url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.compression_improved | [174, 1] | [195, 22] | rw [card_erase_of_mem (min'_mem _ _), card_erase_of_mem Hx, same_size] | case intro.intro.intro.intro.inr.refine'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset α)
h₂ : ∀ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ → U₁.card < U.card → IsCompressed U₁ V₁ 𝒜
UVd : Disjoint U V
same_size : U.card = V.card
hU : U.Nonempty
hV : V.Nonempty
max_lt : U.ma... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.compression_improved | [174, 1] | [195, 22] | rwa [← card_pos, card_erase_of_mem Hx, tsub_pos_iff_lt] | case intro.intro.intro.intro.inr.refine'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset α)
h₂ : ∀ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ → U₁.card < U.card → IsCompressed U₁ V₁ 𝒜
UVd : Disjoint U V
same_size : U.card = V.card
hU : U.Nonempty
hV : V.Nonempty
max_lt : U.ma... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.compression_improved | [174, 1] | [195, 22] | rwa [← Finset.card_pos, card_erase_of_mem (min'_mem _ _), ← same_size, tsub_pos_iff_lt] | case intro.intro.intro.intro.inr.refine'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset α)
h₂ : ∀ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ → U₁.card < U.card → IsCompressed U₁ V₁ 𝒜
UVd : Disjoint U V
same_size : U.card = V.card
hU : U.Nonempty
hV : V.Nonempty
max_lt : U.ma... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.compression_improved | [174, 1] | [195, 22] | refine' (Finset.max'_subset _ $ erase_subset _ _).trans_lt (max_lt.trans_le $
le_max' _ _ $ mem_erase.2 ⟨(min'_lt_max'_of_card _ _).ne', max'_mem _ _⟩) | case intro.intro.intro.intro.inr.refine'_4
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset α)
h₂ : ∀ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ → U₁.card < U.card → IsCompressed U₁ V₁ 𝒜
UVd : Disjoint U V
same_size : U.card = V.card
hU : U.Nonempty
hV : V.Nonempty
max_lt : U.ma... | case intro.intro.intro.intro.inr.refine'_4
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset α)
h₂ : ∀ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ → U₁.card < U.card → IsCompressed U₁ V₁ 𝒜
UVd : Disjoint U V
same_size : U.card = V.card
hU : U.Nonempty
hV : V.Nonempty
max_lt : U.ma... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.compression_improved | [174, 1] | [195, 22] | rwa [← same_size] | case intro.intro.intro.intro.inr.refine'_4
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
𝒜 : Finset (Finset α)
h₂ : ∀ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ → U₁.card < U.card → IsCompressed U₁ V₁ 𝒜
UVd : Disjoint U V
same_size : U.card = V.card
hU : U.Nonempty
hV : V.Nonempty
max_lt : U.ma... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | refine' ⟨h₁, _⟩ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
⊢ IsInitSeg ℬ r | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
⊢ ∀ ⦃s t : Finset α⦄, s ∈ ℬ → { ofColex := t } < { ofColex := s } ∧ t.card = r → t ∈ ℬ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | rintro A B hA ⟨hBA, sizeA⟩ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
⊢ ∀ ⦃s t : Finset α⦄, s ∈ ℬ → { ofColex := t } < { ofColex := s } ∧ t.card = r → t ∈ ℬ | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
⊢ B ∈ ℬ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | by_contra hB | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
⊢ B ∈ ℬ | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | have hAB : A ≠ B := ne_of_mem_of_not_mem hA hB | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
⊢ False | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | have hAB' : A.card = B.card := (h₁ hA).trans sizeA.symm | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
⊢ False | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | have hU : (A \ B).Nonempty := sdiff_nonempty.2 fun h ↦ hAB $ eq_of_subset_of_card_le h hAB'.ge | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | have hV : (B \ A).Nonempty :=
sdiff_nonempty.2 fun h ↦ hAB.symm $ eq_of_subset_of_card_le h hAB'.le | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | have disj : Disjoint (B \ A) (A \ B) := disjoint_sdiff.mono_left sdiff_subset | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | refine' hB _ | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | rw [← (h₂ _ _ ⟨disj, card_sdiff_comm hAB'.symm, hV, hU, smaller⟩).eq] | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | exact mem_compression.2 (Or.inr ⟨hB, A, hA, compress_sdiff_sdiff _ _⟩) | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card =... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | obtain hlt | heq | hgt := lt_trichotomy (max' _ hU) (max' _ hV) | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card = B.card
