).Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b
⊢ ¬G.CliqueFree n

Tactic:
not_cliqueFree_iff_top_isContained

Goal state after:
α : Type u_1
G : SimpleGraph α
n : ℕ
inst✝ : DecidableEq α
s t : α
h : ¬(G.replaceVertex s t).CliqueFree n
φ : Fin n ↪ α
hφ : ∀ {a b : Fin n}, (G.replaceVertex s t).Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b
⊢ completeGraph (Fin n) ⊑ G
Module: Mathlib.Combinatorics.SimpleGraph.Paths

Goal state before:
pp: case refine_2
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
q : G.Walk v u
hp : p.IsPath
hq : q.IsPath
h : p.support.tail.Disjoint q.support.tail
hn : 1 < p.length ∨ 1 < q.length
⊢ p.append q ≠ Walk.nil

pp: case refine_3
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
q : G.Walk v u
hp : p.IsPath
hq : q.IsPath
h : p.support.tail.Disjoint q.support.tail
hn : 1 < p.length ∨ 1 < q.length
⊢ (p.append q).support.tail.Nodup

Tactic:
· grind [nil_append_iff]

Goal state after:
case refine_3
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
q : G.Walk v u
hp : p.IsPath
hq : q.IsPath
h : p.support.tail.Disjoint q.support.tail
hn : 1 < p.length ∨ 1 < q.length
⊢ (p.append q).support.tail.Nodup
Module: Mathlib.Combinatorics.SimpleGraph.Clique

Goal state before:
pp: case pos
α : Type u_1
G : SimpleGraph α
n : ℕ
inst✝ : DecidableEq α
s t : α
φ : Fin n ↪ α
hφ : ∀ {a b : Fin n}, (G.replaceVertex s t).Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b
x : Fin n
hx : φ x = t
y : Fin n
h : ¬G.CliqueFree n
hy : φ y = s
e : x = y
⊢ completeGraph (Fin n) ⊑ G

Tactic:
not_cliqueFree_iff_top_isContained

Goal state after:
case pos
α : Type u_1
G : SimpleGraph α
n : ℕ
inst✝ : DecidableEq α
s t : α
φ : Fin n ↪ α
hφ : ∀ {a b : Fin n}, (G.replaceVertex s t).Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b
x : Fin n
hx : φ x = t
y : Fin n
h : completeGraph (Fin n) ⊑ G
hy : φ y = s
e : x = y
⊢ completeGraph (Fin n) ⊑ G
Module: Mathlib.Combinatorics.SimpleGraph.Clique

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
s : Set α
n : ℕ
⊢ (induce s G).CliqueFree n ↔ G.CliqueFreeOn s n

Tactic:
simp only [CliqueFree, isNClique_induce_iff]
  refine ⟨fun h t ht ↦ ?_, (· <| map_subtype_subset ·)⟩
  have := h <| t.subtype _
  rwa [← filter_eq_self.mpr ht, ← subtype_map]

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

Goal state before:
pp: α : Type u_3
G : SimpleGraph α
inst✝ : Finite α
s : Set α
⊢ (induce s G).cliqueNum ≤ G.cliqueNum

Tactic:
by
  have ⟨t', tc⟩ := (G.induce s).exists_isNClique_cliqueNum
  rw [isNClique_induce_iff] at tc
  exact tc.card_eq ▸ tc.isClique.card_le_cliqueNum

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

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u v : V
p : G.Walk u v
hinj : Injective ⇑f
⊢ (Walk.map f p).support.Nodup ↔ p.support.Nodup

Tactic:
support_map,

Goal state after:
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u v : V
p : G.Walk u v
hinj : Injective ⇑f
⊢ (List.map (⇑f) p.support).Nodup ↔ p.support.Nodup
Module: Mathlib.Combinatorics.SimpleGraph.Clique

Goal state before:
pp: α : Type u_3
G : SimpleGraph α
inst✝ : Finite α
t : Finset α
tc : G.IsIndepSet ↑t
⊢ #t ≤ G.indepNum

Tactic:
by
  rw [← isClique_compl] at tc
  simp_rw [indepNum, ← isNClique_compl]
  exact tc.card_le_cliqueNum

