url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | have hAB : A â B := ne_of_mem_of_not_mem hA hB | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
⢠False | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
⢠False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
⢠False
TACTIC:
|
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 := (hâ hA).trans sizeA.symm | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
⢠False | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
hAB' : A.card = B.card
⢠False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
⢠False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | have hU : (A \ B).Nonempty := sdiff_nonempty.2 fun h ⊠hAB $ eq_of_subset_of_card_le h hAB'.ge | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
hAB' : A.card = B.card
⢠False | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
hAB' : A.card = B.card
hU : (A \ B).Nonempty
⢠False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
hAB' : A.card = B.card
⢠False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | have hV : (B \ A).Nonempty :=
sdiff_nonempty.2 fun h ⊠hAB.symm $ eq_of_subset_of_card_le h hAB'.le | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
hAB' : A.card = B.card
hU : (A \ B).Nonempty
⢠False | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
hAB' : A.card = B.card
hU : (A \ B).Nonempty
hV : (B \ A).Nonempty
⢠False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
hAB' : A.card = B.card
hU : (A \ B).Nonempty
⢠False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | have disj : Disjoint (B \ A) (A \ B) := disjoint_sdiff.mono_left sdiff_subset | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
hAB' : A.card = B.card
hU : (A \ B).Nonempty
hV : (B \ A).Nonempty
⢠False | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
hAB' : A.card = B.card
hU : (A \ B).Nonempty
hV : (B \ A).Nonempty
disj : Disjoint (B \ A) (A \ B)
⢠False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â B
hAB' : A.card = B.card
hU : (A \ B).Nonempty
hV : (B \ A).Nonempty
⢠False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | refine' hB _ | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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 â ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | rw [â (hâ _ _ âšdisj, card_sdiff_comm hAB'.symm, hV, hU, smallerâ©).eq] | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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 â ⬠| case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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 â ð (B \ A) (A \ B) ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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 â â¬
TACTIC:
|
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 âšhB, A, hA, compress_sdiff_sdiff _ _â©) | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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 â ð (B \ A) (A \ B) ⬠| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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 â ð (B \ A) (A \ B) â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | obtain hlt | heq | hgt := lt_trichotomy (max' _ hU) (max' _ hV) | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | rw [â compress_sdiff_sdiff A B] at hAB hBA | case inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := compress (A \ B) (B \ A) B }
sizeA : B.card = r
hB : B â â¬
hAB : compress (A \ B) (B \ A) B â 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | cases hBA.not_lt $ toColex_compress_lt_toColex hlt hAB | case inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := compress (A \ B) (B \ A) B }
sizeA : B.card = r
hB : B â â¬
hAB : compress (A \ B) (B \ A) B â 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := compress (A \ B) (B \ A) B }
sizeA : B.card = r
hB : B â â¬
hAB : compress (A \ B) (B \ A) B â 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
TACTIC:
|
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 âž max'_mem _ _).elim | case inr.inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.isInitSeg_of_compressed | [197, 1] | [218, 73] | assumption | case inr.inr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n : â
⬠: Finset (Finset α)
r : â
hâ : Set.Sized r ââ¬
hâ : â (U V : Finset α), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
A B : Finset α
hA : A â â¬
hBA : { ofColex := B } < { ofColex := A }
sizeA : B.card = r
hB : B â â¬
hAB : A â 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
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [compression] at a ⢠| α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : ð U V ð â ð
⢠Finset.UV.familyMeasure (ð U V ð) < Finset.UV.familyMeasure ð | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : ð U V ð â ð
⢠Finset.UV.familyMeasure (ð U V ð) < Finset.UV.familyMeasure ð
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | have q : â Q â ð.filter fun A ⊠compress U V A â ð, compress U V Q â Q := by
simp_rw [mem_filter]
intro Q hQ h
rw [h] at hQ
exact hQ.2 hQ.1 | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | have uA : (ð.filter fun A => compress U V A â ð) ⪠ð.filter (fun A ⊠compress U V A â ð) = ð :=
filter_union_filter_neg_eq _ _ | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | have neâ : (ð.filter fun A ⊠compress U V A â ð).Nonempty := by
refine' nonempty_iff_ne_empty.2 fun z ⊠a _
rw [filter_image, z, image_empty, union_empty]
rwa [z, union_empty] at uA | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [familyMeasure, familyMeasure, sum_union compress_disjoint] | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠â x â filter (fun x => compress U V x â ð) ð, â a â x, 2 ^ âa +
â x â filter (fun x => x â ð) (image (compress U V) ð), â a â x, 2 ^ âa <
â A â ð, â a â A, 2 ^ âa | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠Finset.UV.familyMeasure (filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð)) <
Finset.UV.familyMeasure ð
