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101
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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/Colex.lean | Finset.Colex.cons_le_cons | [11, 1] | [16, 27] | obtain rfl | hab := eq_or_ne a b | α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder α
inst✝ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
ha : a ∉ s
hb : b ∉ s
⊢ { ofColex := cons a s ha } ≤ { ofColex := cons b s hb } ↔ a ≤ b | case inl
α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder α
inst✝ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a : α
ha hb : a ∉ s
⊢ { ofColex := cons a s ha } ≤ { ofColex := cons a s hb } ↔ a ≤ a
case inr
α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder α
inst✝ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
ha : a ∉ s
hb : b ∉ s
hab : a ≠ b
⊢ { ofColex := cons a s ha } ≤ { ofColex := cons b s hb } ↔ a ≤ b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.cons_le_cons | [11, 1] | [16, 27] | classical
rw [← toColex_sdiff_le_toColex_sdiff', cons_sdiff_cons hab, cons_sdiff_cons hab.symm,
singleton_le_singleton] | case inr
α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder α
inst✝ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
ha : a ∉ s
hb : b ∉ s
hab : a ≠ b
⊢ { ofColex := cons a s ha } ≤ { ofColex := cons b s hb } ↔ a ≤ b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.cons_le_cons | [11, 1] | [16, 27] | simp | case inl
α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder α
inst✝ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a : α
ha hb : a ∉ s
⊢ { ofColex := cons a s ha } ≤ { ofColex := cons a s hb } ↔ a ≤ a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.cons_le_cons | [11, 1] | [16, 27] | rw [← toColex_sdiff_le_toColex_sdiff', cons_sdiff_cons hab, cons_sdiff_cons hab.symm,
singleton_le_singleton] | case inr
α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder α
inst✝ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
ha : a ∉ s
hb : b ∉ s
hab : a ≠ b
⊢ { ofColex := cons a s ha } ≤ { ofColex := cons b s hb } ↔ a ≤ b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.insert_le_insert | [23, 1] | [25, 70] | rw [← cons_eq_insert _ _ ha, ← cons_eq_insert _ _ hb, cons_le_cons] | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
inst✝ : DecidableEq α
ha : a ∉ s
hb : b ∉ s
⊢ { ofColex := insert a s } ≤ { ofColex := insert b s } ↔ a ≤ b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.insert_lt_insert | [27, 1] | [29, 70] | rw [← cons_eq_insert _ _ ha, ← cons_eq_insert _ _ hb, cons_lt_cons] | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
inst✝ : DecidableEq α
ha : a ∉ s
hb : b ∉ s
⊢ { ofColex := insert a s } < { ofColex := insert b s } ↔ a < b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase | [31, 1] | [37, 27] | obtain rfl | hab := eq_or_ne a b | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
inst✝ : DecidableEq α
ha : a ∈ s
hb : b ∈ s
⊢ { ofColex := s.erase a } ≤ { ofColex := s.erase b } ↔ b ≤ a | case inl
α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a : α
inst✝ : DecidableEq α
ha hb : a ∈ s
⊢ { ofColex := s.erase a } ≤ { ofColex := s.erase a } ↔ a ≤ a
case inr
α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
inst✝ : DecidableEq α
ha : a ∈ s
hb : b ∈ s
hab : a ≠ b
⊢ { ofColex := s.erase a } ≤ { ofColex := s.erase b } ↔ b ≤ a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase | [31, 1] | [37, 27] | classical
rw [← toColex_sdiff_le_toColex_sdiff', erase_sdiff_erase hab hb, erase_sdiff_erase hab.symm ha,
singleton_le_singleton] | case inr
α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
inst✝ : DecidableEq α
ha : a ∈ s
hb : b ∈ s
hab : a ≠ b
⊢ { ofColex := s.erase a } ≤ { ofColex := s.erase b } ↔ b ≤ a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase | [31, 1] | [37, 27] | simp | case inl
α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a : α
inst✝ : DecidableEq α
ha hb : a ∈ s
⊢ { ofColex := s.erase a } ≤ { ofColex := s.erase a } ↔ a ≤ a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase | [31, 1] | [37, 27] | rw [← toColex_sdiff_le_toColex_sdiff', erase_sdiff_erase hab hb, erase_sdiff_erase hab.symm ha,
singleton_le_singleton] | case inr
α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
inst✝ : DecidableEq α
ha : a ∈ s
