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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h✝ : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
h : (⊤.deleteVerts G.universalVerts).coe.oddComponents.ncard ≤ G.universalVerts.ncard
⊢ G.deleteUniversalVerts.coe.oddComponents.ncard ≤ G.universalVerts.ncard

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h✝ : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
h : (⊤.deleteVerts G.universalVerts).coe.oddComponents.ncard ≤ G.universalVerts.ncard
⊢ G.deleteUniversalVerts.coe.oddComponents.ncard ≤ G.universalVerts.ncard
Module: Mathlib.Combinatorics.SimpleGraph.Tutte

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
K : G.deleteUniversalVerts.coe.ConnectedComponent
this : G.IsClique (Subtype.val '' K.supp \ M1.verts)
⊢ Even (Subtype.val '' K.supp \ (Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t)).ncard

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
K : G.deleteUniversalVerts.coe.ConnectedComponent
this : G.IsClique (Subtype.val '' K.supp \ M1.verts)
⊢ Even (Subtype.val '' K.supp \ (Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t)).ncard
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits

Goal state before:
pp: case refine_4
ε : ℝ
hε : 0 < ε
r t : ℕ
hr_pos : 0 < r
ht_pos : 0 < t
ε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε
hε' : 0 < ε'
t' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1
ht_lt_rt'ε : ↑t < ↑r * ↑t' * ε
ht'_pos : 0 < t'
N' : ℕ
N : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ε - ↑t)⌉₊
n : ℕ
hn : N ≤ n
G : SimpleGraph (Fin n)
x✝ : DecidableRel G.Adj
hδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n
this : Nonempty (Fin n)
hrε_lt_1 : ↑r * ε < 1
ht_lt_t' : t < t'
ih : completeEquipartiteGraph r t' ⊑ G
K : G.CompleteEquipartiteSubgraph r t'
y : (a : Finset (Fin n)) → a ∈ K.parts → Finset (Fin n)
hy✝ : y ∈ K.parts.pi fun x ↦ powersetCard t x
hy : ∀ (a : Finset (Fin n)) (h : a ∈ K.parts), y a h ∈ powersetCard t a
ht_le_card_filter : t ≤ #{w | ErdosStone.filter.pi K w = ⟨y, hy✝⟩}
s : Finset ↥(ErdosStone.filter K t)
hs_subset : s ⊆ {w | ErdosStone.filter.pi K w = ⟨y, hy✝⟩}
hcards : #s = t
p₁✝ p₂✝ : Finset (Fin n)
v₁ v₂ : Fin n
p₁ : Finset (Fin n)
hp₁ : p₁ ∈ K.parts
hv₁ : v₁ ∈ ↑(y p₁ ⋯)
hyp₁ : y p₁ ⋯ = p₁✝
hy₁' : y p₁ hp₁ ⊆ p₁ ∧ #(y p₁ hp₁) = t
p₂ : Finset (Fin n)
hp₂ : p₂ ∈ K.parts
hne : y p₁ ⋯ ≠ y p₂ ⋯
hv₂ : v₂ ∈ ↑(y p₂ ⋯)
hyp₂ : y p₂ ⋯ = p₂✝
⊢ G.Adj v₁ v₂

Tactic:
]

