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https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | exact (aux hG hGH').image _ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
⊢ (Sym2.map ⇑(Hom.ofLE ⋯) '' H'.edgeSet).Nonempty | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | obtain rfl | hG := eq_or_ne G ⊥ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
⊢ H.edgeFinset.card - G.copyCount H ≤ (G.kill H).edgeFinset.card | case inl
α : Type u_1
β : Type u_2
γ : Type u_3
G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
⊢ H.edgeFinset.card - ⊥.copyCount H ≤ (⊥.kill H).edgeFinset.card
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
⊢ H.edgeFinset.card - G.copyCount H ≤ (G.kill H).edgeFinset.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | let f (H' : {H' : H.Subgraph // Nonempty (G ≃g H'.coe)}) := (aux hG H'.2).some | case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
⊢ H.edgeFinset.card - G.copyCount H ≤ (G.kill H).edgeFinset.card | case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ H.edgeFinset.card - G.copyCount H ≤ (G.kill H).edgeFinset.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | calc
_ = H.edgeFinset.card - card {H' : H.Subgraph // Nonempty (G ≃g H'.coe)} := ?_
_ ≤ H.edgeFinset.card - (univ.image f).card := Nat.sub_le_sub_left card_image_le _
_ = H.edgeFinset.card - (Set.range f).toFinset.card := by rw [Set.toFinset_range]
_ ≤ (H.edgeFinset \ (Set.range f).toFinset).card := le_card_sdiff ..
_ = (G.kill H).edgeFinset.card := ?_ | case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ H.edgeFinset.card - G.copyCount H ≤ (G.kill H).edgeFinset.card | case inr.calc_1
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ H.edgeFinset.card - G.copyCount H = H.edgeFinset.card - Fintype.card { H' // Nonempty (G ≃g H'.coe) }
case inr.calc_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ (H.edgeFinset \ (Set.range f).toFinset).card = (G.kill H).edgeFinset.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | simp only [kill_of_ne_bot, hG, Ne, not_false_iff, Set.iUnion_singleton_eq_range,
Set.toFinset_card, Fintype.card_ofFinset, edgeSet_deleteEdges] | case inr.calc_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ (H.edgeFinset \ (Set.range f).toFinset).card = (G.kill H).edgeFinset.card | case inr.calc_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ (H.edgeFinset \ (Set.range f).toFinset).card =
Fintype.card ↑(H.edgeSet \ ⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | simp only [Finset.sdiff_eq_inter_compl, Set.diff_eq, ← Set.iUnion_singleton_eq_range, coe_sdiff,
Set.coe_toFinset, coe_filter, Set.sep_mem_eq, Set.iUnion_subtype, ← Fintype.card_coe,
← Finset.coe_sort_coe, coe_inter, coe_compl, Set.coe_toFinset, Set.compl_iUnion,
Fintype.card_ofFinset] | case inr.calc_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ (H.edgeFinset \ (Set.range f).toFinset).card =
Fintype.card ↑(H.edgeSet \ ⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | simp | case inl
α : Type u_1
β : Type u_2
γ : Type u_3
G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
⊢ H.edgeFinset.card - ⊥.copyCount H ≤ (⊥.kill H).edgeFinset.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | rw [Set.toFinset_range] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ H.edgeFinset.card - (image f univ).card = H.edgeFinset.card - (Set.range f).toFinset.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | simp only [Set.toFinset_card] | case inr.calc_1
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ H.edgeFinset.card - G.copyCount H = H.edgeFinset.card - Fintype.card { H' // Nonempty (G ≃g H'.coe) } | case inr.calc_1
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ Fintype.card ↑H.edgeSet - G.copyCount H = Fintype.card ↑H.edgeSet - Fintype.card { H' // Nonempty (G ≃g H'.coe) } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | rw [← Set.toFinset_card, ← edgeFinset, copyCount, ← card_subtype, subtype_univ, card_univ] | case inr.calc_1
