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https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | apply Set.mul_mem_mul hxy (by simpa) | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x y : α
hys : y ∈ s
hxy : x * y ∈ t
⊢ x * y * y⁻¹ ∈ t * ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | simpa | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x y : α
hys : y ∈ s
hxy : x * y ∈ t
⊢ y⁻¹ ∈ ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.pairwiseDisjoint_smul | [53, 1] | [60, 80] | rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ hab | α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
⊢ (range fun a => a • ↑s).PairwiseDisjoint id | case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : (fun a => a • ↑s) a ≠ (fun a => a • ↑s) b
⊢ (Disjoint on id) ((fun a => a • ↑s) a) ((fun a => a • ↑s) b) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.pairwiseDisjoint_smul | [53, 1] | [60, 80] | simp only [Function.onFun, id_eq, disjoint_left] at hab ⊢ | case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : (fun a => a • ↑s) a ≠ (fun a => a • ↑s) b
⊢ (Disjoint on id) ((fun a => a • ↑s) a) ((fun a => a • ↑s) b) | case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
⊢ ∀ ⦃a_1 : β⦄, a_1 ∈ a • ↑s → a_1 ∉ b • ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.pairwiseDisjoint_smul | [53, 1] | [60, 80] | rintro _ ⟨c, hc, rfl⟩ ⟨d, hd, (hcd : b • d = a • c)⟩ | case intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
⊢ ∀ ⦃a_1 : β⦄, a_1 ∈ a • ↑s → a_1 ∉ b • ↑s | case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
c : β
hc : c ∈ ↑s
d : β
hd : d ∈ ↑s
hcd : b • d = a • c
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.pairwiseDisjoint_smul | [53, 1] | [60, 80] | refine' hab _ | case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
c : β
hc : c ∈ ↑s
d : β
hd : d ∈ ↑s
hcd : b • d = a • c
⊢ False | case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
c : β
hc : c ∈ ↑s
d : β
hd : d ∈ ↑s
hcd : b • d = a • c
⊢ a • ↑s = b • ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.pairwiseDisjoint_smul | [53, 1] | [60, 80] | rw [← smul_coe hc, ← smul_assoc, ← hcd, smul_assoc, smul_coe hc, smul_coe hd] | case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝³ : Group α
inst✝² : Group β
inst✝¹ : MulAction α β
inst✝ : IsScalarTower α β β
s : Subgroup β
a b : α
hab : a • ↑s ≠ b • ↑s
c : β
hc : c ∈ ↑s
d : β
hd : d ∈ ↑s
hcd : b • d = a • c
⊢ a • ↑s = b • ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_empty | [29, 1] | [30, 86] | simp [mulImpact] | α : Type u_1
β : Type u_2
inst✝² : DecidableEq α
inst✝¹ : DecidableEq β
inst✝ : Mul α
n✝ n : ℕ
⊢ ∅.mulImpact n = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_singleton | [37, 1] | [41, 55] | simp only [mulImpact, singleton_mul', card_smul_finset] | α : Type u_1
β : Type u_2
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : Group α
n✝ : ℕ
inst✝ : Infinite α
a : α
n : ℕ
⊢ {a}.mulImpact n = n | α : Type u_1
β : Type u_2
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : Group α
n✝ : ℕ
inst✝ : Infinite α
a : α
n : ℕ
⊢ ⨅ t, (↑t).card = n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_singleton | [37, 1] | [41, 55] | haveI : Nonempty {t : Finset α // t.card = n} := nonempty_subtype.2 (exists_card_eq _) | α : Type u_1
β : Type u_2
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : Group α
n✝ : ℕ
inst✝ : Infinite α
a : α
n : ℕ
⊢ ⨅ t, (↑t).card = n | α : Type u_1
β : Type u_2
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : Group α
n✝ : ℕ
inst✝ : Infinite α
a : α
n : ℕ
this : Nonempty { t // t.card = n }
⊢ ⨅ t, (↑t).card = n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_singleton | [37, 1] | [41, 55] | exact Eq.trans (iInf_congr Subtype.prop) ciInf_const | α : Type u_1
β : Type u_2
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : Group α
n✝ : ℕ
inst✝ : Infinite α
a : α
n : ℕ
this : Nonempty { t // t.card = n }
⊢ ⨅ t, (↑t).card = n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_map_of_infinite | [56, 1] | [66, 10] | haveI : Infinite β := sorry | α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n = s.mulImpact n | α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this : Infinite β
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n = s.mulImpact n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_map_of_infinite | [56, 1] | [66, 10] | haveI : Nonempty {t : Finset α // t.card = n} := nonempty_subtype.2 (exists_card_eq _) | α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this : Infinite β