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

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u : V
p : G.Walk u u
hp : (Walk.map f p).IsTrail ∧ ¬p.Nil ∧ (Walk.map f p).support.tail.Nodup
⊢ p.IsTrail ∧ ¬p.Nil ∧ p.support.tail.Nodup

Tactic:
support_map,

Goal state after:
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u : V
p : G.Walk u u
hp : (Walk.map f p).IsTrail ∧ ¬p.Nil ∧ (List.map (⇑f) p.support).tail.Nodup
⊢ p.IsTrail ∧ ¬p.Nil ∧ p.support.tail.Nodup
Module: Mathlib.Combinatorics.SimpleGraph.Paths

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u : V
p : G.Walk u u
hinj : Injective ⇑f
⊢ p.IsTrail ∧ ¬p.Nil ∧ (Walk.map f p).support.tail.Nodup ↔ p.IsTrail ∧ ¬p.Nil ∧ p.support.tail.Nodup

Tactic:
support_map,

Goal state after:
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u : V
p : G.Walk u u
hinj : Injective ⇑f
⊢ p.IsTrail ∧ ¬p.Nil ∧ (List.map (⇑f) p.support).tail.Nodup ↔ p.IsTrail ∧ ¬p.Nil ∧ p.support.tail.Nodup
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
pp: α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
hUV : U ≠ V
hunif : ¬G.IsUniform ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ 4 / 5 * ε ≤ ↑(#(star hP G ε hU V)) / 4 ^ #P.parts

Tactic:
have hε₀ : 0 < ε := by sz_positivity

Goal state after:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
hUV : U ≠ V
hunif : ¬G.IsUniform ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
hε₀ : 0 < ε
⊢ 4 / 5 * ε ≤ ↑(#(star hP G ε hU V)) / 4 ^ #P.parts
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: n : ℕ
ε : ℝ
hε : 0 < ε
hn : cornersTheoremBound (ε / 9) ≤ n
A : Finset (ℕ × ℕ)
hAn : ↑A ⊆ ↑(range n) ×ˢ ↑(range n)
hAε : ε * ↑n ^ 2 ≤ ↑(#A)
hA : IsCornerFree ↑A
a b : ℕ
hab : (a, b) ∈ A
this : a < n ∧ b < n
⊢ a < 2 * n + 1 ∧ b < 2 * n + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: n : ℕ
ε : ℝ
hε : 0 < ε
hn : cornersTheoremBound (ε / 9) ≤ n
A : Finset (ℕ × ℕ)
hAn : ↑A ⊆ ↑(range n) ×ˢ ↑(range n)
hAε : ε * ↑n ^ 2 ≤ ↑(#A)
hA : IsCornerFree (Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A)
this✝ : ↑A = Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A
this : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
x : ℕ × ℕ
hx : x.1 < n ∧ x.2 < n
⊢ n ≤ 2 * n

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: n : ℕ
ε : ℝ
hε : 0 < ε
hn : cornersTheoremBound (ε / 9) ≤ n
A : Finset (ℕ × ℕ)
hAn : ↑A ⊆ ↑(range n) ×ˢ ↑(range n)
hAε : ε * ↑n ^ 2 ≤ ↑(#A)
hA : IsCornerFree (Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A)
this✝¹ : ↑A = Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : IsCornerFree (Prod.map Nat.cast Nat.cast '' ↑A)
⊢ False

Tactic:
← coe_image

Goal state after:
n : ℕ
ε : ℝ
hε : 0 < ε
hn : cornersTheoremBound (ε / 9) ≤ n
A : Finset (ℕ × ℕ)
hAn : ↑A ⊆ ↑(range n) ×ˢ ↑(range n)
hAε : ε * ↑n ^ 2 ≤ ↑(#A)
hA : IsCornerFree (Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A)
this✝¹ : ↑A = Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : IsCornerFree ↑(image (Prod.map Nat.cast Nat.cast) A)
⊢ False
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: n : ℕ
ε : ℝ
hε : 0 < ε
hn : cornersTheoremBound (ε / 9) ≤ n
A : Finset (ℕ × ℕ)
hAn : ↑A ⊆ ↑(range n) ×ˢ ↑(range n)
hAε : ε * ↑n ^ 2 ≤ ↑(#A)
hA : IsCornerFree (Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A)
this✝¹ : ↑A = Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : IsCornerFree ↑(image (Prod.map Nat.cast Nat.cast) A)
⊢ cornersTheoremBound (ε / 9) ≤ 2 * n + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: case h₂
n : ℕ
ε : ℝ
hε : 0 < ε
hn : ⌊(triangleRemovalBound (ε / 9 / 9) * 27)⁻¹⌋₊ + 1 ≤ n
A : Finset (ℕ × ℕ)
hAn : ↑A ⊆ ↑(range n) ×ˢ ↑(range n)
hAε : ε * ↑n ^ 2 ≤ ↑(#A)
hA : IsCornerFree (Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A)
this✝¹ : ↑A = Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : IsCornerFree ↑(image (Prod.map Nat.cast Nat.cast) A)
⊢ 1 ≤ n