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ Fintype.card ↑H.edgeSet - G.copyCount H = Fintype.card ↑H.edgeSet - Fintype.card { H' // Nonempty (G ≃g H'.coe) } | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab | [37, 1] | [41, 97] | rw [mulStab, mem_filter, mem_div, and_iff_right_of_imp] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ a ∈ s.mulStab ↔ a • s = s | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ a • s = s → ∃ b ∈ s, ∃ c ∈ s, b / c = a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab | [37, 1] | [41, 97] | obtain ⟨b, hb⟩ := hs | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ a • s = s → ∃ b ∈ s, ∃ c ∈ s, b / c = a | case intro
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ s
⊢ a • s = s → ∃ b ∈ s, ∃ c ∈ s, b / c = a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab | [37, 1] | [41, 97] | exact fun h ↦ ⟨_, by rw [← h]; exact smul_mem_smul_finset hb, _, hb, mul_div_cancel_right _ _⟩ | case intro
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ s
⊢ a • s = s → ∃ b ∈ s, ∃ c ∈ s, b / c = a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab | [37, 1] | [41, 97] | rw [← h] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ s
h : a • s = s
⊢ a * b ∈ s | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ s
h : a • s = s
⊢ a * b ∈ a • s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab | [37, 1] | [41, 97] | exact smul_mem_smul_finset hb | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ s
h : a • s = s
⊢ a * b ∈ a • s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_subset_div_right | [46, 1] | [51, 32] | refine fun b hb ↦ mem_div.2 ⟨_, ?_, _, mem_singleton_self _, mul_div_cancel_right _ _⟩ | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
⊢ s.mulStab ⊆ s / {a} | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
b : α
hb : b ∈ s.mulStab
⊢ b * a ∈ s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_subset_div_right | [46, 1] | [51, 32] | rw [mem_mulStab ⟨a, ha⟩] at hb | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
b : α
hb : b ∈ s.mulStab
⊢ b * a ∈ s | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
b : α
hb : b • s = s
⊢ b * a ∈ s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_subset_div_right | [46, 1] | [51, 32] | rw [← hb] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
b : α
hb : b • s = s
⊢ b * a ∈ s | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
b : α
hb : b • s = s
⊢ b * a ∈ b • s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_subset_div_right | [46, 1] | [51, 32] | exact smul_mem_smul_finset ha | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
b : α
hb : b • s = s
⊢ b * a ∈ b • s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.coe_mulStab | [53, 1] | [55, 29] | ext | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ ↑s.mulStab = ↑(stabilizer α s) | case h
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
x✝ : α
⊢ x✝ ∈ ↑s.mulStab ↔ x✝ ∈ ↑(stabilizer α s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.coe_mulStab | [53, 1] | [55, 29] | simp [mem_mulStab hs] | case h
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
x✝ : α
⊢ x✝ ∈ ↑s.mulStab ↔ x✝ ∈ ↑(stabilizer α s) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab_iff_subset_smul_finset | [57, 1] | [59, 96] | rw [← mem_coe, coe_mulStab hs, SetLike.mem_coe, mem_stabilizer_finset_iff_subset_smul_finset] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ a ∈ s.mulStab ↔ s ⊆ a • s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab_iff_smul_finset_subset | [61, 1] | [63, 96] | rw [← mem_coe, coe_mulStab hs, SetLike.mem_coe, mem_stabilizer_finset_iff_smul_finset_subset] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ a ∈ s.mulStab ↔ a • s ⊆ s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab' | [65, 1] | [67, 74] | rw [← mem_coe, coe_mulStab hs, SetLike.mem_coe, mem_stabilizer_finset'] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ a ∈ s.mulStab ↔ ∀ ⦃b : α⦄, b ∈ s → a • b ∈ s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_empty | [69, 1] | [70, 70] | simp [mulStab] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ ∅.mulStab = ∅ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_singleton | [72, 1] | [74, 52] | simp [mulStab, singleton_one, filter_true_of_mem] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a✝ a : α