hU ... | case inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card = B... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | rw [← compress_sdiff_sdiff A B] at hAB hBA | case inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card = B... | case inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := compress (A \ B) (B \ A) B }
sizeA : B.card = r
hB : B ∉ ℬ
hAB ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | cases hBA.not_lt $ toColex_compress_lt_toColex hlt hAB | case inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := compress (A \ B) (B \ A) B }
sizeA : B.card = r
hB : B ∉ ℬ
hAB ... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | exact (disjoint_right.1 disj (max'_mem _ hU) $ heq.symm ▸ max'_mem _ _).elim | case inr.inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | assumption | case inr.inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n : ℕ
ℬ : Finset (Finset α)
r : ℕ
h₁ : Set.Sized r ↑ℬ
h₂ : ∀ (U V : Finset α), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
A B : Finset α
hA : A ∈ ℬ
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B ∉ ℬ
hAB : A ≠ B
hAB' : A.card... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [compression] at a ⊢ | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : 𝓒 U V 𝒜 ≠ 𝒜
⊢ Finset.UV.familyMeasure (𝓒 U V 𝒜) < Finset.UV.familyMeasure 𝒜 | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
⊢ Finset.UV.familyMeasure (filter (fun x => compr... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | have q : ∀ Q ∈ 𝒜.filter fun A ↦ compress U V A ∉ 𝒜, compress U V Q ≠ Q := by
simp_rw [mem_filter]
intro Q hQ h
rw [h] at hQ
exact hQ.2 hQ.1 | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
⊢ Finset.UV.familyMeasure (filter (fun x => compr... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | have uA : (𝒜.filter fun A => compress U V A ∈ 𝒜) ∪ 𝒜.filter (fun A ↦ compress U V A ∉ 𝒜) = 𝒜 :=
filter_union_filter_neg_eq _ _ | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | have ne₂ : (𝒜.filter fun A ↦ compress U V A ∉ 𝒜).Nonempty := by
refine' nonempty_iff_ne_empty.2 fun z ↦ a _
rw [filter_image, z, image_empty, union_empty]
rwa [z, union_empty] at uA | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [familyMeasure, familyMeasure, sum_union compress_disjoint] | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | conv_rhs => rw [← uA] | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [sum_union (disjoint_filter_filter_neg _ _ _), add_lt_add_iff_left, filter_image,
sum_image compress_injOn] | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | refine' sum_lt_sum_of_nonempty ne₂ fun A hA ↦ _ | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | simp_rw [← sum_image Fin.val_injective.injOn] | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [geomSum_lt_geomSum_iff_toColex_lt_toColex le_rfl,
toColex_image_lt_toColex_image Fin.val_strictMono] | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | exact toColex_compress_lt_toColex h $ q _ hA | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | simp_rw [mem_filter] | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
⊢ ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) 𝒜,... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
⊢ ∀ (Q : Finset (Fin n)), Q ∈ 𝒜 ∧ compress U V Q... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | intro Q hQ h | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
⊢ ∀ (Q : Finset (Fin n)), Q ∈ 𝒜 ∧ compress U V Q... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h✝ : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
Q : Finset (Fin n)
hQ : Q ∈ 𝒜 ∧ compress U V Q ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [h] at hQ | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h✝ : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
Q : Finset (Fin n)
hQ : Q ∈ 𝒜 ∧ compress U V Q ... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h✝ : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
Q : Finset (Fin n)
hQ : Q ∈ 𝒜 ∧ Q ∉ 𝒜
