Datasets:
internlm
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Lean-Github

Dataset card Data Studio Files Community
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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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• π’œ : Finset (Finset Ξ±) hβ‚‚ : βˆ€ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ π’œ UVd : Disjoint U V same_size : U.card = V.card hU : U.Nonempty hV : V.Nonempty max_lt : U.max' hU < V.max' hV x : Ξ± Hx : x ∈ U hU' : succ 0 < U.card ⊒ (U.erase x).card = (V.erase (V.min' hV)).card
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• π’œ : Finset (Finset Ξ±) hβ‚‚ : βˆ€ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ π’œ UVd : Disjoint U V same_size : U.card = V.card hU : U.Nonempty hV : V.Nonempty max_lt : U.max' hU < V.max' hV x : Ξ± Hx : x ∈ U hU' : succ 0 < U.card ⊒ (U.erase x).Nonempty
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• π’œ : Finset (Finset Ξ±) hβ‚‚ : βˆ€ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ π’œ UVd : Disjoint U V same_size : U.card = V.card hU : U.Nonempty hV : V.Nonempty max_lt : U.max' hU < V.max' hV x : Ξ± Hx : x ∈ U hU' : succ 0 < U.card ⊒ (V.erase (V.min' hV)).Nonempty
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• π’œ : Finset (Finset Ξ±) hβ‚‚ : βˆ€ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ π’œ UVd : Disjoint U V same_size : U.card = V.card hU : U.Nonempty hV : V.Nonempty max_lt : U.max' hU < V.max' hV x : Ξ± Hx : x ∈ U hU' : succ 0 < U.card ⊒ (U.erase x).max' β‹― < (V.erase (V.min' hV)).max' β‹―
case intro.intro.intro.intro.inr.refine'_4 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• π’œ : Finset (Finset Ξ±) hβ‚‚ : βˆ€ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ π’œ UVd : Disjoint U V same_size : U.card = V.card hU : U.Nonempty hV : V.Nonempty max_lt : U.max' hU < V.max' hV x : Ξ± Hx : x ∈ U hU' : succ 0 < U.card ⊒ 1 < V.card
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• π’œ : Finset (Finset Ξ±) hβ‚‚ : βˆ€ ⦃U₁ V₁ : Finset α⦄, Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ π’œ UVd : Disjoint U V same_size : U.card = V.card hU : U.Nonempty hV : V.Nonempty max_lt : U.max' hU < V.max' hV x : Ξ± Hx : x ∈ U hU' : succ 0 < U.card ⊒ 1 < V.card
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.UV.isInitSeg_of_compressed
[197, 1]
[218, 73]
refine' ⟨h₁, _⟩
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ ⊒ IsInitSeg ℬ r
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ ⊒ βˆ€ ⦃s t : Finset α⦄, s ∈ ℬ β†’ { ofColex := t } < { ofColex := s } ∧ t.card = r β†’ 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⟩
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ ⊒ βˆ€ ⦃s t : Finset α⦄, s ∈ ℬ β†’ { ofColex := t } < { ofColex := s } ∧ t.card = r β†’ t ∈ ℬ
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± 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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r ⊒ B ∈ ℬ
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ ⊒ False
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ ⊒ False
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B ⊒ False
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card ⊒ False
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card ⊒ False
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty ⊒ False
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty ⊒ False
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty ⊒ False
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty ⊒ False
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) ⊒ False
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) smaller : (B \ A).max' hV < (A \ B).max' hU ⊒ False
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) smaller : (B \ A).max' hV < (A \ B).max' hU ⊒ B ∈ ℬ
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.UV.isInitSeg_of_compressed
[197, 1]
[218, 73]
rw [← (hβ‚‚ _ _ ⟨disj, card_sdiff_comm hAB'.symm, hV, hU, smaller⟩).eq]
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) smaller : (B \ A).max' hV < (A \ B).max' hU ⊒ B ∈ ℬ
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) smaller : (B \ A).max' hV < (A \ B).max' hU ⊒ B ∈ 𝓒 (B \ A) (A \ B) ℬ
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) smaller : (B \ A).max' hV < (A \ B).max' hU ⊒ B ∈ 𝓒 (B \ A) (A \ B) ℬ
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) ⊒ (B \ A).max' hV < (A \ B).max' hU
case inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) hlt : (A \ B).max' hU < (B \ A).max' hV ⊒ (B \ A).max' hV < (A \ B).max' hU case inr.inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) heq : (A \ B).max' hU = (B \ A).max' hV ⊒ (B \ A).max' hV < (A \ B).max' hU case inr.inr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) hgt : (B \ A).max' hV < (A \ B).max' hU ⊒ (B \ A).max' hV < (A \ B).max' hU
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) hlt : (A \ B).max' hU < (B \ A).max' hV ⊒ (B \ A).max' hV < (A \ B).max' hU
case inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := compress (A \ B) (B \ A) B } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : compress (A \ B) (B \ A) B β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) hlt : (A \ B).max' hU < (B \ A).max' hV ⊒ (B \ A).max' hV < (A \ B).max' hU
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := compress (A \ B) (B \ A) B } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : compress (A \ B) (B \ A) B β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) hlt : (A \ B).max' hU < (B \ A).max' hV ⊒ (B \ A).max' hV < (A \ B).max' hU
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 β–Έ max'_mem _ _).elim
case inr.inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) heq : (A \ B).max' hU = (B \ A).max' hV ⊒ (B \ A).max' hV < (A \ B).max' hU
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n : β„• ℬ : Finset (Finset Ξ±) r : β„• h₁ : Set.Sized r ↑ℬ hβ‚‚ : βˆ€ (U V : Finset Ξ±), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ A B : Finset Ξ± hA : A ∈ ℬ hBA : { ofColex := B } < { ofColex := A } sizeA : B.card = r hB : B βˆ‰ ℬ hAB : A β‰  B hAB' : A.card = B.card hU : (A \ B).Nonempty hV : (B \ A).Nonempty disj : Disjoint (B \ A) (A \ B) hgt : (B \ A).max' hV < (A \ B).max' hU ⊒ (B \ A).max' hV < (A \ B).max' hU
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 ⊒
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : 𝓒 U V π’œ β‰  π’œ ⊒ Finset.UV.familyMeasure (𝓒 U V π’œ) < Finset.UV.familyMeasure π’œ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ ⊒ Finset.UV.familyMeasure (filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ)) < Finset.UV.familyMeasure π’œ
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.UV.familyMeasure_compression_lt_familyMeasure
[231, 1]
[256, 47]
have q : βˆ€ Q ∈ π’œ.filter fun A ↦ compress U V A βˆ‰ π’œ, 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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ ⊒ Finset.UV.familyMeasure (filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ)) < Finset.UV.familyMeasure π’œ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q ⊒ Finset.UV.familyMeasure (filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ)) < Finset.UV.familyMeasure π’œ
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.UV.familyMeasure_compression_lt_familyMeasure
[231, 1]
[256, 47]
have uA : (π’œ.filter fun A => compress U V A ∈ π’œ) βˆͺ π’œ.filter (fun A ↦ compress U V A βˆ‰ π’œ) = π’œ := filter_union_filter_neg_eq _ _
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q ⊒ Finset.UV.familyMeasure (filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ)) < Finset.UV.familyMeasure π’œ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ ⊒ Finset.UV.familyMeasure (filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ)) < Finset.UV.familyMeasure π’œ
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.UV.familyMeasure_compression_lt_familyMeasure
[231, 1]
[256, 47]
have neβ‚‚ : (π’œ.filter fun A ↦ compress U V A βˆ‰ π’œ).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
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ ⊒ Finset.UV.familyMeasure (filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ)) < Finset.UV.familyMeasure π’œ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty ⊒ Finset.UV.familyMeasure (filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ)) < Finset.UV.familyMeasure π’œ
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty ⊒ Finset.UV.familyMeasure (filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ)) < Finset.UV.familyMeasure π’œ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty ⊒ βˆ‘ x ∈ filter (fun x => compress U V x ∈ π’œ) π’œ, βˆ‘ a ∈ x, 2 ^ ↑a + βˆ‘ x ∈ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ), βˆ‘ a ∈ x, 2 ^ ↑a < βˆ‘ A ∈ π’œ, βˆ‘ a ∈ A, 2 ^ ↑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]
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty ⊒ βˆ‘ x ∈ filter (fun x => compress U V x ∈ π’œ) π’œ, βˆ‘ a ∈ x, 2 ^ ↑a + βˆ‘ x ∈ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ), βˆ‘ a ∈ x, 2 ^ ↑a < βˆ‘ A ∈ π’œ, βˆ‘ a ∈ A, 2 ^ ↑a
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty ⊒ βˆ‘ x ∈ filter (fun x => compress U V x ∈ π’œ) π’œ, βˆ‘ a ∈ x, 2 ^ ↑a + βˆ‘ x ∈ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ), βˆ‘ a ∈ x, 2 ^ ↑a < βˆ‘ A ∈ filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, βˆ‘ a ∈ A, 2 ^ ↑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]
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty ⊒ βˆ‘ x ∈ filter (fun x => compress U V x ∈ π’œ) π’œ, βˆ‘ a ∈ x, 2 ^ ↑a + βˆ‘ x ∈ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ), βˆ‘ a ∈ x, 2 ^ ↑a < βˆ‘ A ∈ filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, βˆ‘ a ∈ A, 2 ^ ↑a
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty ⊒ βˆ‘ x ∈ filter (fun x => compress U V x βˆ‰ π’œ) π’œ, βˆ‘ a ∈ compress U V x, 2 ^ ↑a < βˆ‘ x ∈ filter (fun a => compress U V a βˆ‰ π’œ) π’œ, βˆ‘ a ∈ x, 2 ^ ↑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 neβ‚‚ fun A hA ↦ _
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty ⊒ βˆ‘ x ∈ filter (fun x => compress U V x βˆ‰ π’œ) π’œ, βˆ‘ a ∈ compress U V x, 2 ^ ↑a < βˆ‘ x ∈ filter (fun a => compress U V a βˆ‰ π’œ) π’œ, βˆ‘ a ∈ x, 2 ^ ↑a
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty A : Finset (Fin n) hA : A ∈ filter (fun x => compress U V x βˆ‰ π’œ) π’œ ⊒ βˆ‘ a ∈ compress U V A, 2 ^ ↑a < βˆ‘ a ∈ A, 2 ^ ↑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]
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty A : Finset (Fin n) hA : A ∈ filter (fun x => compress U V x βˆ‰ π’œ) π’œ ⊒ βˆ‘ a ∈ compress U V A, 2 ^ ↑a < βˆ‘ a ∈ A, 2 ^ ↑a
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty A : Finset (Fin n) hA : A ∈ filter (fun x => compress U V x βˆ‰ π’œ) π’œ ⊒ βˆ‘ x ∈ image (fun x => ↑x) (compress U V A), 2 ^ x < βˆ‘ x ∈ image (fun x => ↑x) A, 2 ^ x
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty A : Finset (Fin n) hA : A ∈ filter (fun x => compress U V x βˆ‰ π’œ) π’œ ⊒ βˆ‘ x ∈ image (fun x => ↑x) (compress U V A), 2 ^ x < βˆ‘ x ∈ image (fun x => ↑x) A, 2 ^ x
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty A : Finset (Fin n) hA : A ∈ filter (fun x => compress U V x βˆ‰ π’œ) π’œ ⊒ { ofColex := compress U V A } < { ofColex := 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
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ neβ‚‚ : (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty A : Finset (Fin n) hA : A ∈ filter (fun x => compress U V x βˆ‰ π’œ) π’œ ⊒ { ofColex := compress U V A } < { ofColex := 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]
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ ⊒ βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ ⊒ βˆ€ (Q : Finset (Fin n)), Q ∈ π’œ ∧ compress U V Q βˆ‰ π’œ β†’ compress U V Q β‰  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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ ⊒ βˆ€ (Q : Finset (Fin n)), Q ∈ π’œ ∧ compress U V Q βˆ‰ π’œ β†’ compress U V Q β‰  Q
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h✝ : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ Q : Finset (Fin n) hQ : Q ∈ π’œ ∧ compress U V Q βˆ‰ π’œ h : compress U V Q = Q ⊒ False
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h✝ : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ Q : Finset (Fin n) hQ : Q ∈ π’œ ∧ compress U V Q βˆ‰ π’œ h : compress U V Q = Q ⊒ False
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h✝ : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ Q : Finset (Fin n) hQ : Q ∈ π’œ ∧ Q βˆ‰ π’œ h : compress U V Q = Q ⊒ False
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h✝ : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ Q : Finset (Fin n) hQ : Q ∈ π’œ ∧ Q βˆ‰ π’œ h : compress U V Q = Q ⊒ False
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 _
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ ⊒ (filter (fun A => compress U V A βˆ‰ π’œ) π’œ).Nonempty
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ z : filter (fun A => compress U V A βˆ‰ π’œ) π’œ = βˆ… ⊒ filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) = π’œ
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ z : filter (fun A => compress U V A βˆ‰ π’œ) π’œ = βˆ… ⊒ filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) = π’œ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ z : filter (fun A => compress U V A βˆ‰ π’œ) π’œ = βˆ… ⊒ filter (fun x => compress U V x ∈ π’œ) π’œ = π’œ
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n : β„• U V : Finset (Fin n) hU : U.Nonempty hV : V.Nonempty h : U.max' hU < V.max' hV π’œ : Finset (Finset (Fin n)) a : filter (fun x => compress U V x ∈ π’œ) π’œ βˆͺ filter (fun x => x βˆ‰ π’œ) (image (compress U V) π’œ) β‰  π’œ q : βˆ€ Q ∈ filter (fun A => compress U V A βˆ‰ π’œ) π’œ, compress U V Q β‰  Q uA : filter (fun A => compress U V A ∈ π’œ) π’œ βˆͺ filter (fun A => compress U V A βˆ‰ π’œ) π’œ = π’œ z : filter (fun A => compress U V A βˆ‰ π’œ) π’œ = βˆ… ⊒ filter (fun x => compress U V x ∈ π’œ) π’œ = π’œ
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ : Finset (Finset (Fin n)) h : Set.Sized r β†‘π’œ ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card ∧ π’œ.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ : Finset (Finset (Fin n)) ⊒ Set.Sized r β†‘π’œ β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card ∧ π’œ.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ 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) π’œ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ : Finset (Finset (Fin n)) ⊒ Set.Sized r β†‘π’œ β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card ∧ π’œ.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ : Finset (Finset (Fin n)) ⊒ βˆ€ (x : Finset (Finset (Fin n))), (βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y x β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ) β†’ Set.Sized r ↑x β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ x).card ∧ x.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ : Finset (Finset (Fin n)) ⊒ βˆ€ (x : Finset (Finset (Fin n))), (βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y x β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ) β†’ Set.Sized r ↑x β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ x).card ∧ x.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
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 ↦ UsefulCompression t.1 t.2 ∧ Β¬ IsCompressed t.1 t.2 A
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
case inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable = βˆ… ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ case inr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
case inr.intro.mk.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ usable t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ usable t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
case inr.intro.mk.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.UV.kruskal_katona_helper
[258, 1]
[290, 73]
have hβ‚‚ : βˆ€ U₁ V₁, UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A := by rintro U₁ V₁ huseful hUcard by_contra h exact hUcard.not_le $ t ⟨U₁, Vβ‚βŸ© $ mem_filter.2 ⟨mem_univ _, huseful, h⟩
case inr.intro.mk.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
case inr.intro.mk.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card hβ‚‚ : βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.UV.kruskal_katona_helper
[258, 1]
[290, 73]
have p1 : (βˆ‚ (𝓒 U V A)).card ≀ (βˆ‚ A).card := compression_improved _ hUV.2.1 hβ‚‚
case inr.intro.mk.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card hβ‚‚ : βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
case inr.intro.mk.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card hβ‚‚ : βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A p1 : (βˆ‚ (𝓒 U V A)).card ≀ (βˆ‚ A).card ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card hβ‚‚ : βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A p1 : (βˆ‚ (𝓒 U V A)).card ≀ (βˆ‚ A).card ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
case inr.intro.mk.intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card hβ‚‚ : βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A p1 : (βˆ‚ (𝓒 U V A)).card ≀ (βˆ‚ A).card hUV' : (U, V).1.card = (U, V).2.card hu : (U, V).1.Nonempty hv : (U, V).2.Nonempty hmax : (U, V).1.max' hu < (U, V).2.max' hv ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card hβ‚‚ : βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A p1 : (βˆ‚ (𝓒 U V A)).card ≀ (βˆ‚ A).card hUV' : (U, V).1.card = (U, V).2.card hu : (U, V).1.Nonempty hv : (U, V).2.Nonempty hmax : (U, V).1.max' hu < (U, V).2.max' hv ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
case inr.intro.mk.intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), (fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card hβ‚‚ : βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A p1 : (βˆ‚ (𝓒 U V A)).card ≀ (βˆ‚ A).card hUV' : (U, V).1.card = (U, V).2.card hu : (U, V).1.Nonempty hv : (U, V).2.Nonempty hmax : (U, V).1.max' hu < (U, V).2.max' hv ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), (fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card hβ‚‚ : βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A p1 : (βˆ‚ (𝓒 U V A)).card ≀ (βˆ‚ A).card hUV' : (U, V).1.card = (U, V).2.card hu : (U, V).1.Nonempty hv : (U, V).2.Nonempty hmax : (U, V).1.max' hu < (U, V).2.max' hv ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
case inr.intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), (fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t✝ : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card hβ‚‚ : βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A p1 : (βˆ‚ (𝓒 U V A)).card ≀ (βˆ‚ A).card hUV' : (U, V).1.card = (U, V).2.card hu : (U, V).1.Nonempty hv : (U, V).2.Nonempty hmax : (U, V).1.max' hu < (U, V).2.max' hv t : Finset (Finset (Fin n)) q1 : (βˆ‚ t).card ≀ (βˆ‚ (𝓒 U V A)).card q2 : (𝓒 U V A).card = t.card q3 : Set.Sized r ↑t q4 : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V t ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), (fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t✝ : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card hβ‚‚ : βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A p1 : (βˆ‚ (𝓒 U V A)).card ≀ (βˆ‚ A).card hUV' : (U, V).1.card = (U, V).2.card hu : (U, V).1.Nonempty hv : (U, V).2.Nonempty hmax : (U, V).1.max' hu < (U, V).2.max' hv t : Finset (Finset (Fin n)) q1 : (βˆ‚ t).card ≀ (βˆ‚ (𝓒 U V A)).card q2 : (𝓒 U V A).card = t.card q3 : Set.Sized r ↑t q4 : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V t ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable = βˆ… ⊒ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ A).card ∧ A.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ
case inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable = βˆ… U V : Finset (Fin n) hUV : Finset.UV.UsefulCompression U V ⊒ IsCompressed U V A
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable = βˆ… U V : Finset (Fin n) hUV : Finset.UV.UsefulCompression U V ⊒ IsCompressed U V A
case inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : βˆ€ (x : Finset (Fin n) Γ— Finset (Fin n)), x βˆ‰ usable U V : Finset (Fin n) hUV : Finset.UV.UsefulCompression U V ⊒ IsCompressed U V A
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : βˆ€ (x : Finset (Fin n) Γ— Finset (Fin n)), x βˆ‰ usable U V : Finset (Fin n) hUV : Finset.UV.UsefulCompression U V ⊒ IsCompressed U V A
case inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h✝ : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : βˆ€ (x : Finset (Fin n) Γ— Finset (Fin n)), x βˆ‰ usable U V : Finset (Fin n) hUV : Finset.UV.UsefulCompression U V h : Β¬IsCompressed U V A ⊒ False
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h✝ : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : βˆ€ (x : Finset (Fin n) Γ— Finset (Fin n)), x βˆ‰ usable U V : Finset (Fin n) hUV : Finset.UV.UsefulCompression U V h : Β¬IsCompressed U V A ⊒ False
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
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card ⊒ βˆ€ (U₁ V₁ : Finset (Fin n)), Finset.UV.UsefulCompression U₁ V₁ β†’ U₁.card < U.card β†’ IsCompressed U₁ V₁ A
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card U₁ V₁ : Finset (Fin n) huseful : Finset.UV.UsefulCompression U₁ V₁ hUcard : U₁.card < U.card ⊒ IsCompressed U₁ V₁ A
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card U₁ V₁ : Finset (Fin n) huseful : Finset.UV.UsefulCompression U₁ V₁ hUcard : U₁.card < U.card ⊒ IsCompressed U₁ V₁ A
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h✝ : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card U₁ V₁ : Finset (Fin n) huseful : Finset.UV.UsefulCompression U₁ V₁ hUcard : U₁.card < U.card h : Β¬IsCompressed U₁ V₁ A ⊒ False
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, Vβ‚βŸ© $ mem_filter.2 ⟨mem_univ _, huseful, h⟩
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r : β„• π’œ A : Finset (Finset (Fin n)) ih : βˆ€ (y : Finset (Finset (Fin n))), InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A β†’ Set.Sized r ↑y β†’ βˆƒ ℬ, (βˆ‚ ℬ).card ≀ (βˆ‚ y).card ∧ y.card = ℬ.card ∧ Set.Sized r ↑ℬ ∧ βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ h✝ : Set.Sized r ↑A usable : Finset (Finset (Fin n) Γ— Finset (Fin n)) := filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ Β¬IsCompressed t.1 t.2 A) univ husable : usable.Nonempty U V : Finset (Fin n) hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ Β¬IsCompressed (U, V).1 (U, V).2 A t : βˆ€ x' ∈ usable, (U, V).1.card ≀ x'.1.card U₁ V₁ : Finset (Fin n) huseful : Finset.UV.UsefulCompression U₁ V₁ hUcard : U₁.card < U.card h : Β¬IsCompressed U₁ V₁ A ⊒ False
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.kruskal_katona
[299, 1]
[314, 53]
obtain βŸ¨β„¬, card_le, t, hℬ, fully_comp⟩ := UV.kruskal_katona_helper π’œ h₁
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ⊒ (βˆ‚ π’ž).card ≀ (βˆ‚ π’œ).card
case intro.intro.intro.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ ⊒ (βˆ‚ π’ž).card ≀ (βˆ‚ π’œ).card
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ ⊒ (βˆ‚ π’ž).card ≀ (βˆ‚ π’œ).card
case h.e'_3.h.e'_2.h.e'_3 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ ⊒ π’ž = ℬ
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.kruskal_katona
[299, 1]
[314, 53]
have hcard : card ℬ = card π’ž := t.symm.trans hβ‚‚
case h.e'_3.h.e'_2.h.e'_3 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ ⊒ π’ž = ℬ
case h.e'_3.h.e'_2.h.e'_3 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ hcard : ℬ.card = π’ž.card ⊒ π’ž = ℬ
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ hcard : ℬ.card = π’ž.card ⊒ π’ž = ℬ
case h.e'_3.h.e'_2.h.e'_3.inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ hcard : ℬ.card = π’ž.card CB : π’ž βŠ† ℬ ⊒ π’ž = ℬ case h.e'_3.h.e'_2.h.e'_3.inr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ hcard : ℬ.card = π’ž.card BC : ℬ βŠ† π’ž ⊒ π’ž = ℬ
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 inst✝ : LinearOrder Ξ± s U✝ V✝ : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ hcard : ℬ.card = π’ž.card U V : Finset (Fin n) hUV : Finset.UV.UsefulCompression U V ⊒ IsCompressed U V ℬ
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ hcard : ℬ.card = π’ž.card CB : π’ž βŠ† ℬ ⊒ π’ž = ℬ
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’œ.card = π’ž.card h₃ : IsInitSeg π’ž r ℬ : Finset (Finset (Fin n)) card_le : (βˆ‚ ℬ).card ≀ (βˆ‚ π’œ).card t : π’œ.card = ℬ.card hℬ : Set.Sized r ↑ℬ fully_comp : βˆ€ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V β†’ IsCompressed U V ℬ hcard : ℬ.card = π’ž.card BC : ℬ βŠ† π’ž ⊒ π’ž = ℬ
no goals
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.strengthened_kk
[316, 1]
[323, 29]
rcases exists_smaller_set π’œ π’ž.card hβ‚‚ with βŸ¨π’œ', prop, size⟩
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r ⊒ (βˆ‚ π’ž).card ≀ (βˆ‚ π’œ).card
case intro.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r π’œ' : Finset (Finset (Fin n)) prop : π’œ' βŠ† π’œ size : π’œ'.card = π’ž.card ⊒ (βˆ‚ π’ž).card ≀ (βˆ‚ π’œ).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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r π’œ' : Finset (Finset (Fin n)) prop : π’œ' βŠ† π’œ size : π’œ'.card = π’ž.card ⊒ (βˆ‚ π’ž).card ≀ (βˆ‚ π’œ).card
case intro.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r π’œ' : Finset (Finset (Fin n)) prop : π’œ' βŠ† π’œ size : π’œ'.card = π’ž.card ⊒ βˆ‚ π’œ' βŠ† βˆ‚ π’œ
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r π’œ' : Finset (Finset (Fin n)) prop : π’œ' βŠ† π’œ size : π’œ'.card = π’ž.card ⊒ βˆ‚ π’œ' βŠ† βˆ‚ π’œ
case intro.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r π’œ' : Finset (Finset (Fin n)) prop : π’œ' βŠ† π’œ size : π’œ'.card = π’ž.card ⊒ (π’œ'.sup fun s => image s.erase s) βŠ† π’œ.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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r π’œ' : Finset (Finset (Fin n)) prop : π’œ' βŠ† π’œ size : π’œ'.card = π’ž.card ⊒ (π’œ'.sup fun s => image s.erase s) βŠ† π’œ.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’œ π’ž
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r ⊒ (βˆ‚ ^[k] π’ž).card ≀ (βˆ‚ ^[k] π’œ).card
case zero Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n i r : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r ⊒ (βˆ‚ ^[0] π’ž).card ≀ (βˆ‚ ^[0] π’œ).card case succ Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n i _k : β„• ih : βˆ€ {r : β„•} {π’œ π’ž : Finset (Finset (Fin n))}, Set.Sized r β†‘π’œ β†’ π’ž.card ≀ π’œ.card β†’ IsInitSeg π’ž r β†’ (βˆ‚ ^[_k] π’ž).card ≀ (βˆ‚ ^[_k] π’œ).card r : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r ⊒ (βˆ‚ ^[_k + 1] π’ž).card ≀ (βˆ‚ ^[_k + 1] π’œ).card
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.iterated_kk
[325, 1]
[332, 22]
simpa
case zero Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n i r : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r ⊒ (βˆ‚ ^[0] π’ž).card ≀ (βˆ‚ ^[0] π’œ).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 hβ‚‚ h₃) _
case succ Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n i _k : β„• ih : βˆ€ {r : β„•} {π’œ π’ž : Finset (Finset (Fin n))}, Set.Sized r β†‘π’œ β†’ π’ž.card ≀ π’œ.card β†’ IsInitSeg π’ž r β†’ (βˆ‚ ^[_k] π’ž).card ≀ (βˆ‚ ^[_k] π’œ).card r : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r ⊒ (βˆ‚ ^[_k + 1] π’ž).card ≀ (βˆ‚ ^[_k + 1] π’œ).card
case succ Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n i _k : β„• ih : βˆ€ {r : β„•} {π’œ π’ž : Finset (Finset (Fin n))}, Set.Sized r β†‘π’œ β†’ π’ž.card ≀ π’œ.card β†’ IsInitSeg π’ž r β†’ (βˆ‚ ^[_k] π’ž).card ≀ (βˆ‚ ^[_k] π’œ).card r : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r ⊒ IsInitSeg (βˆ‚ π’ž) (r - 1)
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.iterated_kk
[325, 1]
[332, 22]
convert h₃.shadow
case succ Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n i _k : β„• ih : βˆ€ {r : β„•} {π’œ π’ž : Finset (Finset (Fin n))}, Set.Sized r β†‘π’œ β†’ π’ž.card ≀ π’œ.card β†’ IsInitSeg π’ž r β†’ (βˆ‚ ^[_k] π’ž).card ≀ (βˆ‚ ^[_k] π’œ).card r : β„• π’œ π’ž : Finset (Finset (Fin n)) h₁ : Set.Sized r β†‘π’œ hβ‚‚ : π’ž.card ≀ π’œ.card h₃ : IsInitSeg π’ž r ⊒ IsInitSeg (βˆ‚ π’ž) (r - 1)
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
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card ⊒ k.choose (r - i) ≀ (βˆ‚ ^[i] π’œ).card
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― ⊒ k.choose (r - i) ≀ (βˆ‚ ^[i] π’œ).card
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
set π’ž : Finset (Finset (Fin n)) := powersetCard r range'k
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― ⊒ k.choose (r - i) ≀ (βˆ‚ ^[i] π’œ).card
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k ⊒ k.choose (r - i) ≀ (βˆ‚ ^[i] π’œ).card
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
have Ccard : π’ž.card = k.choose r := by rw [card_powersetCard, card_attachFin, card_range]
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k ⊒ k.choose (r - i) ≀ (βˆ‚ ^[i] π’œ).card
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r ⊒ k.choose (r - i) ≀ (βˆ‚ ^[i] π’œ).card
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
have : (π’ž : Set (Finset (Fin n))).Sized r := Set.sized_powersetCard _ _
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r ⊒ k.choose (r - i) ≀ (βˆ‚ ^[i] π’œ).card
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž ⊒ k.choose (r - i) ≀ (βˆ‚ ^[i] π’œ).card
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
suffices this : (βˆ‚^[i] π’ž).card = k.choose (r - i) by rw [← this] apply iterated_kk h₁ _ _ rwa [Ccard] refine' βŸ¨β€Ή_β€Ί, _⟩ rintro A B hA ⟨HB₁, HBβ‚‚βŸ© rw [mem_powersetCard] refine' ⟨fun t ht ↦ _, β€Ή_β€ΊβŸ© rw [mem_attachFin, mem_range] have : toColex (image Fin.val B) < toColex (image Fin.val A) := by rwa [toColex_image_lt_toColex_image Fin.val_strictMono] apply Colex.forall_lt_mono this.le _ t (mem_image.2 ⟨t, ht, rfl⟩) simp_rw [mem_image] rintro _ ⟨a, ha, q⟩ rw [mem_powersetCard] at hA rw [← q, ← mem_range] have := hA.1 ha rwa [mem_attachFin] at this
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž ⊒ k.choose (r - i) ≀ (βˆ‚ ^[i] π’œ).card
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž ⊒ (βˆ‚ ^[i] π’ž).card = k.choose (r - i)
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
suffices βˆ‚^[i] π’ž = powersetCard (r - i) range'k by rw [this, card_powersetCard, card_attachFin, card_range]
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž ⊒ (βˆ‚ ^[i] π’ž).card = k.choose (r - i)
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž ⊒ βˆ‚ ^[i] π’ž = powersetCard (r - i) range'k
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
ext B
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž ⊒ βˆ‚ ^[i] π’ž = powersetCard (r - i) range'k
case a Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) ⊒ B ∈ βˆ‚ ^[i] π’ž ↔ B ∈ powersetCard (r - i) range'k
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) ⊒ B ∈ βˆ‚ ^[i] π’ž ↔ B ∈ powersetCard (r - i) range'k
case a Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) ⊒ (βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i) ↔ B βŠ† range'k ∧ B.card = r - i
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
constructor
case a Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) ⊒ (βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i) ↔ B βŠ† range'k ∧ B.card = r - i
case a.mp Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) ⊒ (βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i) β†’ B βŠ† range'k ∧ B.card = r - i case a.mpr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) ⊒ B βŠ† range'k ∧ B.card = r - i β†’ βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
rintro ⟨hBk, hB⟩
case a.mpr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) ⊒ B βŠ† range'k ∧ B.card = r - i β†’ βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i
case a.mpr.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i
case a.mpr.intro.refine_2 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i this : βˆƒ C, B βŠ† C ∧ C βŠ† range'k ∧ C.card = i + B.card ⊒ βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i this : βˆƒ C, B βŠ† C ∧ C βŠ† range'k ∧ C.card = i + B.card ⊒ βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
case a.mpr.intro.refine_2.intro.intro.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = i + B.card ⊒ βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = i + B.card ⊒ βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
case a.mpr.intro.refine_2.intro.intro.intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ βˆƒ s ∈ π’ž, B βŠ† s ∧ (s \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
case a.mpr.intro.refine_2.intro.intro.intro.refine'_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ C ∈ π’ž case a.mpr.intro.refine_2.intro.intro.intro.refine'_2 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ (C \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ C ∈ π’ž case a.mpr.intro.refine_2.intro.intro.intro.refine'_2 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ (C \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
case a.mpr.intro.refine_2.intro.intro.intro.refine'_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ C βŠ† range'k ∧ C.card = r case a.mpr.intro.refine_2.intro.intro.intro.refine'_2 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ (C \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ C βŠ† range'k ∧ C.card = r case a.mpr.intro.refine_2.intro.intro.intro.refine'_2 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ (C \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
case a.mpr.intro.refine_2.intro.intro.intro.refine'_2 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i C : Finset (Fin n) BsubC : B βŠ† C hCrange : C βŠ† range'k hcard : C.card = r ⊒ (C \ B).card = i case a.mpr.intro.refine_1 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this : Set.Sized r β†‘π’ž B : Finset (Fin n) hBk : B βŠ† range'k hB : B.card = r - i ⊒ i + B.card ≀ range'k.card
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
rw [mem_range]
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card m : β„• ⊒ m ∈ range k β†’ m < n
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card m : 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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’ž : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card m : 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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k ⊒ π’ž.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]
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ k.choose (r - i) ≀ (βˆ‚ ^[i] π’œ).card
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ (βˆ‚ ^[i] π’ž).card ≀ (βˆ‚ ^[i] π’œ).card
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
apply iterated_kk h₁ _ _
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ (βˆ‚ ^[i] π’ž).card ≀ (βˆ‚ ^[i] π’œ).card
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ π’ž.card ≀ π’œ.card Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ IsInitSeg π’ž r
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
rwa [Ccard]
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ π’ž.card ≀ π’œ.card Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ IsInitSeg π’ž r
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ IsInitSeg π’ž r
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
refine' βŸ¨β€Ή_β€Ί, _⟩
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ IsInitSeg π’ž r
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ βˆ€ ⦃s t : Finset (Fin n)⦄, s ∈ π’ž β†’ { ofColex := t } < { ofColex := s } ∧ t.card = r β†’ t ∈ π’ž
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
rintro A B hA ⟨HB₁, HBβ‚‚βŸ©
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) ⊒ βˆ€ ⦃s t : Finset (Fin n)⦄, s ∈ π’ž β†’ { ofColex := t } < { ofColex := s } ∧ t.card = r β†’ t ∈ π’ž
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) A B : Finset (Fin n) hA : A ∈ π’ž HB₁ : { ofColex := B } < { ofColex := A } HBβ‚‚ : B.card = r ⊒ B ∈ π’ž
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 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) A B : Finset (Fin n) hA : A ∈ π’ž HB₁ : { ofColex := B } < { ofColex := A } HBβ‚‚ : B.card = r ⊒ B ∈ π’ž
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) A B : Finset (Fin n) hA : A ∈ π’ž HB₁ : { ofColex := B } < { ofColex := A } HBβ‚‚ : B.card = r ⊒ B βŠ† range'k ∧ B.card = r
https://github.com/YaelDillies/LeanCamCombi.git
034199694e3b91536d03bc4a8b0cdbd659cdf50f
LeanCamCombi/KruskalKatona.lean
Finset.lovasz_form
[334, 1]
[382, 91]
refine' ⟨fun t ht ↦ _, β€Ή_β€ΊβŸ©
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) A B : Finset (Fin n) hA : A ∈ π’ž HB₁ : { ofColex := B } < { ofColex := A } HBβ‚‚ : B.card = r ⊒ B βŠ† range'k ∧ B.card = r
case intro Ξ± : Type u_1 inst✝ : LinearOrder Ξ± s U V : Finset Ξ± n r k i : β„• π’œ π’žβœ : Finset (Finset (Fin n)) hir : i ≀ r hrk : r ≀ k hkn : k ≀ n h₁ : Set.Sized r β†‘π’œ hβ‚‚ : k.choose r ≀ π’œ.card range'k : Finset (Fin n) := (range k).attachFin β‹― π’ž : Finset (Finset (Fin n)) := powersetCard r range'k Ccard : π’ž.card = k.choose r this✝ : Set.Sized r β†‘π’ž this : (βˆ‚ ^[i] π’ž).card = k.choose (r - i) A B : Finset (Fin n) hA : A ∈ π’ž HB₁ : { ofColex := B } < { ofColex := A } HBβ‚‚ : B.card = r t : Fin n ht : t ∈ B ⊒ t ∈ range'k