Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas_apply	[66, 1]	[71, 9]	simp	case h α : Type u_1 Ω : Type u_2 inst✝ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ a : α ω : Ω ⊢ ω ∈ {ω \| a ∈ X ω} ↔ ω ∈ (fun ω => a ∈ X ω) ⁻¹' {True}	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	simp_rw [ext_iff, setOf_forall]	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α ⊢ μ {ω \| {a \| a ∈ X ω} = ↑s} = ↑p ^ s.card * (1 - ↑p) ^ (card α - s.card)	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α ⊢ μ (⋂ i, {x \| i ∈ {a \| a ∈ X x} ↔ i ∈ ↑s}) = ↑p ^ s.card * (1 - ↑p) ^ (card α - s.card)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	rw [hX.iIndepFun.meas_iInter, ← s.prod_mul_prod_compl, Finset.prod_eq_pow_card, Finset.prod_eq_pow_card, Finset.card_compl]	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α ⊢ μ (⋂ i, {x \| i ∈ {a \| a ∈ X x} ↔ i ∈ ↑s}) = ↑p ^ s.card * (1 - ↑p) ^ (card α - s.card)	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α ⊢ ∀ a ∈ sᶜ, μ {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s} = 1 - ↑p α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	rintro a	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α ⊢ ∀ (i : α), MeasurableSet {x \| i ∈ {a \| a ∈ X x} ↔ i ∈ ↑s}	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α ⊢ MeasurableSet {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s}
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	by_cases a ∈ s	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α ⊢ MeasurableSet {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s}	case pos α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∈ s ⊢ MeasurableSet {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s} case neg α : Type u_1 Ω : Type u_2 i...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	rintro a hi	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α ⊢ ∀ a ∈ sᶜ, μ {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s} = 1 - ↑p	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α hi : a ∈ sᶜ ⊢ μ {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s} = 1 - ↑p
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	rw [Finset.mem_compl] at hi	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α hi : a ∈ sᶜ ⊢ μ {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s} = 1 - ↑p	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α hi : a ∉ s ⊢ μ {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s} = 1 - ↑p
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	simp only [hi, ← compl_setOf, prob_compl_eq_one_sub₀, mem_setOf_eq, Finset.mem_coe, iff_false_iff, hX.nullMeasurableSet, hX.meas_apply]	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α hi : a ∉ s ⊢ μ {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s} = 1 - ↑p	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	rintro a hi	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α ⊢ ∀ a ∈ s, μ {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s} = ↑p	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α hi : a ∈ s ⊢ μ {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s} = ↑p
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	simp only [hi, mem_setOf_eq, Finset.mem_coe, iff_true_iff, hX.meas_apply]	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α hi : a ∈ s ⊢ μ {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s} = ↑p	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	simp only [mem_setOf_eq, Finset.mem_coe, iff_true_iff, *]	case pos α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∈ s ⊢ MeasurableSet {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s}	case pos α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∈ s ⊢ MeasurableSet {x \| a ∈ X x}