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n = s.mulImpact n | α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝ : Infinite β
this : Nonempty { t // t.card = n }
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n = s.mulImpact n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_map_of_infinite | [56, 1] | [66, 10] | haveI : Nonempty {t : Finset β // t.card = n} := nonempty_subtype.2 (exists_card_eq _) | α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝ : Infinite β
this : Nonempty { t // t.card = n }
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n = s.mulImpact n | α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝¹ : Infinite β
this✝ : Nonempty { t // t.card = n }
this : Nonempty { t // t.card = n }
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n = s.mulImpact n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_map_of_infinite | [56, 1] | [66, 10] | refine' le_antisymm _ _ | α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝¹ : Infinite β
this✝ : Nonempty { t // t.card = n }
this : Nonempty { t // t.card = n }
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n = s.mulImpact n | case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝¹ : Infinite β
this✝ : Nonempty { t // t.card = n }
this : Nonempty { t // t.card = n }
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n ≤ s.mulImpact n
case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝¹ : Infinite β
this✝ : Nonempty { t // t.card = n }
this : Nonempty { t // t.card = n }
⊢ s.mulImpact n ≤ (map { toFun := ⇑f, inj' := hf } s).mulImpact n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_map_of_infinite | [56, 1] | [66, 10] | refine' le_ciInf fun t => _ | case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝¹ : Infinite β
this✝ : Nonempty { t // t.card = n }
this : Nonempty { t // t.card = n }
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n ≤ s.mulImpact n | case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝¹ : Infinite β
this✝ : Nonempty { t // t.card = n }
this : Nonempty { t // t.card = n }
t : { t // t.card = n }
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n ≤ (s * ↑t).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_map_of_infinite | [56, 1] | [66, 10] | sorry | case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝¹ : Infinite β
this✝ : Nonempty { t // t.card = n }
this : Nonempty { t // t.card = n }
t : { t // t.card = n }
⊢ (map { toFun := ⇑f, inj' := hf } s).mulImpact n ≤ (s * ↑t).card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_map_of_infinite | [56, 1] | [66, 10] | refine' le_ciInf fun t => _ | case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝¹ : Infinite β
this✝ : Nonempty { t // t.card = n }
this : Nonempty { t // t.card = n }
⊢ s.mulImpact n ≤ (map { toFun := ⇑f, inj' := hf } s).mulImpact n | case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝¹ : Infinite β
this✝ : Nonempty { t // t.card = n }
this : Nonempty { t // t.card = n }
t : { t // t.card = n }
⊢ s.mulImpact n ≤ (map { toFun := ⇑f, inj' := hf } s * ↑t).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Impact.lean | Finset.mulImpact_map_of_infinite | [56, 1] | [66, 10] | sorry | case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : CommGroup α
inst✝¹ : CommGroup β
n : ℕ
inst✝ : Infinite α
s : Finset α
f : α →* β
hf : Injective ⇑f
this✝¹ : Infinite β
this✝ : Nonempty { t // t.card = n }
this : Nonempty { t // t.card = n }
t : { t // t.card = n }
⊢ s.mulImpact n ≤ (map { toFun := ⇑f, inj' := hf } s * ↑t).card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Archive/CauchyDavenportFromKneser.lean | ZMod.min_le_card_add' | [27, 1] | [37, 24] | haveI : Fact p.Prime := ⟨hp⟩ | p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
⊢ min p (s.card + t.card - 1) ≤ (s + t).card | p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
⊢ min p (s.card + t.card - 1) ≤ (s + t).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Archive/CauchyDavenportFromKneser.lean | ZMod.min_le_card_add' | [27, 1] | [37, 24] | obtain h | h := eq_bot_or_eq_top (AddAction.stabilizer (ZMod p) (s + t)) | p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
⊢ min p (s.card + t.card - 1) ≤ (s + t).card | case inl
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : AddAction.stabilizer (ZMod p) (s + t) = ⊥
