url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
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stringlengths 1
11.2k
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stringlengths 3
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2.09M
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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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (mu 𝕜) a b = -∑ x ∈ Ico a b, (mu 𝕜) a x
hab : ¬a ≤ b
x : α
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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
⊢ IncidenceAlgebra.mu'Aux 𝕜 b a = if a = b then 1 else -∑ x ∈ (Ioc a b).attach, IncidenceAlgebra.mu'Aux 𝕜 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
⊢ (IncidenceAlgebra.mu' 𝕜) a b = if a = b then 1 else -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
⊢ (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a = b
⊢ (IncidenceAlgebra.mu' 𝕜) 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' 𝕜 a b = _ := mu'_apply_of_ne h | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
⊢ ∑ x ∈ Icc a b, (IncidenceAlgebra.mu' 𝕜) x b = 0 | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
⊢ ∑ x ∈ Icc a b, (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
⊢ ∑ x ∈ Icc a b, (IncidenceAlgebra.mu' 𝕜) x b = 0 | case pos
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : a ≤ b
⊢ ∑ x ∈ Icc a b, (IncidenceAlgebra.mu' 𝕜) x b = 0
case neg
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
⊢ ∑ x ∈ Icc a b, (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : a ≤ b
⊢ ∑ x ∈ Icc a b, (IncidenceAlgebra.mu' 𝕜) x b = 0 | case pos
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : a ≤ b
⊢ ∑ x ∈ cons a (Ioc a b) ⋯, (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : a ≤ b
⊢ ∑ x ∈ cons a (Ioc a b) ⋯, (IncidenceAlgebra.mu' 𝕜) 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 : ∀ x ∈ Icc a b, ¬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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
⊢ ∑ x ∈ Icc a b, (IncidenceAlgebra.mu' 𝕜) x b = 0 | case neg
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this✝ : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
this : ∀ x ∈ Icc a b, ¬x ≤ b
⊢ ∑ x ∈ Icc a b, (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this✝ : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
this : ∀ x ∈ Icc a b, ¬x ≤ b
⊢ ∑ x ∈ Icc a b, (IncidenceAlgebra.mu' 𝕜) x b = 0 | case h.e'_2.a
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this✝ : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
this : ∀ x ∈ Icc a b, ¬x ≤ b
x✝ : α
a✝ : x✝ ∈ Icc a b
⊢ (IncidenceAlgebra.mu' 𝕜) 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 ‹_› ‹_›)] | case h.e'_2.a
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this✝ : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
this : ∀ x ∈ Icc a b, ¬x ≤ b
x✝ : α
a✝ : x✝ ∈ Icc a b
⊢ (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
⊢ ∀ x ∈ Icc a b, ¬x ≤ b | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
x : α
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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
x : α
hx : x ∈ Icc a b
hn : x ≤ b
⊢ False | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
x : α
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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
x : α
hx : x ∈ Icc a b
hn : x ≤ b
⊢ a ≤ b | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
x : α
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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
this : (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) x b