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	exact ⟨{True}, trivial, by ext; simp⟩	case pos α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∈ s ⊢ MeasurableSet {x \| a ∈ X x}	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	ext	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∈ s ⊢ (fun ω => a ∈ X ω) ⁻¹' {True} = {x \| a ∈ X x}	case h α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∈ s x✝ : Ω ⊢ x✝ ∈ (fun ω => a ∈ X ω) ⁻¹' {True} ↔ x✝ ∈ {x \| a ∈ X x}
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	simp	case h α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∈ s x✝ : Ω ⊢ x✝ ∈ (fun ω => a ∈ X ω) ⁻¹' {True} ↔ x✝ ∈ {x \| a ∈ X x}	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	simp only [mem_setOf_eq, Finset.mem_coe, iff_false_iff, *]	case neg α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∉ s ⊢ MeasurableSet {x \| a ∈ {a \| a ∈ X x} ↔ a ∈ ↑s}	case neg α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∉ s ⊢ MeasurableSet {x \| a ∉ X x}
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	exact ⟨{False}, trivial, by ext; simp⟩	case neg α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∉ s ⊢ MeasurableSet {x \| a ∉ X x}	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	ext	α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∉ s ⊢ (fun ω => a ∈ X ω) ⁻¹' {False} = {x \| a ∉ X x}	case h α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∉ s x✝ : Ω ⊢ x✝ ∈ (fun ω => a ∈ X ω) ⁻¹' {False} ↔ x✝ ∈ {x \| a ∉ X x}
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.meas	[75, 1]	[92, 43]	simp	case h α : Type u_1 Ω : Type u_2 inst✝² : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝¹ : IsProbabilityMeasure μ inst✝ : Fintype α s : Finset α a : α h✝ : a ∉ s x✝ : Ω ⊢ x✝ ∈ (fun ω => a ∈ X ω) ⁻¹' {False} ↔ x✝ ∈ {x \| a ∉ X x}	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.compl	[94, 1]	[104, 66]	simp only [iIndepFun_iff, mem_compl_iff, MeasurableSpace.comap_not]	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ ⊢ iIndepFun inferInstance (fun a ω => a ∈ (X ω)ᶜ) μ	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ ⊢ ∀ (s : Finset α) {f' : α → Set Ω}, (∀ i ∈ s, MeasurableSet (f' i)) → μ (⋂ i ∈ s, f' i) = ∏ i ∈ s, μ (f' i)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.compl	[94, 1]	[104, 66]	exact (iIndepFun_iff _ _ _).1 hX.iIndepFun	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ ⊢ ∀ (s : Finset α) {f' : α → Set Ω}, (∀ i ∈ s, MeasurableSet (f' i)) → μ (⋂ i ∈ s, f' i) = ∏ i ∈ s, μ (f' i)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.compl	[94, 1]	[104, 66]	have : Measurable Not := fun _ _ ↦ trivial	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ a : α ⊢ Measure.map (fun ω => a ∈ (X ω)ᶜ) μ = (PMF.bernoulli' (1 - p) ⋯).toMeasure	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ a : α this : Measurable Not ⊢ Measure.map (fun ω => a ∈ (X ω)ᶜ) μ = (PMF.bernoulli' (1 - p) ⋯).toMeasure
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.compl	[94, 1]	[104, 66]	refine' (this.aemeasurable.map_map_of_aemeasurable (hX.aemeasurable _)).symm.trans _	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ a : α this : Measurable Not ⊢ Measure.map (fun ω => a ∈ (X ω)ᶜ) μ = (PMF.bernoulli' (1 - p) ⋯).toMeasure	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ a : α this : Measurable Not ⊢ Measure.map Not (Measure.map (fun ω => a ∈ X ω) μ) = (PMF.bernoulli' (1 - p) ⋯).toMeasure