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: n : ℕ
ε : ℝ
hε : 0 < ε
hn : cornersTheoremBound (ε / 9) ≤ n
A : Finset (ℕ × ℕ)
hAn : ↑A ⊆ ↑(range n) ×ˢ ↑(range n)
hAε : ε * ↑n ^ 2 ≤ ↑(#A)
hA : IsCornerFree (Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A)
this✝¹ : ↑A = Prod.map Fin.val Fin.val '' Prod.map Nat.cast Nat.cast '' ↑A
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : IsCornerFree ↑(image (Prod.map Nat.cast Nat.cast) A)
⊢ n ≤ 2 * n + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: n : ℕ
ε : ℝ
hε : 0 < ε
hG : cornersTheoremBound (ε / 3) ≤ n
A : Finset ℕ
hAn : ↑A ⊆ Set.Iio n
hAε : ε * ↑n ≤ ↑(#A)
hA : ThreeAPFree ↑A
a : ℕ
ha : a ∈ A
this : a < n
⊢ a < 2 * n + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: n : ℕ
ε : ℝ
hε : 0 < ε
hG : cornersTheoremBound (ε / 3) ≤ n
A : Finset ℕ
hAn : ↑A ⊆ Set.Iio n
hAε : ε * ↑n ≤ ↑(#A)
hA : ThreeAPFree (Fin.val '' Nat.cast '' ↑A)
this✝ : ↑A = Fin.val '' Nat.cast '' ↑A
this : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
x : ℕ
hx : x ∈ Set.Iio n
⊢ n ≤ 2 * n

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: n : ℕ
ε : ℝ
hε : 0 < ε
hG : cornersTheoremBound (ε / 3) ≤ n
A : Finset ℕ
hAn : ↑A ⊆ Set.Iio n
hAε : ε * ↑n ≤ ↑(#A)
hA : ThreeAPFree (Fin.val '' Nat.cast '' ↑A)
this✝¹ : ↑A = Fin.val '' Nat.cast '' ↑A
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : ThreeAPFree ((fun x ↦ ↑x) '' ↑A)
⊢ False

Tactic:
← coe_image

Goal state after:
n : ℕ
ε : ℝ
hε : 0 < ε
hG : cornersTheoremBound (ε / 3) ≤ n
A : Finset ℕ
hAn : ↑A ⊆ Set.Iio n
hAε : ε * ↑n ≤ ↑(#A)
hA : ThreeAPFree (Fin.val '' Nat.cast '' ↑A)
this✝¹ : ↑A = Fin.val '' Nat.cast '' ↑A
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : ThreeAPFree ↑(image (fun x ↦ ↑x) A)
⊢ False
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: n : ℕ
ε : ℝ
hε : 0 < ε
hG : cornersTheoremBound (ε / 3) ≤ n
A : Finset ℕ
hAn : ↑A ⊆ Set.Iio n
hAε : ε * ↑n ≤ ↑(#A)
hA : ThreeAPFree (Fin.val '' Nat.cast '' ↑A)
this✝¹ : ↑A = Fin.val '' Nat.cast '' ↑A
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : ThreeAPFree ↑(image (fun x ↦ ↑x) A)
⊢ cornersTheoremBound (ε / 3) ≤ 2 * n + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.Energy

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : CommGroup α
inst✝ : Fintype α
t : Finset α
f : α × α × α → (α × α) × α × α := fun x ↦ ((x.1 * x.2.2, x.1 * x.2.1), x.2)
this : Set.InjOn f ↑(univ ×ˢ t ×ˢ t)
a : (α × α) × α × α
⊢ (∃ a_1 a_2 b, (a_2 ∈ t ∧ b ∈ t) ∧ f (a_1, a_2, b) = a) → (a.2.1 ∈ t ∧ a.2.2 ∈ t) ∧ a.1.1 * a.2.1 = a.1.2 * a.2.2