hb : b ∈ s
hab : a ≠ b
⊢ { ofColex := s.erase a } ≤ { ofColex := s.erase b } ↔ b ≤ a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | simp only [lt_iff_exists_forall_lt] | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
⊢ { ofColex := s } < { ofColex := t } ↔ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
⊢ (∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a) ↔ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | refine ⟨fun h ↦ ?_, ?_⟩ | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
⊢ (∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a) ↔ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
case refine_2
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
⊢ (∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)) → ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | let u := (t \ s).filter fun w ↦ ∀ a ∈ s, a ∉ t → a < w | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | have mem_u {w : α} : w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w := by simp [u, and_assoc] | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | have hu : u.Nonempty := h.imp fun _ ↦ mem_u.2 | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | let m := max' _ hu | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
m : α := u.max' hu
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | have ⟨hmt, hms, hm⟩ : m ∈ t ∧ m ∉ s ∧ ∀ a ∈ s, a ∉ t → a < m := mem_u.1 $ max'_mem _ _ | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
m : α := u.max' hu
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
m : α := u.max' hu
hmt : m ∈ t
hms : m ∉ s
hm : ∀ a ∈ s, a ∉ t → a < m
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | refine ⟨m, hmt, hms, fun a hma ↦ ⟨fun has ↦ not_imp_comm.1 (hm _ has) hma.asymm, fun hat ↦ ?_⟩⟩ | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
m : α := u.max' hu
hmt : m ∈ t
hms : m ∉ s
hm : ∀ a ∈ s, a ∉ t → a < m
⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t) | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a✝ : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
m : α := u.max' hu
hmt : m ∈ t
hms : m ∉ s
hm : ∀ a ∈ s, a ∉ t → a < m
a : α
hma : m < a
hat : a ∈ t
⊢ a ∈ s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | by_contra has | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a✝ : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
m : α := u.max' hu
hmt : m ∈ t
hms : m ∉ s
hm : ∀ a ∈ s, a ∉ t → a < m
a : α
hma : m < a
hat : a ∈ t
⊢ a ∈ s | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a✝ : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
m : α := u.max' hu
hmt : m ∈ t
hms : m ∉ s
hm : ∀ a ∈ s, a ∉ t → a < m
a : α
hma : m < a
hat : a ∈ t
has : a ∉ s
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | have hau : a ∈ u := mem_u.2 ⟨hat, has, fun b hbs hbt ↦ (hm _ hbs hbt).trans hma⟩ | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a✝ : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
m : α := u.max' hu
hmt : m ∈ t
hms : m ∉ s
hm : ∀ a ∈ s, a ∉ t → a < m
a : α
hma : m < a
hat : a ∈ t
has : a ∉ s
⊢ False | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a✝ : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
m : α := u.max' hu
hmt : m ∈ t
hms : m ∉ s
hm : ∀ a ∈ s, a ∉ t → a < m
a : α
hma : m < a
hat : a ∈ t
has : a ∉ s
hau : a ∈ u
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | exact hma.not_le $ le_max' _ _ hau | case refine_1
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a✝ : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
mem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w
hu : u.Nonempty
m : α := u.max' hu
hmt : m ∈ t
hms : m ∉ s
hm : ∀ a ∈ s, a ∉ t → a < m
a : α
hma : m < a
hat : a ∈ t
has : a ∉ s
hau : a ∈ u
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | simp [u, and_assoc] | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
h : ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a
u : Finset α := filter (fun w => ∀ a ∈ s, a ∉ t → a < w) (t \ s)
w : α
⊢ w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | rintro ⟨w, hwt, hws, hw⟩ | case refine_2
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
⊢ (∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)) → ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a | case refine_2.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