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.compl	[94, 1]	[104, 66]	rw [hX.map, PMF.map_toMeasure _ this, PMF.map_not_bernoulli']	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ a : α this : Measurable Not ⊢ Measure.map Not (Measure.map (fun ω => a ∈ X ω) μ) = (PMF.bernoulli' (1 - p) ⋯).toMeasure	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	refine' iIndepSet.Indep_comap ((iIndepSet_iff_meas_iInter fun i ↦ _).2 _)	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ ⊢ iIndepFun inferInstance (fun a ω => a ∈ X ω ∩ Y ω) μ	case refine'_1 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ i : α ⊢ MeasurableSet fun ω => (X ω ∩ Y ω) i case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpac...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	refine' MeasurableSet.inter _ _	case refine'_1 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ i : α ⊢ MeasurableSet fun ω => (X ω ∩ Y ω) i case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpac...	case refine'_1.refine'_1 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ i : α ⊢ MeasurableSet fun ω => X ω i case refine'_1.refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : Me...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	sorry	case refine'_1.refine'_1 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ i : α ⊢ MeasurableSet fun ω => X ω i case refine'_1.refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : Me...	case refine'_1.refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ i : α ⊢ MeasurableSet fun ω => Y ω i case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSp...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	sorry	case refine'_1.refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ i : α ⊢ MeasurableSet fun ω => Y ω i case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSp...	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ ⊢ ∀ (t : Finset α), μ (⋂ i ∈ t, fun ω => (X ω ∩ Y ω) i) = ∏ i ∈ t, μ fun ω => (X ω ∩ Y ω) i
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	refine' fun s ↦ _	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ ⊢ ∀ (t : Finset α), μ (⋂ i ∈ t, fun ω => (X ω ∩ Y ω) i) = ∏ i ∈ t, μ fun ω => (X ω ∩ Y ω) i	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ μ (⋂ i ∈ s, fun ω => (X ω ∩ Y ω) i) = ∏ i ∈ s, μ fun ω => (X ω ∩ Y ω) i
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	change μ (⋂ i ∈ s, {ω \| X ω i} ∩ {ω \| Y ω i}) = s.prod fun i ↦ μ ({ω \| X ω i} ∩ {ω \| Y ω i})	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ μ (⋂ i ∈ s, fun ω => (X ω ∩ Y ω) i) = ∏ i ∈ s, μ fun ω => (X ω ∩ Y ω) i	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ μ (⋂ i ∈ s, {ω \| X ω i} ∩ {ω \| Y ω i}) = ∏ i ∈ s, μ ({ω \| X ω i} ∩ {ω \| Y ω i})
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	simp_rw [iInter_inter_distrib]	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ μ (⋂ i ∈ s, {ω \| X ω i} ∩ {ω \| Y ω i}) = ∏ i ∈ s, μ ({ω \| X ω i} ∩ {ω \| Y ω i})	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ μ ((⋂ i ∈ s, {ω \| X ω i}) ∩ ⋂ i ∈ s, {ω \| Y ω i}) = ∏ i ∈ s, μ ({ω \| X ω i} ∩ {ω \| Y ω i})
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	rw [h.meas_inter, hX.iIndepFun.meas_biInter, hY.iIndepFun.meas_biInter, ← Finset.prod_mul_distrib]	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ μ ((⋂ i ∈ s, {ω \| X ω i}) ∩ ⋂ i ∈ s, {ω \| Y ω i}) = ∏ i ∈ s, μ ({ω \| X ω i} ∩ {ω \| Y ω i})	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ ∏ x ∈ s, μ {ω \| X ω x} * μ {ω \| Y ω x} = ∏ i ∈ s, μ ({ω \| X ω i} ∩ {ω \| Y ω i}) case refine'_2...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	refine' Finset.prod_congr rfl fun i hi ↦ (h.meas_inter _ _).symm	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ ∏ x ∈ s, μ {ω \| X ω x} * μ {ω \| Y ω x} = ∏ i ∈ s, μ ({ω \| X ω i} ∩ {ω \| Y ω i}) case refine'_2...	case refine'_2.refine'_1 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α i : α hi : i ∈ s ⊢ MeasurableSet {ω \| X ω i} case refine'_2.refine'_2 α : Type u_1 Ω :...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	sorry	case refine'_2.refine'_1 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α i : α hi : i ∈ s ⊢ MeasurableSet {ω \| X ω i} case refine'_2.refine'_2 α : Type u_1 Ω :...	