h : comp... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | exact hQ.2 hQ.1 | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h✝ : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
Q : Finset (Fin n)
hQ : Q ∈ 𝒜 ∧ Q ∉ 𝒜
h : comp... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | refine' nonempty_iff_ne_empty.2 fun z ↦ a _ | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [filter_image, z, image_empty, union_empty] | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rwa [z, union_empty] at uA | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n : ℕ
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
𝒜 : Finset (Finset (Fin n))
a : filter (fun x => compress U V x ∈ 𝒜) 𝒜 ∪ filter (fun x => x ∉ 𝒜) (image (compress U V) 𝒜) ≠ 𝒜
q : ∀ Q ∈ filter (fun A => compress U V A ∉ 𝒜) �... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | revert h | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 : Finset (Finset (Fin n))
h : Set.Sized r ↑𝒜
⊢ ∃ ℬ,
(∂ ℬ).card ≤ (∂ 𝒜).card ∧
𝒜.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 : Finset (Finset (Fin n))
⊢ Set.Sized r ↑𝒜 →
∃ ℬ,
(∂ ℬ).card ≤ (∂ 𝒜).card ∧
𝒜.card = ℬ.card ∧
Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | apply WellFounded.recursion (InvImage.wf familyMeasure wellFounded_lt) 𝒜 | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 : Finset (Finset (Fin n))
⊢ Set.Sized r ↑𝒜 →
∃ ℬ,
(∂ ℬ).card ≤ (∂ 𝒜).card ∧
𝒜.card = ℬ.card ∧
Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 : Finset (Finset (Fin n))
⊢ ∀ (x : Finset (Finset (Fin n))),
(∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y x →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | intro A ih h | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 : Finset (Finset (Fin n))
⊢ ∀ (x : Finset (Finset (Fin n))),
(∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y x →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
Set.Siz... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | set usable : Finset (Finset (Fin n) × Finset (Fin n)) :=
univ.filter fun t ↦ UsefulCompression t.1 t.2 ∧ ¬ IsCompressed t.1 t.2 A | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
Set.Siz... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
Set.Siz... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain husable | husable := usable.eq_empty_or_nonempty | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
Set.Siz... | case inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain ⟨⟨U, V⟩, hUV, t⟩ := exists_min_image usable (fun t ↦ t.1.card) husable | case inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
... | case inr.intro.mk.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | rw [mem_filter] at hUV | case inr.intro.mk.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.ca... | case inr.intro.mk.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | have h₂ : ∀ U₁ V₁, UsefulCompression U₁ V₁ → U₁.card < U.card → IsCompressed U₁ V₁ A := by
rintro U₁ V₁ huseful hUcard
by_contra h
exact hUcard.not_le $ t ⟨U₁, V₁⟩ $ mem_filter.2 ⟨mem_univ _, huseful, h⟩ | case inr.intro.mk.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.ca... | case inr.intro.mk.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | have p1 : (∂ (𝓒 U V A)).card ≤ (∂ A).card := compression_improved _ hUV.2.1 h₂ | case inr.intro.mk.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.ca... | case inr.intro.mk.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain ⟨-, hUV', hu, hv, hmax⟩ := hUV.2.1 | case inr.intro.mk.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.ca... | case inr.intro.mk.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | unfold InvImage at ih | case inr.intro.mk.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
... | case inr.intro.mk.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain ⟨t, q1, q2, q3, q4⟩ := ih (𝓒 U V A)
(familyMeasure_compression_lt_familyMeasure hmax hUV.2.2) (h.uvCompression hUV') | case inr.intro.mk.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).ca... | case inr.intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) →
Set.Sized r ↑y →
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | exact ⟨t, q1.trans p1, (card_compression _ _ _).symm.trans q2, q3, q4⟩ | case inr.intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) →
Set.Sized r ↑y →
... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | refine' ⟨A, le_rfl, rfl, h, fun U V hUV ↦ _⟩ | case inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