⊢ {a}.mulStab = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.Nonempty.of_mulStab | [76, 1] | [78, 80] | simp_rw [nonempty_iff_ne_empty, not_imp_not] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ s.mulStab.Nonempty → s.Nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ s = ∅ → s.mulStab = ∅ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.Nonempty.of_mulStab | [76, 1] | [78, 80] | rintro rfl | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ s = ∅ → s.mulStab = ∅ | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
t : Finset α
a : α
⊢ ∅.mulStab = ∅ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.Nonempty.of_mulStab | [76, 1] | [78, 80] | exact mulStab_empty | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
t : Finset α
a : α
⊢ ∅.mulStab = ∅ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_eq_one | [92, 1] | [98, 36] | refine ⟨fun h ↦ ?_, fun h ↦ by rw [h, card_one]⟩ | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ s.mulStab.card = 1 ↔ s.mulStab = 1 | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
h : s.mulStab.card = 1
⊢ s.mulStab = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_eq_one | [92, 1] | [98, 36] | obtain ⟨a, ha⟩ := card_eq_one.1 h | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
h : s.mulStab.card = 1
⊢ s.mulStab = 1 | case intro
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a✝ : α
h : s.mulStab.card = 1
a : α
ha : s.mulStab = {a}
⊢ s.mulStab = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_eq_one | [92, 1] | [98, 36] | rw [ha] | case intro
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a✝ : α
h : s.mulStab.card = 1
a : α
ha : s.mulStab = {a}
⊢ s.mulStab = 1 | case intro
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a✝ : α
h : s.mulStab.card = 1
a : α
ha : s.mulStab = {a}
⊢ {a} = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_eq_one | [92, 1] | [98, 36] | rw [eq_singleton_iff_nonempty_unique_mem, mulStab_nonempty, ← one_mem_mulStab] at ha | case intro
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a✝ : α
h : s.mulStab.card = 1
a : α
ha : s.mulStab = {a}
⊢ {a} = 1 | case intro
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a✝ : α
h : s.mulStab.card = 1
a : α
ha : 1 ∈ s.mulStab ∧ ∀ x ∈ s.mulStab, x = a
⊢ {a} = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_eq_one | [92, 1] | [98, 36] | rw [← ha.2 _ ha.1, singleton_one] | case intro
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a✝ : α
h : s.mulStab.card = 1
a : α
ha : 1 ∈ s.mulStab ∧ ∀ x ∈ s.mulStab, x = a
⊢ {a} = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_eq_one | [92, 1] | [98, 36] | rw [h, card_one] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
h : s.mulStab = 1
⊢ s.mulStab.card = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.subset_mulStab_mul_left | [104, 1] | [110, 27] | obtain rfl | hs := s.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ht : t.Nonempty
⊢ s.mulStab ⊆ (s * t).mulStab | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
t : Finset α
a : α
ht : t.Nonempty
⊢ ∅.mulStab ⊆ (∅ * t).mulStab
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ht : t.Nonempty
hs : s.Nonempty
⊢ s.mulStab ⊆ (s * t).mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.subset_mulStab_mul_left | [104, 1] | [110, 27] | simp_rw [subset_iff, mem_mulStab hs, mem_mulStab (hs.mul ht)] | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ht : t.Nonempty
hs : s.Nonempty
⊢ s.mulStab ⊆ (s * t).mulStab | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ht : t.Nonempty
hs : s.Nonempty
⊢ ∀ ⦃x : α⦄, x • s = s → x • (s * t) = s * t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.subset_mulStab_mul_left | [104, 1] | [110, 27] | rintro a h | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ht : t.Nonempty
hs : s.Nonempty
⊢ ∀ ⦃x : α⦄, x • s = s → x • (s * t) = s * t | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a✝ : α
ht : t.Nonempty
hs : s.Nonempty
a : α