hab : ¬a ≤ b
x : α
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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
⊢ mu 𝕜 * zeta 𝕜 = 1 | case h
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
⊢ (mu 𝕜 * zeta 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
⊢ (mu 𝕜 * zeta 𝕜) a b = 1 a b | case h
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
⊢ ∑ x ∈ Icc a b, (mu 𝕜) a x * (zeta 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
⊢ ∑ x ∈ Icc a b, (mu 𝕜) a x * (zeta 𝕜) x b = if a = b then 1 else 0 | case pos
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : a = b
⊢ ∑ x ∈ Icc a b, (mu 𝕜) a x * (zeta 𝕜) x b = 1
case neg
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ ∑ x ∈ Icc a b, (mu 𝕜) a x * (zeta 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : a = b
⊢ ∑ x ∈ Icc a b, (mu 𝕜) a x * (zeta 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ ∑ x ∈ Icc a b, (mu 𝕜) a x * (zeta 𝕜) x b = 0 | case neg
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ (∑ x ∈ Icc a b, if x ≤ b then (mu 𝕜) a x else (mu 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ (∑ x ∈ Icc a b, if x ≤ b then (mu 𝕜) a x else (mu 𝕜) a x * 0) = 0 | case h.e'_2.a
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝¹ : a ≤ b
he : ¬a = b
x✝ : α
a✝ : x✝ ∈ Icc a b
⊢ (if x✝ ≤ b then (mu 𝕜) a x✝ else (mu 𝕜) a x✝ * 0) = (mu 𝕜) a x✝
case neg.convert_3
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ One 𝕜 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.mu_mul_zeta | [496, 1] | [503, 35] | rw [if_pos (mem_Icc.mp ‹_›).2] | case h.e'_2.a
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝¹ : a ≤ b
he : ¬a = b
x✝ : α
a✝ : x✝ ∈ Icc a b
⊢ (if x✝ ≤ b then (mu 𝕜) a x✝ else (mu 𝕜) a x✝ * 0) = (mu 𝕜) a x✝
case neg.convert_3
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ One 𝕜 | 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
⊢ zeta 𝕜 * IncidenceAlgebra.mu' 𝕜 = 1 | case h
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
⊢ (zeta 𝕜 * IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
⊢ (zeta 𝕜 * IncidenceAlgebra.mu' 𝕜) a b = 1 a b | case h
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
⊢ ∑ x ∈ Icc a b, (zeta 𝕜) a x * (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
⊢ ∑ x ∈ Icc a b, (zeta 𝕜) a x * (IncidenceAlgebra.mu' 𝕜) x b = if a = b then 1 else 0 | case pos
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : a = b
⊢ ∑ x ∈ Icc a b, (zeta 𝕜) a x * (IncidenceAlgebra.mu' 𝕜) x b = 1
case neg
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ ∑ x ∈ Icc a b, (zeta 𝕜) a x * (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : a = b
⊢ ∑ x ∈ Icc a b, (zeta 𝕜) a x * (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ ∑ x ∈ Icc a b, (zeta 𝕜) a x * (IncidenceAlgebra.mu' 𝕜) x b = 0 | case neg
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ (∑ x ∈ Icc a b, if a ≤ x then (IncidenceAlgebra.mu' 𝕜) x b else 0 * (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ (∑ x ∈ Icc a b, if a ≤ x then (IncidenceAlgebra.mu' 𝕜) x b else 0 * (IncidenceAlgebra.mu' 𝕜) x b) = 0 | case h.e'_2.a
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝¹ : a ≤ b
he : ¬a = b
x✝ : α
a✝ : x✝ ∈ Icc a b
⊢ (if a ≤ x✝ then (IncidenceAlgebra.mu' 𝕜) x✝ b else 0 * (IncidenceAlgebra.mu' 𝕜) x✝ b) = (IncidenceAlgebra.mu' 𝕜) x✝ b
case neg.convert_3
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ One 𝕜 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.zeta_mul_mu' | [505, 1] | [512, 35] | rw [if_pos (mem_Icc.mp ‹_›).1] | case h.e'_2.a
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝¹ : a ≤ b
he : ¬a = b
x✝ : α
a✝ : x✝ ∈ Icc a b
⊢ (if a ≤ x✝ then (IncidenceAlgebra.mu' 𝕜) x✝ b else 0 * (IncidenceAlgebra.mu' 𝕜) x✝ b) = (IncidenceAlgebra.mu' 𝕜) x✝ b
case neg.convert_3
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : MulOneClass 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α
a✝ : a ≤ b
he : ¬a = b
⊢ One 𝕜 | 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
⊢ mu 𝕜 = IncidenceAlgebra.mu' 𝕜 | 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
⊢ mu 𝕜 = IncidenceAlgebra.mu' 𝕜 | 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
⊢ (mu 𝕜) a b = -∑ x ∈ Ioc a b, (mu 𝕜) x b | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
⊢ (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
⊢ (IncidenceAlgebra.mu' 𝕜) a b = -∑ x ∈ Ioc a b, (IncidenceAlgebra.mu' 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
⊢ zeta 𝕜 * mu 𝕜 = 1 | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
⊢ zeta 𝕜 * IncidenceAlgebra.mu' 𝕜 = 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
⊢ zeta 𝕜 * IncidenceAlgebra.mu' 𝕜 = 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
h : a ≠ b
⊢ ∑ x ∈ Icc a b, (mu 𝕜) 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 𝕜 αᵒᵈ :=
{ toFun := fun a b ↦ mu 𝕜 (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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ (mu 𝕜) (toDual a) (toDual b) = (mu 𝕜) b a | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
⊢ (mu 𝕜) (toDual a) (toDual b) = (mu 𝕜) b a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.mu_toDual | [539, 1] | [559, 10] | suffices mu 𝕜 = mud by rw [this]; rfl | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
⊢ (mu 𝕜) (toDual a) (toDual b) = (mu 𝕜) b a | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
⊢ mu 𝕜 = 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
⊢ mud * zeta 𝕜 = 1 | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
⊢ mud * zeta 𝕜 = 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
⊢ mud * zeta 𝕜 = 1 | case h
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
⊢ (mud * zeta 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
⊢ (mud * zeta 𝕜) a b = 1 a b | case h
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
⊢ (∑ 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
⊢ (∑ 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a : αᵒᵈ
a✝ : a ≤ a
⊢ (∑ 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
h : a ≠ b
⊢ (∑ 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a✝ b✝ : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
a b : αᵒᵈ
hab : ¬a ≤ b
⊢ ¬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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this✝ : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
this : mu 𝕜 = mud
⊢ (mu 𝕜) (toDual a) (toDual b) = (mu 𝕜) b a | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this✝ : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
this : mu 𝕜 = mud
⊢ mud (toDual a) (toDual b) = (mu 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this✝ : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
this : mu 𝕜 = mud
⊢ mud (toDual a) (toDual b) = (mu 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this✝ : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
this : mud * zeta 𝕜 = 1
⊢ mu 𝕜 = mud | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this✝ : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
this : mud * zeta 𝕜 = mu 𝕜 * zeta 𝕜
⊢ mu 𝕜 = 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this✝ : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
this : mud * zeta 𝕜 * mu 𝕜 = mu 𝕜 * zeta 𝕜 * mu 𝕜