... | case inl
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | rw [eq_empty_iff_forall_not_mem] at husable | case inl
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
... | case inl
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | by_contra h | case inl
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
... | case inl
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | exact husable ⟨U, V⟩ $ mem_filter.2 ⟨mem_univ _, hUV, h⟩ | case inl
α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | rintro U₁ V₁ huseful hUcard | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
Set.S... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
Set.S... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | by_contra h | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
Set.S... | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
Set.S... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | exact hUcard.not_le $ t ⟨U₁, V₁⟩ $ mem_filter.2 ⟨mem_univ _, huseful, h⟩ | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r : ℕ
𝒜 A : Finset (Finset (Fin n))
ih :
∀ (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A →
Set.Sized r ↑y →
∃ ℬ,
(∂ ℬ).card ≤ (∂ y).card ∧
y.card = ℬ.card ∧
Set.S... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | obtain ⟨ℬ, card_le, t, hℬ, fully_comp⟩ := UV.kruskal_katona_helper 𝒜 h₁ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
⊢ (∂ 𝒞).card ≤ (∂ 𝒜).card | case intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
ℬ : Finset (Finset (Fin n))
card_le : (∂ ℬ).card ≤ (∂ 𝒜).card
t : 𝒜.card = ℬ.card
hℬ : Set.Sized r ↑ℬ
fully_comp : ∀ (U V : Finse... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | convert card_le | case intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
ℬ : Finset (Finset (Fin n))
card_le : (∂ ℬ).card ≤ (∂ 𝒜).card
t : 𝒜.card = ℬ.card
hℬ : Set.Sized r ↑ℬ
fully_comp : ∀ (U V : Finse... | case h.e'_3.h.e'_2.h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
ℬ : Finset (Finset (Fin n))
card_le : (∂ ℬ).card ≤ (∂ 𝒜).card
t : 𝒜.card = ℬ.card
hℬ : Set.Sized r ↑ℬ
fully_comp : ∀ (U V : Finset (... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | have hcard : card ℬ = card 𝒞 := t.symm.trans h₂ | case h.e'_3.h.e'_2.h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
ℬ : Finset (Finset (Fin n))
card_le : (∂ ℬ).card ≤ (∂ 𝒜).card
t : 𝒜.card = ℬ.card
hℬ : Set.Sized r ↑ℬ
fully_comp : ∀ (U V : Finset (... | case h.e'_3.h.e'_2.h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
ℬ : Finset (Finset (Fin n))
card_le : (∂ ℬ).card ≤ (∂ 𝒜).card
t : 𝒜.card = ℬ.card
hℬ : Set.Sized r ↑ℬ
fully_comp : ∀ (U V : Finset (... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | obtain CB | BC :=
h₃.total (UV.isInitSeg_of_compressed hℬ fun U V hUV ↦ by convert fully_comp U V hUV) | case h.e'_3.h.e'_2.h.e'_3
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
ℬ : Finset (Finset (Fin n))
card_le : (∂ ℬ).card ≤ (∂ 𝒜).card
t : 𝒜.card = ℬ.card
hℬ : Set.Sized r ↑ℬ
fully_comp : ∀ (U V : Finset (... | case h.e'_3.h.e'_2.h.e'_3.inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
ℬ : Finset (Finset (Fin n))
card_le : (∂ ℬ).card ≤ (∂ 𝒜).card
t : 𝒜.card = ℬ.card
hℬ : Set.Sized r ↑ℬ
fully_comp : ∀ (U V : Fins... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | convert fully_comp U V hUV | α : Type u_1
inst✝ : LinearOrder α
s U✝ V✝ : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
ℬ : Finset (Finset (Fin n))
card_le : (∂ ℬ).card ≤ (∂ 𝒜).card
t : 𝒜.card = ℬ.card
hℬ : Set.Sized r ↑ℬ
fully_comp : ∀ (U V : Finset (Fin n)), Finset.UV.Usefu... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | exact eq_of_subset_of_card_le CB hcard.le | case h.e'_3.h.e'_2.h.e'_3.inl
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
ℬ : Finset (Finset (Fin n))
card_le : (∂ ℬ).card ≤ (∂ 𝒜).card
t : 𝒜.card = ℬ.card
hℬ : Set.Sized r ↑ℬ
fully_comp : ∀ (U V : Fins... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | exact (eq_of_subset_of_card_le BC hcard.ge).symm | case h.e'_3.h.e'_2.h.e'_3.inr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒜.card = 𝒞.card
h₃ : IsInitSeg 𝒞 r
ℬ : Finset (Finset (Fin n))
card_le : (∂ ℬ).card ≤ (∂ 𝒜).card
t : 𝒜.card = ℬ.card
hℬ : Set.Sized r ↑ℬ
fully_comp : ∀ (U V : Fins... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.strengthened_kk | [316, 1] | [323, 29] | rcases exists_smaller_set 𝒜 𝒞.card h₂ with ⟨𝒜', prop, size⟩ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
⊢ (∂ 𝒞).card ≤ (∂ 𝒜).card | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
𝒜' : Finset (Finset (Fin n))
prop : 𝒜' ⊆ 𝒜
size : 𝒜'.card = 𝒞.card
⊢ (∂ 𝒞).card ≤ (∂ 𝒜).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.strengthened_kk | [316, 1] | [323, 29] | refine' (kruskal_katona (fun A hA ↦ h₁ (prop hA)) size h₃).trans (card_le_card _) | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
𝒜' : Finset (Finset (Fin n))
prop : 𝒜' ⊆ 𝒜
size : 𝒜'.card = 𝒞.card
⊢ (∂ 𝒞).card ≤ (∂ 𝒜).card | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
𝒜' : Finset (Finset (Fin n))
prop : 𝒜' ⊆ 𝒜
size : 𝒜'.card = 𝒞.card
⊢ ∂ 𝒜' ⊆ ∂ 𝒜 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.strengthened_kk | [316, 1] | [323, 29] | rw [shadow, shadow] | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
𝒜' : Finset (Finset (Fin n))
prop : 𝒜' ⊆ 𝒜
size : 𝒜'.card = 𝒞.card
⊢ ∂ 𝒜' ⊆ ∂ 𝒜 | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
𝒜' : Finset (Finset (Fin n))
prop : 𝒜' ⊆ 𝒜
size : 𝒜'.card = 𝒞.card
⊢ (𝒜'.sup fun s => image s.erase s) ⊆ 𝒜.sup fun s => image s.erase s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.strengthened_kk | [316, 1] | [323, 29] | apply shadow_monotone prop | case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
𝒜' : Finset (Finset (Fin n))
prop : 𝒜' ⊆ 𝒜
size : 𝒜'.card = 𝒞.card
⊢ (𝒜'.sup fun s => image s.erase s) ⊆ 𝒜.sup fun s => image s.erase s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | induction' k with _k ih generalizing r 𝒜 𝒞 | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
⊢ (∂ ^[k] 𝒞).card ≤ (∂ ^[k] 𝒜).card | case zero
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n i r : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
⊢ (∂ ^[0] 𝒞).card ≤ (∂ ^[0] 𝒜).card
case succ
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n i _k : ℕ
ih :
∀ {r : ℕ} {𝒜 𝒞 : Finset (Finset... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | simpa | case zero
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n i r : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
⊢ (∂ ^[0] 𝒞).card ≤ (∂ ^[0] 𝒜).card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | refine' ih h₁.shadow (strengthened_kk h₁ h₂ h₃) _ | case succ
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n i _k : ℕ
ih :
∀ {r : ℕ} {𝒜 𝒞 : Finset (Finset (Fin n))},
Set.Sized r ↑𝒜 → 𝒞.card ≤ 𝒜.card → IsInitSeg 𝒞 r → (∂ ^[_k] 𝒞).card ≤ (∂ ^[_k] 𝒜).card
r : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg �... | case succ
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n i _k : ℕ
ih :
∀ {r : ℕ} {𝒜 𝒞 : Finset (Finset (Fin n))},
Set.Sized r ↑𝒜 → 𝒞.card ≤ 𝒜.card → IsInitSeg 𝒞 r → (∂ ^[_k] 𝒞).card ≤ (∂ ^[_k] 𝒜).card
r : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | convert h₃.shadow | case succ
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n i _k : ℕ
ih :
∀ {r : ℕ} {𝒜 𝒞 : Finset (Finset (Fin n))},
Set.Sized r ↑𝒜 → 𝒞.card ≤ 𝒜.card → IsInitSeg 𝒞 r → (∂ ^[_k] 𝒞).card ≤ (∂ ^[_k] 𝒜).card
r : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg �... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | set range'k : Finset (Fin n) :=
attachFin (range k) fun m ↦ by rw [mem_range]; apply forall_lt_iff_le.2 hkn | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
⊢ k.choose (r - i) ≤ (∂ ^[i] 𝒜).card | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
⊢ k.choose (r - i) ≤ (∂ ^[i] 𝒜).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | set 𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
⊢ k.choose (r - i) ≤ (∂ ^[i] 𝒜).card | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
⊢ k.choose (r - i) ≤ (∂ ^[i] 𝒜).ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have Ccard : 𝒞.card = k.choose r := by rw [card_powersetCard, card_attachFin, card_range] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
⊢ k.choose (r - i) ≤ (∂ ^[i] 𝒜).ca... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
⊢ k.ch... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have : (𝒞 : Set (Finset (Fin n))).Sized r := Set.sized_powersetCard _ _ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
⊢ k.ch... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this :... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | suffices this : (∂^[i] 𝒞).card = k.choose (r - i) by
rw [← this]
apply iterated_kk h₁ _ _
rwa [Ccard]
refine' ⟨‹_›, _⟩
rintro A B hA ⟨HB₁, HB₂⟩
rw [mem_powersetCard]
refine' ⟨fun t ht ↦ _, ‹_›⟩
rw [mem_attachFin, mem_range]
have : toColex (image Fin.val B) < toColex (image Fin.val A) := by
rwa [... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this :... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this :... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | suffices ∂^[i] 𝒞 = powersetCard (r - i) range'k by
rw [this, card_powersetCard, card_attachFin, card_range] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this :... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this :... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | ext B | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this :... | case a
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_powersetCard, mem_shadow_iterate_iff_exists_sdiff] | case a
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r... | case a
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | constructor | case a
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r... | case a.mp
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choos... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rintro ⟨hBk, hB⟩ | case a.mpr
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choo... | case a.mpr.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have := exists_intermediate_set i ?_ hBk | case a.mpr.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = ... | case a.mpr.intro.refine_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : �... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | obtain ⟨C, BsubC, hCrange, hcard⟩ := this | case a.mpr.intro.refine_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : �... | case a.mpr.intro.refine_2.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [hB, ← Nat.add_sub_assoc hir, Nat.add_sub_cancel_left] at hcard | case a.mpr.intro.refine_2.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r... | case a.mpr.intro.refine_2.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | refine' ⟨C, _, BsubC, _⟩ | case a.mpr.intro.refine_2.intro.intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r... | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powe... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_powersetCard] | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powe... | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powe... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | exact ⟨hCrange, hcard⟩ | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powe... | case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powe... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_range] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
m : ℕ
⊢ m ∈ range k → m < n | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
m : ℕ
⊢ m < k → m < n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | apply forall_lt_iff_le.2 hkn | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
m : ℕ
⊢ m < k → m < n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [card_powersetCard, card_attachFin, card_range] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
⊢ 𝒞.card = k.choose r | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [← this] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ ... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | apply iterated_kk h₁ _ _ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ ... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rwa [Ccard] | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ ... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | refine' ⟨‹_›, _⟩ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ ... | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rintro A B hA ⟨HB₁, HB₂⟩ | α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choose r
this✝ ... | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choo... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_powersetCard] | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choo... | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choo... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | refine' ⟨fun t ht ↦ _, ‹_›⟩ | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choo... | case intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞✝ : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ 𝒜.card
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : 𝒞.card = k.choo... |
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