h : a • s = s
⊢ a • (s * t) = s * t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.subset_mulStab_mul_left | [104, 1] | [110, 27] | rw [← smul_mul_assoc, h] | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a✝ : α
ht : t.Nonempty
hs : s.Nonempty
a : α
h : a • s = s
⊢ a • (s * t) = s * t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.subset_mulStab_mul_left | [104, 1] | [110, 27] | simp | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
t : Finset α
a : α
ht : t.Nonempty
⊢ ∅.mulStab ⊆ (∅ * t).mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_mul | [112, 1] | [116, 98] | obtain rfl | hs := s.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ s.mulStab * s = s | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ ∅.mulStab * ∅ = ∅
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ s.mulStab * s = s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_mul | [112, 1] | [116, 98] | exact mul_empty _ | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ ∅.mulStab * ∅ = ∅ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_mul | [112, 1] | [116, 98] | simp only [← coe_inj, hs, coe_mul, coe_mulStab, ← stabilizer_coe_finset, stabilizer_mul_self] | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ s.mulStab * s = s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mul_subset_right_iff | [118, 1] | [121, 57] | simp_rw [← smul_eq_mul, ← biUnion_smul_finset, biUnion_subset,
← mem_mulStab_iff_smul_finset_subset ht, subset_iff] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ht : t.Nonempty
⊢ s * t ⊆ t ↔ s ⊆ t.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mul_subset_right | [123, 1] | [127, 38] | obtain rfl | ht := t.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ s ⊆ t.mulStab → s * t ⊆ t | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s : Finset α
a : α
⊢ s ⊆ ∅.mulStab → s * ∅ ⊆ ∅
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ht : t.Nonempty
⊢ s ⊆ t.mulStab → s * t ⊆ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mul_subset_right | [123, 1] | [127, 38] | simp | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s : Finset α
a : α
⊢ s ⊆ ∅.mulStab → s * ∅ ⊆ ∅ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mul_subset_right | [123, 1] | [127, 38] | exact (mul_subset_right_iff ht).2 | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ht : t.Nonempty
⊢ s ⊆ t.mulStab → s * t ⊆ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.smul_mulStab | [129, 1] | [134, 72] | obtain rfl | hs := s.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s.mulStab
⊢ a • s.mulStab = s.mulStab | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
t : Finset α
a : α
ha : a ∈ ∅.mulStab
⊢ a • ∅.mulStab = ∅.mulStab
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s.mulStab
hs : s.Nonempty
⊢ a • s.mulStab = s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.smul_mulStab | [129, 1] | [134, 72] | rw [← mem_coe, coe_mulStab hs, SetLike.mem_coe] at ha | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s.mulStab
hs : s.Nonempty
⊢ a • s.mulStab = s.mulStab | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ stabilizer α s
hs : s.Nonempty
⊢ a • s.mulStab = s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.smul_mulStab | [129, 1] | [134, 72] | rw [← coe_inj, coe_smul_finset, coe_mulStab hs, Subgroup.smul_coe ha] | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ stabilizer α s
hs : s.Nonempty
⊢ a • s.mulStab = s.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.smul_mulStab | [129, 1] | [134, 72] | simp | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
t : Finset α
a : α
ha : a ∈ ∅.mulStab
⊢ a • ∅.mulStab = ∅.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_mul_mulStab | [136, 1] | [141, 46] | obtain rfl | hs := s.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ s.mulStab * s.mulStab = s.mulStab | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ ∅.mulStab * ∅.mulStab = ∅.mulStab