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | conv_rhs => rw [â uA] | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠â x â filter (fun x => compress U V x â ð) ð, â a â x, 2 ^ âa +
â x â filter (fun x => x â ð) (image (compress U V) ð), â a â x, 2 ^ âa <
â A â ð, â a â A, 2 ^ âa | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠â x â filter (fun x => compress U V x â ð) ð, â a â x, 2 ^ âa +
â x â filter (fun x => x â ð) (image (compress U V) ð), â a â x, 2 ^ âa <
â A â filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð, â a â A, 2 ^ âa | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠â x â filter (fun x => compress U V x â ð) ð, â a â x, 2 ^ âa +
â x â filter (fun x => x â ð) (image (compress U V) ð), â a â x, 2 ^ âa <
â A â ð, â a â A, 2 ^ âa
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [sum_union (disjoint_filter_filter_neg _ _ _), add_lt_add_iff_left, filter_image,
sum_image compress_injOn] | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠â x â filter (fun x => compress U V x â ð) ð, â a â x, 2 ^ âa +
â x â filter (fun x => x â ð) (image (compress U V) ð), â a â x, 2 ^ âa <
â A â filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð, â a â A, 2 ^ âa | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠â x â filter (fun x => compress U V x â ð) ð, â a â compress U V x, 2 ^ âa <
â x â filter (fun a => compress U V a â ð) ð, â a â x, 2 ^ âa | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠â x â filter (fun x => compress U V x â ð) ð, â a â x, 2 ^ âa +
â x â filter (fun x => x â ð) (image (compress U V) ð), â a â x, 2 ^ âa <
â A â filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð, â a â A, 2 ^ âa
TACTIC:
|
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 neâ fun A hA ⊠_ | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠â x â filter (fun x => compress U V x â ð) ð, â a â compress U V x, 2 ^ âa <
â x â filter (fun a => compress U V a â ð) ð, â a â x, 2 ^ âa | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
A : Finset (Fin n)
hA : A â filter (fun x => compress U V x â ð) ð
⢠â a â compress U V A, 2 ^ âa < â a â A, 2 ^ âa | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
⢠â x â filter (fun x => compress U V x â ð) ð, â a â compress U V x, 2 ^ âa <
â x â filter (fun a => compress U V a â ð) ð, â a â x, 2 ^ âa
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | simp_rw [â sum_image Fin.val_injective.injOn] | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
A : Finset (Fin n)
hA : A â filter (fun x => compress U V x â ð) ð
⢠â a â compress U V A, 2 ^ âa < â a â A, 2 ^ âa | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
A : Finset (Fin n)
hA : A â filter (fun x => compress U V x â ð) ð
⢠â x â image (fun x => âx) (compress U V A), 2 ^ x < â x â image (fun x => âx) A, 2 ^ x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
A : Finset (Fin n)
hA : A â filter (fun x => compress U V x â ð) ð
⢠â a â compress U V A, 2 ^ âa < â a â A, 2 ^ âa
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [geomSum_lt_geomSum_iff_toColex_lt_toColex le_rfl,
toColex_image_lt_toColex_image Fin.val_strictMono] | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
A : Finset (Fin n)
hA : A â filter (fun x => compress U V x â ð) ð
⢠â x â image (fun x => âx) (compress U V A), 2 ^ x < â x â image (fun x => âx) A, 2 ^ x | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
A : Finset (Fin n)
hA : A â filter (fun x => compress U V x â ð) ð
⢠{ ofColex := compress U V A } < { ofColex := A } | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
A : Finset (Fin n)
hA : A â filter (fun x => compress U V x â ð) ð
⢠â x â image (fun x => âx) (compress U V A), 2 ^ x < â x â image (fun x => âx) A, 2 ^ x
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | exact toColex_compress_lt_toColex h $ q _ hA | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
A : Finset (Fin n)
hA : A â filter (fun x => compress U V x â ð) ð
⢠{ ofColex := compress U V A } < { ofColex := A } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
neâ : (filter (fun A => compress U V A â ð) ð).Nonempty
A : Finset (Fin n)
hA : A â filter (fun x => compress U V x â ð) ð
⢠{ ofColex := compress U V A } < { ofColex := A }
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | simp_rw [mem_filter] | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
⢠â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
⢠â (Q : Finset (Fin n)), Q â ð ⧠compress U V Q â ð â compress U V Q â Q | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
⢠â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | intro Q hQ h | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
⢠â (Q : Finset (Fin n)), Q â ð ⧠compress U V Q â ð â compress U V Q â Q | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
hâ : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
Q : Finset (Fin n)
hQ : Q â ð ⧠compress U V Q â ð
h : compress U V Q = Q
⢠False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
⢠â (Q : Finset (Fin n)), Q â ð ⧠compress U V Q â ð â compress U V Q â Q
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [h] at hQ | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
hâ : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
Q : Finset (Fin n)
hQ : Q â ð ⧠compress U V Q â ð
h : compress U V Q = Q
⢠False | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
hâ : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
Q : Finset (Fin n)
hQ : Q â ð ⧠Q â ð
h : compress U V Q = Q
⢠False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
hâ : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
Q : Finset (Fin n)
hQ : Q â ð ⧠compress U V Q â ð
h : compress U V Q = Q
⢠False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | exact hQ.2 hQ.1 | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
hâ : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
Q : Finset (Fin n)
hQ : Q â ð ⧠Q â ð
h : compress U V Q = Q
⢠False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
hâ : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
Q : Finset (Fin n)
hQ : Q â ð ⧠Q â ð