⊢ mu 𝕜 = mud | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this✝ : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
this : mud * zeta 𝕜 * mu 𝕜 = mu 𝕜 * zeta 𝕜 * mu 𝕜
⊢ mud = mu 𝕜 |
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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this✝ : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
this : mud * zeta 𝕜 * mu 𝕜 = mu 𝕜 * zeta 𝕜 * mu 𝕜
⊢ mud = mu 𝕜 | 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a : αᵒᵈ
a✝ : a ≤ a
⊢ (∑ 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
h : a ≠ b
⊢ (∑ 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
h : a ≠ b
⊢ (∑ 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 ∑ x in Icc (ofDual b) (ofDual a), mu 𝕜 x a = 0 | case h.inr
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
h : a ≠ b
⊢ (∑ 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
h : a ≠ b
⊢ (∑ x ∈ Icc a b, if x ≤ b then mud a x else 0) = ∑ x ∈ Icc (ofDual b) (ofDual a), (mu 𝕜) x a
case h.inr
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
h : a ≠ b
⊢ ∑ x ∈ Icc (ofDual b) (ofDual a), (mu 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
h : a ≠ b
⊢ (∑ x ∈ Icc a b, if x ≤ b then mud a x else 0) = ∑ x ∈ Icc (ofDual b) (ofDual a), (mu 𝕜) x a
case h.inr
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
h : a ≠ b
⊢ ∑ x ∈ Icc (ofDual b) (ofDual a), (mu 𝕜) x a = 0 | case h.inr
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
h : a ≠ b
⊢ ∑ x ∈ Icc (ofDual b) (ofDual a), (mu 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
h : a ≠ b
⊢ ∑ x ∈ Icc (ofDual b) (ofDual a), (mu 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ f x = ∑ y ∈ Ici x, (mu 𝕜) x y * g y | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ y ∈ Ici x, (mu 𝕜) 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
∑ y in Ici x, mu 𝕜 x y * g y = ∑ y in Ici x, mu 𝕜 x y * ∑ z in Ici y, f z := by simp_rw [h]
_ = ∑ y in Ici x, mu 𝕜 x y * ∑ z in Ici y, zeta 𝕜 y z * f z := by
congr with y
rw [sum_congr rfl fun z hz ↦ ?_]
rw [zeta_apply, if_pos (mem_Ici.mp ‹_›), one_mul]
_ = ∑ y in Ici x, ∑ z in Ici y, mu 𝕜 x y * zeta 𝕜 y z * f z := by simp [mul_sum]
_ = ∑ z in Ici x, ∑ y in Icc x z, mu 𝕜 x y * zeta 𝕜 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)
_ = ∑ z in Ici x, (mu 𝕜 * zeta 𝕜 : IncidenceAlgebra 𝕜 α) x z * f z := by
simp_rw [mul_apply, sum_mul]
_ = ∑ y in Ici x, ∑ z in Ici y, (1 : IncidenceAlgebra 𝕜 α) x z * f z := by
simp [mu_mul_zeta 𝕜, ← 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ y ∈ Ici x, (mu 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ y ∈ Ici x, (mu 𝕜) x y * g y = ∑ y ∈ Ici x, (mu 𝕜) x y * ∑ 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ y ∈ Ici x, (mu 𝕜) x y * ∑ z ∈ Ici y, f z = ∑ y ∈ Ici x, (mu 𝕜) x y * ∑ z ∈ Ici y, (zeta 𝕜) y z * f z | case e_f.h
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
y : α
⊢ (mu 𝕜) x y * ∑ z ∈ Ici y, f z = (mu 𝕜) x y * ∑ z ∈ Ici y, (zeta 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
y : α
⊢ (mu 𝕜) x y * ∑ z ∈ Ici y, f z = (mu 𝕜) x y * ∑ z ∈ Ici y, (zeta 𝕜) y z * f z | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
y z : α
hz : z ∈ Ici y
⊢ f z = (zeta 𝕜) 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 ‹_›), one_mul] | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
y z : α
hz : z ∈ Ici y
⊢ f z = (zeta 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ y ∈ Ici x, (mu 𝕜) x y * ∑ z ∈ Ici y, (zeta 𝕜) y z * f z = ∑ y ∈ Ici x, ∑ z ∈ Ici y, (mu 𝕜) x y * (zeta 𝕜) 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
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ y ∈ Ici x, ∑ z ∈ Ici y, (mu 𝕜) x y * (zeta 𝕜) y z * f z =