Goal state after:
case refine_4
ε : ℝ
hε : 0 < ε
r t : ℕ
hr_pos : 0 < r
ht_pos : 0 < t
ε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε
hε' : 0 < ε'
t' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1
ht_lt_rt'ε : ↑t < ↑r * ↑t' * ε
ht'_pos : 0 < t'
N' : ℕ
N : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ε - ↑t)⌉₊
n : ℕ
hn : N ≤ n
G : SimpleGraph (Fin n)
x✝ : DecidableRel G.Adj
hδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n
this : Nonempty (Fin n)
hrε_lt_1 : ↑r * ε < 1
ht_lt_t' : t < t'
ih : completeEquipartiteGraph r t' ⊑ G
K : G.CompleteEquipartiteSubgraph r t'
y : (a : Finset (Fin n)) → a ∈ K.parts → Finset (Fin n)
hy✝ : y ∈ K.parts.pi fun x ↦ powersetCard t x
hy : ∀ (a : Finset (Fin n)) (h : a ∈ K.parts), y a h ∈ powersetCard t a
ht_le_card_filter : t ≤ #{w | ErdosStone.filter.pi K w = ⟨y, hy✝⟩}
s : Finset ↥(ErdosStone.filter K t)
hs_subset : s ⊆ {w | ErdosStone.filter.pi K w = ⟨y, hy✝⟩}
hcards : #s = t
p₁✝ p₂✝ : Finset (Fin n)
v₁ v₂ : Fin n
p₁ : Finset (Fin n)
hp₁ : p₁ ∈ K.parts
hv₁ : v₁ ∈ ↑(y p₁ ⋯)
hyp₁ : y p₁ ⋯ = p₁✝
hy₁' : y p₁ hp₁ ⊆ p₁ ∧ #(y p₁ hp₁) = t
p₂ : Finset (Fin n)
hp₂ : p₂ ∈ K.parts
hne : y p₁ ⋯ ≠ y p₂ ⋯
hv₂ : v₂ ∈ ↑(y p₂ ⋯)
hyp₂ : y p₂ ⋯ = p₂✝
⊢ G.Adj v₁ v₂
Module: Mathlib.Combinatorics.SimpleGraph.Tutte

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
complMatch : G.deleteUniversalVerts.coe.ConnectedComponent → G.Subgraph
hcomplMatch_compl :
  ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).verts = Subtype.val '' K.supp \ M1.verts
hcomplMatch_match : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).IsMatching
M2 : G.Subgraph := ⨆ K, complMatch K
i j : G.deleteUniversalVerts.coe.ConnectedComponent
hij : i ≠ j
⊢ Disjoint (Subtype.val '' i.supp \ M1.verts) (Subtype.val '' j.supp \ M1.verts)

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
complMatch : G.deleteUniversalVerts.coe.ConnectedComponent → G.Subgraph
hcomplMatch_compl :
  ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).verts = Subtype.val '' K.supp \ M1.verts
hcomplMatch_match : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).IsMatching
M2 : G.Subgraph := ⨆ K, complMatch K
i j : G.deleteUniversalVerts.coe.ConnectedComponent
hij : i ≠ j
⊢ Disjoint (Subtype.val '' i.supp \ M1.verts) (Subtype.val '' j.supp \ M1.verts)
Module: Mathlib.Combinatorics.SimpleGraph.Tutte

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
complMatch : G.deleteUniversalVerts.coe.ConnectedComponent → G.Subgraph
hcomplMatch_compl :
  ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).verts = Subtype.val '' K.supp \ M1.verts
hcomplMatch_match : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).IsMatching
M2 : G.Subgraph := ⨆ K, complMatch K
hM2 : M2.IsMatching
⊢ ∀ (i : G.deleteUniversalVerts.coe.ConnectedComponent), Disjoint M1.verts (complMatch i).verts

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
complMatch : G.deleteUniversalVerts.coe.ConnectedComponent → G.Subgraph
hcomplMatch_compl :
  ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).verts = Subtype.val '' K.supp \ M1.verts
hcomplMatch_match : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).IsMatching
M2 : G.Subgraph := ⨆ K, complMatch K
hM2 : M2.IsMatching
⊢ ∀ (i : G.deleteUniversalVerts.coe.ConnectedComponent), Disjoint M1.verts (complMatch i).verts
Module: Mathlib.Combinatorics.SimpleGraph.Tutte