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ s.mulStab * s.mulStab = s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_mul_mulStab | [136, 1] | [141, 46] | simp | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ ∅.mulStab * ∅.mulStab = ∅.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_mul_mulStab | [136, 1] | [141, 46] | simp_rw [← smul_eq_mul, ← biUnion_smul_finset, biUnion_congr rfl fun _ ↦ smul_mulStab,
← sup_eq_biUnion, sup_const hs.mulStab] | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ s.mulStab * s.mulStab = s.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.inter_mulStab_subset_mulStab_union | [143, 1] | [152, 53] | obtain rfl | hs := s.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ s.mulStab ∩ t.mulStab ⊆ (s ∪ t).mulStab | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
t : Finset α
a : α
⊢ ∅.mulStab ∩ t.mulStab ⊆ (∅ ∪ t).mulStab
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ s.mulStab ∩ t.mulStab ⊆ (s ∪ t).mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.inter_mulStab_subset_mulStab_union | [143, 1] | [152, 53] | obtain rfl | ht := t.eq_empty_or_nonempty | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ s.mulStab ∩ t.mulStab ⊆ (s ∪ t).mulStab | case inr.inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s : Finset α
a : α
hs : s.Nonempty
⊢ s.mulStab ∩ ∅.mulStab ⊆ (s ∪ ∅).mulStab
case inr.inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
ht : t.Nonempty
⊢ s.mulStab ∩ t.mulStab ⊆ (s ∪ t).mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.inter_mulStab_subset_mulStab_union | [143, 1] | [152, 53] | intro x hx | case inr.inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
ht : t.Nonempty
⊢ s.mulStab ∩ t.mulStab ⊆ (s ∪ t).mulStab | case inr.inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
ht : t.Nonempty
x : α
hx : x ∈ s.mulStab ∩ t.mulStab
⊢ x ∈ (s ∪ t).mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.inter_mulStab_subset_mulStab_union | [143, 1] | [152, 53] | rw [mem_mulStab (hs.mono subset_union_left), smul_finset_union,
(mem_mulStab hs).mp (mem_of_mem_inter_left hx),
(mem_mulStab ht).mp (mem_of_mem_inter_right hx)] | case inr.inr
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
ht : t.Nonempty
x : α
hx : x ∈ s.mulStab ∩ t.mulStab
⊢ x ∈ (s ∪ t).mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.inter_mulStab_subset_mulStab_union | [143, 1] | [152, 53] | simp | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
t : Finset α
a : α
⊢ ∅.mulStab ∩ t.mulStab ⊆ (∅ ∪ t).mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.inter_mulStab_subset_mulStab_union | [143, 1] | [152, 53] | simp | case inr.inl
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s : Finset α
a : α
hs : s.Nonempty
⊢ s.mulStab ∩ ∅.mulStab ⊆ (s ∪ ∅).mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_subset_div_left | [158, 1] | [162, 68] | refine fun b hb ↦ mem_div.2 ⟨_, mem_singleton_self _, _, ?_, div_div_cancel _ _⟩ | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
⊢ s.mulStab ⊆ {a} / s | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
b : α
hb : b ∈ s.mulStab
⊢ a / b ∈ s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_subset_div_left | [158, 1] | [162, 68] | rw [mem_mulStab ⟨a, ha⟩] at hb | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
b : α
hb : b ∈ s.mulStab
⊢ a / b ∈ s | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
b : α
hb : b • s = s
⊢ a / b ∈ s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_subset_div_left | [158, 1] | [162, 68] | rwa [← hb, ← inv_smul_mem_iff, smul_eq_mul, inv_mul_eq_div] at ha | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
ha : a ∈ s
b : α
hb : b • s = s