⊢ ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | refine ⟨w, hwt, hws, fun a has hat ↦ ?_⟩ | case refine_2.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
⊢ ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a | case refine_2.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a✝ w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
a : α
has : a ∈ s
hat : a ∉ t
⊢ a < w |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | by_contra! hwa | case refine_2.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a✝ w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
a : α
has : a ∈ s
hat : a ∉ t
⊢ a < w | case refine_2.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a✝ w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
a : α
has : a ∈ s
hat : a ∉ t
hwa : w ≤ a
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.lt_iff_exists_forall_lt_mem_iff_mem | [55, 1] | [71, 73] | exact hat $ (hw $ hwa.lt_of_ne $ ne_of_mem_of_not_mem hwt hat).1 has | case refine_2.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a✝ w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
a : α
has : a ∈ s
hat : a ∉ t
hwa : w ≤ a
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | generalize_proofs ht | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hst : { ofColex := s } ≤ { ofColex := t }
hcard : s.card ≤ t.card
ha : a ∈ s
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase (t.min' ⋯) } | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hst : { ofColex := s } ≤ { ofColex := t }
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase (t.min' ht) } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | set m := min' t ht | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hst : { ofColex := s } ≤ { ofColex := t }
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase (t.min' ht) } | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hst : { ofColex := s } ≤ { ofColex := t }
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | obtain rfl | h' := eq_or_ne s t | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hst : { ofColex := s } ≤ { ofColex := t }
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s : Finset α
a : α
ha : a ∈ s
hst : { ofColex := s } ≤ { ofColex := s }
hcard : s.card ≤ s.card
ht : s.Nonempty
m : α := s.min' ht
⊢ { ofColex := s.erase a } ≤ { ofColex := s.erase m }
case inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hst : { ofColex := s } ≤ { ofColex := t }
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | replace hst := hst.lt_of_ne $ toColex_inj.not.2 h' | case inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hst : { ofColex := s } ≤ { ofColex := t }
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hst : { ofColex := s } < { ofColex := t }
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | simp only [lt_iff_exists_forall_lt_mem_iff_mem] at hst | case inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hst : { ofColex := s } < { ofColex := t }
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hst : ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | obtain ⟨w, hwt, hws, hw⟩ := hst | case inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hst : ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inr.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | obtain hwa | haw := (ne_of_mem_of_not_mem ha hws).symm.lt_or_lt | case inr.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inr.intro.intro.intro.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m }
case inr.intro.intro.intro.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | obtain rfl | hmw : m = w ∨ m < w := (min'_le _ _ hwt).eq_or_lt | case inr.intro.intro.intro.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inr.intro.intro.intro.inr.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m }
case inr.intro.intro.intro.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | exact (erase_le_erase ha $ min'_mem _ _).2 $ min'_le _ _ $ ha | case inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s : Finset α
a : α
ha : a ∈ s
hst : { ofColex := s } ≤ { ofColex := s }
hcard : s.card ≤ s.card
ht : s.Nonempty
m : α := s.min' ht