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
complMatch : G.deleteUniversalVerts.coe.ConnectedComponent → G.Subgraph
hcomplMatch_compl :
  ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).verts = Subtype.val '' K.supp \ M1.verts
hcomplMatch_match : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).IsMatching
M2 : G.Subgraph := ⨆ K, complMatch K
hM2 : M2.IsMatching
disjointM12 : Disjoint M1.support M2.support
⊢ Set.univ \ G.universalVerts ⊆ M1.verts ∪ M2.verts

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
complMatch : G.deleteUniversalVerts.coe.ConnectedComponent → G.Subgraph
hcomplMatch_compl :
  ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).verts = Subtype.val '' K.supp \ M1.verts
hcomplMatch_match : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).IsMatching
M2 : G.Subgraph := ⨆ K, complMatch K
hM2 : M2.IsMatching
disjointM12 : Disjoint M1.support M2.support
⊢ Set.univ \ G.universalVerts ⊆ M1.verts ∪ M2.verts
Module: Mathlib.Combinatorics.SimpleGraph.Tutte

Goal state before:
pp: case intro.refine_2
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
hveven : Even (Nat.card V)
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
val✝ : Fintype V
M : G.Subgraph
hM : M.IsMatching
hsub : M.vertsᶜ ⊆ G.universalVerts
M' : G.Subgraph
hM' : M'.verts = M.vertsᶜ ∧ M'.IsMatching
⊢ M.verts ∪ M.vertsᶜ = Set.univ

Tactic:
]

Goal state after:
case intro.refine_2
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
hveven : Even (Nat.card V)
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
val✝ : Fintype V
M : G.Subgraph
hM : M.IsMatching
hsub : M.vertsᶜ ⊆ G.universalVerts
M' : G.Subgraph
hM' : M'.verts = M.vertsᶜ ∧ M'.IsMatching
⊢ M.verts ∪ M.vertsᶜ = Set.univ
Module: Mathlib.Combinatorics.SimpleGraph.Tutte

Goal state before:
pp: case intro.refine_2
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
hveven : Even (Nat.card V)
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
val✝ : Fintype V
M : G.Subgraph
hM : M.IsMatching
hsub : M.vertsᶜ ⊆ G.universalVerts
M' : G.Subgraph
hM' : M'.verts = M.vertsᶜ ∧ M'.IsMatching
⊢ M.verts ∪ M.vertsᶜ = Set.univ

Tactic:
exact M.verts.union_compl_self

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
hodd : Odd (Nat.card V)
⊢ 0 < (⊤.deleteVerts ∅).coe.oddComponents.ncard

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
hodd : Odd (Nat.card V)
⊢ 0 < (⊤.deleteVerts ∅).coe.oddComponents.ncard
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ ν : YoungDiagram
h : (fun y ↦ ↑y.cells) μ = (fun y ↦ ↑y.cells) ν
⊢ ↑μ.cells = ↑ν.cells

Tactic:
]

Goal state after:
μ ν : YoungDiagram
h : (fun y ↦ ↑y.cells) μ = (fun y ↦ ↑y.cells) ν
⊢ ↑μ.cells = ↑ν.cells
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ ν : YoungDiagram
⊢ IsLowerSet (↑μ.cells ∪ ↑ν.cells)

Tactic:
]

Goal state after:
μ ν : YoungDiagram
⊢ IsLowerSet (↑μ.cells ∪ ↑ν.cells)
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ ν : YoungDiagram
⊢ IsLowerSet (↑μ.cells ∩ ↑ν.cells)

Tactic:
]

Goal state after:
μ ν : YoungDiagram
⊢ IsLowerSet (↑μ.cells ∩ ↑ν.cells)
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ ν : YoungDiagram
h : μ.cells = ν.cells
⊢ μ.cells = ν.cells

Tactic:
]

Goal state after:
μ ν : YoungDiagram
h : μ.cells = ν.cells
⊢ μ.cells = ν.cells
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ ν : YoungDiagram
⊢ μ = ν.transpose.transpose ↔ μ = ν

Tactic:
]