⊢ a / b ∈ s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.subset_mulStab_mul_right | [164, 1] | [166, 50] | rw [mul_comm] | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ t.mulStab ⊆ (s * t).mulStab | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ t.mulStab ⊆ (t * s).mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.subset_mulStab_mul_right | [164, 1] | [166, 50] | exact subset_mulStab_mul_left hs | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ t.mulStab ⊆ (t * s).mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mul_mulStab | [168, 1] | [169, 94] | rw [mul_comm] | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ s * s.mulStab = s | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ s.mulStab * s = s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mul_mulStab | [168, 1] | [169, 94] | exact mulStab_mul _ | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ s.mulStab * s = s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mul_mulStab_mul_mul_mul_mulStab_mul | [171, 1] | [174, 58] | rw [mul_mul_mul_comm, mulStab_mul_mulStab, mul_mulStab] | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ s * (s * t).mulStab * (t * (s * t).mulStab) = s * t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mul_subset_left_iff | [180, 1] | [182, 41] | rw [mul_comm, mul_subset_right_iff hs] | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ s * t ⊆ s ↔ t ⊆ s.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mul_subset_left | [184, 1] | [185, 94] | rw [mul_comm] | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ t ⊆ s.mulStab → s * t ⊆ s | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ t ⊆ s.mulStab → t * s ⊆ s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mul_subset_left | [184, 1] | [185, 94] | exact mul_subset_right | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ t ⊆ s.mulStab → t * s ⊆ s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_idem | [187, 1] | [192, 7] | obtain rfl | hs := s.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ s.mulStab.mulStab = s.mulStab | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ ∅.mulStab.mulStab = ∅.mulStab
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ s.mulStab.mulStab = s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_idem | [187, 1] | [192, 7] | rw [← coe_inj, coe_mulStab hs, coe_mulStab hs.mulStab, ← stabilizer_coe_finset, coe_mulStab hs] | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ s.mulStab.mulStab = s.mulStab | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ ↑(stabilizer α ↑(stabilizer α s)) = ↑(stabilizer α s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_idem | [187, 1] | [192, 7] | simp | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ ↑(stabilizer α ↑(stabilizer α s)) = ↑(stabilizer α s) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_idem | [187, 1] | [192, 7] | simp | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ ∅.mulStab.mulStab = ∅.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_smul | [194, 1] | [198, 89] | obtain rfl | hs := s.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a✝ a : α
s : Finset α
⊢ (a • s).mulStab = s.mulStab | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a✝ a : α
⊢ (a • ∅).mulStab = ∅.mulStab
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a✝ a : α
s : Finset α
hs : s.Nonempty
⊢ (a • s).mulStab = s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_smul | [194, 1] | [198, 89] | simp | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a✝ a : α
⊢ (a • ∅).mulStab = ∅.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_smul | [194, 1] | [198, 89] | rw [← coe_inj, coe_mulStab hs, coe_mulStab hs.smul_finset, stabilizer_smul_eq_right] | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a✝ a : α
s : Finset α
hs : s.Nonempty
⊢ (a • s).mulStab = s.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mulStab_image_coe_quotient | [202, 1] | [206, 92] | rw [← coe_inj, coe_mulStab (hs.image _), ← stabilizer_coe_finset, ← stabilizer_coe_finset,
coe_image, coe_one, stabilizer_image_coe_quotient, Subgroup.coe_bot, Set.singleton_one] | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ (image QuotientGroup.mk s).mulStab = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.preimage_image_quotientMk_stabilizer_eq_mul_mulStab | [208, 1] | [213, 47] | rw [QuotientGroup.preimage_image_mk_eq_iUnion_smul, mul_comm, Set.iUnion_subtype] | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
ht : t.Nonempty
s : Finset α
⊢ QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s) = ↑s * ↑t.mulStab | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
ht : t.Nonempty