Tactic:
rintro ⟨b, c, d, hcd, rfl⟩

Goal state after:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : CommGroup α
inst✝ : Fintype α
t : Finset α
f : α × α × α → (α × α) × α × α := fun x ↦ ((x.1 * x.2.2, x.1 * x.2.1), x.2)
this : Set.InjOn f ↑(univ ×ˢ t ×ˢ t)
b c d : α
hcd : c ∈ t ∧ d ∈ t
⊢ ((f (b, c, d)).2.1 ∈ t ∧ (f (b, c, d)).2.2 ∈ t) ∧
    (f (b, c, d)).1.1 * (f (b, c, d)).2.1 = (f (b, c, d)).1.2 * (f (b, c, d)).2.2
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: case h₂
n : ℕ
ε : ℝ
hε : 0 < ε
hG : ⌊(triangleRemovalBound (ε / 3 / 9) * 27)⁻¹⌋₊ + 1 ≤ n
A : Finset ℕ
hAn : ↑A ⊆ Set.Iio n
hAε : ε * ↑n ≤ ↑(#A)
hA : ThreeAPFree (Fin.val '' Nat.cast '' ↑A)
this✝¹ : ↑A = Fin.val '' Nat.cast '' ↑A
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : ThreeAPFree ↑(image (fun x ↦ ↑x) A)
⊢ 1 ≤ n

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Roth

Goal state before:
pp: n : ℕ
ε : ℝ
hε : 0 < ε
hG : cornersTheoremBound (ε / 3) ≤ n
A : Finset ℕ
hAn : ↑A ⊆ Set.Iio n
hAε : ε * ↑n ≤ ↑(#A)
hA : ThreeAPFree (Fin.val '' Nat.cast '' ↑A)
this✝¹ : ↑A = Fin.val '' Nat.cast '' ↑A
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : ThreeAPFree ↑(image (fun x ↦ ↑x) A)
⊢ n ≤ 2 * n + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.ErdosGinzburgZiv

Goal state before:
pp: ι : Type u_1
p : ℕ
inst✝ : Fact (Nat.Prime p)
s : Finset ι
a : ι → ZMod p
this : 2 ≤ p
⊢ p - 1 + (p - 1) < 2 * p - 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.SubsetSum

Goal state before:
pp: M : Type u_1
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
A✝ : Finset M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a : M
A : Finset M
A_lt_a : ∀ x ∈ A, x < a
ih : (∀ x ∈ A, 0 < x) → (#A + 1).choose 2 < #A.subsetSum
A_pos : ∀ x ∈ insert a A, 0 < x
A_pos' : ∀ x ∈ A, 0 < x
⊢ 0 < 2

Tactic:
lia

Goal state after:
Module: Mathlib.Data.Finset.Slice

Goal state before:
pp: α : Type u_1
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
A : Finset α
⊢ A ∈ 𝒜 → A ∈ powersetCard r univ ↔ A ∈ ↑𝒜 → #A = r

Tactic:
mem_powersetCard_univ,

Goal state after:
α : Type u_1
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
A : Finset α
⊢ A ∈ 𝒜 → #A = r ↔ A ∈ ↑𝒜 → #A = r
Module: Mathlib.Combinatorics.Additive.ErdosGinzburgZiv

Goal state before:
pp: case h₁
m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
n : ℕ
hn : 2 ≤ n
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * (m * n) - 1 ≤ #s
k : ℕ
ih : k ≤ 2 * m - 1 → ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
hk : k + 1 ≤ 2 * m - 1
𝒜 : Finset (Finset ι)
h𝒜card : #𝒜 = k
h𝒜disj : (↑𝒜).Pairwise Disjoint
h𝒜 : ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
⊢ 2 ≤ 2 * m - k