∑ z ∈ Ici x, ∑ y ∈ Icc x z, (mu 𝕜) x y * (zeta 𝕜) y z * f z | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ x_1 ∈ (Ici x).sigma fun y => Ici y, (mu 𝕜) x x_1.fst * (zeta 𝕜) x_1.fst x_1.snd * f x_1.snd =
∑ z ∈ Ici x, ∑ y ∈ Icc x z, (mu 𝕜) x y * (zeta 𝕜) y z * f z |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.moebius_inversion_top | [572, 1] | [598, 65] | erw [sum_sigma' (Ici x) fun z ↦ Icc x z] | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ x_1 ∈ (Ici x).sigma fun y => Ici y, (mu 𝕜) x x_1.fst * (zeta 𝕜) x_1.fst x_1.snd * f x_1.snd =
∑ z ∈ Ici x, ∑ y ∈ Icc x z, (mu 𝕜) x y * (zeta 𝕜) y z * f z | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ x_1 ∈ (Ici x).sigma fun y => Ici y, (mu 𝕜) x x_1.fst * (zeta 𝕜) x_1.fst x_1.snd * f x_1.snd =
∑ x_1 ∈ (Ici x).sigma fun z => Icc x z, (mu 𝕜) x x_1.snd * (zeta 𝕜) x_1.snd x_1.fst * f x_1.fst |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.moebius_inversion_top | [572, 1] | [598, 65] | simp only [mul_boole, MulZeroClass.zero_mul, ite_mul, zeta_apply] | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ x_1 ∈ (Ici x).sigma fun y => Ici y, (mu 𝕜) x x_1.fst * (zeta 𝕜) x_1.fst x_1.snd * f x_1.snd =
∑ x_1 ∈ (Ici x).sigma fun z => Icc x z, (mu 𝕜) x x_1.snd * (zeta 𝕜) x_1.snd x_1.fst * f x_1.fst | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ (∑ x_1 ∈ (Ici x).sigma fun y => Ici y, if x_1.fst ≤ x_1.snd then (mu 𝕜) x x_1.fst * f x_1.snd else 0) =
∑ x_1 ∈ (Ici x).sigma fun z => Icc x z, if x_1.snd ≤ x_1.fst then (mu 𝕜) x x_1.snd * f x_1.fst else 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.moebius_inversion_top | [572, 1] | [598, 65] | apply sum_nbij' (fun ⟨a, b⟩ ↦ ⟨b, a⟩) (fun ⟨a, b⟩ ↦ ⟨b, a⟩) <;>
aesop (add simp mul_assoc) (add unsafe le_trans) | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ (∑ x_1 ∈ (Ici x).sigma fun y => Ici y, if x_1.fst ≤ x_1.snd then (mu 𝕜) x x_1.fst * f x_1.snd else 0) =
∑ x_1 ∈ (Ici x).sigma fun z => Icc x z, if x_1.snd ≤ x_1.fst then (mu 𝕜) x x_1.snd * f x_1.fst else 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.moebius_inversion_top | [572, 1] | [598, 65] | simp_rw [mul_apply, sum_mul] | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ z ∈ Ici x, ∑ y ∈ Icc x z, (mu 𝕜) x y * (zeta 𝕜) y z * f z = ∑ z ∈ Ici x, (mu 𝕜 * zeta 𝕜) x z * f z | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.moebius_inversion_top | [572, 1] | [598, 65] | simp [mu_mul_zeta 𝕜, ← add_sum_Ioi] | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ z ∈ Ici x, (mu 𝕜 * zeta 𝕜) x z * f z = ∑ y ∈ Ici x, ∑ z ∈ Ici y, 1 x z * f z | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ (∑ x_1 ∈ Ioi x, if x_1 ≤ x then f x else 0) = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.moebius_inversion_top | [572, 1] | [598, 65] | exact sum_eq_zero fun y hy ↦ if_neg (mem_Ioi.mp hy).not_le | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ (∑ x_1 ∈ Ioi x, if x_1 ≤ x then f x else 0) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.moebius_inversion_top | [572, 1] | [598, 65] | simp [one_apply, ← add_sum_Ioi] | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ ∑ y ∈ Ici x, ∑ z ∈ Ici y, 1 x z * f z = f x | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderTop α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Ici x, f y
x : α
this : DecidableRel fun x x_1 => x ≤ x_1 := Classical.decRel fun x x_1 => x ≤ x_1
⊢ (∑ x_1 ∈ Ioi x, if x_1 ≤ x then f x else 0) = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.moebius_inversion_bot | [606, 1] | [610, 76] | convert moebius_inversion_top (α := αᵒᵈ) f g h x using 3 | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderBot α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Iic x, f y