s : Finset α
⊢ ⋃ x, ⋃ (hx : x ∈ stabilizer α t), ⟨x, hx⟩ • ↑s = ↑t.mulStab * ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.preimage_image_quotientMk_stabilizer_eq_mul_mulStab | [208, 1] | [213, 47] | simp_rw [Subgroup.mk_smul, ← SetLike.mem_coe, ← coe_mulStab ht, ← coe_smul_finset, ← coe_biUnion,
biUnion_smul_finset, smul_eq_mul, coe_mul] | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
ht : t.Nonempty
s : Finset α
⊢ ⋃ x, ⋃ (hx : x ∈ stabilizer α t), ⟨x, hx⟩ • ↑s = ↑t.mulStab * ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.preimage_image_quotientMk_mulStabilizer | [215, 1] | [220, 90] | obtain rfl | hs := s.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s) = ↑s | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑∅) = ↑∅
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s) = ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.preimage_image_quotientMk_mulStabilizer | [215, 1] | [220, 90] | simp | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑∅) = ↑∅ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.preimage_image_quotientMk_mulStabilizer | [215, 1] | [220, 90] | rw [preimage_image_quotientMk_stabilizer_eq_mul_mulStab hs s, ← coe_mul, mul_mulStab] | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s) = ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.pairwiseDisjoint_smul_finset_mulStab | [222, 1] | [230, 87] | obtain rfl | hs := s.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ (Set.range fun a => a • s.mulStab).PairwiseDisjoint id | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ (Set.range fun a => a • ∅.mulStab).PairwiseDisjoint id
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ (Set.range fun a => a • s.mulStab).PairwiseDisjoint id |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.pairwiseDisjoint_smul_finset_mulStab | [222, 1] | [230, 87] | rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
hs : s.Nonempty
⊢ (Set.range fun a => a • s.mulStab).PairwiseDisjoint id | case inr.intro.intro
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a✝ : α
s : Finset α
hs : s.Nonempty
a b : α
⊢ (fun a => a • s.mulStab) a ≠ (fun a => a • s.mulStab) b →
(Disjoint on id) ((fun a => a • s.mulStab) a) ((fun a => a • s.mulStab) b) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.pairwiseDisjoint_smul_finset_mulStab | [222, 1] | [230, 87] | simp only [onFun, id_eq] | case inr.intro.intro
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a✝ : α
s : Finset α
hs : s.Nonempty
a b : α
⊢ (fun a => a • s.mulStab) a ≠ (fun a => a • s.mulStab) b →
(Disjoint on id) ((fun a => a • s.mulStab) a) ((fun a => a • s.mulStab) b) | case inr.intro.intro
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a✝ : α
s : Finset α
hs : s.Nonempty
a b : α
⊢ a • s.mulStab ≠ b • s.mulStab → Disjoint (a • s.mulStab) (b • s.mulStab) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.pairwiseDisjoint_smul_finset_mulStab | [222, 1] | [230, 87] | simp_rw [← disjoint_coe, ← coe_injective.ne_iff, coe_smul_finset, coe_mulStab hs] | case inr.intro.intro
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a✝ : α
s : Finset α
hs : s.Nonempty
a b : α
⊢ a • s.mulStab ≠ b • s.mulStab → Disjoint (a • s.mulStab) (b • s.mulStab) | case inr.intro.intro
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a✝ : α
s : Finset α
hs : s.Nonempty
a b : α
⊢ a • ↑(stabilizer α s) ≠ b • ↑(stabilizer α s) → Disjoint (a • ↑(stabilizer α s)) (b • ↑(stabilizer α s)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.pairwiseDisjoint_smul_finset_mulStab | [222, 1] | [230, 87] | exact Subgroup.pairwiseDisjoint_smul _ (Set.mem_range_self _) (Set.mem_range_self _) | case inr.intro.intro
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a✝ : α
s : Finset α
hs : s.Nonempty
a b : α
⊢ a • ↑(stabilizer α s) ≠ b • ↑(stabilizer α s) → Disjoint (a • ↑(stabilizer α s)) (b • ↑(stabilizer α s)) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.pairwiseDisjoint_smul_finset_mulStab | [222, 1] | [230, 87] | simp | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ (Set.range fun a => a • ∅.mulStab).PairwiseDisjoint id | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.disjoint_smul_finset_mulStab_mul_mulStab | [232, 1] | [239, 49] | simp_rw [@not_imp_comm (_ ⊆ _), ← smul_eq_mul, ← biUnion_smul_finset, disjoint_biUnion_right,