case refine'_2.refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α i : α hi : i ∈ s ⊢ MeasurableSet {ω \| Y ω i} case refine'_2 α : Type u_1 Ω : Type u_2 ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	sorry	case refine'_2.refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α i : α hi : i ∈ s ⊢ MeasurableSet {ω \| Y ω i} case refine'_2 α : Type u_1 Ω : Type u_2 ...	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ ∀ i ∈ s, MeasurableSet {ω \| Y ω i} case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : Measurabl...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	sorry	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ ∀ i ∈ s, MeasurableSet {ω \| Y ω i} case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : Measurabl...	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ ∀ i ∈ s, MeasurableSet {ω \| X ω i} case refine'_2.hs α : Type u_1 Ω : Type u_2 inst✝¹ : Measur...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	sorry	case refine'_2 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ ∀ i ∈ s, MeasurableSet {ω \| X ω i} case refine'_2.hs α : Type u_1 Ω : Type u_2 inst✝¹ : Measur...	case refine'_2.hs α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ MeasurableSet (⋂ i ∈ s, {ω \| X ω i}) case refine'_2.ht α : Type u_1 Ω : Type u_2 inst✝¹ : M...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	sorry	case refine'_2.hs α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ MeasurableSet (⋂ i ∈ s, {ω \| X ω i}) case refine'_2.ht α : Type u_1 Ω : Type u_2 inst✝¹ : M...	case refine'_2.ht α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ MeasurableSet (⋂ i ∈ s, {ω \| Y ω i})
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.inter	[106, 1]	[128, 17]	sorry	case refine'_2.ht α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ s : Finset α ⊢ MeasurableSet (⋂ i ∈ s, {ω \| Y ω i})	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.union	[130, 1]	[141, 68]	convert (hX.compl.inter hY.compl _).compl using 1	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ ⊢ IsBernoulliSeq (fun ω => X ω ∪ Y ω) (p + q - p * q) μ	case h.e'_4 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ ⊢ (fun ω => X ω ∪ Y ω) = fun ω => ((X ω)ᶜ ∩ (Y ω)ᶜ)ᶜ case h.e'_5 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSp...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.union	[130, 1]	[141, 68]	simp [compl_inter]	case h.e'_4 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ ⊢ (fun ω => X ω ∪ Y ω) = fun ω => ((X ω)ᶜ ∩ (Y ω)ᶜ)ᶜ	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.union	[130, 1]	[141, 68]	rw [mul_tsub, mul_one, tsub_tsub, tsub_tsub_cancel_of_le, tsub_mul, one_mul, add_tsub_assoc_of_le (mul_le_of_le_one_left' $ hX.le_one)]	case h.e'_5 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ ⊢ p + q - p * q = 1 - (1 - p) * (1 - q)	case h.e'_5 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ ⊢ p + (1 - p) * q ≤ 1
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.union	[130, 1]	[141, 68]	exact (add_le_add_left (mul_le_of_le_one_right' $ hY.le_one) _).trans_eq (add_tsub_cancel_of_le hX.le_one)	case h.e'_5 α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ ⊢ p + (1 - p) * q ≤ 1	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/BernoulliSeq.lean	ProbabilityTheory.IsBernoulliSeq.union	[130, 1]	[141, 68]	rwa [IndepFun_iff, MeasurableSpace.comap_compl measurable_compl, MeasurableSpace.comap_compl measurable_compl, ← IndepFun_iff]	α : Type u_1 Ω : Type u_2 inst✝¹ : MeasurableSpace Ω X Y : Ω → Set α μ : Measure Ω p q : ℝ≥0 hX : IsBernoulliSeq X p μ hY : IsBernoulliSeq Y q μ inst✝ : IsProbabilityMeasure μ h : IndepFun X Y μ ⊢ IndepFun (fun ω => (X ω)ᶜ) (fun ω => (Y ω)ᶜ) μ	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	refine' ⟨fun h => ⟨h.1.superset, fun s hs => _⟩, fun h => ⟨h.1.antisymm' fun x hx => _, fun s₁ hs₁ => _⟩⟩	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E ⊢ K₁.Subdivides K₂ ↔ K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂	case refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ ⊢ ∃ s₂ ∈ K₂, combiInterior ℝ s ⊆ combiInterior ℝ s₂ case refine'_2 𝕜 : Type u_1 E : Type u_2 inst✝² :...