⊢ { ofColex := s.erase a } ≤ { ofColex := s.erase m } | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | have hma : m < a := (min'_le _ _ hwt).trans_lt hwa | case inr.intro.intro.intro.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inr.intro.intro.intro.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | refine (lt_iff_exists_forall_lt.2 ⟨a, mem_erase.2 ⟨hma.ne', (hw hwa).1 ha⟩,
not_mem_erase _ _, fun b hbs hbt ↦ ?_⟩).le | case inr.intro.intro.intro.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inr.intro.intro.intro.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ { ofColex := { val := t.val.erase m, nodup := ⋯ } }.ofColex
⊢ b < a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | change b ∉ t.erase m at hbt | case inr.intro.intro.intro.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ { ofColex := { val := t.val.erase m, nodup := ⋯ } }.ofColex
⊢ b < a | case inr.intro.intro.intro.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t.erase m
⊢ b < a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | rw [mem_erase, not_and_or, not_ne_iff] at hbt | case inr.intro.intro.intro.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t.erase m
⊢ b < a | case inr.intro.intro.intro.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b = m ∨ b ∉ t
⊢ b < a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | obtain rfl | hbt := hbt | case inr.intro.intro.intro.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b = m ∨ b ∉ t
⊢ b < a | case inr.intro.intro.intro.inl.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
hbs : m ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
⊢ m < a
case inr.intro.intro.intro.inl.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t
⊢ b < a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | assumption | case inr.intro.intro.intro.inl.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
hbs : m ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
⊢ m < a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | by_contra! hab | case inr.intro.intro.intro.inl.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t
⊢ b < a | case inr.intro.intro.intro.inl.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t
hab : a ≤ b
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | exact hbt $ (hw $ hwa.trans_le hab).1 $ mem_of_mem_erase hbs | case inr.intro.intro.intro.inl.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t
hab : a ≤ b
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | have : erase t m ⊆ erase s a := by
rintro b hb
rw [mem_erase] at hb ⊢
exact ⟨(haw.trans_le $ min'_le _ _ hb.2).ne', (hw $ hb.1.lt_of_le' $ min'_le _ _ hb.2).2 hb.2⟩ | case inr.intro.intro.intro.inr.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inr.intro.intro.intro.inr.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
this : t.erase m ⊆ s.erase a
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | rw [eq_of_subset_of_card_le this] | case inr.intro.intro.intro.inr.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
this : t.erase m ⊆ s.erase a
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inr.intro.intro.intro.inr.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
this : t.erase m ⊆ s.erase a
⊢ (s.erase a).card ≤ (t.erase m).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | rw [card_erase_of_mem ha, card_erase_of_mem (min'_mem _ _)] | case inr.intro.intro.intro.inr.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
this : t.erase m ⊆ s.erase a
⊢ (s.erase a).card ≤ (t.erase m).card | case inr.intro.intro.intro.inr.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
this : t.erase m ⊆ s.erase a
⊢ s.card - 1 ≤ t.card - 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | exact tsub_le_tsub_right hcard _ | case inr.intro.intro.intro.inr.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
this : t.erase m ⊆ s.erase a
⊢ s.card - 1 ≤ t.card - 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | rintro b hb | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
⊢ t.erase m ⊆ s.erase a | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
b : α
hb : b ∈ t.erase m
⊢ b ∈ s.erase a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | rw [mem_erase] at hb ⊢ | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
b : α
hb : b ∈ t.erase m
⊢ b ∈ s.erase a | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
b : α
hb : b ≠ m ∧ b ∈ t