⊢ min p (s.card + t.card - 1) ≤ (s + t).card
case inr
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : AddAction.stabilizer (ZMod p) (s + t) = ⊤
⊢ min p (s.card + t.card - 1) ≤ (s + t).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Archive/CauchyDavenportFromKneser.lean | ZMod.min_le_card_add' | [27, 1] | [37, 24] | refine' min_le_of_right_le _ | case inl
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : AddAction.stabilizer (ZMod p) (s + t) = ⊥
⊢ min p (s.card + t.card - 1) ≤ (s + t).card | case inl
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : AddAction.stabilizer (ZMod p) (s + t) = ⊥
⊢ s.card + t.card - 1 ≤ (s + t).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Archive/CauchyDavenportFromKneser.lean | ZMod.min_le_card_add' | [27, 1] | [37, 24] | rw [← AddSubgroup.coe_eq_zero, ← coe_addStab (hs.add ht), coe_eq_zero] at h | case inl
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : AddAction.stabilizer (ZMod p) (s + t) = ⊥
⊢ s.card + t.card - 1 ≤ (s + t).card | case inl
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : (s + t).addStab = 0
⊢ s.card + t.card - 1 ≤ (s + t).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Archive/CauchyDavenportFromKneser.lean | ZMod.min_le_card_add' | [27, 1] | [37, 24] | simpa [*] using add_kneser s t | case inl
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : (s + t).addStab = 0
⊢ s.card + t.card - 1 ≤ (s + t).card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Archive/CauchyDavenportFromKneser.lean | ZMod.min_le_card_add' | [27, 1] | [37, 24] | rw [← AddSubgroup.coe_eq_univ, ← coe_addStab (hs.add ht), coe_eq_univ] at h | case inr
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : AddAction.stabilizer (ZMod p) (s + t) = ⊤
⊢ min p (s.card + t.card - 1) ≤ (s + t).card | case inr
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : (s + t).addStab = univ
⊢ min p (s.card + t.card - 1) ≤ (s + t).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Archive/CauchyDavenportFromKneser.lean | ZMod.min_le_card_add' | [27, 1] | [37, 24] | refine' card_addStab_le_card.trans' _ | case inr
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : (s + t).addStab = univ
⊢ min p (s.card + t.card - 1) ≤ (s + t).card | case inr
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : (s + t).addStab = univ
⊢ min p (s.card + t.card - 1) ≤ (s + t).addStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Archive/CauchyDavenportFromKneser.lean | ZMod.min_le_card_add' | [27, 1] | [37, 24] | simp [*, card_univ] | case inr
p : ℕ
hp : p.Prime
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
this : Fact p.Prime
h : (s + t).addStab = univ
⊢ min p (s.card + t.card - 1) ≤ (s + t).addStab.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Nat/Defs.lean | Nat.eq_of_dvd_of_lt_two_mul | [6, 1] | [11, 67] | obtain ⟨_ | _ | c, rfl⟩ := hdvd | a b : ℕ
ha : a ≠ 0
hdvd : b ∣ a
hlt : a < 2 * b
⊢ a = b | case intro.zero
b : ℕ
ha : b * 0 ≠ 0
hlt : b * 0 < 2 * b
⊢ b * 0 = b
case intro.succ.zero
b : ℕ
ha : b * (0 + 1) ≠ 0
hlt : b * (0 + 1) < 2 * b
⊢ b * (0 + 1) = b
case intro.succ.succ
b c : ℕ
ha : b * (c + 1 + 1) ≠ 0
hlt : b * (c + 1 + 1) < 2 * b
⊢ b * (c + 1 + 1) = b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Nat/Defs.lean | Nat.eq_of_dvd_of_lt_two_mul | [6, 1] | [11, 67] | simp at ha | case intro.zero
b : ℕ
ha : b * 0 ≠ 0
hlt : b * 0 < 2 * b
⊢ b * 0 = b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Nat/Defs.lean | Nat.eq_of_dvd_of_lt_two_mul | [6, 1] | [11, 67] | exact Nat.mul_one _ | case intro.succ.zero
b : ℕ
ha : b * (0 + 1) ≠ 0
hlt : b * (0 + 1) < 2 * b
⊢ b * (0 + 1) = b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Nat/Defs.lean | Nat.eq_of_dvd_of_lt_two_mul | [6, 1] | [11, 67] | rw [Nat.mul_comm] at hlt | case intro.succ.succ
b c : ℕ
ha : b * (c + 1 + 1) ≠ 0
hlt : b * (c + 1 + 1) < 2 * b
⊢ b * (c + 1 + 1) = b | case intro.succ.succ
b c : ℕ
ha : b * (c + 1 + 1) ≠ 0
hlt : (c + 1 + 1) * b < 2 * b
⊢ b * (c + 1 + 1) = b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Nat/Defs.lean | Nat.eq_of_dvd_of_lt_two_mul | [6, 1] | [11, 67] | cases Nat.not_le_of_lt hlt (Nat.mul_le_mul_right _ (by omega)) | case intro.succ.succ
b c : ℕ
ha : b * (c + 1 + 1) ≠ 0
hlt : (c + 1 + 1) * b < 2 * b
⊢ b * (c + 1 + 1) = b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Nat/Defs.lean | Nat.eq_of_dvd_of_lt_two_mul | [6, 1] | [11, 67] | omega | b c : ℕ