h : compress U V Q = Q
⢠False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | refine' nonempty_iff_ne_empty.2 fun z ⊠a _ | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
⢠(filter (fun A => compress U V A â ð) ð).Nonempty | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
z : filter (fun A => compress U V A â ð) ð = â
⢠filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) = ð | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
⢠(filter (fun A => compress U V A â ð) ð).Nonempty
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rw [filter_image, z, image_empty, union_empty] | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
z : filter (fun A => compress U V A â ð) ð = â
⢠filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) = ð | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
z : filter (fun A => compress U V A â ð) ð = â
⢠filter (fun x => compress U V x â ð) ð = ð | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
z : filter (fun A => compress U V A â ð) ð = â
⢠filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) = ð
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.familyMeasure_compression_lt_familyMeasure | [231, 1] | [256, 47] | rwa [z, union_empty] at uA | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
z : filter (fun A => compress U V A â ð) ð = â
⢠filter (fun x => compress U V x â ð) ð = ð | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n : â
U V : Finset (Fin n)
hU : U.Nonempty
hV : V.Nonempty
h : U.max' hU < V.max' hV
ð : Finset (Finset (Fin n))
a : filter (fun x => compress U V x â ð) ð ⪠filter (fun x => x â ð) (image (compress U V) ð) â ð
q : â Q â filter (fun A => compress U V A â ð) ð, compress U V Q â Q
uA : filter (fun A => compress U V A â ð) ð ⪠filter (fun A => compress U V A â ð) ð = ð
z : filter (fun A => compress U V A â ð) ð = â
⢠filter (fun x => compress U V x â ð) ð = ð
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | revert h | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð : Finset (Finset (Fin n))
h : Set.Sized r âð
⢠â â¬,
(â â¬).card †(â ð).card â§
ð.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð : Finset (Finset (Fin n))
⢠Set.Sized r âð â
â â¬,
(â â¬).card †(â ð).card â§
ð.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð : Finset (Finset (Fin n))
h : Set.Sized r âð
⢠â â¬,
(â â¬).card †(â ð).card â§
ð.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
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) ð | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð : Finset (Finset (Fin n))
⢠Set.Sized r âð â
â â¬,
(â â¬).card †(â ð).card â§
ð.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð : Finset (Finset (Fin n))
⢠â (x : Finset (Finset (Fin n))),
(â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y x â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬) â
Set.Sized r âx â
â â¬,
(â â¬).card †(â x).card â§
x.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð : Finset (Finset (Fin n))
⢠Set.Sized r âð â
â â¬,
(â â¬).card †(â ð).card â§
ð.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | intro A ih h | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð : Finset (Finset (Fin n))
⢠â (x : Finset (Finset (Fin n))),
(â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y x â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬) â
Set.Sized r âx â
â â¬,
(â â¬).card †(â x).card â§
x.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð : Finset (Finset (Fin n))
⢠â (x : Finset (Finset (Fin n))),
(â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y x â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬) â
Set.Sized r âx â
â â¬,
(â â¬).card †(â x).card â§
x.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
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) Ã Finset (Fin n)) :=
univ.filter fun t ⊠UsefulCompression t.1 t.2 ⧠¬ IsCompressed t.1 t.2 A | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain husable | husable := usable.eq_empty_or_nonempty | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| case inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable = â
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
case inr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain âšâšU, Vâ©, hUV, tâ© := exists_min_image usable (fun t ⊠t.1.card) husable | case inr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â usable
t : â x' â usable, (U, V).1.card †x'.1.card
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | rw [mem_filter] at hUV | case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â usable
t : â x' â usable, (U, V).1.card †x'.1.card
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â usable
t : â x' â usable, (U, V).1.card †x'.1.card
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | have hâ : â Uâ Vâ, UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A := by
rintro Uâ Vâ huseful hUcard
by_contra h
exact hUcard.not_le $ t âšUâ, Vââ© $ mem_filter.2 âšmem_univ _, huseful, hâ© | case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | have p1 : (â (ð U V A)).card †(â A).card := compression_improved _ hUV.2.1 hâ | case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â A).card
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain âš-, hUV', hu, hv, hmaxâ© := hUV.2.1 | case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â A).card
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| case inr.intro.mk.intro.intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â 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
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.mk.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â A).card
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | unfold InvImage at ih | case inr.intro.mk.intro.intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â 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
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| case inr.intro.mk.intro.intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â 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