x : α
⊢ f x = ∑ y ∈ Iic x, (mu 𝕜) y x * g y | case h.e'_3.a.h.e'_5
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderBot α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Iic x, f y
x x✝ : α
a✝ : x✝ ∈ Ici x
⊢ (mu 𝕜) x✝ x = (mu 𝕜) x x✝ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.moebius_inversion_bot | [606, 1] | [610, 76] | erw [mu_toDual] | case h.e'_3.a.h.e'_5
𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder α
inst✝² : OrderBot α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
f g : α → 𝕜
h : ∀ (x : α), g x = ∑ y ∈ Iic x, f y
x x✝ : α
a✝ : x✝ ∈ Ici x
⊢ (mu 𝕜) x✝ x = (mu 𝕜) x x✝ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Incidence.lean | IncidenceAlgebra.zeta_prod_apply | [626, 1] | [627, 42] | simp [← ite_and, Prod.le_def, and_comm] | 𝕄 : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕝 : Type u_4
𝕞 : Type u_5
α : Type u_6
β : Type u_7
inst✝⁴ : Ring 𝕜
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
a b : α × β
⊢ (zeta 𝕜) a b = (zeta 𝕜) a.1 b.1 * (zeta 𝕜) a.2 b.2 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.deleteEdges_eq_self | [13, 1] | [15, 56] | rw [deleteEdges, sdiff_eq_left, disjoint_fromEdgeSet] | α : Type u_1
β : Type u_2
G H : SimpleGraph α
s✝ s : Set (Sym2 α)
⊢ G.deleteEdges s = G ↔ Disjoint G.edgeSet s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_empty | [30, 1] | [30, 74] | ext | α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a b : α
G : SimpleGraph α
⊢ G.ind ∅ = ⊥ | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a b : α
G : SimpleGraph α
x✝¹ x✝ : α
⊢ (G.ind ∅).Adj x✝¹ x✝ ↔ ⊥.Adj x✝¹ x✝ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_empty | [30, 1] | [30, 74] | simp | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a b : α
G : SimpleGraph α
x✝¹ x✝ : α
⊢ (G.ind ∅).Adj x✝¹ x✝ ↔ ⊥.Adj x✝¹ x✝ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_univ | [31, 1] | [31, 76] | ext | α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a b : α
G : SimpleGraph α
⊢ G.ind univ = G | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a b : α
G : SimpleGraph α
x✝¹ x✝ : α
⊢ (G.ind univ).Adj x✝¹ x✝ ↔ G.Adj x✝¹ x✝ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_univ | [31, 1] | [31, 76] | simp | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a b : α
G : SimpleGraph α
x✝¹ x✝ : α
⊢ (G.ind univ).Adj x✝¹ x✝ ↔ G.Adj x✝¹ x✝ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_singleton | [33, 1] | [35, 39] | ext | α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a✝ b : α
G : SimpleGraph α
a : α
⊢ G.ind {a} = ⊥ | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a✝ b : α
G : SimpleGraph α
a x✝¹ x✝ : α
⊢ (G.ind {a}).Adj x✝¹ x✝ ↔ ⊥.Adj x✝¹ x✝ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_singleton | [33, 1] | [35, 39] | simp | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a✝ b : α
G : SimpleGraph α
a x✝¹ x✝ : α
⊢ (G.ind {a}).Adj x✝¹ x✝ ↔ ⊥.Adj x✝¹ x✝ | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a✝ b : α
G : SimpleGraph α
a x✝¹ x✝ : α
⊢ G.Adj x✝¹ x✝ → x✝¹ = a → ¬x✝ = a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_singleton | [33, 1] | [35, 39] | rintro h rfl | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a✝ b : α
G : SimpleGraph α
a x✝¹ x✝ : α
⊢ G.Adj x✝¹ x✝ → x✝¹ = a → ¬x✝ = a | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a b : α
G : SimpleGraph α
x✝¹ x✝ : α
h : G.Adj x✝¹ x✝