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.ErdosGinzburgZiv

Goal state before:
pp: m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
n : ℕ
hn : 2 ≤ n
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * (m * n) - 1 ≤ #s
k : ℕ
ih : k ≤ 2 * m - 1 → ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
hk : k + 1 ≤ 2 * m - 1
𝒜 : Finset (Finset ι)
h𝒜card : #𝒜 = k
h𝒜disj : (↑𝒜).Pairwise Disjoint
h𝒜 : ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
this : 2 * n - 1 ≤ #(s \ 𝒜.biUnion id)
t₀ : Finset ι
ht₀ : t₀ ⊆ s \ 𝒜.biUnion id
ht₀card : #t₀ = n
ht₀sum : ↑n ∣ ∑ i ∈ t₀, a i
h : t₀ ∈ 𝒜
⊢ n = 0

Tactic:
by
        simpa [← card_eq_zero, ht₀card] using sdiff_disjoint.mono ht₀ <| subset_biUnion_of_mem id h

Goal state after:
Module: Mathlib.Combinatorics.Additive.ErdosGinzburgZiv

Goal state before:
pp: m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
k : ℕ
hk : k + 1 ≤ 2 * m - 1
𝒜 : Finset (Finset ι)
h𝒜card : #𝒜 = k
h𝒜disj : (↑𝒜).Pairwise Disjoint
t₀ : Finset ι
ht₀ : t₀ ⊆ s \ 𝒜.biUnion id
h : t₀ ∈ 𝒜
hn : 2 ≤ 0
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * 0 - 1 ≤ #s → ∃ t ⊆ s, #t = 0 ∧ ↑0 ∣ ∑ i ∈ t, a i
hs : 2 * (m * 0) - 1 ≤ #s
ih : k ≤ 2 * m - 1 → ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = 0 ∧ ↑0 ∣ ∑ i ∈ t, a i
h𝒜 : ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = 0 ∧ ↑0 ∣ ∑ i ∈ t, a i
this : 2 * 0 - 1 ≤ #(s \ 𝒜.biUnion id)
ht₀card : #t₀ = 0
ht₀sum : ↑0 ∣ ∑ i ∈ t₀, a i
⊢ False

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Colex

Goal state before:
pp: case inr
α : Type u_1
inst✝ : PartialOrder α
s : Finset α
a b : α
ha : a ∉ s
hb : b ∉ s
hab : a ≠ b
⊢ toColex {a} ≤ toColex (cons b s hb \ cons a s ha) ↔ a ≤ b

Tactic:
cons_sdiff_cons hab.symm,

Goal state after:
case inr
α : Type u_1
inst✝ : PartialOrder α
s : Finset α
a b : α
ha : a ∉ s
hb : b ∉ s
hab : a ≠ b
⊢ toColex {a} ≤ toColex {b} ↔ a ≤ b
Module: Mathlib.Combinatorics.Colex

Goal state before:
pp: n : ℕ
hn : 2 ≤ n
s t : Colex (Finset ℕ)
a : ℕ
hat : a ∈ ofColex t
has : a ∉ ofColex s
ha : ∀ b ∈ ofColex s, b ∉ ofColex t → b < a
x✝¹ : ℕ
x✝ : x✝¹ ∈ ofColex t \ ofColex s
⊢ 0 ≤ n ^ x✝¹

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Derangements.Finite

Goal state before:
pp: case ind.succ.succ.a
n : ℕ
hyp : ∀ m < n + 1 + 1, card ↑(derangements (Fin m)) = numDerangements m
⊢ n + 1 < n + 1 + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Derangements.Finite

Goal state before:
pp: case ind.succ.succ.a
n : ℕ
hyp : ∀ m < n + 1 + 1, card ↑(derangements (Fin m)) = numDerangements m
⊢ n < n + 1 + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.VerySmallDoubling

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
A✝ B S : Finset G
a✝ b✝ c✝ d✝ x y : G
A : Finset G
h : ↑(#(A * A)) < 3 / 2 * ↑(#A)
h₁ : ∀ x ∈ A, ∀ y ∈ A, 1 / 2 * ↑(#A) < ↑(#(x •> A ∩ y •> A))
a : G
ha : a⁻¹ ∈ A
b : G
hb : b ∈ A
c : G
hc : c⁻¹ ∈ A
d : G
hd : d ∈ A
⊢ 1 / 2 * ↑(#A) < ↑(#(A ∩ (a * b)⁻¹ •> A))