⊢ b ≠ a ∧ b ∈ s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | exact ⟨(haw.trans_le $ min'_le _ _ hb.2).ne', (hw $ hb.1.lt_of_le' $ min'_le _ _ hb.2).2 hb.2⟩ | α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
b : α
hb : b ≠ m ∧ b ∈ t
⊢ b ≠ a ∧ b ∈ s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | refine (lt_iff_exists_forall_lt.2 ⟨w, mem_erase.2 ⟨hmw.ne', hwt⟩, mt mem_of_mem_erase hws,
fun b hbs hbt ↦ ?_⟩).le | case inr.intro.intro.intro.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | case inr.intro.intro.intro.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ { ofColex := { val := t.val.erase m, nodup := ⋯ } }.ofColex
⊢ b < w |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | change b ∉ t.erase m at hbt | case inr.intro.intro.intro.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ { ofColex := { val := t.val.erase m, nodup := ⋯ } }.ofColex
⊢ b < w | case inr.intro.intro.intro.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t.erase m
⊢ b < w |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | rw [mem_erase, not_and_or, not_ne_iff] at hbt | case inr.intro.intro.intro.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t.erase m
⊢ b < w | case inr.intro.intro.intro.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b = m ∨ b ∉ t
⊢ b < w |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | obtain rfl | hbt := hbt | case inr.intro.intro.intro.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b = m ∨ b ∉ t
⊢ b < w | case inr.intro.intro.intro.inr.inr.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
hbs : m ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
⊢ m < w
case inr.intro.intro.intro.inr.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t
⊢ b < w |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | assumption | case inr.intro.intro.intro.inr.inr.inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
hbs : m ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
⊢ m < w | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | by_contra! hwb | case inr.intro.intro.intro.inr.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t
⊢ b < w | case inr.intro.intro.intro.inr.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t
hwb : w ≤ b
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Colex.lean | Finset.Colex.erase_le_erase_min' | [73, 1] | [115, 94] | exact hbt $ (hw $ hwb.lt_of_ne $ ne_of_mem_of_not_mem hwt hbt).1 $ mem_of_mem_erase hbs | case inr.intro.intro.intro.inr.inr.inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : s.card ≤ t.card
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
haw : a < w
hmw : m < w
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ⋯ } }.ofColex
hbt : b ∉ t
hwb : w ≤ b
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | one_le_schnirelmannDensity_iff | [10, 1] | [12, 76] | rw [schnirelmannDensity_le_one.ge_iff_eq, schnirelmannDensity_eq_one_iff] | A B : Set ℕ
n : ℕ
⊢ 1 ≤ schnirelmannDensity A ↔ {0}ᶜ ⊆ A | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | one_le_schnirelmannDensity_iff_of_zero_mem | [14, 1] | [17, 91] | rw [schnirelmannDensity_le_one.ge_iff_eq, schnirelmannDensity_eq_one_iff_of_zero_mem hA] | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
⊢ 1 ≤ schnirelmannDensity A ↔ A = Set.univ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_nonneg | [22, 1] | [22, 90] | positivity | A✝ B : Set ℕ
n✝ : ℕ
A : Set ℕ
n : ℕ
⊢ 0 ≤ countelements A n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | card_Icc_one_n_n | [24, 1] | [25, 47] | rw [Nat.card_Icc 1 n, add_tsub_cancel_right] | A B : Set ℕ
n✝ n : ℕ
⊢ (Icc 1 n).card = n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_le_n | [27, 1] | [28, 57] | simpa [countelements] using card_filter_le (Icc 1 n) _ | A✝ B : Set ℕ
n✝ : ℕ
A : Set ℕ
n : ℕ
⊢ countelements A n ≤ n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | repeat rw [countelements] | A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ countelements A (n - 1) = countelements A n | A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ (filter (fun x => x ∈ A) (Icc 1 (n - 1))).card = (filter (fun x => x ∈ A) (Icc 1 n)).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | refine card_le_card fun x hx ↦ ?_ | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ (filter (fun x => x ∈ A) (Icc 1 n)).card ≤ (filter (fun x => x ∈ A) (Icc 1 (n - 1))).card | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : x ∈ filter (fun x => x ∈ A) (Icc 1 n)