Goal state after:
μ ν : YoungDiagram
⊢ μ = ν.transpose.transpose ↔ μ = ν
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ ν : YoungDiagram
h : μ.transpose.transpose ≤ ν
⊢ μ.transpose ≤ ν.transpose

Tactic:
]

Goal state after:
μ ν : YoungDiagram
h : μ.transpose.transpose ≤ ν
⊢ μ.transpose ≤ ν.transpose
Module: Mathlib.Combinatorics.Tiling.Tile

Goal state before:
pp: case refine_1
G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
pt₁ : PlacedTile ps
⊢ pt₁.groupElts = pt₁.groupElts

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ : YoungDiagram
i j : ℕ
hj : (i, j) ∉ μ.cells
⊢ ∃ j, (i, j) ∉ μ

Tactic:
]

Goal state after:
μ : YoungDiagram
i j : ℕ
hj : (i, j) ∉ μ.cells
⊢ ∃ j, (i, j) ∉ μ
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ : YoungDiagram
i j : ℕ
⊢ (i, j) ∈ μ ↔ ∀ m ≤ j, ¬(i, m) ∉ μ

Tactic:
]

Goal state after:
μ : YoungDiagram
i j : ℕ
⊢ (i, j) ∈ μ ↔ ∀ m ≤ j, ¬(i, m) ∉ μ
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ : YoungDiagram
i1 i2 : ℕ
hi : i1 ≤ i2
h_lt : μ.rowLen i1 < μ.rowLen i2
⊢ μ.rowLen i1 < μ.rowLen i1

Tactic:
]

Goal state after:
μ : YoungDiagram
i1 i2 : ℕ
hi : i1 ≤ i2
h_lt : μ.rowLen i1 < μ.rowLen i2
⊢ μ.rowLen i1 < μ.rowLen i1
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ : YoungDiagram
i1 i2 : ℕ
hi : i1 ≤ i2
h_lt : (i2, μ.rowLen i1) ∈ μ
⊢ (i1, μ.rowLen i1) ∈ μ

Tactic:
]

Goal state after:
μ : YoungDiagram
i1 i2 : ℕ
hi : i1 ≤ i2
h_lt : (i2, μ.rowLen i1) ∈ μ
⊢ (i1, μ.rowLen i1) ∈ μ
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ : YoungDiagram
i j : ℕ
⊢ (i, j) ∈ μ ↔ (j, i) ∈ μ.transpose

Tactic:
]

Goal state after:
μ : YoungDiagram
i j : ℕ
⊢ (i, j) ∈ μ ↔ (j, i) ∈ μ.transpose
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ : YoungDiagram
x : ℕ
hx : ∃ a ∈ List.range (μ.colLen 0), μ.rowLen a = x
⊢ 0 < x

Tactic:
]

Goal state after:
μ : YoungDiagram
x : ℕ
hx : ∃ a ∈ List.range (μ.colLen 0), μ.rowLen a = x
⊢ 0 < x
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: μ : YoungDiagram
i : ℕ
hi : 0 < μ.rowLen i
⊢ 0 < μ.rowLen i

Tactic:
]

Goal state after:
μ : YoungDiagram
i : ℕ
hi : 0 < μ.rowLen i
⊢ 0 < μ.rowLen i
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: case nil
c : ℕ × ℕ
⊢ c ∈ ∅ ↔ ∃ (h : c.1 < [].length), c.2 < [][c.1]

Tactic:
]

Goal state after:
case nil
c : ℕ × ℕ
⊢ c ∈ ∅ ↔ ∃ (h : c.1 < [].length), c.2 < [][c.1]
Module: Mathlib.Combinatorics.Tiling.Tile

Goal state before:
pp: G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
pt : PlacedTile ps
a b : G
r : a⁻¹ * b ∈ Subgroup.map (MulAction.stabilizer G ↑(↑ps pt.index)).subtype (↑ps pt.index).symmetries
⊢ a • ↑(↑ps pt.index) = b • ↑(↑ps pt.index)

