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.isInitSeg_of_compressed | [197, 1] | [218, 73] | have hAB : A β B := ne_of_mem_of_not_mem hA hB | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
β’ False | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β B
β’ 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 := (hβ hA).trans sizeA.symm | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β B
β’ False | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β 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 β¦ hAB $ eq_of_subset_of_card_le h hAB'.ge | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β B
hAB' : A.card =... | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β 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 β¦ hAB.symm $ eq_of_subset_of_card_le h hAB'.le | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β B
hAB' : A.card =... | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β B
hAB' : A.card =... | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β B
hAB' : A.card =... | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β B
hAB' : A.card =... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | rw [β (hβ _ _ β¨disj, card_sdiff_comm hAB'.symm, hV, hU, smallerβ©).eq] | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β B
hAB' : A.card =... | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β 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 β¨hB, A, hA, compress_sdiff_sdiff _ _β©) | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β 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) | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β B
hAB' : A.card = B.card
hU ... | case inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β 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 [β compress_sdiff_sdiff A B] at hAB hBA | case inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β B
hAB' : A.card = B... | case inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := compress (A \ B) (B \ A) B }
sizeA : B.card = r
hB : B β β¬
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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := compress (A \ B) (B \ A) B }
sizeA : B.card = r
hB : B β β¬
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 βΈ max'_mem _ _).elim | case inr.inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n : β
β¬ : Finset (Finset Ξ±)
r : β
hβ : Set.Sized r ββ¬
hβ : β (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β IsCompressed U V β¬
A B : Finset Ξ±
hA : A β β¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B β β¬
hAB : A β 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 β’ | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : π U V π β π
β’ Finset.UV.familyMeasure (π U V π) < Finset.UV.familyMeasure π | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
β’ 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 : β Q β π.filter fun A β¦ compress U V A β π, compress U V Q β Q := by
simp_rw [mem_filter]
intro Q hQ h
rw [h] at hQ
exact hQ.2 hQ.1 | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
β’ Finset.UV.familyMeasure (filter (fun x => compr... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | have uA : (π.filter fun A => compress U V A β π) βͺ π.filter (fun A β¦ compress U V A β π) = π :=
filter_union_filter_neg_eq _ _ | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | have neβ : (π.filter fun A β¦ compress U V A β π).Nonempty := by
refine' nonempty_iff_ne_empty.2 fun z β¦ a _
rw [filter_image, z, image_empty, union_empty]
rwa [z, union_empty] at uA | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
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] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | conv_rhs => rw [β uA] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
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] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
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 neβ fun A hA β¦ _ | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | simp_rw [β sum_image Fin.val_injective.injOn] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
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] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
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 | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | 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] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
β’ β Q β filter (fun A => compress U V A β π) π,... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
β’ β (Q : Finset (Fin n)), Q β π β§ 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 | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
β’ β (Q : Finset (Fin n)), Q β π β§ compress U V Q... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
hβ : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
Q : Finset (Fin n)
hQ : Q β π β§ 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 | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
hβ : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
Q : Finset (Fin n)
hQ : Q β π β§ compress U V Q ... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
hβ : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
Q : Finset (Fin n)
hQ : Q β π β§ Q β π
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 | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
hβ : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
Q : Finset (Fin n)
hQ : Q β π β§ Q β π
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 β¦ a _ | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
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] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... |
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 | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n : β
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
π : Finset (Finset (Fin n))
a : filter (fun x => compress U V x β π) π βͺ filter (fun x => x β π) (image (compress U V) π) β π
q : β Q β filter (fun A => compress U V A β π) οΏ½... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | revert h | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π : Finset (Finset (Fin n))
h : Set.Sized r βπ
β’ β β¬,
(β β¬).card β€ (β π).card β§
π.card = β¬.card β§ Set.Sized r ββ¬ β§ β (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β IsCompressed U V β¬ | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π : Finset (Finset (Fin n))
β’ Set.Sized r βπ β
β β¬,
(β β¬).card β€ (β π).card β§
π.card = β¬.card β§
Set.Sized r ββ¬ β§ β (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β IsCompressed U V β¬ |
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) π | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π : Finset (Finset (Fin n))
β’ Set.Sized r βπ β
β β¬,
(β β¬).card β€ (β π).card β§
π.card = β¬.card β§
Set.Sized r ββ¬ β§ β (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β IsCompressed U V β¬ | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π : Finset (Finset (Fin n))
β’ β (x : Finset (Finset (Fin n))),
(β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y x β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | intro A ih h | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π : Finset (Finset (Fin n))
β’ β (x : Finset (Finset (Fin n))),
(β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y x β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
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) Γ Finset (Fin n)) :=
univ.filter fun t β¦ UsefulCompression t.1 t.2 β§ Β¬ IsCompressed t.1 t.2 A | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
Set.Siz... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
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 | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
Set.Siz... | case inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain β¨β¨U, Vβ©, hUV, tβ© := exists_min_image usable (fun t β¦ t.1.card) husable | case inr
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
... | case inr.intro.mk.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.ca... | case inr.intro.mk.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | have hβ : β Uβ Vβ, UsefulCompression Uβ Vβ β Uβ.card < U.card β IsCompressed Uβ Vβ A := by
rintro Uβ Vβ huseful hUcard
by_contra h
exact hUcard.not_le $ t β¨Uβ, Vββ© $ mem_filter.2 β¨mem_univ _, huseful, hβ© | case inr.intro.mk.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.ca... | case inr.intro.mk.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | have p1 : (β (π U V A)).card β€ (β A).card := compression_improved _ hUV.2.1 hβ | case inr.intro.mk.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.ca... | case inr.intro.mk.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain β¨-, hUV', hu, hv, hmaxβ© := hUV.2.1 | case inr.intro.mk.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.ca... | case inr.intro.mk.intro.intro.intro.intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
... |
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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
... | case inr.intro.mk.intro.intro.intro.intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) β
Set.Sized r βy β
β β¬,
(β β¬).ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain β¨t, q1, q2, q3, q4β© := ih (π U V A)
(familyMeasure_compression_lt_familyMeasure hmax hUV.2.2) (h.uvCompression hUV') | case inr.intro.mk.intro.intro.intro.intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) β
Set.Sized r βy β
β β¬,
(β β¬).ca... | case inr.intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) β
Set.Sized r βy β
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | exact β¨t, q1.trans p1, (card_compression _ _ _).symm.trans q2, q3, q4β© | case inr.intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) β
Set.Sized r βy β
... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | refine' β¨A, le_rfl, rfl, h, fun U V hUV β¦ _β© | case inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
... | case inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
... |
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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
... | case inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | by_contra h | case inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
... | case inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | exact husable β¨U, Vβ© $ mem_filter.2 β¨mem_univ _, hUV, hβ© | case inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | rintro Uβ Vβ huseful hUcard | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
Set.S... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
Set.S... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | by_contra h | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
Set.S... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
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 β¨Uβ, Vββ© $ mem_filter.2 β¨mem_univ _, huseful, hβ© | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r : β
π A : Finset (Finset (Fin n))
ih :
β (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β
Set.Sized r βy β
β β¬,
(β β¬).card β€ (β y).card β§
y.card = β¬.card β§
Set.S... | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | obtain β¨β¬, card_le, t, hβ¬, fully_compβ© := UV.kruskal_katona_helper π hβ | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β’ (β π).card β€ (β π).card | case intro.intro.intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β¬ : Finset (Finset (Fin n))
card_le : (β β¬).card β€ (β π).card
t : π.card = β¬.card
hβ¬ : Set.Sized r ββ¬
fully_comp : β (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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β¬ : Finset (Finset (Fin n))
card_le : (β β¬).card β€ (β π).card
t : π.card = β¬.card
hβ¬ : Set.Sized r ββ¬
fully_comp : β (U V : Finse... | case h.e'_3.h.e'_2.h.e'_3
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β¬ : Finset (Finset (Fin n))
card_le : (β β¬).card β€ (β π).card
t : π.card = β¬.card
hβ¬ : Set.Sized r ββ¬
fully_comp : β (U V : Finset (... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | have hcard : card β¬ = card π := t.symm.trans hβ | case h.e'_3.h.e'_2.h.e'_3
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β¬ : Finset (Finset (Fin n))
card_le : (β β¬).card β€ (β π).card
t : π.card = β¬.card
hβ¬ : Set.Sized r ββ¬
fully_comp : β (U V : Finset (... | case h.e'_3.h.e'_2.h.e'_3
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β¬ : Finset (Finset (Fin n))
card_le : (β β¬).card β€ (β π).card
t : π.card = β¬.card
hβ¬ : Set.Sized r ββ¬
fully_comp : β (U V : Finset (... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | obtain CB | BC :=
hβ.total (UV.isInitSeg_of_compressed hβ¬ fun U V hUV β¦ by convert fully_comp U V hUV) | case h.e'_3.h.e'_2.h.e'_3
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β¬ : Finset (Finset (Fin n))
card_le : (β β¬).card β€ (β π).card
t : π.card = β¬.card
hβ¬ : Set.Sized r ββ¬
fully_comp : β (U V : Finset (... | case h.e'_3.h.e'_2.h.e'_3.inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β¬ : Finset (Finset (Fin n))
card_le : (β β¬).card β€ (β π).card
t : π.card = β¬.card
hβ¬ : Set.Sized r ββ¬
fully_comp : β (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 | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s Uβ Vβ : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β¬ : Finset (Finset (Fin n))
card_le : (β β¬).card β€ (β π).card
t : π.card = β¬.card
hβ¬ : Set.Sized r ββ¬
fully_comp : β (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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β¬ : Finset (Finset (Fin n))
card_le : (β β¬).card β€ (β π).card
t : π.card = β¬.card
hβ¬ : Set.Sized r ββ¬
fully_comp : β (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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card = π.card
hβ : IsInitSeg π r
β¬ : Finset (Finset (Fin n))
card_le : (β β¬).card β€ (β π).card
t : π.card = β¬.card
hβ¬ : Set.Sized r ββ¬
fully_comp : β (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 π π.card hβ with β¨π', prop, sizeβ© | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg π r
β’ (β π).card β€ (β π).card | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg π r
π' : Finset (Finset (Fin n))
prop : π' β π
size : π'.card = π.card
β’ (β π).card β€ (β π).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.strengthened_kk | [316, 1] | [323, 29] | refine' (kruskal_katona (fun A hA β¦ hβ (prop hA)) size hβ).trans (card_le_card _) | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg π r
π' : Finset (Finset (Fin n))
prop : π' β π
size : π'.card = π.card
β’ (β π).card β€ (β π).card | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg π r
π' : Finset (Finset (Fin n))
prop : π' β π
size : π'.card = π.card
β’ β π' β β π |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.strengthened_kk | [316, 1] | [323, 29] | rw [shadow, shadow] | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg π r
π' : Finset (Finset (Fin n))
prop : π' β π
size : π'.card = π.card
β’ β π' β β π | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg π r
π' : Finset (Finset (Fin n))
prop : π' β π
size : π'.card = π.card
β’ (π'.sup fun s => image s.erase s) β π.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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg π r
π' : Finset (Finset (Fin n))
prop : π' β π
size : π'.card = π.card
β’ (π'.sup fun s => image s.erase s) β π.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 π π | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg π r
β’ (β ^[k] π).card β€ (β ^[k] π).card | case zero
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n i r : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg π r
β’ (β ^[0] π).card β€ (β ^[0] π).card
case succ
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n i _k : β
ih :
β {r : β} {π π : Finset (Finset... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | simpa | case zero
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n i r : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg π r
β’ (β ^[0] π).card β€ (β ^[0] π).card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | refine' ih hβ.shadow (strengthened_kk hβ hβ hβ) _ | case succ
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n i _k : β
ih :
β {r : β} {π π : Finset (Finset (Fin n))},
Set.Sized r βπ β π.card β€ π.card β IsInitSeg π r β (β ^[_k] π).card β€ (β ^[_k] π).card
r : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg οΏ½... | case succ
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n i _k : β
ih :
β {r : β} {π π : Finset (Finset (Fin n))},
Set.Sized r βπ β π.card β€ π.card β IsInitSeg π r β (β ^[_k] π).card β€ (β ^[_k] π).card
r : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg οΏ½... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | convert hβ.shadow | case succ
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n i _k : β
ih :
β {r : β} {π π : Finset (Finset (Fin n))},
Set.Sized r βπ β π.card β€ π.card β IsInitSeg π r β (β ^[_k] π).card β€ (β ^[_k] π).card
r : β
π π : Finset (Finset (Fin n))
hβ : Set.Sized r βπ
hβ : π.card β€ π.card
hβ : IsInitSeg οΏ½... | 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 β¦ by rw [mem_range]; apply forall_lt_iff_le.2 hkn | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
β’ k.choose (r - i) β€ (β ^[i] π).card | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
β’ k.choose (r - i) β€ (β ^[i] π).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | set π : Finset (Finset (Fin n)) := powersetCard r range'k | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
β’ k.choose (r - i) β€ (β ^[i] π).card | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
β’ k.choose (r - i) β€ (β ^[i] π).ca... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have Ccard : π.card = k.choose r := by rw [card_powersetCard, card_attachFin, card_range] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
β’ k.choose (r - i) β€ (β ^[i] π).ca... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
β’ k.ch... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have : (π : Set (Finset (Fin n))).Sized r := Set.sized_powersetCard _ _ | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
β’ k.ch... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
this :... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | suffices this : (β^[i] π).card = k.choose (r - i) by
rw [β this]
apply iterated_kk hβ _ _
rwa [Ccard]
refine' β¨βΉ_βΊ, _β©
rintro A B hA β¨HBβ, HBββ©
rw [mem_powersetCard]
refine' β¨fun t ht β¦ _, βΉ_βΊβ©
rw [mem_attachFin, mem_range]
have : toColex (image Fin.val B) < toColex (image Fin.val A) := by
rwa [... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
this :... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
this :... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | suffices β^[i] π = powersetCard (r - i) range'k by
rw [this, card_powersetCard, card_attachFin, card_range] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
this :... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
this :... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | ext B | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
this :... | case a
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r... | case a
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | constructor | case a
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r... | case a.mp
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choos... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rintro β¨hBk, hBβ© | case a.mpr
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choo... | case a.mpr.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = ... | case a.mpr.intro.refine_2
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : οΏ½... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | obtain β¨C, BsubC, hCrange, hcardβ© := this | case a.mpr.intro.refine_2
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : οΏ½... | case a.mpr.intro.refine_2.intro.intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : 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, β Nat.add_sub_assoc hir, Nat.add_sub_cancel_left] at hcard | case a.mpr.intro.refine_2.intro.intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r... | case a.mpr.intro.refine_2.intro.intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | refine' β¨C, _, BsubC, _β© | case a.mpr.intro.refine_2.intro.intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r... | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powe... | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powe... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | exact β¨hCrange, hcardβ© | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powe... | case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : 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] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
m : β
β’ m β range k β m < n | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
m : β
β’ m < k β 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 | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π π : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
m : β
β’ m < k β 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] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
β’ π.card = k.choose r | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [β this] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβ ... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | apply iterated_kk hβ _ _ | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβ ... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rwa [Ccard] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβ ... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | refine' β¨βΉ_βΊ, _β© | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβ ... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rintro A B hA β¨HBβ, HBββ© | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβ ... | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choo... | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choo... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | refine' β¨fun t ht β¦ _, βΉ_βΊβ© | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choo... | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choo... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_attachFin, mem_range] | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choo... | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choo... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have : toColex (image Fin.val B) < toColex (image Fin.val A) := by
rwa [toColex_image_lt_toColex_image Fin.val_strictMono] | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choo... | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choo... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | apply Colex.forall_lt_mono this.le _ t (mem_image.2 β¨t, ht, rflβ©) | case intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choo... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβΒΉ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | simp_rw [mem_image] | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβΒΉ... | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβΒΉ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rintro _ β¨a, ha, qβ© | Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = k.choose r
thisβΒΉ... | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_powersetCard] at hA | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = ... | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [β q, β mem_range] | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = ... | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have := hA.1 ha | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = ... | case intro.intro
Ξ± : Type u_1
instβ : LinearOrder Ξ±
s U V : Finset Ξ±
n r k i : β
π πβ : Finset (Finset (Fin n))
hir : i β€ r
hrk : r β€ k
hkn : k β€ n
hβ : Set.Sized r βπ
hβ : k.choose r β€ π.card
range'k : Finset (Fin n) := (range k).attachFin β―
π : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : π.card = ... |
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