⊢ x ∈ filter (fun x => x ∈ A) (Icc 1 (n - 1)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | rw [mem_filter] | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : x ∈ filter (fun x => x ∈ A) (Icc 1 n)
⊢ x ∈ filter (fun x => x ∈ A) (Icc 1 (n - 1)) | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : x ∈ filter (fun x => x ∈ A) (Icc 1 n)
⊢ x ∈ Icc 1 (n - 1) ∧ x ∈ A |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | rw [mem_filter, mem_Icc] at hx | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : x ∈ filter (fun x => x ∈ A) (Icc 1 n)
⊢ x ∈ Icc 1 (n - 1) ∧ x ∈ A | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : (1 ≤ x ∧ x ≤ n) ∧ x ∈ A
⊢ x ∈ Icc 1 (n - 1) ∧ x ∈ A |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | refine ⟨mem_Icc.2 ⟨hx.1.1, Nat.le_pred_of_lt $ hx.1.2.lt_of_ne ?_⟩, hx.2⟩ | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : (1 ≤ x ∧ x ≤ n) ∧ x ∈ A
⊢ x ∈ Icc 1 (n - 1) ∧ x ∈ A | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : (1 ≤ x ∧ x ≤ n) ∧ x ∈ A
⊢ x ≠ n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | rintro rfl | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : (1 ≤ x ∧ x ≤ n) ∧ x ∈ A
⊢ x ≠ n | case refine_2
A B : Set ℕ
x : ℕ
hn : x ∉ A
hx : (1 ≤ x ∧ x ≤ x) ∧ x ∈ A
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | exact hn hx.2 | case refine_2
A B : Set ℕ
x : ℕ
hn : x ∉ A
hx : (1 ≤ x ∧ x ≤ x) ∧ x ∈ A
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | rw [countelements] | A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ (filter (fun x => x ∈ A) (Icc 1 (n - 1))).card = countelements A n | A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ (filter (fun x => x ∈ A) (Icc 1 (n - 1))).card = (filter (fun x => x ∈ A) (Icc 1 n)).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | simp only [tsub_le_iff_right, le_add_iff_nonneg_right, zero_le_one] | case refine_1
A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ n - 1 ≤ n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | by_contra! h | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
⊢ n ∈ A + B | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hnA : n ∉ A := Set.not_mem_subset (Set.subset_add_left _ hB) h | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
⊢ False | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hnB : n ∉ B := Set.not_mem_subset (Set.subset_add_right _ hA) h | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
⊢ False | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hca : countelements A (n - 1) = countelements A n := countelements_pred hnA | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
⊢ False | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hcb : countelements B (n - 1) = countelements B n := countelements_pred hnB | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
⊢ False | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | obtain rfl | hn1 := n.eq_zero_or_pos | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
⊢ False | case inl
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : 0 ≤ countelements A 0 + countelements B 0
h : 0 ∉ A + B
hnA : 0 ∉ A
hnB : 0 ∉ B
hca : countelements A (0 - 1) = countelements A 0
hcb : countelements B (0 - 1) = countelements B 0
⊢ False
case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | apply h | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ False | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ n ∈ A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | simp only [Nat.lt_one_iff, tsub_eq_zero_iff_le, mem_Ioo, and_imp, Set.singleton_sub,
Set.mem_image, ne_eq] at lem3 | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).Nonempty
⊢ n ∈ A + B | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => ∃ x_1 ∈ B, n - x_1 = x) (Ioo 0 n)).Nonempty
⊢ n ∈ A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have lem31 : (A ∩ ({n} - B) ∩ Set.Ioo 0 n).Nonempty := by
rw [← filter_and, ← coe_nonempty, coe_filter, Set.setOf_and, Set.setOf_and, Set.setOf_mem_eq,
Set.inter_comm] at lem3
convert lem3 using 3 <;> ext <;> simp | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => ∃ x_1 ∈ B, n - x_1 = x) (Ioo 0 n)).Nonempty
⊢ n ∈ A + B | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => ∃ x_1 ∈ B, n - x_1 = x) (Ioo 0 n)).Nonempty