Tactic:
]

Goal state after:
G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
pt : PlacedTile ps
a b : G
r : a⁻¹ * b ∈ Subgroup.map (MulAction.stabilizer G ↑(↑ps pt.index)).subtype (↑ps pt.index).symmetries
⊢ a • ↑(↑ps pt.index) = b • ↑(↑ps pt.index)
Module: Mathlib.Combinatorics.Tiling.Tile

Goal state before:
pp: G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
pt : PlacedTile ps
a b : G
r : a⁻¹ * b ∈ Subgroup.map (MulAction.stabilizer G ↑(↑ps pt.index)).subtype (↑ps pt.index).symmetries
⊢ a⁻¹ * b ∈ MulAction.stabilizer G ↑(↑ps pt.index)

Tactic:
]

Goal state after:
G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
pt : PlacedTile ps
a b : G
r : a⁻¹ * b ∈ Subgroup.map (MulAction.stabilizer G ↑(↑ps pt.index)).subtype (↑ps pt.index).symmetries
⊢ a⁻¹ * b ∈ MulAction.stabilizer G ↑(↑ps pt.index)
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: case cons
head✝ : ℕ
tail✝ : List ℕ
tail_ih✝ : ∀ {c : ℕ × ℕ}, c ∈ YoungDiagram.cellsOfRowLens tail✝ ↔ ∃ (h : c.1 < tail✝.length), c.2 < tail✝[c.1]
c : ℕ × ℕ
⊢ c ∈
      {0} ×ˢ Finset.range head✝ ∪
        Finset.map ({ toFun := Nat.succ, inj' := Nat.succ_injective }.prodMap (Embedding.refl ℕ))
          (YoungDiagram.cellsOfRowLens tail✝) ↔
    ∃ (h : c.1 < (head✝ :: tail✝).length), c.2 < (head✝ :: tail✝)[c.1]

Tactic:
]

Goal state after:
case cons
head✝ : ℕ
tail✝ : List ℕ
tail_ih✝ : ∀ {c : ℕ × ℕ}, c ∈ YoungDiagram.cellsOfRowLens tail✝ ↔ ∃ (h : c.1 < tail✝.length), c.2 < tail✝[c.1]
c : ℕ × ℕ
⊢ c ∈
      {0} ×ˢ Finset.range head✝ ∪
        Finset.map ({ toFun := Nat.succ, inj' := Nat.succ_injective }.prodMap (Embedding.refl ℕ))
          (YoungDiagram.cellsOfRowLens tail✝) ↔
    ∃ (h : c.1 < (head✝ :: tail✝).length), c.2 < (head✝ :: tail✝)[c.1]
Module: Mathlib.Combinatorics.Tiling.Tile

Goal state before:
pp: G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
index : ιₚ
groupElts : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps index)).subtype (↑ps index).symmetries
⊢ (Quotient.out groupElts • ↑(↑ps index)).Nonempty ↔ (↑(↑ps index)).Nonempty

Tactic:
]

Goal state after:
G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
index : ιₚ
groupElts : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps index)).subtype (↑ps index).symmetries
⊢ (Quotient.out groupElts • ↑(↑ps index)).Nonempty ↔ (↑(↑ps index)).Nonempty
Module: Mathlib.Combinatorics.Tiling.Tile

Goal state before:
pp: G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
index : ιₚ
groupElts : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps index)).subtype (↑ps index).symmetries
⊢ (Quotient.out groupElts • ↑(↑ps index)).Finite ↔ (↑(↑ps index)).Finite

Tactic:
]

Goal state after:
G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
index : ιₚ
groupElts : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps index)).subtype (↑ps index).symmetries
⊢ (Quotient.out groupElts • ↑(↑ps index)).Finite ↔ (↑(↑ps index)).Finite
Module: Mathlib.Combinatorics.Tiling.Tile

