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

Goal state before:
pp: case convert_4
n k ℓ μ : ℕ
V : Type u
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
h : G.IsSRGWith n k ℓ μ
this : DecidableEq V := Classical.decEq V
v w : V
hw : w ∈ Gᶜ.neighborFinset v
⊢ #(bipartiteBelow G.Adj (G.neighborFinset v) w) = μ

Tactic:
simp_rw [neighborFinset_compl, mem_sdiff, mem_compl, mem_singleton, mem_neighborFinset,
      ← Ne.eq_def] at hw

Goal state after:
case convert_4
n k ℓ μ : ℕ
V : Type u
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
h : G.IsSRGWith n k ℓ μ
this : DecidableEq V := Classical.decEq V
v w : V
hw : ¬G.Adj v w ∧ w ≠ v
⊢ #(bipartiteBelow G.Adj (G.neighborFinset v) w) = μ
Module: Mathlib.Combinatorics.SimpleGraph.StronglyRegular

Goal state before:
pp: case convert_4
n k ℓ μ : ℕ
V : Type u
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
h : G.IsSRGWith n k ℓ μ
this : DecidableEq V := Classical.decEq V
v w : V
hw : ¬G.Adj v w ∧ w ≠ v
⊢ #(bipartiteBelow G.Adj (G.neighborFinset v) w) = μ

Tactic:
[

Goal state after:
case convert_4
n k ℓ μ : ℕ
V : Type u
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
h : G.IsSRGWith n k ℓ μ
this : DecidableEq V := Classical.decEq V
v w : V
hw : ¬G.Adj v w ∧ w ≠ v
⊢ #(bipartiteBelow G.Adj (G.neighborFinset v) w) = μ
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits

Goal state before:
pp: case refine_2
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
⊢ #(Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ) = r ∨ t = 0

Tactic:
[

Goal state after:
case refine_2
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
⊢ #(Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ) = r ∨ t = 0
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits

Goal state before:
pp: case refine_3
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
p : Finset (Fin n)
hp : p ∈ Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ
⊢ #p = t

Tactic:
[

Goal state after:
case refine_3
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
p : Finset (Fin n)
hp : p ∈ Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ
⊢ #p = t
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 : Finset (Fin n)
hp₁ : p₁ ∈ ↑(Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ)
p₂ : Finset (Fin n)
hp₂ : p₂ ∈ ↑(Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ)
hne : p₁ ≠ p₂
v₁ : Fin n
hv₁ : v₁ ∈ ↑p₁
v₂ : Fin n
hv₂ : v₂ ∈ ↑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 : Finset (Fin n)
hp₁ : p₁ ∈ ↑(Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ)
p₂ : Finset (Fin n)
hp₂ : p₂ ∈ ↑(Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ)
hne : p₁ ≠ p₂
v₁ : Fin n
hv₁ : v₁ ∈ ↑p₁
v₂ : Fin n
hv₂ : v₂ ∈ ↑p₂
⊢ G.Adj v₁ v₂
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits

Goal state before:
pp: case inr
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
K' : G.CompleteEquipartiteSubgraph r t :=
  { parts := Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ, card_parts := ⋯, card_mem_parts := ⋯,
    isCompleteBetween := ⋯ }
p' : Finset (Fin n)
hp' : p' ∈ K'.parts
v : Fin n
hv : v ∈ ↑p'
w : Fin n
hw : w ∈ ↑(Finset.map (Function.Embedding.subtype fun x ↦ x ∈ ErdosStone.filter K t) s)
w' : ↥(ErdosStone.filter K t)
hw'_mem : w' ∈ s
hw'_eq : ↑w' = w
⊢ G.Adj v w

Tactic:
[

Goal state after:
case inr
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
K' : G.CompleteEquipartiteSubgraph r t :=
  { parts := Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ, card_parts := ⋯, card_mem_parts := ⋯,
    isCompleteBetween := ⋯ }
p' : Finset (Fin n)
hp' : p' ∈ K'.parts
v : Fin n
hv : v ∈ ↑p'
w : Fin n
hw : w ∈ ↑(Finset.map (Function.Embedding.subtype fun x ↦ x ∈ ErdosStone.filter K t) s)
w' : ↥(ErdosStone.filter K t)
hw'_mem : w' ∈ s
hw'_eq : ↑w' = w
⊢ G.Adj v w
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits

Goal state before:
pp: case inr
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
K' : G.CompleteEquipartiteSubgraph r t :=
  { parts := Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ, card_parts := ⋯, card_mem_parts := ⋯,
    isCompleteBetween := ⋯ }
p' : Finset (Fin n)
v : Fin n
hv : v ∈ ↑p'
w : Fin n
hw : w ∈ ↑(Finset.map (Function.Embedding.subtype fun x ↦ x ∈ ErdosStone.filter K t) s)
w' : ↥(ErdosStone.filter K t)
hw'_eq : ↑w' = w
p : Finset (Fin n)
hp : p ∈ K.parts
hp'_eq : y p hp = p'
hw'_mem : w' ∈ {w | ErdosStone.filter.pi K w = ⟨y, hy✝⟩}
⊢ G.Adj v w

Tactic:
[

Goal state after:
case inr
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
K' : G.CompleteEquipartiteSubgraph r t :=
  { parts := Finset.map { toFun := fun p ↦ y ↑p ⋯, inj' := ⋯ } univ, card_parts := ⋯, card_mem_parts := ⋯,
    isCompleteBetween := ⋯ }
p' : Finset (Fin n)
v : Fin n
hv : v ∈ ↑p'
w : Fin n
hw : w ∈ ↑(Finset.map (Function.Embedding.subtype fun x ↦ x ∈ ErdosStone.filter K t) s)
w' : ↥(ErdosStone.filter K t)
hw'_eq : ↑w' = w
p : Finset (Fin n)
hp : p ∈ K.parts
hp'_eq : y p hp = p'
hw'_mem : w' ∈ {w | ErdosStone.filter.pi K w = ⟨y, hy✝⟩}
⊢ G.Adj v w
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 ∈ g • pt ↔ g⁻¹ • x ∈ pt

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
x : X
pt : PlacedTile ps
⊢ x ∈ g • pt ↔ g⁻¹ • x ∈ pt
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 ∈ g⁻¹ • pt ↔ g • x ∈ pt

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
x : X
pt : PlacedTile ps
⊢ x ∈ g⁻¹ • pt ↔ g • x ∈ pt
Module: Mathlib.Combinatorics.Young.SemistandardTableau

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

Tactic:
· exact T.row_weak h cell

Goal state after:
Module: Mathlib.Computability.Primrec.Basic

Goal state before:
pp: case some
α : Type ?u.4
β : Type ?u.6
inst✝¹ : Primcodable α
inst✝ : Primcodable β
n : ℕ
val✝ : α
⊢ Nat.rec 0 (fun n_1 n_ih ↦ Nat.rec 0 (fun n n_ih ↦ (Nat.pair n_1 n).succ) (encode (decode (Nat.unpair n).2)))
      (encode (some val✝)) =
    encode ((some val✝).bind fun a ↦ Option.map (Prod.mk a) (decode (Nat.unpair n).2))

Tactic:
cases @decode β _ n.unpair.2 <;> simp

Goal state after:
Module: Mathlib.Computability.Primrec.Basic

Goal state before:
pp: case some
α : Type u_2
β : Type u_3
inst✝¹ : Primcodable α
inst✝ : Primcodable β
n : ℕ
val✝ : α
⊢ Nat.rec 0 (fun n n_ih ↦ encode val✝ + 1) (encode (decode (Nat.unpair n).2)) =
    encode (Option.map (Function.const β val✝) (decode (Nat.unpair n).2))

Tactic:
cases @decode β _ n.unpair.2 <;> simp

Goal state after:
Module: Mathlib.Computability.Primrec.Basic

Goal state before:
pp: case some
α : Type u_2
β : Type u_3
inst✝¹ : Primcodable α
inst✝ : Primcodable β
n : ℕ
val✝ : α
⊢ Nat.rec 0 (fun n n_ih ↦ n + 1) (encode (decode (Nat.unpair n).2)) = encode (decode (Nat.unpair n).2)

Tactic:
cases @decode β _ n.unpair.2 <;> simp

Goal state after:
Module: Mathlib.Computability.Primrec.List

Goal state before:
pp: α : Type u_1
β : Type u_2
σ : Type u_4
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
f : α → List β
g : α → σ
h : α → β × List β × σ → σ
hf : Primrec f
hg : Primrec g
hh : Primrec₂ h
F : α → List β × σ := fun a ↦ List.foldr (fun b s ↦ (b :: s.1, h a (b, s))) ([], g a) (f a)
this✝ : Primrec F
a : α
this : F a = (f a, List.recOn (f a) (g a) fun b l IH ↦ h a (b, l, IH))
⊢ (F a).2 = List.recOn (f a) (g a) fun b l IH ↦ h a (b, l, IH)