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	obtain ⟨t, ht, hst⟩ := h.2 hs	case refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ ⊢ ∃ s₂ ∈ K₂, combiInterior ℝ s ⊆ combiInterior ℝ s₂	case refine'_1.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ t : Finset E ht : t ∈ K₂ hst : (convexHull ℝ) ↑s ⊆ (convexHull ℝ) ↑t ⊢ ∃ s₂ ∈ K₂, combiInt...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	obtain ⟨u, hut, hu, hsu⟩ := simplex_combiInteriors_split_interiors_nonempty (K₁.nonempty hs) (K₂.indep ht) hst	case refine'_1.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ t : Finset E ht : t ∈ K₂ hst : (convexHull ℝ) ↑s ⊆ (convexHull ℝ) ↑t ⊢ ∃ s₂ ∈ K₂, combiInt...	case refine'_1.intro.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ t : Finset E ht : t ∈ K₂ hst : (convexHull ℝ) ↑s ⊆ (convexHull ℝ) ↑t u :...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	exact ⟨u, K₂.down_closed' ht hut hu, hsu⟩	case refine'_1.intro.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₁.Subdivides K₂ s : Finset E hs : s ∈ K₁ t : Finset E ht : t ∈ K₂ hst : (convexHull ℝ) ↑s ⊆ (convexHull ℝ) ↑t u :...	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	obtain ⟨s₁, hs₁, hx⟩ := mem_space_iff.1 hx	case refine'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx : x ∈ K₁.space ⊢ x ∈ K₂.space	case refine'_2.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E hs₁ : s₁ ∈ ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	obtain ⟨s₂, hs₂, hs₁s₂⟩ := h.2 _ hs₁	case refine'_2.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E hs₁ : s₁ ∈ ...	case refine'_2.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	rw [mem_space_iff]	case refine'_2.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E...	case refine'_2.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	exact ⟨s₂, hs₂, convexHull_subset_convexHull_of_combiInterior_subset_combiInterior (K₁.indep hs₁) (K₂.indep hs₂) hs₁s₂ hx⟩	case refine'_2.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ x : E hx✝ : x ∈ K₁.space s₁ : Finset E...	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	obtain ⟨s₂, hs₂, hs₁s₂⟩ := h.2 _ hs₁	case refine'_3 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ s₁ : Finset E hs₁ : s₁ ∈ K₁ ⊢ ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ ...	case refine'_3.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ s₁ : Finset E hs₁ : s₁ ∈ K₁ s₂ : Finset E hs₂ : s₂...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_combiInteriors_subset_combiInteriors	[50, 1]	[66, 28]	exact ⟨_, hs₂, convexHull_subset_convexHull_of_combiInterior_subset_combiInterior (K₁.indep hs₁) (K₂.indep hs₂) hs₁s₂⟩	case refine'_3.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E h : K₂.space ⊆ K₁.space ∧ ∀ s₁ ∈ K₁, ∃ s₂ ∈ K₂, combiInterior ℝ s₁ ⊆ combiInterior ℝ s₂ s₁ : Finset E hs₁ : s₁ ∈ K₁ s₂ : Finset E hs₂ : s₂...	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	constructor	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E ⊢ K₁.Subdivides K₂ ↔ (K₁.faces.Nonempty → K₂.faces.Nonempty) ∧ K₁.space ⊆ K₂.space ∧ ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterio...	case mp 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E ⊢ K₁.Subdivides K₂ → (K₁.faces.Nonempty → K₂.faces.Nonempty) ∧ K₁.space ⊆ K₂.space ∧ ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rintro ⟨hspace, hsubdiv⟩	case mp 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E ⊢ K₁.Subdivides K₂ → (K₁.faces.Nonempty → K₂.faces.Nonempty) ∧ K₁.space ⊆ K₂.space ∧ ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...	case mp.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ ⊢ (K₁.faces.Nonempty → K₂.faces....
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	refine' ⟨_, hspace.le, fun s hs => _⟩	case mp.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ ⊢ (K₁.faces.Nonempty → K₂.faces....	case mp.intro.refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ ⊢ K₁.faces.Nonempty → ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	refine' ⟨{t \| t ∈ K₁ ∧ combiInterior ℝ t ⊆ combiInterior ℝ s}, fun t ht => ht.1, _⟩	case mp.intro.refine'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s ∈...	case mp.intro.refine'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s ∈...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	ext x	case mp.intro.refine'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s ∈...	case mp.intro.refine'_2.h 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	refine' ⟨fun hxs => _, _⟩	case mp.intro.refine'_2.h 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s...	case mp.intro.refine'_2.h.refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finse...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rintro ⟨s₁, hs₁⟩	case mp.intro.refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ ⊢ K₁.faces.Nonempty → ...	case mp.intro.refine'_1.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s₁ : Finset E hs...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨s₂, hs₂, -⟩ := hsubdiv hs₁	case mp.intro.refine'_1.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s₁ : Finset E hs...	case mp.intro.refine'_1.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s₁ :...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	exact ⟨s₂, hs₂⟩	case mp.intro.refine'_1.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s₁ :...	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	have hxspace := mem_space_iff.2 ⟨s, hs, hxs.1⟩	case mp.intro.refine'_2.h.refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finse...	case mp.intro.refine'_2.h.refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finse...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rw [← hspace, ← combiInteriors_cover, mem_iUnion₂] at hxspace	case mp.intro.refine'_2.h.refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finse...	case mp.intro.refine'_2.h.refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finse...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨t, ht, hxt⟩ := hxspace	case mp.intro.refine'_2.h.refine'_1 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finse...	case mp.intro.refine'_2.h.refine'_1.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	refine' mem_iUnion₂_of_mem ⟨ht, fun y hyt => _⟩ hxt	case mp.intro.refine'_2.h.refine'_1.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ...	case mp.intro.refine'_2.h.refine'_1.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨u, hu, htu⟩ := hsubdiv ht	case mp.intro.refine'_2.h.refine'_1.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ...	case mp.intro.refine'_2.h.refine'_1.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (co...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨W, hWu, htW⟩ := simplex_combiInteriors_split_interiors (K₂.indep hu) htu	case mp.intro.refine'_2.h.refine'_1.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (co...	case mp.intro.refine'_2.h.refine'_1.intro.intro.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rw [disjoint_interiors hs (K₂.down_closed hu hWu _) hxs (htW hxt)]	case mp.intro.refine'_2.h.refine'_1.intro.intro.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ...	case mp.intro.refine'_2.h.refine'_1.intro.intro.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	exact htW hyt	case mp.intro.refine'_2.h.refine'_1.intro.intro.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ...	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s ∈ K₂ x : E hxs : x ∈ com...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rintro rfl	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s ∈ K₂ x : E hxs : x ∈ com...	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s ∈ K₂ x : E hxs : x ∈ com...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	have := htW hxt	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s ∈ K₂ x : E hxs : x ∈ com...	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s ∈ K₂ x : E hxs : x ∈ com...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	simp at this	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finset E hs : s ∈ K₂ x : E hxs : x ∈ com...	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rw [mem_iUnion₂]	case mp.intro.refine'_2.h.refine'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finse...	case mp.intro.refine'_2.h.refine'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finse...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rintro ⟨t, ⟨-, hts⟩, hxt⟩	case mp.intro.refine'_2.h.refine'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHull ℝ) ↑s₂ s : Finse...	case mp.intro.refine'_2.h.refine'_2.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHu...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	exact hts hxt	case mp.intro.refine'_2.h.refine'_2.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hspace : K₁.space = K₂.space hsubdiv : ∀ ⦃s₁ : Finset E⦄, s₁ ∈ K₁ → ∃ s₂ ∈ K₂, (convexHull ℝ) ↑s₁ ⊆ (convexHu...	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rintro ⟨hempty, hspace, hpartition⟩	case mpr 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E ⊢ ((K₁.faces.Nonempty → K₂.faces.Nonempty) ∧ K₁.space ⊆ K₂.space ∧ ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ s₁) → ...	