Tactic:
refine (h₁ b hb _ ha).trans_le ?_

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
A✝ B S : Finset G
a✝ b✝ c✝ d✝ x y : G
A : Finset G
h : ↑(#(A * A)) < 3 / 2 * ↑(#A)
h₁ : ∀ x ∈ A, ∀ y ∈ A, 1 / 2 * ↑(#A) < ↑(#(x •> A ∩ y •> A))
a : G
ha : a⁻¹ ∈ A
b : G
hb : b ∈ A
c : G
hc : c⁻¹ ∈ A
d : G
hd : d ∈ A
⊢ ↑(#(b •> A ∩ a⁻¹ •> A)) ≤ ↑(#(A ∩ (a * b)⁻¹ •> A))
Module: Mathlib.Combinatorics.Configuration

Goal state before:
pp: case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
⊢ lineCount L p = lineCount L q

Tactic:
obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _

Goal state after:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
⊢ lineCount L p = lineCount L q
Module: Mathlib.Combinatorics.Enumerative.Catalan.Tree

Goal state before:
pp: n : ℕ
ijh : ↥(antidiagonal n)
this : (↑ijh).1 ≤ n
⊢ (↑ijh).1 < n + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.Catalan.Tree

Goal state before:
pp: n : ℕ
ijh : ↥(antidiagonal n)
this : (↑ijh).2 ≤ n
⊢ (↑ijh).2 < n + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.Catalan.Basic

Goal state before:
pp: d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑m.centralBinom / (↑m + 1)
x : Fin d.succ
⊢ ↑x ≤ d

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.Bell

Goal state before:
pp: x : ℕ
hx : x ≠ 0
c : ℕ
⊢ x ! * (c + 1) * x ! ^ c * c ! * ∏ j ∈ Finset.range (c + 1), (j * x + x - 1).choose (x - 1) =
    (x ! ^ c * c ! * ∏ j ∈ Finset.range c, (j * x + x - 1).choose (x - 1)) * (c * x + x - 1).choose (x - 1) * x ! *
      (c + 1)

Tactic:
by
        rw [Finset.prod_range_succ]; ring

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DyckWord

Goal state before:
pp: p : DyckWord
h : ↑p ≠ []
f : ¬(↑p).getLast h = D
s : count U (take ((↑p).length - 1) ↑p) + 1 = count D (take ((↑p).length - 1) ↑p) + 0
this : count D (take ((↑p).length - 1) ↑p) ≤ count U (take ((↑p).length - 1) ↑p)
⊢ False

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DyckWord

Goal state before:
pp: p q : DyckWord
h : p ≠ 0
i : ℕ
hi : count U (List.take i ↑p) = count D (List.take i ↑p)
this : count U (List.take i ↑p) + count U (List.drop i ↑p) = count D (List.take i ↑p ++ List.drop i ↑p)
⊢ count U (List.drop i ↑p) = count D (List.drop i ↑p)

Tactic:
count_append

Goal state after:
p q : DyckWord
h : p ≠ 0
i : ℕ
hi : count U (List.take i ↑p) = count D (List.take i ↑p)
this : count U (List.take i ↑p) + count U (List.drop i ↑p) = count D (List.take i ↑p) + count D (List.drop i ↑p)
⊢ count U (List.drop i ↑p) = count D (List.drop i ↑p)
Module: Mathlib.Combinatorics.Enumerative.DyckWord

Goal state before:
pp: p q : DyckWord
h : p ≠ 0
i : ℕ
hi : count U (List.take i ↑p) = count D (List.take i ↑p)
this : count U (List.take i ↑p) + count U (List.drop i ↑p) = count D (List.take i ↑p) + count D (List.drop i ↑p)
⊢ count U (List.drop i ↑p) = count D (List.drop i ↑p)

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DyckWord

Goal state before:
pp: p : DyckWord
⊢ p.nest ≠ 0

Tactic:
by simp [← toList_ne_nil, nest]

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DyckWord

Goal state before:
pp: p : DyckWord
i : ℕ
lb : 0 < i
ub : i < 1 + (↑p).length + 1
⊢ i ≤ (↑p).length + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DyckWord

Goal state before:
pp: p : DyckWord
i : ℕ
lb : 0 < i
ub : i < 1 + (↑p).length + 1
⊢ 1 ≤ i

Tactic:
lia

Goal state after: