Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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stringclasses
147 values
commit
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147 values
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7
101
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2.09M
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]
Ξ± : Type u_1 Ξ© : Type u_2 inst✝¹ : MeasurableSpace Ξ© X Y : Ξ© β†’ Set Ξ± ΞΌ : Measure Ξ© p q : ℝβ‰₯0 hX : IsBernoulliSeq X p ΞΌ hY : IsBernoulliSeq Y q ΞΌ inst✝ : IsProbabilityMeasure ΞΌ h : IndepFun X Y ΞΌ ⊒ IndepFun (fun Ο‰ => (X Ο‰)ᢜ) (fun Ο‰ => (Y Ο‰)ᢜ) ΞΌ
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.exists_littlewood_offord_partition
[20, 1]
[35, 8]
induction' s using Finset.induction with i s hi ih
ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r hf : βˆ€ i ∈ s, r ≀ β€–f iβ€– ⊒ βˆƒ P, P.parts.card = s.card.choose (s.card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
case empty ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r hf : βˆ€ i ∈ βˆ…, r ≀ β€–f iβ€– ⊒ βˆƒ P, P.parts.card = βˆ….card.choose (βˆ….card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† βˆ…) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) case insert ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s✝ : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r i : ΞΉ s : Finset ΞΉ hi : i βˆ‰ s ih : (βˆ€ i ∈ s, r ≀ β€–f iβ€–) β†’ βˆƒ P, P.parts.card = s.card.choose (s.card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) hf : βˆ€ i_1 ∈ insert i s, r ≀ β€–f i_1β€– ⊒ βˆƒ P, P.parts.card = (insert i s).card.choose ((insert i s).card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† insert i s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.exists_littlewood_offord_partition
[20, 1]
[35, 8]
obtain ⟨P, hP, hs, hPr⟩ := ih fun j hj ↦ hf _ $ mem_insert_of_mem hj
case insert ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s✝ : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r i : ΞΉ s : Finset ΞΉ hi : i βˆ‰ s ih : (βˆ€ i ∈ s, r ≀ β€–f iβ€–) β†’ βˆƒ P, P.parts.card = s.card.choose (s.card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) hf : βˆ€ i_1 ∈ insert i s, r ≀ β€–f i_1β€– ⊒ βˆƒ P, P.parts.card = (insert i s).card.choose ((insert i s).card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† insert i s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
case insert.intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s✝ : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r i : ΞΉ s : Finset ΞΉ hi : i βˆ‰ s ih : (βˆ€ i ∈ s, r ≀ β€–f iβ€–) β†’ βˆƒ P, P.parts.card = s.card.choose (s.card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) hf : βˆ€ i_1 ∈ insert i s, r ≀ β€–f i_1β€– P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hPr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) ⊒ βˆƒ P, P.parts.card = (insert i s).card.choose ((insert i s).card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† insert i s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.exists_littlewood_offord_partition
[20, 1]
[35, 8]
clear ih
case insert.intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s✝ : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r i : ΞΉ s : Finset ΞΉ hi : i βˆ‰ s ih : (βˆ€ i ∈ s, r ≀ β€–f iβ€–) β†’ βˆƒ P, P.parts.card = s.card.choose (s.card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) hf : βˆ€ i_1 ∈ insert i s, r ≀ β€–f i_1β€– P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hPr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) ⊒ βˆƒ P, P.parts.card = (insert i s).card.choose ((insert i s).card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† insert i s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
case insert.intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s✝ : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r i : ΞΉ s : Finset ΞΉ hi : i βˆ‰ s hf : βˆ€ i_1 ∈ insert i s, r ≀ β€–f i_1β€– P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hPr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) ⊒ βˆƒ P, P.parts.card = (insert i s).card.choose ((insert i s).card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† insert i s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.exists_littlewood_offord_partition
[20, 1]
[35, 8]
obtain ⟨g, hg, hgf⟩ := exists_dual_vector ℝ (f i) (norm_pos_iff.1 $ hr.trans_le $ hf _ $ mem_insert_self _ _)
case insert.intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s✝ : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r i : ΞΉ s : Finset ΞΉ hi : i βˆ‰ s hf : βˆ€ i_1 ∈ insert i s, r ≀ β€–f i_1β€– P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hPr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) ⊒ βˆƒ P, P.parts.card = (insert i s).card.choose ((insert i s).card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† insert i s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
case insert.intro.intro.intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s✝ : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r i : ΞΉ s : Finset ΞΉ hi : i βˆ‰ s hf : βˆ€ i_1 ∈ insert i s, r ≀ β€–f i_1β€– P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hPr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) g : E β†’L[ℝ] ℝ hg : β€–gβ€– = 1 hgf : g (f i) = ↑‖f iβ€– ⊒ βˆƒ P, P.parts.card = (insert i s).card.choose ((insert i s).card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† insert i s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.exists_littlewood_offord_partition
[20, 1]
[35, 8]
choose t ht using fun π’œ (hπ’œ : π’œ ∈ P.parts) ↦ Finset.exists_max_image _ (fun t ↦ g (βˆ‘ i in t, f i)) (P.nonempty_of_mem_parts hπ’œ)
case insert.intro.intro.intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s✝ : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r i : ΞΉ s : Finset ΞΉ hi : i βˆ‰ s hf : βˆ€ i_1 ∈ insert i s, r ≀ β€–f i_1β€– P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hPr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) g : E β†’L[ℝ] ℝ hg : β€–gβ€– = 1 hgf : g (f i) = ↑‖f iβ€– ⊒ βˆƒ P, P.parts.card = (insert i s).card.choose ((insert i s).card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† insert i s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
case insert.intro.intro.intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s✝ : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r i : ΞΉ s : Finset ΞΉ hi : i βˆ‰ s hf : βˆ€ i_1 ∈ insert i s, r ≀ β€–f i_1β€– P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hPr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) g : E β†’L[ℝ] ℝ hg : β€–gβ€– = 1 hgf : g (f i) = ↑‖f iβ€– t : (π’œ : Finset (Finset ΞΉ)) β†’ π’œ ∈ P.parts β†’ Finset ΞΉ ht : βˆ€ (π’œ : Finset (Finset ΞΉ)) (hπ’œ : π’œ ∈ P.parts), t π’œ hπ’œ ∈ π’œ ∧ βˆ€ x' ∈ π’œ, g (βˆ‘ i ∈ x', f i) ≀ g (βˆ‘ i ∈ t π’œ hπ’œ, f i) ⊒ βˆƒ P, P.parts.card = (insert i s).card.choose ((insert i s).card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† insert i s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.exists_littlewood_offord_partition
[20, 1]
[35, 8]
sorry
case insert.intro.intro.intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s✝ : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r i : ΞΉ s : Finset ΞΉ hi : i βˆ‰ s hf : βˆ€ i_1 ∈ insert i s, r ≀ β€–f i_1β€– P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hPr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) g : E β†’L[ℝ] ℝ hg : β€–gβ€– = 1 hgf : g (f i) = ↑‖f iβ€– t : (π’œ : Finset (Finset ΞΉ)) β†’ π’œ ∈ P.parts β†’ Finset ΞΉ ht : βˆ€ (π’œ : Finset (Finset ΞΉ)) (hπ’œ : π’œ ∈ P.parts), t π’œ hπ’œ ∈ π’œ ∧ βˆ€ x' ∈ π’œ, g (βˆ‘ i ∈ x', f i) ≀ g (βˆ‘ i ∈ t π’œ hπ’œ, f i) ⊒ βˆƒ P, P.parts.card = (insert i s).card.choose ((insert i s).card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† insert i s) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.exists_littlewood_offord_partition
[20, 1]
[35, 8]
exact ⟨Finpartition.indiscrete $ singleton_ne_empty _, by simp⟩
case empty ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r hf : βˆ€ i ∈ βˆ…, r ≀ β€–f iβ€– ⊒ βˆƒ P, P.parts.card = βˆ….card.choose (βˆ….card / 2) ∧ (βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† βˆ…) ∧ βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.exists_littlewood_offord_partition
[20, 1]
[35, 8]
simp
ΞΉ : Type u_1 E : Type u_2 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s : Finset ΞΉ f : ΞΉ β†’ E r : ℝ inst✝ : DecidableEq ΞΉ hr : 0 < r hf : βˆ€ i ∈ βˆ…, r ≀ β€–f iβ€– ⊒ (Finpartition.indiscrete β‹―).parts.card = βˆ….card.choose (βˆ….card / 2) ∧ (βˆ€ π’œ ∈ (Finpartition.indiscrete β‹―).parts, βˆ€ t ∈ π’œ, t βŠ† βˆ…) ∧ βˆ€ π’œ ∈ (Finpartition.indiscrete β‹―).parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i)
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.card_le_of_forall_dist_sum_le
[37, 1]
[47, 73]
obtain ⟨P, hP, _hs, hr⟩ := exists_littlewood_offord_partition hr hf
ΞΉ : Type u_1 E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s : Finset ΞΉ f : ΞΉ β†’ E r : ℝ hr : 0 < r hπ’œ : βˆ€ t ∈ π’œ, t βŠ† s hf : βˆ€ i ∈ s, r ≀ β€–f iβ€– hπ’œr : βˆ€ u ∈ π’œ, βˆ€ v ∈ π’œ, dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) < r ⊒ π’œ.card ≀ s.card.choose (s.card / 2)
case intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s : Finset ΞΉ f : ΞΉ β†’ E r : ℝ hr✝ : 0 < r hπ’œ : βˆ€ t ∈ π’œ, t βŠ† s hf : βˆ€ i ∈ s, r ≀ β€–f iβ€– hπ’œr : βˆ€ u ∈ π’œ, βˆ€ v ∈ π’œ, dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) < r P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) _hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) ⊒ π’œ.card ≀ s.card.choose (s.card / 2)
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.card_le_of_forall_dist_sum_le
[37, 1]
[47, 73]
rw [← hP]
case intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s : Finset ΞΉ f : ΞΉ β†’ E r : ℝ hr✝ : 0 < r hπ’œ : βˆ€ t ∈ π’œ, t βŠ† s hf : βˆ€ i ∈ s, r ≀ β€–f iβ€– hπ’œr : βˆ€ u ∈ π’œ, βˆ€ v ∈ π’œ, dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) < r P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) _hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) ⊒ π’œ.card ≀ s.card.choose (s.card / 2)
case intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s : Finset ΞΉ f : ΞΉ β†’ E r : ℝ hr✝ : 0 < r hπ’œ : βˆ€ t ∈ π’œ, t βŠ† s hf : βˆ€ i ∈ s, r ≀ β€–f iβ€– hπ’œr : βˆ€ u ∈ π’œ, βˆ€ v ∈ π’œ, dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) < r P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) _hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) ⊒ π’œ.card ≀ P.parts.card
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/LittlewoodOfford.lean
Finset.card_le_of_forall_dist_sum_le
[37, 1]
[47, 73]
simpa only [exists_prop] using P.exists_mem (mem_powerset.2 $ hπ’œ _ ht)
case intro.intro.intro ΞΉ : Type u_1 E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E π’œ : Finset (Finset ΞΉ) s : Finset ΞΉ f : ΞΉ β†’ E r : ℝ hr✝ : 0 < r hπ’œ : βˆ€ t ∈ π’œ, t βŠ† s hf : βˆ€ i ∈ s, r ≀ β€–f iβ€– hπ’œr : βˆ€ u ∈ π’œ, βˆ€ v ∈ π’œ, dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) < r P : Finpartition s.powerset hP : P.parts.card = s.card.choose (s.card / 2) _hs : βˆ€ π’œ ∈ P.parts, βˆ€ t ∈ π’œ, t βŠ† s hr : βˆ€ π’œ ∈ P.parts, (β†‘π’œ).Pairwise fun u v => r ≀ dist (βˆ‘ i ∈ u, f i) (βˆ‘ i ∈ v, f i) t : Finset ΞΉ ht : t ∈ π’œ ⊒ βˆƒ b ∈ P.parts, (fun x x_1 => x ∈ x_1) t b
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.ext
[97, 1]
[102, 66]
refine' DFunLike.coe_injective' (funextβ‚‚ fun a b ↦ _)
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝¹ : Zero π•œ inst✝ : LE Ξ± a b : Ξ± f g : IncidenceAlgebra π•œ Ξ± h : βˆ€ (a b : Ξ±), a ≀ b β†’ f a b = g a b ⊒ f = g
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝¹ : Zero π•œ inst✝ : LE Ξ± a✝ b✝ : Ξ± f g : IncidenceAlgebra π•œ Ξ± h : βˆ€ (a b : Ξ±), a ≀ b β†’ f a b = g a b a b : Ξ± ⊒ f a b = g a b
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.ext
[97, 1]
[102, 66]
by_cases hab : a ≀ b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝¹ : Zero π•œ inst✝ : LE Ξ± a✝ b✝ : Ξ± f g : IncidenceAlgebra π•œ Ξ± h : βˆ€ (a b : Ξ±), a ≀ b β†’ f a b = g a b a b : Ξ± ⊒ f a b = g a b
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝¹ : Zero π•œ inst✝ : LE Ξ± a✝ b✝ : Ξ± f g : IncidenceAlgebra π•œ Ξ± h : βˆ€ (a b : Ξ±), a ≀ b β†’ f a b = g a b a b : Ξ± hab : a ≀ b ⊒ f a b = g a b case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝¹ : Zero π•œ inst✝ : LE Ξ± a✝ b✝ : Ξ± f g : IncidenceAlgebra π•œ Ξ± h : βˆ€ (a b : Ξ±), a ≀ b β†’ f a b = g a b a b : Ξ± hab : Β¬a ≀ b ⊒ f a b = g a b
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.ext
[97, 1]
[102, 66]
exact h _ _ hab
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝¹ : Zero π•œ inst✝ : LE Ξ± a✝ b✝ : Ξ± f g : IncidenceAlgebra π•œ Ξ± h : βˆ€ (a b : Ξ±), a ≀ b β†’ f a b = g a b a b : Ξ± hab : a ≀ b ⊒ f a b = g a b
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.ext
[97, 1]
[102, 66]
rw [apply_eq_zero_of_not_le hab, apply_eq_zero_of_not_le hab]
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝¹ : Zero π•œ inst✝ : LE Ξ± a✝ b✝ : Ξ± f g : IncidenceAlgebra π•œ Ξ± h : βˆ€ (a b : Ξ±), a ≀ b β†’ f a b = g a b a b : Ξ± hab : Β¬a ≀ b ⊒ f a b = g a b
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_zeta
[345, 1]
[350, 49]
rw [mul_apply, card_eq_sum_ones, Nat.cast_sum, Nat.cast_one]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± ⊒ (zeta π•œ * zeta π•œ) a b = ↑(Icc a b).card
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± ⊒ βˆ‘ x ∈ Icc a b, (zeta π•œ) a x * (zeta π•œ) x b = βˆ‘ x ∈ Icc a b, 1
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_zeta
[345, 1]
[350, 49]
refine' sum_congr rfl fun x hx ↦ _
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± ⊒ βˆ‘ x ∈ Icc a b, (zeta π•œ) a x * (zeta π•œ) x b = βˆ‘ x ∈ Icc a b, 1
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b x : Ξ± hx : x ∈ Icc a b ⊒ (zeta π•œ) a x * (zeta π•œ) x b = 1
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_zeta
[345, 1]
[350, 49]
rw [mem_Icc] at hx
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b x : Ξ± hx : x ∈ Icc a b ⊒ (zeta π•œ) a x * (zeta π•œ) x b = 1
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b x : Ξ± hx : a ≀ x ∧ x ≀ b ⊒ (zeta π•œ) a x * (zeta π•œ) x b = 1
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_zeta
[345, 1]
[350, 49]
rw [zeta_of_le hx.1, zeta_of_le hx.2, one_mul]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b x : Ξ± hx : a ≀ x ∧ x ≀ b ⊒ (zeta π•œ) a x * (zeta π•œ) x b = 1
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_kappa
[352, 1]
[357, 49]
rw [mul_apply, card_eq_sum_ones, Nat.cast_sum, Nat.cast_one]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± ⊒ (zeta π•œ * zeta π•œ) a b = ↑(Icc a b).card
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± ⊒ βˆ‘ x ∈ Icc a b, (zeta π•œ) a x * (zeta π•œ) x b = βˆ‘ x ∈ Icc a b, 1
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_kappa
[352, 1]
[357, 49]
refine' sum_congr rfl fun x hx ↦ _
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± ⊒ βˆ‘ x ∈ Icc a b, (zeta π•œ) a x * (zeta π•œ) x b = βˆ‘ x ∈ Icc a b, 1
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b x : Ξ± hx : x ∈ Icc a b ⊒ (zeta π•œ) a x * (zeta π•œ) x b = 1
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_kappa
[352, 1]
[357, 49]
rw [mem_Icc] at hx
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b x : Ξ± hx : x ∈ Icc a b ⊒ (zeta π•œ) a x * (zeta π•œ) x b = 1
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b x : Ξ± hx : a ≀ x ∧ x ≀ b ⊒ (zeta π•œ) a x * (zeta π•œ) x b = 1
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_kappa
[352, 1]
[357, 49]
rw [zeta_of_le hx.1, zeta_of_le hx.2, one_mul]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Semiring π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b x : Ξ± hx : a ≀ x ∧ x ≀ b ⊒ (zeta π•œ) a x * (zeta π•œ) x b = 1
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.muAux_apply
[375, 1]
[377, 13]
rw [muAux]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± ⊒ muAux π•œ a b = if a = b then 1 else -βˆ‘ x ∈ (Ico a b).attach, muAux π•œ a ↑x
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_apply
[392, 1]
[393, 43]
rw [mu, coe_mk, muAux_apply, sum_attach]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± ⊒ (mu π•œ) a b = if a = b then 1 else -βˆ‘ x ∈ Ico a b, (mu π•œ) a x
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_apply_of_eq
[395, 1]
[395, 88]
rw [mu_apply, if_pos h]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a = b ⊒ (mu π•œ) a b = 1
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_apply_of_ne
[399, 1]
[400, 26]
rw [mu_apply, if_neg h]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b ⊒ (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
have : mu π•œ a b = _ := mu_apply_of_ne h
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x = 0
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
by_cases hab : a ≀ b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x = 0
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x = 0 case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
rw [Icc_eq_cons_Ico hab]
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x = 0
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : a ≀ b ⊒ βˆ‘ x ∈ cons b (Ico a b) β‹―, (mu π•œ) a x = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
simp [this, neg_add_self]
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : a ≀ b ⊒ βˆ‘ x ∈ cons b (Ico a b) β‹―, (mu π•œ) a x = 0
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
have : βˆ€ x ∈ Icc a b, Β¬a ≀ x := by intro x hx hn apply hab rw [mem_Icc] at hx exact le_trans hn hx.2
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x = 0
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this✝ : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b this : βˆ€ x ∈ Icc a b, Β¬a ≀ x ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
convert sum_const_zero
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this✝ : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b this : βˆ€ x ∈ Icc a b, Β¬a ≀ x ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x = 0
case h.e'_2.a 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this✝ : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b this : βˆ€ x ∈ Icc a b, Β¬a ≀ x x✝ : Ξ± a✝ : x✝ ∈ Icc a b ⊒ (mu π•œ) a x✝ = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
simp [apply_eq_zero_of_not_le (this β€Ή_β€Ί β€Ή_β€Ί)]
case h.e'_2.a 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this✝ : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b this : βˆ€ x ∈ Icc a b, Β¬a ≀ x x✝ : Ξ± a✝ : x✝ ∈ Icc a b ⊒ (mu π•œ) a x✝ = 0
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
intro x hx hn
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b ⊒ βˆ€ x ∈ Icc a b, Β¬a ≀ x
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b x : Ξ± hx : x ∈ Icc a b hn : a ≀ x ⊒ False
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
apply hab
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b x : Ξ± hx : x ∈ Icc a b hn : a ≀ x ⊒ False
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b x : Ξ± hx : x ∈ Icc a b hn : a ≀ x ⊒ a ≀ b
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
rw [mem_Icc] at hx
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b x : Ξ± hx : x ∈ Icc a b hn : a ≀ x ⊒ a ≀ b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b x : Ξ± hx : a ≀ x ∧ x ≀ b hn : a ≀ x ⊒ a ≀ b
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_right
[408, 1]
[419, 50]
exact le_trans hn hx.2
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (mu π•œ) a b = -βˆ‘ x ∈ Ico a b, (mu π•œ) a x hab : Β¬a ≀ b x : Ξ± hx : a ≀ x ∧ x ≀ b hn : a ≀ x ⊒ a ≀ b
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'Aux_apply
[441, 1]
[443, 14]
rw [mu'Aux]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± ⊒ IncidenceAlgebra.mu'Aux π•œ b a = if a = b then 1 else -βˆ‘ x ∈ (Ioc a b).attach, IncidenceAlgebra.mu'Aux π•œ b ↑x
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_apply
[458, 1]
[459, 45]
rw [mu', coe_mk, mu'Aux_apply, sum_attach]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± ⊒ (IncidenceAlgebra.mu' π•œ) a b = if a = b then 1 else -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_apply_of_ne
[461, 1]
[462, 27]
rw [mu'_apply, if_neg h]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b ⊒ (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_apply_of_eq
[464, 1]
[464, 91]
rw [mu'_apply, if_pos h]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : Preorder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a = b ⊒ (IncidenceAlgebra.mu' π•œ) a b = 1
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
have : mu' π•œ a b = _ := mu'_apply_of_ne h
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b ⊒ βˆ‘ x ∈ Icc a b, (IncidenceAlgebra.mu' π•œ) x b = 0
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b ⊒ βˆ‘ x ∈ Icc a b, (IncidenceAlgebra.mu' π•œ) x b = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
by_cases hab : a ≀ b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b ⊒ βˆ‘ x ∈ Icc a b, (IncidenceAlgebra.mu' π•œ) x b = 0
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (IncidenceAlgebra.mu' π•œ) x b = 0 case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (IncidenceAlgebra.mu' π•œ) x b = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
rw [Icc_eq_cons_Ioc hab]
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (IncidenceAlgebra.mu' π•œ) x b = 0
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : a ≀ b ⊒ βˆ‘ x ∈ cons a (Ioc a b) β‹―, (IncidenceAlgebra.mu' π•œ) x b = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
simp [this, neg_add_self]
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : a ≀ b ⊒ βˆ‘ x ∈ cons a (Ioc a b) β‹―, (IncidenceAlgebra.mu' π•œ) x b = 0
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
have : βˆ€ x ∈ Icc a b, Β¬x ≀ b := by intro x hx hn apply hab rw [mem_Icc] at hx exact le_trans hx.1 hn
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (IncidenceAlgebra.mu' π•œ) x b = 0
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this✝ : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b this : βˆ€ x ∈ Icc a b, Β¬x ≀ b ⊒ βˆ‘ x ∈ Icc a b, (IncidenceAlgebra.mu' π•œ) x b = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
convert sum_const_zero
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this✝ : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b this : βˆ€ x ∈ Icc a b, Β¬x ≀ b ⊒ βˆ‘ x ∈ Icc a b, (IncidenceAlgebra.mu' π•œ) x b = 0
case h.e'_2.a 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this✝ : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b this : βˆ€ x ∈ Icc a b, Β¬x ≀ b x✝ : Ξ± a✝ : x✝ ∈ Icc a b ⊒ (IncidenceAlgebra.mu' π•œ) x✝ b = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
simp [apply_eq_zero_of_not_le (this β€Ή_β€Ί β€Ή_β€Ί)]
case h.e'_2.a 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this✝ : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b this : βˆ€ x ∈ Icc a b, Β¬x ≀ b x✝ : Ξ± a✝ : x✝ ∈ Icc a b ⊒ (IncidenceAlgebra.mu' π•œ) x✝ b = 0
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
intro x hx hn
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b ⊒ βˆ€ x ∈ Icc a b, Β¬x ≀ b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b x : Ξ± hx : x ∈ Icc a b hn : x ≀ b ⊒ False
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
apply hab
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b x : Ξ± hx : x ∈ Icc a b hn : x ≀ b ⊒ False
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b x : Ξ± hx : x ∈ Icc a b hn : x ≀ b ⊒ a ≀ b
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
rw [mem_Icc] at hx
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b x : Ξ± hx : x ∈ Icc a b hn : x ≀ b ⊒ a ≀ b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b x : Ξ± hx : a ≀ x ∧ x ≀ b hn : x ≀ b ⊒ a ≀ b
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu'_spec_of_ne_left
[476, 1]
[487, 50]
exact le_trans hx.1 hn
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : AddCommGroup π•œ inst✝³ : One π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b this : (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b hab : Β¬a ≀ b x : Ξ± hx : a ≀ x ∧ x ≀ b hn : x ≀ b ⊒ a ≀ b
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_mul_zeta
[496, 1]
[503, 35]
ext a b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 ⊒ mu π•œ * zeta π•œ = 1
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b ⊒ (mu π•œ * zeta π•œ) a b = 1 a b
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_mul_zeta
[496, 1]
[503, 35]
rw [mul_apply, one_apply]
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b ⊒ (mu π•œ * zeta π•œ) a b = 1 a b
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x * (zeta π•œ) x b = if a = b then 1 else 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_mul_zeta
[496, 1]
[503, 35]
split_ifs with he
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x * (zeta π•œ) x b = if a = b then 1 else 0
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : a = b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x * (zeta π•œ) x b = 1 case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x * (zeta π•œ) x b = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_mul_zeta
[496, 1]
[503, 35]
simp [he]
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : a = b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x * (zeta π•œ) x b = 1
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_mul_zeta
[496, 1]
[503, 35]
simp only [mul_one, zeta_apply, mul_ite]
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) a x * (zeta π•œ) x b = 0
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ (βˆ‘ x ∈ Icc a b, if x ≀ b then (mu π•œ) a x else (mu π•œ) a x * 0) = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_mul_zeta
[496, 1]
[503, 35]
convert mu_spec_of_ne_right he
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ (βˆ‘ x ∈ Icc a b, if x ≀ b then (mu π•œ) a x else (mu π•œ) a x * 0) = 0
case h.e'_2.a 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝¹ : a ≀ b he : Β¬a = b x✝ : Ξ± a✝ : x✝ ∈ Icc a b ⊒ (if x✝ ≀ b then (mu π•œ) a x✝ else (mu π•œ) a x✝ * 0) = (mu π•œ) a x✝ case neg.convert_3 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ One π•œ
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_mul_zeta
[496, 1]
[503, 35]
rw [if_pos (mem_Icc.mp β€Ή_β€Ί).2]
case h.e'_2.a 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝¹ : a ≀ b he : Β¬a = b x✝ : Ξ± a✝ : x✝ ∈ Icc a b ⊒ (if x✝ ≀ b then (mu π•œ) a x✝ else (mu π•œ) a x✝ * 0) = (mu π•œ) a x✝ case neg.convert_3 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ One π•œ
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_mu'
[505, 1]
[512, 35]
ext a b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 ⊒ zeta π•œ * IncidenceAlgebra.mu' π•œ = 1
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b ⊒ (zeta π•œ * IncidenceAlgebra.mu' π•œ) a b = 1 a b
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_mu'
[505, 1]
[512, 35]
rw [mul_apply, one_apply]
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b ⊒ (zeta π•œ * IncidenceAlgebra.mu' π•œ) a b = 1 a b
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (zeta π•œ) a x * (IncidenceAlgebra.mu' π•œ) x b = if a = b then 1 else 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_mu'
[505, 1]
[512, 35]
split_ifs with he
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b ⊒ βˆ‘ x ∈ Icc a b, (zeta π•œ) a x * (IncidenceAlgebra.mu' π•œ) x b = if a = b then 1 else 0
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : a = b ⊒ βˆ‘ x ∈ Icc a b, (zeta π•œ) a x * (IncidenceAlgebra.mu' π•œ) x b = 1 case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ βˆ‘ x ∈ Icc a b, (zeta π•œ) a x * (IncidenceAlgebra.mu' π•œ) x b = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_mu'
[505, 1]
[512, 35]
simp [he]
case pos 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : a = b ⊒ βˆ‘ x ∈ Icc a b, (zeta π•œ) a x * (IncidenceAlgebra.mu' π•œ) x b = 1
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_mu'
[505, 1]
[512, 35]
simp only [zeta_apply, one_mul, ite_mul]
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ βˆ‘ x ∈ Icc a b, (zeta π•œ) a x * (IncidenceAlgebra.mu' π•œ) x b = 0
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ (βˆ‘ x ∈ Icc a b, if a ≀ x then (IncidenceAlgebra.mu' π•œ) x b else 0 * (IncidenceAlgebra.mu' π•œ) x b) = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_mu'
[505, 1]
[512, 35]
convert mu'_spec_of_ne_left he
case neg 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ (βˆ‘ x ∈ Icc a b, if a ≀ x then (IncidenceAlgebra.mu' π•œ) x b else 0 * (IncidenceAlgebra.mu' π•œ) x b) = 0
case h.e'_2.a 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝¹ : a ≀ b he : Β¬a = b x✝ : Ξ± a✝ : x✝ ∈ Icc a b ⊒ (if a ≀ x✝ then (IncidenceAlgebra.mu' π•œ) x✝ b else 0 * (IncidenceAlgebra.mu' π•œ) x✝ b) = (IncidenceAlgebra.mu' π•œ) x✝ b case neg.convert_3 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ One π•œ
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_mu'
[505, 1]
[512, 35]
rw [if_pos (mem_Icc.mp β€Ή_β€Ί).1]
case h.e'_2.a 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝¹ : a ≀ b he : Β¬a = b x✝ : Ξ± a✝ : x✝ ∈ Icc a b ⊒ (if a ≀ x✝ then (IncidenceAlgebra.mu' π•œ) x✝ b else 0 * (IncidenceAlgebra.mu' π•œ) x✝ b) = (IncidenceAlgebra.mu' π•œ) x✝ b case neg.convert_3 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁡ : AddCommGroup π•œ inst✝⁴ : MulOneClass π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 a b : Ξ± a✝ : a ≀ b he : Β¬a = b ⊒ One π•œ
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_eq_mu'
[520, 1]
[522, 67]
classical exact left_inv_eq_right_inv (mu_mul_zeta _ _) (zeta_mul_mu' _ _)
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± ⊒ mu π•œ = IncidenceAlgebra.mu' π•œ
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_eq_mu'
[520, 1]
[522, 67]
exact left_inv_eq_right_inv (mu_mul_zeta _ _) (zeta_mul_mu' _ _)
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± ⊒ mu π•œ = IncidenceAlgebra.mu' π•œ
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_apply_of_ne'
[524, 1]
[525, 42]
rw [mu_eq_mu']
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b ⊒ (mu π•œ) a b = -βˆ‘ x ∈ Ioc a b, (mu π•œ) x b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b ⊒ (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_apply_of_ne'
[524, 1]
[525, 42]
exact mu'_apply_of_ne h
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b ⊒ (IncidenceAlgebra.mu' π•œ) a b = -βˆ‘ x ∈ Ioc a b, (IncidenceAlgebra.mu' π•œ) x b
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_mu
[527, 1]
[528, 41]
rw [mu_eq_mu']
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 ⊒ zeta π•œ * mu π•œ = 1
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 ⊒ zeta π•œ * IncidenceAlgebra.mu' π•œ = 1
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.zeta_mul_mu
[527, 1]
[528, 41]
exact zeta_mul_mu' _ _
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : LocallyFiniteOrder Ξ± inst✝¹ : DecidableEq Ξ± inst✝ : DecidableRel fun x x_1 => x ≀ x_1 ⊒ zeta π•œ * IncidenceAlgebra.mu' π•œ = 1
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_spec_of_ne_left
[530, 1]
[531, 40]
rw [mu_eq_mu', mu'_spec_of_ne_left h]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± h : a β‰  b ⊒ βˆ‘ x ∈ Icc a b, (mu π•œ) x b = 0
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
let mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b ↦ mu π•œ (ofDual b) (ofDual a) eq_zero_of_not_le' := fun a b hab ↦ apply_eq_zero_of_not_le (by exact hab) _ }
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 ⊒ (mu π•œ) (toDual a) (toDual b) = (mu π•œ) b a
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } ⊒ (mu π•œ) (toDual a) (toDual b) = (mu π•œ) b a
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
suffices mu π•œ = mud by rw [this]; rfl
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } ⊒ (mu π•œ) (toDual a) (toDual b) = (mu π•œ) b a
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } ⊒ mu π•œ = mud
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
clear a b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } ⊒ mud * zeta π•œ = 1
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } ⊒ mud * zeta π•œ = 1
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
ext a b
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } ⊒ mud * zeta π•œ = 1
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b ⊒ (mud * zeta π•œ) a b = 1 a b
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
simp only [mul_boole, one_apply, mul_apply, coe_mk, zeta_apply]
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b ⊒ (mud * zeta π•œ) a b = 1 a b
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b ⊒ (βˆ‘ x ∈ Icc a b, if x ≀ b then mud a x else 0) = if a = b then 1 else 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
obtain rfl | h := eq_or_ne a b
case h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b ⊒ (βˆ‘ x ∈ Icc a b, if x ≀ b then mud a x else 0) = if a = b then 1 else 0
case h.inl 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a : Ξ±α΅’α΅ˆ a✝ : a ≀ a ⊒ (βˆ‘ x ∈ Icc a a, if x ≀ a then mud a x else 0) = if a = a then 1 else 0 case h.inr 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b h : a β‰  b ⊒ (βˆ‘ x ∈ Icc a b, if x ≀ b then mud a x else 0) = if a = b then 1 else 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
exact hab
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a✝ b✝ : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 a b : Ξ±α΅’α΅ˆ hab : Β¬a ≀ b ⊒ Β¬ofDual b ≀ ofDual a
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
rw [this]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this✝ : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } this : mu π•œ = mud ⊒ (mu π•œ) (toDual a) (toDual b) = (mu π•œ) b a
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this✝ : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } this : mu π•œ = mud ⊒ mud (toDual a) (toDual b) = (mu π•œ) b a
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
rfl
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this✝ : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } this : mu π•œ = mud ⊒ mud (toDual a) (toDual b) = (mu π•œ) b a
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
rw [← mu_mul_zeta] at this
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this✝ : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } this : mud * zeta π•œ = 1 ⊒ mu π•œ = mud
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this✝ : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } this : mud * zeta π•œ = mu π•œ * zeta π•œ ⊒ mu π•œ = mud
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
symm
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this✝ : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } this : mud * zeta π•œ * mu π•œ = mu π•œ * zeta π•œ * mu π•œ ⊒ mu π•œ = mud
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this✝ : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } this : mud * zeta π•œ * mu π•œ = mu π•œ * zeta π•œ * mu π•œ ⊒ mud = mu π•œ
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
simpa [mul_assoc, zeta_mul_mu] using this
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± this✝ : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } this : mud * zeta π•œ * mu π•œ = mu π•œ * zeta π•œ * mu π•œ ⊒ mud = mu π•œ
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
simp [mud]
case h.inl 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a : Ξ±α΅’α΅ˆ a✝ : a ≀ a ⊒ (βˆ‘ x ∈ Icc a a, if x ≀ a then mud a x else 0) = if a = a then 1 else 0
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
rw [if_neg h]
case h.inr 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b h : a β‰  b ⊒ (βˆ‘ x ∈ Icc a b, if x ≀ b then mud a x else 0) = if a = b then 1 else 0
case h.inr 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b h : a β‰  b ⊒ (βˆ‘ x ∈ Icc a b, if x ≀ b then mud a x else 0) = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
convert_to βˆ‘ x in Icc (ofDual b) (ofDual a), mu π•œ x a = 0
case h.inr 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b h : a β‰  b ⊒ (βˆ‘ x ∈ Icc a b, if x ≀ b then mud a x else 0) = 0
case h.e'_2 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b h : a β‰  b ⊒ (βˆ‘ x ∈ Icc a b, if x ≀ b then mud a x else 0) = βˆ‘ x ∈ Icc (ofDual b) (ofDual a), (mu π•œ) x a case h.inr 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b h : a β‰  b ⊒ βˆ‘ x ∈ Icc (ofDual b) (ofDual a), (mu π•œ) x a = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