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.mk.intro.intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â 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
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | obtain âšt, q1, q2, q3, q4â© := ih (ð U V A)
(familyMeasure_compression_lt_familyMeasure hmax hUV.2.2) (h.uvCompression hUV') | case inr.intro.mk.intro.intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â 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
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| case inr.intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
tâ : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â 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 : (â t).card †(â (ð U V A)).card
q2 : (ð U V A).card = t.card
q3 : Set.Sized r ât
q4 : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V t
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.mk.intro.intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â 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
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | exact âšt, q1.trans p1, (card_compression _ _ _).symm.trans q2, q3, q4â© | case inr.intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
tâ : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â 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 : (â t).card †(â (ð U V A)).card
q2 : (ð U V A).card = t.card
q3 : Set.Sized r ât
q4 : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V t
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
(fun x x_1 => x < x_1) (Finset.UV.familyMeasure y) (Finset.UV.familyMeasure A) â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
tâ : â x' â usable, (U, V).1.card †x'.1.card
hâ : â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
p1 : (â (ð U V A)).card †(â 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 : (â t).card †(â (ð U V A)).card
q2 : (ð U V A).card = t.card
q3 : Set.Sized r ât
q4 : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V t
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | refine' âšA, le_rfl, rfl, h, fun U V hUV ⊠_â© | case inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable = â
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V ⬠| case inl
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable = â
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
⢠IsCompressed U V A | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable = â
⢠â â¬,
(â â¬).card †(â A).card â§
A.card = â¬.card ⧠Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | rw [eq_empty_iff_forall_not_mem] at husable | case inl
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable = â
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
⢠IsCompressed U V A | case inl
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : â (x : Finset (Fin n) Ã Finset (Fin n)), x â usable
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
⢠IsCompressed U V A | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable = â
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
⢠IsCompressed U V A
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | by_contra h | case inl
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : â (x : Finset (Fin n) Ã Finset (Fin n)), x â usable
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
⢠IsCompressed U V A | case inl
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hâ : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : â (x : Finset (Fin n) Ã Finset (Fin n)), x â usable
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
h : ¬IsCompressed U V A
⢠False | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : â (x : Finset (Fin n) Ã Finset (Fin n)), x â usable
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
⢠IsCompressed U V A
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | exact husable âšU, Vâ© $ mem_filter.2 âšmem_univ _, hUV, hâ© | case inl
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hâ : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : â (x : Finset (Fin n) Ã Finset (Fin n)), x â usable
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
h : ¬IsCompressed U V A
⢠False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hâ : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : â (x : Finset (Fin n) Ã Finset (Fin n)), x â usable
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
h : ¬IsCompressed U V A
⢠False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | rintro Uâ Vâ huseful hUcard | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
⢠â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
Uâ Vâ : Finset (Fin n)
huseful : Finset.UV.UsefulCompression Uâ Vâ
hUcard : Uâ.card < U.card
⢠IsCompressed Uâ Vâ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
⢠â (Uâ Vâ : Finset (Fin n)), Finset.UV.UsefulCompression Uâ Vâ â Uâ.card < U.card â IsCompressed Uâ Vâ A
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | by_contra h | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
Uâ Vâ : Finset (Fin n)
huseful : Finset.UV.UsefulCompression Uâ Vâ
hUcard : Uâ.card < U.card
⢠IsCompressed Uâ Vâ A | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hâ : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
Uâ Vâ : Finset (Fin n)
huseful : Finset.UV.UsefulCompression Uâ Vâ
hUcard : Uâ.card < U.card
h : ¬IsCompressed Uâ Vâ A
⢠False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
h : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
Uâ Vâ : Finset (Fin n)
huseful : Finset.UV.UsefulCompression Uâ Vâ
hUcard : Uâ.card < U.card
⢠IsCompressed Uâ Vâ A
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.UV.kruskal_katona_helper | [258, 1] | [290, 73] | exact hUcard.not_le $ t âšUâ, Vââ© $ mem_filter.2 âšmem_univ _, huseful, hâ© | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hâ : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
Uâ Vâ : Finset (Fin n)
huseful : Finset.UV.UsefulCompression Uâ Vâ
hUcard : Uâ.card < U.card
h : ¬IsCompressed Uâ Vâ A
⢠False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r : â
ð A : Finset (Finset (Fin n))
ih :
â (y : Finset (Finset (Fin n))),
InvImage (fun x x_1 => x < x_1) Finset.UV.familyMeasure y A â
Set.Sized r ây â
â â¬,
(â â¬).card †(â y).card â§
y.card = â¬.card â§
Set.Sized r â⬠⧠â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hâ : Set.Sized r âA
usable : Finset (Finset (Fin n) Ã Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ⧠¬IsCompressed t.1 t.2 A) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) â univ ⧠Finset.UV.UsefulCompression (U, V).1 (U, V).2 ⧠¬IsCompressed (U, V).1 (U, V).2 A
t : â x' â usable, (U, V).1.card †x'.1.card
Uâ Vâ : Finset (Fin n)
huseful : Finset.UV.UsefulCompression Uâ Vâ
hUcard : Uâ.card < U.card
h : ¬IsCompressed Uâ Vâ A
⢠False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | obtain âšâ¬, card_le, t, hâ¬, fully_compâ© := UV.kruskal_katona_helper ð hâ | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⢠(â ð).card †(â ð).card | case intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
⢠(â ð).card †(â ð).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⢠(â ð).card †(â ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | convert card_le | case intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
⢠(â ð).card †(â ð).card | case h.e'_3.h.e'_2.h.e'_3
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
⢠ð = ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
⢠(â ð).card †(â ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | have hcard : card ⬠= card ð := t.symm.trans hâ | case h.e'_3.h.e'_2.h.e'_3
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
⢠ð = ⬠| case h.e'_3.h.e'_2.h.e'_3
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
⢠ð = ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_2.h.e'_3
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
⢠ð = â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | obtain CB | BC :=
hâ.total (UV.isInitSeg_of_compressed h⬠fun U V hUV ⊠by convert fully_comp U V hUV) | case h.e'_3.h.e'_2.h.e'_3
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
⢠ð = ⬠| case h.e'_3.h.e'_2.h.e'_3.inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
CB : ð â â¬
⢠ð = â¬
case h.e'_3.h.e'_2.h.e'_3.inr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
BC : ⬠â ð
⢠ð = ⬠| Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_2.h.e'_3
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
⢠ð = â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | convert fully_comp U V hUV | α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
⢠IsCompressed U V ⬠| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s Uâ Vâ : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
U V : Finset (Fin n)
hUV : Finset.UV.UsefulCompression U V
⢠IsCompressed U V â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | exact eq_of_subset_of_card_le CB hcard.le | case h.e'_3.h.e'_2.h.e'_3.inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
CB : ð â â¬
⢠ð = ⬠| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_2.h.e'_3.inl
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
CB : ð â â¬
⢠ð = â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.kruskal_katona | [299, 1] | [314, 53] | exact (eq_of_subset_of_card_le BC hcard.ge).symm | case h.e'_3.h.e'_2.h.e'_3.inr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
BC : ⬠â ð
⢠ð = ⬠| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_2.h.e'_3.inr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card = ð.card
hâ : IsInitSeg ð r
⬠: Finset (Finset (Fin n))
card_le : (â â¬).card †(â ð).card
t : ð.card = â¬.card
h⬠: Set.Sized r ââ¬
fully_comp : â (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V â IsCompressed U V â¬
hcard : â¬.card = ð.card
BC : ⬠â ð
⢠ð = â¬
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.strengthened_kk | [316, 1] | [323, 29] | rcases exists_smaller_set ð ð.card hâ with âšð', prop, sizeâ© | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠(â ð).card †(â ð).card | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
ð' : Finset (Finset (Fin n))
prop : ð' â ð
size : ð'.card = ð.card
⢠(â ð).card †(â ð).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠(â ð).card †(â ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.strengthened_kk | [316, 1] | [323, 29] | refine' (kruskal_katona (fun A hA ⊠hâ (prop hA)) size hâ).trans (card_le_card _) | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
ð' : Finset (Finset (Fin n))
prop : ð' â ð
size : ð'.card = ð.card
⢠(â ð).card †(â ð).card | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
ð' : Finset (Finset (Fin n))
prop : ð' â ð
size : ð'.card = ð.card
⢠â ð' â â ð | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
ð' : Finset (Finset (Fin n))
prop : ð' â ð
size : ð'.card = ð.card
⢠(â ð).card †(â ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.strengthened_kk | [316, 1] | [323, 29] | rw [shadow, shadow] | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
ð' : Finset (Finset (Fin n))
prop : ð' â ð
size : ð'.card = ð.card
⢠â ð' â â ð | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
ð' : Finset (Finset (Fin n))
prop : ð' â ð
size : ð'.card = ð.card
⢠(ð'.sup fun s => image s.erase s) â ð.sup fun s => image s.erase s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
ð' : Finset (Finset (Fin n))
prop : ð' â ð
size : ð'.card = ð.card
⢠â ð' â â ð
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.strengthened_kk | [316, 1] | [323, 29] | apply shadow_monotone prop | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
ð' : Finset (Finset (Fin n))
prop : ð' â ð
size : ð'.card = ð.card
⢠(ð'.sup fun s => image s.erase s) â ð.sup fun s => image s.erase s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
ð' : Finset (Finset (Fin n))
prop : ð' â ð
size : ð'.card = ð.card
⢠(ð'.sup fun s => image s.erase s) â ð.sup fun s => image s.erase s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | induction' k with _k ih generalizing r ð ð | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠(â ^[k] ð).card †(â ^[k] ð).card | case zero
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n i r : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠(â ^[0] ð).card †(â ^[0] ð).card
case succ
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n i _k : â
ih :
â {r : â} {ð ð : Finset (Finset (Fin n))},
Set.Sized r âð â ð.card †ð.card â IsInitSeg ð r â (â ^[_k] ð).card †(â ^[_k] ð).card
r : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠(â ^[_k + 1] ð).card †(â ^[_k + 1] ð).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠(â ^[k] ð).card †(â ^[k] ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | simpa | case zero
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n i r : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠(â ^[0] ð).card †(â ^[0] ð).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n i r : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠(â ^[0] ð).card †(â ^[0] ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | refine' ih hâ.shadow (strengthened_kk hâ hâ hâ) _ | case succ
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n i _k : â
ih :
â {r : â} {ð ð : Finset (Finset (Fin n))},
Set.Sized r âð â ð.card †ð.card â IsInitSeg ð r â (â ^[_k] ð).card †(â ^[_k] ð).card
r : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠(â ^[_k + 1] ð).card †(â ^[_k + 1] ð).card | case succ
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n i _k : â
ih :
â {r : â} {ð ð : Finset (Finset (Fin n))},
Set.Sized r âð â ð.card †ð.card â IsInitSeg ð r â (â ^[_k] ð).card †(â ^[_k] ð).card
r : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠IsInitSeg (â ð) (r - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n i _k : â
ih :
â {r : â} {ð ð : Finset (Finset (Fin n))},
Set.Sized r âð â ð.card †ð.card â IsInitSeg ð r â (â ^[_k] ð).card †(â ^[_k] ð).card
r : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠(â ^[_k + 1] ð).card †(â ^[_k + 1] ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.iterated_kk | [325, 1] | [332, 22] | convert hâ.shadow | case succ
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n i _k : â
ih :
â {r : â} {ð ð : Finset (Finset (Fin n))},
Set.Sized r âð â ð.card †ð.card â IsInitSeg ð r â (â ^[_k] ð).card †(â ^[_k] ð).card
r : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠IsInitSeg (â ð) (r - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n i _k : â
ih :
â {r : â} {ð ð : Finset (Finset (Fin n))},
Set.Sized r âð â ð.card †ð.card â IsInitSeg ð r â (â ^[_k] ð).card †(â ^[_k] ð).card
r : â
ð ð : Finset (Finset (Fin n))
hâ : Set.Sized r âð
hâ : ð.card †ð.card
hâ : IsInitSeg ð r
⢠IsInitSeg (â ð) (r - 1)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | set range'k : Finset (Fin n) :=
attachFin (range k) fun m ⊠by rw [mem_range]; apply forall_lt_iff_le.2 hkn | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
⢠k.choose (r - i) †(â ^[i] ð).card | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
⢠k.choose (r - i) †(â ^[i] ð).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
⢠k.choose (r - i) †(â ^[i] ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | set ð : Finset (Finset (Fin n)) := powersetCard r range'k | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
⢠k.choose (r - i) †(â ^[i] ð).card | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
⢠k.choose (r - i) †(â ^[i] ð).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
⢠k.choose (r - i) †(â ^[i] ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have Ccard : ð.card = k.choose r := by rw [card_powersetCard, card_attachFin, card_range] | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
⢠k.choose (r - i) †(â ^[i] ð).card | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
⢠k.choose (r - i) †(â ^[i] ð).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
⢠k.choose (r - i) †(â ^[i] ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have : (ð : Set (Finset (Fin n))).Sized r := Set.sized_powersetCard _ _ | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
⢠k.choose (r - i) †(â ^[i] ð).card | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
⢠k.choose (r - i) †(â ^[i] ð).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
⢠k.choose (r - i) †(â ^[i] ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | suffices this : (â^[i] ð).card = k.choose (r - i) by
rw [â this]
apply iterated_kk hâ _ _
rwa [Ccard]
refine' âšâ¹_âº, _â©
rintro A B hA âšHBâ, HBââ©
rw [mem_powersetCard]
refine' âšfun t ht ⊠_, â¹_âºâ©
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 âšt, ht, rflâ©)
simp_rw [mem_image]
rintro _ âša, ha, qâ©
rw [mem_powersetCard] at hA
rw [â q, â mem_range]
have := hA.1 ha
rwa [mem_attachFin] at this | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
⢠k.choose (r - i) †(â ^[i] ð).card | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
⢠(â ^[i] ð).card = k.choose (r - i) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
⢠k.choose (r - i) †(â ^[i] ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | suffices â^[i] ð = powersetCard (r - i) range'k by
rw [this, card_powersetCard, card_attachFin, card_range] | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
⢠(â ^[i] ð).card = k.choose (r - i) | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
⢠â ^[i] ð = powersetCard (r - i) range'k | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
⢠(â ^[i] ð).card = k.choose (r - i)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | ext B | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
⢠â ^[i] ð = powersetCard (r - i) range'k | case a
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
⢠B â â ^[i] ð â B â powersetCard (r - i) range'k | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
⢠â ^[i] ð = powersetCard (r - i) range'k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_powersetCard, mem_shadow_iterate_iff_exists_sdiff] | case a
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
⢠B â â ^[i] ð â B â powersetCard (r - i) range'k | case a
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
⢠(â s â ð, B â s ⧠(s \ B).card = i) â B â range'k ⧠B.card = r - i | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
⢠B â â ^[i] ð â B â powersetCard (r - i) range'k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | constructor | case a
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
⢠(â s â ð, B â s ⧠(s \ B).card = i) â B â range'k ⧠B.card = r - i | case a.mp
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
⢠(â s â ð, B â s ⧠(s \ B).card = i) â B â range'k ⧠B.card = r - i
case a.mpr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
⢠B â range'k ⧠B.card = r - i â â s â ð, B â s ⧠(s \ B).card = i | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
⢠(â s â ð, B â s ⧠(s \ B).card = i) â B â range'k ⧠B.card = r - i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rintro âšhBk, hBâ© | case a.mpr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
⢠B â range'k ⧠B.card = r - i â â s â ð, B â s ⧠(s \ B).card = i | case a.mpr.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠â s â ð, B â s ⧠(s \ B).card = i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
⢠B â range'k ⧠B.card = r - i â â s â ð, B â s ⧠(s \ B).card = i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have := exists_intermediate_set i ?_ hBk | case a.mpr.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠â s â ð, B â s ⧠(s \ B).card = i | case a.mpr.intro.refine_2
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
this : â C, B â C ⧠C â range'k ⧠C.card = i + B.card
⢠â s â ð, B â s ⧠(s \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠â s â ð, B â s ⧠(s \ B).card = i
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | obtain âšC, BsubC, hCrange, hcardâ© := this | case a.mpr.intro.refine_2
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
this : â C, B â C ⧠C â range'k ⧠C.card = i + B.card
⢠â s â ð, B â s ⧠(s \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | case a.mpr.intro.refine_2.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = i + B.card
⢠â s â ð, B â s ⧠(s \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.refine_2
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
this : â C, B â C ⧠C â range'k ⧠C.card = i + B.card
⢠â s â ð, B â s ⧠(s \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [hB, â Nat.add_sub_assoc hir, Nat.add_sub_cancel_left] at hcard | case a.mpr.intro.refine_2.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = i + B.card
⢠â s â ð, B â s ⧠(s \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | case a.mpr.intro.refine_2.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠â s â ð, B â s ⧠(s \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.refine_2.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = i + B.card
⢠â s â ð, B â s ⧠(s \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | refine' âšC, _, BsubC, _â© | case a.mpr.intro.refine_2.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠â s â ð, B â s ⧠(s \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠C â ð
case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠(C \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.refine_2.intro.intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠â s â ð, B â s ⧠(s \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_powersetCard] | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠C â ð
case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠(C \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠C â range'k ⧠C.card = r
case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠(C \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠C â ð
case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠(C \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | exact âšhCrange, hcardâ© | case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠C â range'k ⧠C.card = r
case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠(C \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠(C \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.refine_2.intro.intro.intro.refine'_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠C â range'k ⧠C.card = r
case a.mpr.intro.refine_2.intro.intro.intro.refine'_2
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
C : Finset (Fin n)
BsubC : B â C
hCrange : C â range'k
hcard : C.card = r
⢠(C \ B).card = i
case a.mpr.intro.refine_1
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
this : Set.Sized r âð
B : Finset (Fin n)
hBk : B â range'k
hB : B.card = r - i
⢠i + B.card †range'k.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_range] | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
m : â
⢠m â range k â m < n | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
m : â
⢠m < k â m < n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
m : â
⢠m â range k â m < n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | apply forall_lt_iff_le.2 hkn | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
m : â
⢠m < k â m < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ð : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
m : â
⢠m < k â m < n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [card_powersetCard, card_attachFin, card_range] | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
⢠ð.card = k.choose r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
⢠ð.card = k.choose r
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [â this] | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠k.choose (r - i) †(â ^[i] ð).card | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠(â ^[i] ð).card †(â ^[i] ð).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠k.choose (r - i) †(â ^[i] ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | apply iterated_kk hâ _ _ | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠(â ^[i] ð).card †(â ^[i] ð).card | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠ð.card †ð.card
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠IsInitSeg ð r | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠(â ^[i] ð).card †(â ^[i] ð).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rwa [Ccard] | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠ð.card †ð.card
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠IsInitSeg ð r | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠IsInitSeg ð r | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠ð.card †ð.card
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠IsInitSeg ð r
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | refine' âšâ¹_âº, _â© | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠IsInitSeg ð r | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠â âŠs t : Finset (Fin n)âŠ, s â ð â { ofColex := t } < { ofColex := s } ⧠t.card = r â t â ð | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠IsInitSeg ð r
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rintro A B hA âšHBâ, HBââ© | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠â âŠs t : Finset (Fin n)âŠ, s â ð â { ofColex := t } < { ofColex := s } ⧠t.card = r â t â ð | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
⢠B â ð | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
⢠â âŠs t : Finset (Fin n)âŠ, s â ð â { ofColex := t } < { ofColex := s } ⧠t.card = r â t â ð
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_powersetCard] | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
⢠B â ð | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
⢠B â range'k ⧠B.card = r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
⢠B â ð
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | refine' âšfun t ht ⊠_, â¹_âºâ© | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
⢠B â range'k ⧠B.card = r | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
⢠t â range'k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
⢠B â range'k ⧠B.card = r
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_attachFin, mem_range] | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
⢠t â range'k | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
⢠ât < k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
⢠t â range'k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have : toColex (image Fin.val B) < toColex (image Fin.val A) := by
rwa [toColex_image_lt_toColex_image Fin.val_strictMono] | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
⢠ât < k | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⢠ât < k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ : Set.Sized r âð
this : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
⢠ât < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | apply Colex.forall_lt_mono this.le _ t (mem_image.2 âšt, ht, rflâ©) | case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⢠ât < k | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⢠â b â image Fin.val A, b < k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⢠ât < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | simp_rw [mem_image] | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⢠â b â image Fin.val A, b < k | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⢠â (b : â), (â a â A, âa = b) â b < k | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⢠â b â image Fin.val A, b < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rintro _ âša, ha, qâ© | α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⢠â (b : â), (â a â A, âa = b) â b < k | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
bâ : â
a : Fin n
ha : a â A
q : âa = bâ
⢠bâ < k | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
⢠â (b : â), (â a â A, âa = b) â b < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [mem_powersetCard] at hA | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
bâ : â
a : Fin n
ha : a â A
q : âa = bâ
⢠bâ < k | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â range'k ⧠A.card = r
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
bâ : â
a : Fin n
ha : a â A
q : âa = bâ
⢠bâ < k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â ð
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
bâ : â
a : Fin n
ha : a â A
q : âa = bâ
⢠bâ < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | rw [â q, â mem_range] | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â range'k ⧠A.card = r
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
bâ : â
a : Fin n
ha : a â A
q : âa = bâ
⢠bâ < k | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â range'k ⧠A.card = r
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
bâ : â
a : Fin n
ha : a â A
q : âa = bâ
⢠âa â range k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â range'k ⧠A.card = r
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
bâ : â
a : Fin n
ha : a â A
q : âa = bâ
⢠bâ < k
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/KruskalKatona.lean | Finset.lovasz_form | [334, 1] | [382, 91] | have := hA.1 ha | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â range'k ⧠A.card = r
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
bâ : â
a : Fin n
ha : a â A
q : âa = bâ
⢠âa â range k | case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ² : Set.Sized r âð
thisâ¹ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â range'k ⧠A.card = r
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
thisâ : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
bâ : â
a : Fin n
ha : a â A
q : âa = bâ
this : a â range'k
⢠âa â range k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
instâ : LinearOrder α
s U V : Finset α
n r k i : â
ð ðâ : Finset (Finset (Fin n))
hir : i †r
hrk : r †k
hkn : k †n
hâ : Set.Sized r âð
hâ : k.choose r †ð.card
range'k : Finset (Fin n) := (range k).attachFin â¯
ð : Finset (Finset (Fin n)) := powersetCard r range'k
Ccard : ð.card = k.choose r
thisâ¹ : Set.Sized r âð
thisâ : (â ^[i] ð).card = k.choose (r - i)
A B : Finset (Fin n)
hA : A â range'k ⧠A.card = r
HBâ : { ofColex := B } < { ofColex := A }
HBâ : B.card = r
t : Fin n
ht : t â B
this : { ofColex := image Fin.val B } < { ofColex := image Fin.val A }
bâ : â
a : Fin n
ha : a â A
q : âa = bâ
⢠âa â range k
TACTIC:
|
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