Classical.not_forall] | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ ¬a • s.mulStab ⊆ t * s.mulStab → Disjoint (a • s.mulStab) (t * s.mulStab) | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ (∃ x, ∃ (_ : x ∈ t), ¬Disjoint (a • s.mulStab) (x • s.mulStab)) → a • s.mulStab ⊆ t.biUnion fun x => x • s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.disjoint_smul_finset_mulStab_mul_mulStab | [232, 1] | [239, 49] | rintro ⟨b, hb, h⟩ | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ (∃ x, ∃ (_ : x ∈ t), ¬Disjoint (a • s.mulStab) (x • s.mulStab)) → a • s.mulStab ⊆ t.biUnion fun x => x • s.mulStab | case intro.intro
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ t
h : ¬Disjoint (a • s.mulStab) (b • s.mulStab)
⊢ a • s.mulStab ⊆ t.biUnion fun x => x • s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.disjoint_smul_finset_mulStab_mul_mulStab | [232, 1] | [239, 49] | rw [s.pairwiseDisjoint_smul_finset_mulStab.eq (Set.mem_range_self _) (Set.mem_range_self _) h] | case intro.intro
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ t
h : ¬Disjoint (a • s.mulStab) (b • s.mulStab)
⊢ a • s.mulStab ⊆ t.biUnion fun x => x • s.mulStab | case intro.intro
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ t
h : ¬Disjoint (a • s.mulStab) (b • s.mulStab)
⊢ b • s.mulStab ⊆ t.biUnion fun x => x • s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_dvd_card | [246, 1] | [248, 71] | simpa only [mul_mulStab] using s.card_mulStab_dvd_card_mul_mulStab s | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ s.mulStab.card ∣ s.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_le_card | [250, 1] | [254, 60] | obtain rfl | hs := s.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ s.mulStab.card ≤ s.card | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
t : Finset α
a : α
⊢ ∅.mulStab.card ≤ ∅.card
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ s.mulStab.card ≤ s.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_le_card | [250, 1] | [254, 60] | rfl | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
t : Finset α
a : α
⊢ ∅.mulStab.card ≤ ∅.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_le_card | [250, 1] | [254, 60] | exact Nat.le_of_dvd hs.card_pos s.card_mulStab_dvd_card | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ s.mulStab.card ≤ s.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_dvd_card_mulStab | [265, 1] | [281, 7] | obtain rfl | ht := t.eq_empty_or_nonempty | ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
h : s.mulStab ⊆ t.mulStab
⊢ s.mulStab.card ∣ t.mulStab.card | case inl
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
a : α
hs : s.Nonempty
h : s.mulStab ⊆ ∅.mulStab
⊢ s.mulStab.card ∣ ∅.mulStab.card
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
h : s.mulStab ⊆ t.mulStab
ht : t.Nonempty
⊢ s.mulStab.card ∣ t.mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_dvd_card_mulStab | [265, 1] | [281, 7] | rw [← coe_subset, coe_mulStab hs, coe_mulStab ht, SetLike.coe_subset_coe] at h | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
h : s.mulStab ⊆ t.mulStab
ht : t.Nonempty
⊢ s.mulStab.card ∣ t.mulStab.card | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
h : stabilizer α s ≤ stabilizer α t
ht : t.Nonempty
⊢ s.mulStab.card ∣ t.mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.card_mulStab_dvd_card_mulStab | [265, 1] | [281, 7] | letI : Fintype (stabilizer α s) := fintypeStabilizerOfMulStab hs | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
h : stabilizer α s ≤ stabilizer α t
ht : t.Nonempty
⊢ s.mulStab.card ∣ t.mulStab.card | case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
h : stabilizer α s ≤ stabilizer α t
ht : t.Nonempty
this : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs
⊢ s.mulStab.card ∣ t.mulStab.card |
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