sorry
case h.e'_2 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b h : a β‰  b ⊒ (βˆ‘ x ∈ Icc a b, if x ≀ b then mud a x else 0) = βˆ‘ x ∈ Icc (ofDual b) (ofDual a), (mu π•œ) x a case h.inr 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b h : a β‰  b ⊒ βˆ‘ x ∈ Icc (ofDual b) (ofDual a), (mu π•œ) x a = 0
case h.inr 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b h : a β‰  b ⊒ βˆ‘ x ∈ Icc (ofDual b) (ofDual a), (mu π•œ) x a = 0
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.mu_toDual
[539, 1]
[559, 10]
sorry
case h.inr 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝³ : Ring π•œ inst✝² : PartialOrder Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 mud : IncidenceAlgebra π•œ Ξ±α΅’α΅ˆ := { toFun := fun a b => (mu π•œ) (ofDual b) (ofDual a), eq_zero_of_not_le' := β‹― } a b : Ξ±α΅’α΅ˆ a✝ : a ≀ b h : a β‰  b ⊒ βˆ‘ x ∈ Icc (ofDual b) (ofDual a), (mu π•œ) x a = 0
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.moebius_inversion_top
[572, 1]
[598, 65]
symm
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 ⊒ f x = βˆ‘ y ∈ Ici x, (mu π•œ) x y * g y
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 ⊒ βˆ‘ y ∈ Ici x, (mu π•œ) x y * g y = f x
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.moebius_inversion_top
[572, 1]
[598, 65]
calc βˆ‘ y in Ici x, mu π•œ x y * g y = βˆ‘ y in Ici x, mu π•œ x y * βˆ‘ z in Ici y, f z := by simp_rw [h] _ = βˆ‘ y in Ici x, mu π•œ x y * βˆ‘ z in Ici y, zeta π•œ y z * f z := by congr with y rw [sum_congr rfl fun z hz ↦ ?_] rw [zeta_apply, if_pos (mem_Ici.mp β€Ή_β€Ί), one_mul] _ = βˆ‘ y in Ici x, βˆ‘ z in Ici y, mu π•œ x y * zeta π•œ y z * f z := by simp [mul_sum] _ = βˆ‘ z in Ici x, βˆ‘ y in Icc x z, mu π•œ x y * zeta π•œ y z * f z := by erw [sum_sigma' (Ici x) fun y ↦ Ici y] erw [sum_sigma' (Ici x) fun z ↦ Icc x z] simp only [mul_boole, MulZeroClass.zero_mul, ite_mul, zeta_apply] apply sum_nbij' (fun ⟨a, b⟩ ↦ ⟨b, a⟩) (fun ⟨a, b⟩ ↦ ⟨b, a⟩) <;> aesop (add simp mul_assoc) (add unsafe le_trans) _ = βˆ‘ z in Ici x, (mu π•œ * zeta π•œ : IncidenceAlgebra π•œ Ξ±) x z * f z := by simp_rw [mul_apply, sum_mul] _ = βˆ‘ y in Ici x, βˆ‘ z in Ici y, (1 : IncidenceAlgebra π•œ Ξ±) x z * f z := by simp [mu_mul_zeta π•œ, ← add_sum_Ioi] exact sum_eq_zero fun y hy ↦ if_neg (mem_Ioi.mp hy).not_le _ = f x := by simp [one_apply, ← add_sum_Ioi] exact sum_eq_zero fun y hy ↦ if_neg (mem_Ioi.mp hy).not_le
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 ⊒ βˆ‘ y ∈ Ici x, (mu π•œ) x y * g y = f x
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.moebius_inversion_top
[572, 1]
[598, 65]
simp_rw [h]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 ⊒ βˆ‘ y ∈ Ici x, (mu π•œ) x y * g y = βˆ‘ y ∈ Ici x, (mu π•œ) x y * βˆ‘ z ∈ Ici y, f z
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.moebius_inversion_top
[572, 1]
[598, 65]
congr with y
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 ⊒ βˆ‘ y ∈ Ici x, (mu π•œ) x y * βˆ‘ z ∈ Ici y, f z = βˆ‘ y ∈ Ici x, (mu π•œ) x y * βˆ‘ z ∈ Ici y, (zeta π•œ) y z * f z
case e_f.h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 y : Ξ± ⊒ (mu π•œ) x y * βˆ‘ z ∈ Ici y, f z = (mu π•œ) x y * βˆ‘ z ∈ Ici y, (zeta π•œ) y z * f z
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.moebius_inversion_top
[572, 1]
[598, 65]
rw [sum_congr rfl fun z hz ↦ ?_]
case e_f.h 𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 y : Ξ± ⊒ (mu π•œ) x y * βˆ‘ z ∈ Ici y, f z = (mu π•œ) x y * βˆ‘ z ∈ Ici y, (zeta π•œ) y z * f z
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 y z : Ξ± hz : z ∈ Ici y ⊒ f z = (zeta π•œ) y z * f z
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.moebius_inversion_top
[572, 1]
[598, 65]
rw [zeta_apply, if_pos (mem_Ici.mp β€Ή_β€Ί), one_mul]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 y z : Ξ± hz : z ∈ Ici y ⊒ f z = (zeta π•œ) y z * f z
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.moebius_inversion_top
[572, 1]
[598, 65]
simp [mul_sum]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 ⊒ βˆ‘ y ∈ Ici x, (mu π•œ) x y * βˆ‘ z ∈ Ici y, (zeta π•œ) y z * f z = βˆ‘ y ∈ Ici x, βˆ‘ z ∈ Ici y, (mu π•œ) x y * (zeta π•œ) y z * f z
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/Incidence.lean
IncidenceAlgebra.moebius_inversion_top
[572, 1]
[598, 65]
erw [sum_sigma' (Ici x) fun y ↦ Ici y]
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 ⊒ βˆ‘ y ∈ Ici x, βˆ‘ z ∈ Ici y, (mu π•œ) x y * (zeta π•œ) y z * f z = βˆ‘ z ∈ Ici x, βˆ‘ y ∈ Icc x z, (mu π•œ) x y * (zeta π•œ) y z * f z
𝕄 : Type u_1 F : Type u_2 π•œ : Type u_3 𝕝 : Type u_4 π•ž : Type u_5 Ξ± : Type u_6 Ξ² : Type u_7 inst✝⁴ : Ring π•œ inst✝³ : PartialOrder Ξ± inst✝² : OrderTop Ξ± inst✝¹ : LocallyFiniteOrder Ξ± inst✝ : DecidableEq Ξ± a b : Ξ± f g : Ξ± β†’ π•œ h : βˆ€ (x : Ξ±), g x = βˆ‘ y ∈ Ici x, f y x : Ξ± this : DecidableRel fun x x_1 => x ≀ x_1 := Classical.decRel fun x x_1 => x ≀ x_1 ⊒ βˆ‘ x_1 ∈ (Ici x).sigma fun y => Ici y, (mu π•œ) x x_1.fst * (zeta π•œ) x_1.fst x_1.snd * f x_1.snd = βˆ‘ z ∈ Ici x, βˆ‘ y ∈ Icc x z, (mu π•œ) x y * (zeta π•œ) y z * f z