lem31 : (A ∩ ({n} - B) ∩ Set.Ioo 0 n).Nonempty
⊢ n ∈ A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | obtain ⟨_, ⟨hxA, n, rfl, x, hxB, rfl⟩, hx⟩ := lem31 | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => ∃ x_1 ∈ B, n - x_1 = x) (Ioo 0 n)).Nonempty
lem31 : (A ∩ ({n} - B) ∩ Set.Ioo 0 n).Nonempty
⊢ n ∈ A + B | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
n : ℕ
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => ∃ x_1 ∈ B, n - x_1 = x) (Ioo 0 n)).Nonempty
x : ℕ
hxB : x ∈ B
hxA : (fun x x_1 => x - x_1) n x ∈ A
hx : (fun x x_1 => x - x_1) n x ∈ Set.Ioo 0 n
⊢ n ∈ A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | simp only [Set.mem_Ioo, Nat.succ_le_iff, tsub_pos_iff_lt, tsub_le_iff_right] at hx | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
n : ℕ
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => ∃ x_1 ∈ B, n - x_1 = x) (Ioo 0 n)).Nonempty
x : ℕ
hxB : x ∈ B
hxA : (fun x x_1 => x - x_1) n x ∈ A
hx : (fun x x_1 => x - x_1) n x ∈ Set.Ioo 0 n
⊢ n ∈ A + B | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
n : ℕ
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => ∃ x_1 ∈ B, n - x_1 = x) (Ioo 0 n)).Nonempty
x : ℕ
hxB : x ∈ B
hxA : (fun x x_1 => x - x_1) n x ∈ A
hx : x < n ∧ n - x < n
⊢ n ∈ A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact ⟨_, hxA, x, hxB, tsub_add_cancel_of_le hx.1.le⟩ | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
n : ℕ
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => ∃ x_1 ∈ B, n - x_1 = x) (Ioo 0 n)).Nonempty
x : ℕ
hxB : x ∈ B
hxA : (fun x x_1 => x - x_1) n x ∈ A
hx : x < n ∧ n - x < n
⊢ n ∈ A + B | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | contradiction | case inl
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : 0 ≤ countelements A 0 + countelements B 0
h : 0 ∉ A + B
hnA : 0 ∉ A
hnB : 0 ∉ B
hca : countelements A (0 - 1) = countelements A 0
hcb : countelements B (0 - 1) = countelements B 0
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [countelements] | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1) | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = (filter (fun x => x ∈ B) (Icc 1 (n - 1))).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [← hfim, card_image_of_injOn] | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n)
⊢ (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = (filter (fun x => x ∈ B) (Icc 1 (n - 1))).card | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n)
⊢ (filter (fun x => x ∈ B) (Ioo 0 n)).card = (filter (fun x => x ∈ B) (Icc 1 (n - 1))).card
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n)
⊢ Set.InjOn (fun x => n - x) ↑(filter (fun x => x ∈ B) (Ioo 0 n)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | congr | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n)
⊢ (filter (fun x => x ∈ B) (Ioo 0 n)).card = (filter (fun x => x ∈ B) (Icc 1 (n - 1))).card
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n)
⊢ Set.InjOn (fun x => n - x) ↑(filter (fun x => x ∈ B) (Ioo 0 n)) | case e_s.e_s.e_b
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n)
⊢ n = ((n - 1).add 0).succ
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n)
⊢ Set.InjOn (fun x => n - x) ↑(filter (fun x => x ∈ B) (Ioo 0 n)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact (tsub_add_cancel_of_le $ Nat.succ_le_iff.2 hn1).symm | case e_s.e_s.e_b
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n)
⊢ n = ((n - 1).add 0).succ
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n)
⊢ Set.InjOn (fun x => n - x) ↑(filter (fun x => x ∈ B) (Ioo 0 n)) | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n)
⊢ Set.InjOn (fun x => n - x) ↑(filter (fun x => x ∈ B) (Ioo 0 n)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | ext | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈ {n} - B) (Ioo 0 n) | case a
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
a✝ : ℕ
⊢ a✝ ∈ image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) ↔ a✝ ∈ filter (fun x => x ∈ {n} - B) (Ioo 0 n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | aesop | case a
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
a✝ : ℕ
⊢ a✝ ∈ image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) ↔ a✝ ∈ filter (fun x => x ∈ {n} - B) (Ioo 0 n) | no goals |
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