Tactic:
this

Goal state after:
α : Type u_1
β : Type u_2
σ : Type u_4
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
f : α → List β
g : α → σ
h : α → β × List β × σ → σ
hf : Primrec f
hg : Primrec g
hh : Primrec₂ h
F : α → List β × σ := fun a ↦ List.foldr (fun b s ↦ (b :: s.1, h a (b, s))) ([], g a) (f a)
this✝ : Primrec F
a : α
this : F a = (f a, List.recOn (f a) (g a) fun b l IH ↦ h a (b, l, IH))
⊢ (f a, List.recOn (f a) (g a) fun b l IH ↦ h a (b, l, IH)).2 = List.recOn (f a) (g a) fun b l IH ↦ h a (b, l, IH)
Module: Mathlib.Computability.Partrec

Goal state before:
pp: case false
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
pm : (p m).Dom
e : (p m).get pm = false
⊢ { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }

Tactic:
suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        intro n h
        rcases h.lt_or_eq_dec with h | h
        · exact al _ h
        · rw [h]
          exact ⟨_, e⟩

Goal state after:
Module: Mathlib.Computability.Partrec

Goal state before:
pp: case succ
f : ℕ → ℕ
⊢ Nat.Partrec ↑Nat.succ

Tactic:
| succ =>

Goal state after:
Module: Mathlib.Computability.Primrec.List

Goal state before:
pp: case succ
n : ℕ
f : List.Vector ℕ n → ℕ
⊢ Primrec fun v ↦ v.head.succ

Tactic:
| succ =>

Goal state after:
Module: Mathlib.Computability.Primrec.List

Goal state before:
pp: case succ
n : ℕ
f✝ : List.Vector ℕ n → ℕ
f : ℕ → ℕ
⊢ Primrec' fun v ↦ v.head.succ

Tactic:
| succ =>

Goal state after:
Module: Mathlib.Computability.Ackermann

Goal state before:
pp: case succ
f : ℕ → ℕ
⊢ ∃ m, ∀ (n : ℕ), n.succ < ack m n

Tactic:
| succ =>

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

Goal state before:
pp: case pos
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ∀ (M : G.Subgraph), ¬M.IsPerfectMatching
hvEven : Even (Nat.card V)
val✝ : Fintype V
Gmax : SimpleGraph V
hSubgraph : G ≤ Gmax
hMatchingFree : Gmax.IsMatchingFree
hMaximal : ∀ G' > Gmax, ∃ M, M.IsPerfectMatching
hc : ¬Nat.card ↑Gmax.universalVerts < Fintype.card ↑(⊤.deleteVerts Gmax.universalVerts).coe.oddComponents
h' : ∀ (K : Gmax.deleteUniversalVerts.coe.ConnectedComponent), Gmax.deleteUniversalVerts.coe.IsClique K.supp
⊢ False

Tactic:
[

Goal state after:
case pos
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ∀ (M : G.Subgraph), ¬M.IsPerfectMatching
hvEven : Even (Nat.card V)
val✝ : Fintype V
Gmax : SimpleGraph V
hSubgraph : G ≤ Gmax
hMatchingFree : Gmax.IsMatchingFree
hMaximal : ∀ G' > Gmax, ∃ M, M.IsPerfectMatching
hc : ¬Nat.card ↑Gmax.universalVerts < Fintype.card ↑(⊤.deleteVerts Gmax.universalVerts).coe.oddComponents
h' : ∀ (K : Gmax.deleteUniversalVerts.coe.ConnectedComponent), Gmax.deleteUniversalVerts.coe.IsClique K.supp
⊢ False
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
ih : ∀ (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑k - 1) * x₀) (2 ^ (-↑k) * x₀) → f z = 0
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-↑k - 1 - 1) * x₀) (2 ^ (-↑k - 1) * x₀)
⊢ z ≤ 2 ^ (-↑k - 1) * x₀

Tactic:
exact hz.2

Goal state after:
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: ⊢ (fun n ↦ ↑n / log ↑n ^ 2) = fun n ↦ ↑n * (log ↑n ^ 2)⁻¹

Tactic:
[

Goal state after:
⊢ (fun n ↦ ↑n / log ↑n ^ 2) = fun n ↦ ↑n * (log ↑n ^ 2)⁻¹
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: p : ℝ
hp : p ≠ 0
⊢ (fun z ↦ z ^ (p - 1) / log z ^ 2) =o[atTop] fun z ↦ z ^ (p - 1) / 1

Tactic:
[

Goal state after:
p : ℝ
hp : p ≠ 0
⊢ (fun z ↦ z ^ (p - 1) / log z ^ 2) =o[atTop] fun z ↦ z ^ (p - 1) / 1
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: p : ℝ
hp : p ≠ 0
⊢ (fun z ↦ -(z ^ (p - 1) / log z ^ 2)) =o[atTop] fun z ↦ z ^ (p - 1) / 1

Tactic:
[

Goal state after:
p : ℝ
hp : p ≠ 0
⊢ (fun z ↦ -(z ^ (p - 1) / log z ^ 2)) =o[atTop] fun z ↦ z ^ (p - 1) / 1
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
C c : ℝ
hc_mem : c ∈ Set.Ioo 0 1
hc : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c * ↑n ≤ ↑(r i n)
n : ℕ
hn₁ : ⌈C / c⌉₊ ≤ n
hn₂ : ∀ (i : α), c * ↑n ≤ ↑(r i n)
i : α
⊢ C ≤ ↑(r i n)

Tactic:
have h₁ := hc_mem.1

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
C c : ℝ
hc_mem : c ∈ Set.Ioo 0 1
hc : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c * ↑n ≤ ↑(r i n)
n : ℕ
hn₁ : ⌈C / c⌉₊ ≤ n
hn₂ : ∀ (i : α), c * ↑n ≤ ↑(r i n)
i : α
h₁ : 0 < c
⊢ C ≤ ↑(r i n)
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: case h₁
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c : ℝ := ⋯
h_max_bi_pos : 0 < b (max_bi b)
h_max_bi_lt_one : 0 < 1 - b (max_bi b)
hc_pos : 0 < c
h₁ : 0 < (1 - b (max_bi b)) / 2
hc_lt_one : c < 1
hlo : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ c * ‖↑x‖
hlo' : ∀ᶠ (x : ℕ) in atTop, ‖↑x / log ↑x ^ 2‖ ≤ (1 - b (max_bi b)) / 2 * ‖↑x‖
n : ℕ
hn' : ∀ (i : α), ↑(r i n) ≤ b i * ↑n + ↑n / log ↑n ^ 2
i : α
hn : ↑n / log ↑n ^ 2 ≤ (1 - b (max_bi b)) / 2 * ↑n
⊢ b i ≤ b (max_bi b)

Tactic:
exact max_bi_le _

Goal state after:
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: x : ℝ
hx : 1 < x
this : log x ≠ 0
⊢ DifferentiableAt ℝ (fun z ↦ 1 / log z) x

Tactic:
[

Goal state after:
x : ℝ
hx : 1 < x
this : log x ≠ 0
⊢ DifferentiableAt ℝ (fun z ↦ 1 / log z) x
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: ⊢ (fun x ↦ -x⁻¹ / log x ^ 2) = fun x ↦ (-x * log x ^ 2)⁻¹

Tactic:
[

Goal state after:
⊢ (fun x ↦ -x⁻¹ / log x ^ 2) = fun x ↦ (-x * log x ^ 2)⁻¹
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x ↦ x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x ↦ x ^ (p a b - 1)
⊢ (fun n ↦ ↑n ^ p a b / log ↑n ^ 2) = fun n ↦ ↑n ^ p a b * (1 / log ↑n ^ 2)

Tactic:
[

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x ↦ x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x ↦ x ^ (p a b - 1)
⊢ (fun n ↦ ↑n ^ p a b / log ↑n ^ 2) = fun n ↦ ↑n ^ p a b * (1 / log ↑n ^ 2)
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x ↦ x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x ↦ x ^ (p a b - 1)
⊢ (fun n ↦ b i ^ p a b * ↑n ^ p a b * (1 / log ↑n ^ 2)) =Θ[atTop] fun n ↦ ↑n ^ p a b * (1 / log ↑n ^ 2)

Tactic:
[

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x ↦ x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x ↦ x ^ (p a b - 1)
⊢ (fun n ↦ b i ^ p a b * ↑n ^ p a b * (1 / log ↑n ^ 2)) =Θ[atTop] fun n ↦ ↑n ^ p a b * (1 / log ↑n ^ 2)