case mpr.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	have hspace : K₁.space = K₂.space := by refine' hspace.antisymm fun x hx => _ rw [← combiInteriors_cover, mem_iUnion₂] at hx ⊢ obtain ⟨s, hs, hxs⟩ := hx obtain ⟨F, hF, hsint⟩ := hpartition _ hs rw [hsint, mem_iUnion₂] at hxs obtain ⟨t, ht, hxt⟩ := hxs exact ⟨t, hF ht, hxt⟩	case mpr.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ ...	case mpr.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	refine' ⟨hspace, fun s hs => _⟩	case mpr.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈...	case mpr.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain rfl \| hsnonempty := s.eq_empty_or_nonempty	case mpr.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ...	case mpr.intro.intro.inl 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨x, hx⟩ := hsnonempty.combiInterior (K₁.indep hs)	case mpr.intro.intro.inr 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃...	case mpr.intro.intro.inr.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	have hxspace := mem_space_iff.2 ⟨s, hs, hx.1⟩	case mpr.intro.intro.inr.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ ...	case mpr.intro.intro.inr.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rw [hspace, ← combiInteriors_cover, mem_iUnion₂] at hxspace	case mpr.intro.intro.inr.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ ...	case mpr.intro.intro.inr.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨t, ht, hxt⟩ := hxspace	case mpr.intro.intro.inr.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ ...	case mpr.intro.intro.inr.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, com...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	use t, ht	case mpr.intro.intro.inr.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, com...	case right 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rw [← closure_combiInterior_eq_convexHull (K₁.indep hs)]	case right 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...	case right 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	refine' closure_minimal (fun x' hx' => _) (isClosed_convexHull _)	case right 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...	case right 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	have hxspace := mem_space_iff.2 ⟨s, hs, hx'.1⟩	case right 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...	case right 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rw [hspace, ← combiInteriors_cover, mem_iUnion₂] at hxspace	case right 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...	case right 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨t', ht', hxt'⟩ := hxspace	case right 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, comb...	case right.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	convert hxt'.1	case right.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ ...	case h.e'_5.h.e'_6.h.e'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ =...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨F, hF, hinterior⟩ := hpartition _ ht	case h.e'_5.h.e'_6.h.e'_2 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ =...	case h.e'_5.h.e'_6.h.e'_2.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInt...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨F', hF', hinterior'⟩ := hpartition _ ht'	case h.e'_5.h.e'_6.h.e'_2.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInt...	case h.e'_5.h.e'_6.h.e'_2.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.fac...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	apply disjoint_interiors ht ht' (_ : x ∈ _) _	case h.e'_5.h.e'_6.h.e'_2.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.fac...	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace✝ : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	refine' hspace.antisymm fun x hx => _	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ s₁...	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ s₁...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rw [← combiInteriors_cover, mem_iUnion₂] at hx ⊢	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ s₁...	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ s₁...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨s, hs, hxs⟩ := hx	𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ s₁...	case intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨F, hF, hsint⟩ := hpartition _ hs	case intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, ...	case intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	rw [hsint, mem_iUnion₂] at hxs	case intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ ...	case intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ ...
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/SimplicialComplex/Subdivision.lean	Geometry.SimplicialComplex.subdivides_iff_partition	[68, 1]	[128, 15]	obtain ⟨t, ht, hxt⟩ := hxs	case intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ ...	case intro.intro.intro.intro.intro.intro 𝕜 : Type u_1 E : Type u_2 inst✝² : SeminormedAddCommGroup E inst✝¹ : T2Space E inst✝ : NormedSpace ℝ E s✝ t✝ : Finset E m : ℕ K₁ K₂ : SimplicialComplex ℝ E hempty : K₁.faces.Nonempty → K₂.faces.Nonempty hspace : K₁.space ⊆ K₂.space hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiI...

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.