ha : b * (c + 1 + 1) ≠ 0
hlt : (c + 1 + 1) * b < 2 * b
⊢ 2 ≤ c + 1 + 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | refine' ⟨fun h => ⟨h.1, fun x y hx hy z hz =>
⟨hAconv.openSegment_subset hx hy.1 hz, fun hzB => hy.2 (h.2 hx hy.1 hzB hz).2⟩⟩,
fun h => ⟨h.1, fun x hx y hy z hzB hz => ⟨_, _⟩⟩⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ IsExtreme 𝕜 s t ↔ t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
⊢ x ∈ t
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
⊢ y ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | by_contra hxB | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
⊢ x ∈ t | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
hxB : x ∉ t
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | rw [openSegment_symm] at hz | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
hxB : x ∉ t
⊢ False | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 y x
hxB : x ∉ t
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | exact (h.2 hy ⟨hx, hxB⟩ hz).2 hzB | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 y x
hxB : x ∉ t
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | by_contra hyB | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
⊢ y ∈ t | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
hyB : y ∉ t
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.isExtreme_iff_openSegment_subset_diff | [22, 1] | [31, 38] | exact (h.2 hx ⟨hy, hyB⟩ hz).2 hzB | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hAconv : Convex 𝕜 s
h : t ⊆ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s \ t → openSegment 𝕜 x y ⊆ s \ t
x : E
hx : x ∈ s
y : E
hy : y ∈ s
z : E
hzB : z ∈ t
hz : z ∈ openSegment 𝕜 x y
hyB : y ∉ t
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | refine' ⟨fun h => _, fun hs => _⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s ↔ ConvexIndependent 𝕜 fun p => ↑p | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
h : extremePoints 𝕜 ((convexHull 𝕜) s) = s
⊢ ConvexIndependent 𝕜 fun p => ↑p
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hs : ConvexIndependent 𝕜 fun p => ↑p
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | rw [convexIndependent_set_iff_not_mem_convexHull_diff] at hs | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hs : ConvexIndependent 𝕜 fun p => ↑p
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | refine' extremePoints_convexHull_subset.antisymm fun x hxs => ⟨subset_convexHull 𝕜 _ hxs, _⟩ | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
⊢ extremePoints 𝕜 ((convexHull 𝕜) s) = s | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
⊢ ∀ ⦃x₁ : E⦄, x₁ ∈ (convexHull 𝕜) s → ∀ ⦃x₂ : E⦄, x₂ ∈ (convexHull 𝕜) s → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | by_contra! h | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
⊢ ∀ ⦃x₁ : E⦄, x₁ ∈ (convexHull 𝕜) s → ∀ ⦃x₂ : E⦄, x₂ ∈ (convexHull 𝕜) s → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
h : ∃ x₁ ∈ (convexHull 𝕜) s, ∃ x₂ ∈ (convexHull 𝕜) s, x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | obtain ⟨x₁, hx₁, x₂, hx₂, hx⟩ := h | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
h : ∃ x₁ ∈ (convexHull 𝕜) s, ∃ x₂ ∈ (convexHull 𝕜) s, x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ False | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | suffices h : x₁ ∈ convexHull 𝕜 (s \ {x}) ∧ x₂ ∈ convexHull 𝕜 (s \ {x}) by
exact hs _ hxs (convex_iff_openSegment_subset.1 (convex_convexHull 𝕜 _) h.1 h.2 hx.1) | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ False | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x}) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | have hx₁₂ : segment 𝕜 x₁ x₂ ⊆ convexHull 𝕜 s := (convex_convexHull 𝕜 _).segment_subset hx₁ hx₂ | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
⊢ x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x}) | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
hx₁₂ : segment 𝕜 x₁ x₂ ⊆ (convexHull 𝕜) s
⊢ x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x}) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | sorry | case refine'_2.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
hx₁₂ : segment 𝕜 x₁ x₂ ⊆ (convexHull 𝕜) s
⊢ x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x}) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | exact (convex_convexHull 𝕜 _).convexIndependent_extremePoints | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x : E