⊢ ¬x✝ = x✝¹ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_singleton | [33, 1] | [35, 39] | exact h.ne' | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a b : α
G : SimpleGraph α
x✝¹ x✝ : α
h : G.Adj x✝¹ x✝
⊢ ¬x✝ = x✝¹ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_inter | [37, 1] | [39, 19] | ext | α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝¹ : Set (Sym2 α)
s✝ t✝ : Set α
a b : α
G : SimpleGraph α
s t : Set α
⊢ G.ind (s ∩ t) = G.ind s ⊓ G.ind t | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝¹ : Set (Sym2 α)
s✝ t✝ : Set α
a b : α
G : SimpleGraph α
s t : Set α
x✝¹ x✝ : α
⊢ (G.ind (s ∩ t)).Adj x✝¹ x✝ ↔ (G.ind s ⊓ G.ind t).Adj x✝¹ x✝ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_inter | [37, 1] | [39, 19] | simp | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝¹ : Set (Sym2 α)
s✝ t✝ : Set α
a b : α
G : SimpleGraph α
s t : Set α
x✝¹ x✝ : α
⊢ (G.ind (s ∩ t)).Adj x✝¹ x✝ ↔ (G.ind s ⊓ G.ind t).Adj x✝¹ x✝ | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝¹ : Set (Sym2 α)
s✝ t✝ : Set α
a b : α
G : SimpleGraph α
s t : Set α
x✝¹ x✝ : α
⊢ G.Adj x✝¹ x✝ ∧ (x✝¹ ∈ s ∧ x✝¹ ∈ t) ∧ x✝ ∈ s ∧ x✝ ∈ t ↔
(G.Adj x✝¹ x✝ ∧ x✝¹ ∈ s ∧ x✝ ∈ s) ∧ G.Adj x✝¹ x✝ ∧ x✝¹ ∈ t ∧ x✝ ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.ind_inter | [37, 1] | [39, 19] | tauto | case Adj.h.h.a
α : Type u_1
β : Type u_2
G✝ H : SimpleGraph α
s✝¹ : Set (Sym2 α)
s✝ t✝ : Set α
a b : α
G : SimpleGraph α
s t : Set α
x✝¹ x✝ : α
⊢ G.Adj x✝¹ x✝ ∧ (x✝¹ ∈ s ∧ x✝¹ ∈ t) ∧ x✝ ∈ s ∧ x✝ ∈ t ↔
(G.Adj x✝¹ x✝ ∧ x✝¹ ∈ s ∧ x✝ ∈ s) ∧ G.Adj x✝¹ x✝ ∧ x✝¹ ∈ t ∧ x✝ ∈ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.disjoint_ind | [41, 1] | [43, 77] | rw [disjoint_iff, ← ind_inter, disjoint_iff_inter_eq_empty.1 h, ind_empty] | α : Type u_1
β : Type u_2
G H : SimpleGraph α
s✝ : Set (Sym2 α)
s t : Set α
a b : α
h : Disjoint s t
⊢ Disjoint (G.ind s) (G.ind t) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/FieldTheory/Finite/Basic.lean | ZMod.pow_card_sub_one | [6, 1] | [9, 75] | split_ifs with hx | p : ℕ
inst✝ : Fact p.Prime
x : ZMod p
⊢ x ^ (p - 1) = if x ≠ 0 then 1 else 0 | case pos
p : ℕ
inst✝ : Fact p.Prime
x : ZMod p
hx : x ≠ 0
⊢ x ^ (p - 1) = 1
case neg
p : ℕ
inst✝ : Fact p.Prime
x : ZMod p
hx : ¬x ≠ 0
⊢ x ^ (p - 1) = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/FieldTheory/Finite/Basic.lean | ZMod.pow_card_sub_one | [6, 1] | [9, 75] | exact pow_card_sub_one_eq_one hx | case pos
p : ℕ
inst✝ : Fact p.Prime
x : ZMod p
hx : x ≠ 0
⊢ x ^ (p - 1) = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/FieldTheory/Finite/Basic.lean | ZMod.pow_card_sub_one | [6, 1] | [9, 75] | simp [of_not_not hx, (Fact.out : p.Prime).one_lt, tsub_eq_zero_iff_le] | case neg
p : ℕ
inst✝ : Fact p.Prime
x : ZMod p
hx : ¬x ≠ 0
⊢ x ^ (p - 1) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Independence.lean | AffineIndependent.convexIndependent | [11, 1] | [15, 8] | intro s x hx | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hp : AffineIndependent 𝕜 p
⊢ ConvexIndependent 𝕜 p | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hp : AffineIndependent 𝕜 p
s : Set ι
x : ι
hx : p x ∈ (convexHull 𝕜) (p '' s)
⊢ x ∈ s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Independence.lean | AffineIndependent.convexIndependent | [11, 1] | [15, 8] | by_contra | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hp : AffineIndependent 𝕜 p
s : Set ι
x : ι
hx : p x ∈ (convexHull 𝕜) (p '' s)
⊢ x ∈ s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hp : AffineIndependent 𝕜 p
s : Set ι
x : ι
hx : p x ∈ (convexHull 𝕜) (p '' s)
x✝ : x ∉ s