Goal state before:
pp: G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
g : G
pt : PlacedTile ps
a b : G
r : a⁻¹ * b ∈ Subgroup.map (MulAction.stabilizer G ↑(↑ps pt.index)).subtype (↑ps pt.index).symmetries
⊢ { index := pt.index, groupElts := ↑(g * a) } = { index := pt.index, groupElts := ↑(g * b) }

Tactic:
]

Goal state after:
G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
g : G
pt : PlacedTile ps
a b : G
r : a⁻¹ * b ∈ Subgroup.map (MulAction.stabilizer G ↑(↑ps pt.index)).subtype (↑ps pt.index).symmetries
⊢ { index := pt.index, groupElts := ↑(g * a) } = { index := pt.index, groupElts := ↑(g * b) }
Module: Mathlib.Combinatorics.Tiling.Tile

Goal state before:
pp: G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G X
ps : Protoset G X ιₚ
g : G
x : X
pt : PlacedTile ps
⊢ x ∈ pt ↔ x ∈ pt

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Young.YoungDiagram

Goal state before:
pp: w : List ℕ
hw : w.SortedGE
i2 j2 i1 j1 : ℕ
hi : i1 ≤ i2
hj : j1 ≤ j2
hcell : ∃ (h : (i2, j2).1 < w.length), (i2, j2).2 < w[(i2, j2).1]
⊢ ∃ (h : (i1, j1).1 < w.length), (i1, j1).2 < w[(i1, j1).1]

Tactic:
]

Goal state after:
w : List ℕ
hw : w.SortedGE
i2 j2 i1 j1 : ℕ
hi : i1 ≤ i2
hj : j1 ≤ j2
hcell : ∃ (h : (i2, j2).1 < w.length), (i2, j2).2 < w[(i2, j2).1]
⊢ ∃ (h : (i1, j1).1 < w.length), (i1, j1).2 < w[(i1, j1).1]
Module: Mathlib.Combinatorics.SimpleGraph.Tutte

Goal state before:
pp: case h
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le : (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe ≤ G ⊔ edge a c ⊔ edge x b
hxc : x ∉ (cycles.connectedComponentMk c).supp
⊢ ¬(cycles.connectedComponentMk c).toSimpleGraph.spanningCoe.Adj x b

Tactic:
]

Goal state after:
case h
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le : (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe ≤ G ⊔ edge a c ⊔ edge x b
hxc : x ∉ (cycles.connectedComponentMk c).supp
⊢ ¬(cycles.connectedComponentMk c).toSimpleGraph.spanningCoe.Adj x b
Module: Mathlib.Combinatorics.Young.SemistandardTableau

Goal state before:
pp: case inl
μ : YoungDiagram
T : SemistandardYoungTableau μ
i j1 j2 : ℕ
hj : j1 ≤ j2
cell : (i, j2) ∈ μ
h : j1 = j2
⊢ T i j2 ≤ T i j2

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Young.SemistandardTableau

Goal state before:
pp: case inl
μ : YoungDiagram
T : SemistandardYoungTableau μ
i1 i2 j : ℕ
hi : i1 ≤ i2
cell : (i2, j) ∈ μ
h : i1 = i2
⊢ T i2 j ≤ T i2 j

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Young.SemistandardTableau

Goal state before:
pp: μ : YoungDiagram
i✝ j1✝ j2✝ : ℕ
hj : j1✝ < j2✝
hcell : (i✝, j2✝) ∈ μ
⊢ (if (i✝, j1✝) ∈ μ then i✝ else 0) ≤ if (i✝, j2✝) ∈ μ then i✝ else 0

Tactic:
if_pos hcell,

Goal state after:
μ : YoungDiagram
i✝ j1✝ j2✝ : ℕ
hj : j1✝ < j2✝
hcell : (i✝, j2✝) ∈ μ
⊢ (if (i✝, j1✝) ∈ μ then i✝ else 0) ≤ i✝