h : extremePoints 𝕜 ((convexHull 𝕜) s) = s
⊢ ConvexIndependent 𝕜 fun p => ↑p | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | extremePoints_convexHull_eq_iff_convexIndependent | [33, 1] | [48, 8] | exact hs _ hxs (convex_iff_openSegment_subset.1 (convex_convexHull 𝕜 _) h.1 h.2 hx.1) | 𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
x✝ : E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
x : E
hxs : x ∈ s
x₁ : E
hx₁ : x₁ ∈ (convexHull 𝕜) s
x₂ : E
hx₂ : x₂ ∈ (convexHull 𝕜) s
hx : x ∈ openSegment 𝕜 x₁ x₂ ∧ (x₁ = x → x₂ ≠ x)
h : x₁ ∈ (convexHull 𝕜) (s \ {x}) ∧ x₂ ∈ (convexHull 𝕜) (s \ {x})
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | inter_frontier_self_inter_convexHull_extreme | [69, 1] | [73, 8] | refine' ⟨inter_subset_left, fun x₁ hx₁A x₂ hx₂A x hxs hx => ⟨⟨hx₁A, _⟩, hx₂A, _⟩⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
⊢ IsExtreme 𝕜 (closure s) (closure s ∩ frontier ((convexHull 𝕜) s)) | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₁ ∈ frontier ((convexHull 𝕜) s)
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₂ ∈ frontier ((convexHull 𝕜) s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | inter_frontier_self_inter_convexHull_extreme | [69, 1] | [73, 8] | sorry | case refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₁ ∈ frontier ((convexHull 𝕜) s)
case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₂ ∈ frontier ((convexHull 𝕜) s) | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₂ ∈ frontier ((convexHull 𝕜) s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | inter_frontier_self_inter_convexHull_extreme | [69, 1] | [73, 8] | sorry | case refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ x₁ : E
hx₁A : x₁ ∈ closure s
x₂ : E
hx₂A : x₂ ∈ closure s
x : E
hxs : x ∈ closure s ∩ frontier ((convexHull 𝕜) s)
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₂ ∈ frontier ((convexHull 𝕜) s) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | frontier_extreme | [76, 1] | [81, 87] | convert
(inter_frontier_self_inter_convexHull_extreme :
IsExtreme 𝕜 (closure s) (closure s ∩ frontier (convexHull 𝕜 s))) using 1 | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ IsExtreme 𝕜 s (frontier s) | case h.e'_6
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ s = closure s
case h.e'_7
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ frontier s = closure s ∩ frontier ((convexHull 𝕜) s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | frontier_extreme | [76, 1] | [81, 87] | rw [Convex.convexHull_eq hA₁, inter_eq_self_of_subset_right frontier_subset_closure] | case h.e'_7
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ frontier s = closure s ∩ frontier ((convexHull 𝕜) s) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | frontier_extreme | [76, 1] | [81, 87] | exact (IsClosed.closure_eq hA₂).symm | case h.e'_6
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hA₁ : Convex 𝕜 s
hA₂ : IsClosed s
⊢ s = closure s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.frontier_extreme_to_closure | [84, 1] | [87, 8] | use frontier_subset_closure | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ IsExtreme 𝕜 (closure s) (frontier s) | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ ∀ ⦃x₁ : E⦄,
x₁ ∈ closure s →
∀ ⦃x₂ : E⦄,
x₂ ∈ closure s → ∀ ⦃x : E⦄, x ∈ frontier s → x ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ frontier s ∧ x₂ ∈ frontier s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | Convex.frontier_extreme_to_closure | [84, 1] | [87, 8] | sorry | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAconv : Convex 𝕜 s
⊢ ∀ ⦃x₁ : E⦄,
x₁ ∈ closure s →
∀ ⦃x₂ : E⦄,
x₂ ∈ closure s → ∀ ⦃x : E⦄, x ∈ frontier s → x ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ frontier s ∧ x₂ ∈ frontier s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | rintro x hxB | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
⊢ t ⊆ frontier s | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
⊢ x ∈ frontier s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | obtain ⟨y, hyA, hyB⟩ := nonempty_of_ssubset ⟨hAB.1, hBA⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
⊢ x ∈ frontier s | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ frontier s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | rw [frontier_eq_closure_inter_closure] | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ frontier s | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ closure s ∩ closure sᶜ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | use subset_closure (hAB.1 hxB) | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ closure s ∩ closure sᶜ | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ closure sᶜ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | rw [mem_closure_iff_seq_limit] | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ x ∈ closure sᶜ | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ sᶜ) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | let z : ℕ → E := fun n => (1 + 1 / n.succ : 𝕜) • x - (1 / n.succ : 𝕜) • y | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ sᶜ) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
z : ℕ → E := fun n => (1 + 1 / ↑n.succ) • x - (1 / ↑n.succ) • y
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ sᶜ) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | use z | case right
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
z : ℕ → E := fun n => (1 + 1 / ↑n.succ) • x - (1 / ↑n.succ) • y
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ sᶜ) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | case h
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
z : ℕ → E := fun n => (1 + 1 / ↑n.succ) • x - (1 / ↑n.succ) • y
⊢ (∀ (n : ℕ), z n ∈ sᶜ) ∧ Filter.Tendsto z Filter.atTop (nhds x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.subset_frontier | [90, 1] | [135, 8] | sorry | case h
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s t : Set E
x✝ : E
hAB : IsExtreme 𝕜 s t
hBA : ¬s ⊆ t
x : E
hxB : x ∈ t
y : E
hyA : y ∈ s
hyB : y ∉ t
z : ℕ → E := fun n => (1 + 1 / ↑n.succ) • x - (1 / ↑n.succ) • y
⊢ (∀ (n : ℕ), z n ∈ sᶜ) ∧ Filter.Tendsto z Filter.atTop (nhds x) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | refine' Subset.antisymm (fun x hx => _) (closure_mono interior_subset) | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x : E
s : Set E
hAconv : Convex 𝕜 s
hAnemp : (interior s).Nonempty
⊢ closure s = closure (interior s) | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
hAnemp : (interior s).Nonempty
x : E
hx : x ∈ closure s
⊢ x ∈ closure (interior s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | obtain ⟨y, hy⟩ := hAnemp | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
hAnemp : (interior s).Nonempty
x : E
hx : x ∈ closure s
⊢ x ∈ closure (interior s) | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x : E
hx : x ∈ closure s
y : E
hy : y ∈ interior s
⊢ x ∈ closure (interior s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | rw [mem_closure_iff_seq_limit] at hx ⊢ | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x : E
hx : x ∈ closure s
y : E
hy : y ∈ interior s
⊢ x ∈ closure (interior s) | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x : E
hx : ∃ x_1, (∀ (n : ℕ), x_1 n ∈ s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x)
y : E
hy : y ∈ interior s
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ interior s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | obtain ⟨z, hzA, hzx⟩ := hx | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x : E
hx : ∃ x_1, (∀ (n : ℕ), x_1 n ∈ s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x)
y : E
hy : y ∈ interior s
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ interior s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ interior s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | refine' ⟨fun n => (1 - 1 / (n + 2) : 𝕜) • z n + (1 / (n + 2) : 𝕜) • y, fun n => _, _⟩ | case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ interior s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ interior s
case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | have h : Filter.Tendsto (fun n : ℕ => 1 / ((n : 𝕜) + 2)) Filter.atTop (nhds (0 : 𝕜)) := by sorry | case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds x) | case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | rw [← add_zero x, ← one_smul 𝕜 x, ← zero_smul 𝕜 y] | case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds x) | case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds (1 • x + 0 • y)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | convert ((h.const_sub _).smul hzx).add (h.smul_const _) | case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds (1 • x + 0 • y)) | case h.e'_5.h.e'_3.h.e'_5.h.e'_5
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ 1 = 1 - 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | rw [sub_zero] | case h.e'_5.h.e'_3.h.e'_5.h.e'_5
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ 1 = 1 - 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | rw [← closure_diff_frontier] at hy ⊢ | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ interior s | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | have h₁ : (1 : 𝕜) < ↑n + 2 := by norm_cast; norm_num | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
h₁ : 1 < ↑n + 2
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | have h₀ := zero_lt_one.trans h₁ | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
h₁ : 1 < ↑n + 2
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
h₁ : 1 < ↑n + 2
h₀ : 0 < ↑n + 2
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | exact
(hAconv.closure.isExtreme_iff_openSegment_subset_diff.1
hAconv.frontier_extreme_to_closure).2
(subset_closure (hzA n)) hy
⟨1 - 1 / (n + 2), 1 / (n + 2), sub_pos.2 <| (div_lt_one h₀).2 h₁, div_pos zero_lt_one h₀,
sub_add_cancel _ _, rfl⟩ | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
h₁ : 1 < ↑n + 2
h₀ : 0 < ↑n + 2
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | norm_cast | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ 1 < ↑n + 2 | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ 1 < n + 2 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | norm_num | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ 1 < n + 2 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | sorry | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
⊢ Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | ConvexIndependent.subset_of_convexHull_eq_convexHull | [160, 1] | [166, 59] | rintro x hx | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
⊢ s ⊆ t | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
⊢ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | ConvexIndependent.subset_of_convexHull_eq_convexHull | [160, 1] | [166, 59] | have hxextreme := (extremePoints_convexHull_eq_iff_convexIndependent.2 hs).symm.subset hx | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
⊢ x ∈ t | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
hxextreme : x ∈ extremePoints 𝕜 ((convexHull 𝕜) fun x => x ∈ s.val)
⊢ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | ConvexIndependent.subset_of_convexHull_eq_convexHull | [160, 1] | [166, 59] | erw [h] at hxextreme | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
hxextreme : x ∈ extremePoints 𝕜 ((convexHull 𝕜) fun x => x ∈ s.val)
⊢ x ∈ t | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
hxextreme : x ∈ extremePoints 𝕜 ((convexHull 𝕜) ↑t)
⊢ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | ConvexIndependent.subset_of_convexHull_eq_convexHull | [160, 1] | [166, 59] | exact_mod_cast extremePoints_convexHull_subset hxextreme | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
hxextreme : x ∈ extremePoints 𝕜 ((convexHull 𝕜) ↑t)
⊢ x ∈ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.skeleton_nonempty_iff | [32, 1] | [38, 20] | refine' ⟨Set.Nonempty.mono skeleton_le, _⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
⊢ (K.skeleton k).faces.Nonempty ↔ K.faces.Nonempty | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
⊢ K.faces.Nonempty → (K.skeleton k).faces.Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.skeleton_nonempty_iff | [32, 1] | [38, 20] | rintro ⟨s, hs⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
⊢ K.faces.Nonempty → (K.skeleton k).faces.Nonempty | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
s : Finset E
hs : s ∈ K.faces
⊢ (K.skeleton k).faces.Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.skeleton_nonempty_iff | [32, 1] | [38, 20] | obtain ⟨x, hx⟩ := K.nonempty hs | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
s : Finset E
hs : s ∈ K.faces
⊢ (K.skeleton k).faces.Nonempty | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
s : Finset E
hs : s ∈ K.faces
x : E
hx : x ∈ s
⊢ (K.skeleton k).faces.Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.skeleton_nonempty_iff | [32, 1] | [38, 20] | refine' ⟨{x}, K.down_closed' hs (singleton_subset_iff.2 hx) <| singleton_nonempty _, _⟩ | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
s : Finset E
hs : s ∈ K.faces
x : E
hx : x ∈ s
⊢ (K.skeleton k).faces.Nonempty | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
s : Finset E
hs : s ∈ K.faces
x : E
hx : x ∈ s
⊢ {x}.card ≤ k + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.skeleton_nonempty_iff | [32, 1] | [38, 20] | rw [card_singleton] | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
s : Finset E
hs : s ∈ K.faces
x : E
hx : x ∈ s
⊢ {x}.card ≤ k + 1 | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
s : Finset E
hs : s ∈ K.faces
x : E
hx : x ∈ s
⊢ 1 ≤ k + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.skeleton_nonempty_iff | [32, 1] | [38, 20] | exact le_add_self | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
s : Finset E
hs : s ∈ K.faces
x : E
hx : x ∈ s
⊢ 1 ≤ k + 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.Pure.skeleton_of_le | [40, 1] | [44, 34] | refine' ⟨fun s hs => hs.2, _⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
h : k ≤ n
⊢ (K.skeleton k).Pure k | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
h : k ≤ n
⊢ Set.Sized (k + 1) (K.skeleton k).facets |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.Pure.skeleton_of_le | [40, 1] | [44, 34] | rintro s ⟨⟨hs, hscard⟩, hsmax⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
h : k ≤ n
⊢ Set.Sized (k + 1) (K.skeleton k).facets | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
hK : K.Pure n
h : k ≤ n
s : Finset E
hsmax : ∀ ⦃t : Finset E⦄, t ∈ (K.skeleton k).faces → s ⊆ t → s = t
hs : s ∈ K
hscard : s.card ≤ k + 1
⊢ s.card = k + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.Pure.skeleton_of_le | [40, 1] | [44, 34] | obtain ⟨t, ht, hst, htcard⟩ := hK.exists_face_of_card_le (add_le_add_right h 1) hs hscard | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
hK : K.Pure n
h : k ≤ n
s : Finset E
hsmax : ∀ ⦃t : Finset E⦄, t ∈ (K.skeleton k).faces → s ⊆ t → s = t
hs : s ∈ K
hscard : s.card ≤ k + 1
⊢ s.card = k + 1 | case intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t✝ : Finset E
A : Set (Finset E)
hK : K.Pure n
h : k ≤ n
s : Finset E
hsmax : ∀ ⦃t : Finset E⦄, t ∈ (K.skeleton k).faces → s ⊆ t → s = t
hs : s ∈ K
hscard : s.card ≤ k + 1
t : Finset E
ht : t ∈ K
hst : s ⊆ t
htcard : t.card = k + 1
⊢ s.card = k + 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.Pure.skeleton_of_le | [40, 1] | [44, 34] | rwa [hsmax ⟨ht, htcard.le⟩ hst] | case intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t✝ : Finset E
A : Set (Finset E)
hK : K.Pure n
h : k ≤ n
s : Finset E
hsmax : ∀ ⦃t : Finset E⦄, t ∈ (K.skeleton k).faces → s ⊆ t → s = t
hs : s ∈ K
hscard : s.card ≤ k + 1
t : Finset E
ht : t ∈ K
hst : s ⊆ t
htcard : t.card = k + 1
⊢ s.card = k + 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.Pure.skeleton | [53, 1] | [61, 18] | obtain hn | hn := le_total k n | 𝕜 : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
⊢ (K.skeleton k).Pure (min k n) | case inl
𝕜 : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
hn : k ≤ n
⊢ (K.skeleton k).Pure (min k n)
case inr
𝕜 : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
hn : n ≤ k
⊢ (K.skeleton k).Pure (min k n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.Pure.skeleton | [53, 1] | [61, 18] | rw [min_eq_left hn] | case inl
𝕜 : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
hn : k ≤ n
⊢ (K.skeleton k).Pure (min k n) | case inl
𝕜 : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
hn : k ≤ n
⊢ (K.skeleton k).Pure k |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.Pure.skeleton | [53, 1] | [61, 18] | exact hK.skeleton_of_le hn | case inl
𝕜 : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
hn : k ≤ n
⊢ (K.skeleton k).Pure k | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.Pure.skeleton | [53, 1] | [61, 18] | rw [min_eq_right hn] | case inr
𝕜 : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
hn : n ≤ k
⊢ (K.skeleton k).Pure (min k n) | case inr
𝕜 : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
hn : n ≤ k
⊢ (K.skeleton k).Pure n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Skeleton.lean | Geometry.SimplicialComplex.Pure.skeleton | [53, 1] | [61, 18] | refine' ⟨fun s hs => hK.1 <| skeleton_le hs, fun s hs => _⟩ | case inr
𝕜 : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s t : Finset E
A : Set (Finset E)
hK : K.Pure n
hn : n ≤ k
⊢ (K.skeleton k).Pure n | case inr
𝕜 : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
m n k : ℕ
K : SimplicialComplex 𝕜 E
s✝ t : Finset E
A : Set (Finset E)
hK : K.Pure n
hn : n ≤ k
s : Finset E
hs : s ∈ (K.skeleton k).facets
⊢ s.card = n + 1 |
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