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Independence.lean | AffineIndependent.convexIndependent | [11, 1] | [15, 8] | sorry | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hp : AffineIndependent 𝕜 p
s : Set ι
x : ι
hx : p x ∈ (convexHull 𝕜) (p '' s)
x✝ : x ∉ s
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Finset/PosDiffs.lean | Finset.mem_posDiffs | [55, 1] | [56, 98] | simp_rw [posDiffs, mem_image, mem_filter, mem_product, Prod.exists, and_assoc, exists_and_left] | α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ a ∈ s \₊ t ↔ ∃ b ∈ s, ∃ c ∈ t, c ≤ b ∧ b \ c = a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Finset/PosDiffs.lean | Finset.posDiffs_empty | [58, 1] | [58, 79] | simp [posDiffs] | α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ s \₊ ∅ = ∅ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Finset/PosDiffs.lean | Finset.empty_posDiffs | [59, 1] | [59, 79] | simp [posDiffs] | α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
inst✝ : DecidableEq α
s✝ t : Finset α
a : α
s : Finset α
⊢ ∅ \₊ s = ∅ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Finset/PosDiffs.lean | Finset.posDiffs_subset_diffs | [61, 1] | [63, 57] | simp only [subset_iff, mem_posDiffs, mem_diffs] | α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ s \₊ t ⊆ s \\ t | α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ ∀ ⦃x : α⦄, (∃ b ∈ s, ∃ c ∈ t, c ≤ b ∧ b \ c = x) → ∃ a ∈ s, ∃ b ∈ t, a \ b = x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Finset/PosDiffs.lean | Finset.posDiffs_subset_diffs | [61, 1] | [63, 57] | exact fun a ⟨b, hb, c, hc, _, ha⟩ ↦ ⟨b, hb, c, hc, ha⟩ | α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ ∀ ⦃x : α⦄, (∃ b ∈ s, ∃ c ∈ t, c ≤ b ∧ b \ c = x) → ∃ a ∈ s, ∃ b ∈ t, a \ b = x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Finset/PosDiffs.lean | Finset.card_posDiffs_self_le | [73, 1] | [81, 10] | revert h𝒜 | α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
h𝒜 : (↑𝒜).OrdConnected
⊢ (𝒜 \₊ 𝒜).card ≤ 𝒜.card | α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ (↑𝒜).OrdConnected → (𝒜 \₊ 𝒜).card ≤ 𝒜.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Finset/PosDiffs.lean | Finset.card_posDiffs_self_le | [73, 1] | [81, 10] | refine' Finset.memberFamily_induction_on 𝒜 _ _ _ | α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ (↑𝒜).OrdConnected → (𝒜 \₊ 𝒜).card ≤ 𝒜.card | case refine'_1
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ (↑∅).OrdConnected → (∅ \₊ ∅).card ≤ ∅.card
case refine'_2
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ (↑{∅}).OrdConnected → ({∅} \₊ {∅}).card ≤ {∅}.card
case refine'_3
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ ∀ (a : α) ⦃𝒜 : Finset (Finset α)⦄,
((↑(nonMemberSubfamily a 𝒜)).OrdConnected →
(nonMemberSubfamily a 𝒜 \₊ nonMemberSubfamily a 𝒜).card ≤ (nonMemberSubfamily a 𝒜).card) →
((↑(memberSubfamily a 𝒜)).OrdConnected →
(memberSubfamily a 𝒜 \₊ memberSubfamily a 𝒜).card ≤ (memberSubfamily a 𝒜).card) →
(↑𝒜).OrdConnected → (𝒜 \₊ 𝒜).card ≤ 𝒜.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Finset/PosDiffs.lean | Finset.card_posDiffs_self_le | [73, 1] | [81, 10] | simp | case refine'_1
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ (↑∅).OrdConnected → (∅ \₊ ∅).card ≤ ∅.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Finset/PosDiffs.lean | Finset.card_posDiffs_self_le | [73, 1] | [81, 10] | intro | case refine'_2
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ (↑{∅}).OrdConnected → ({∅} \₊ {∅}).card ≤ {∅}.card | case refine'_2
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
h𝒜✝ : (↑{∅}).OrdConnected
⊢ ({∅} \₊ {∅}).card ≤ {∅}.card |
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