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/Kneser/Kneser.lean | Finset.mul_aux1 | [133, 1] | [158, 69] | linarith [hconv, hnotconv] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hdisj : Disjoint C (s' * t')
H : Finset α := C.mulStab
H' : Finset α := (s' * t').mulStab
hsub : H' ⊆ H
C' : Finset α := C ∪ s' * t'
hCun : C'.mulStab = H'
hconv : ↑(s ∩ t).card + ↑((s ∪ t) * H).card ≤ ↑C.card + ↑H.card
hnotconv : ↑C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hdisj : Disjoint C (s' * t')
H : Finset α := C.mulStab
H' : Finset α := (s' * t').mulStab
hsub : H' ⊆ H
C' : Finset α := C ∪ s' * t'
hCun : C'.mulStab = H'
hconv : ↑(s ∩ t)... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_aux1 | [133, 1] | [158, 69] | rw [card_union_of_disjoint hdisj, Int.ofNat_add] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hdisj : Disjoint C (s' * t')
H : Finset α := C.mulStab
H' : Finset α := (s' * t').mulStab
hsub : H' ⊆ H
C' : Finset α := C ∪ s' * t'
hCun : C'.mulStab = H'
hconv : ↑(s ∩ t).card + ↑((s ∪ t) * H).card ≤ ↑C.card + ↑H.card
hnotconv : ↑C... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hdisj : Disjoint C (s' * t')
H : Finset α := C.mulStab
H' : Finset α := (s' * t').mulStab
hsub : H' ⊆ H
C' : Finset α := C ∪ s' * t'
hCun : C'.mulStab = H'
hconv : ↑(s ∩ t).card + ↑((s ∪ t) * H).card ≤ ↑C.card + ↑H.card
hnotconv : ↑C... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hdisj : Disjoint C (s' * t')
H : Finset α := C.mulStab
H' : Finset α := (s' * t').mulStab
hsub : H' ⊆ H
C' : Finset α := C ∪ s' * t'
hCun : C'.mulStab = H'
hconv : ↑(s ∩ t)... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_aux1 | [133, 1] | [158, 69] | abel | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hdisj : Disjoint C (s' * t')
H : Finset α := C.mulStab
H' : Finset α := (s' * t').mulStab
hsub : H' ⊆ H
C' : Finset α := C ∪ s' * t'
hCun : C'.mulStab = H'
hconv : ↑(s ∩ t).card + ↑((s ∪ t) * H).card ≤ ↑C.card + ↑H.card
hnotconv : ↑C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hdisj : Disjoint C (s' * t')
H : Finset α := C.mulStab
H' : Finset α := (s' * t').mulStab
hsub : H' ⊆ H
C' : Finset α := C ∪ s' * t'
hCun : C'.mulStab = H'
hconv : ↑(s ∩ t)... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_aux1 | [133, 1] | [158, 69] | linarith [ih] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hdisj : Disjoint C (s' * t')
H : Finset α := C.mulStab
H' : Finset α := (s' * t').mulStab
hsub : H' ⊆ H
C' : Finset α := C ∪ s' * t'
hCun : C'.mulStab = H'
hconv : ↑(s ∩ t).card + ↑((s ∪ t) * H).card ≤ ↑C.card + ↑H.card
hnotconv : ↑C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hdisj : Disjoint C (s' * t')
H : Finset α := C.mulStab
H' : Finset α := (s' * t').mulStab
hsub : H' ⊆ H
C' : Finset α := C ∪ s' * t'
hCun : C'.mulStab = H'
hconv : ↑(s ∩ t)... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_smul_mulStab | [160, 1] | [168, 29] | suffices Disjoint (a • s.mulStab) (s * s.mulStab) by
simpa [mul_comm, disjoint_comm, mulStab_mul] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ Disjoint s (a • s.mulStab) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ Disjoint (a • s.mulStab) (s * s.mulStab) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ Disjoint s (a • s.mulStab)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_smul_mulStab | [160, 1] | [168, 29] | apply disjoint_smul_finset_mulStab_mul_mulStab | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ Disjoint (a • s.mulStab) (s * s.mulStab) | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ ¬a • s.mulStab ⊆ s * s.mulStab | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ Disjoint (a • s.mulStab) (s * s.mulStab)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_smul_mulStab | [160, 1] | [168, 29] | rw [mul_comm, mulStab_mul] | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ ¬a • s.mulStab ⊆ s * s.mulStab | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ ¬a • s.mulStab ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ ¬a • s.mulStab ⊆ s * s.mulStab
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_smul_mulStab | [160, 1] | [168, 29] | contrapose! has | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ ¬a • s.mulStab ⊆ s | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : a • s.mulStab ⊆ s
⊢ a • s.mulStab ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
⊢ ¬a • s.mulStab ⊆ s
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_smul_mulStab | [160, 1] | [168, 29] | exact subset_trans has hst | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : a • s.mulStab ⊆ s
⊢ a • s.mulStab ⊆ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : a • s.mulStab ⊆ s
⊢ a • s.mulStab ⊆ t
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_smul_mulStab | [160, 1] | [168, 29] | simpa [mul_comm, disjoint_comm, mulStab_mul] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
this : Disjoint (a • s.mulStab) (s * s.mulStab)
⊢ Disjoint s (a • s.mulStab) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hst : s ⊆ t
has : ¬a • s.mulStab ⊆ t
this : Disjoint (a • s.mulStab) (s * s.mulStab)
⊢ Disjoint s (a • s.mulStab)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | obtain rfl | hC := C.eq_empty_or_nonempty | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab).c... | case inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C : Finset α
a✝ b✝ a b : α
s t : Finset α
has : a ∈ s
hsC : Disjoint t (a • ∅.mulStab)
hst : (s ∩ a • ∅.mulStab * (t ∩ b • ∅.mulStab)).mulStab ⊆ ∅.mulStab
⊢ ↑∅.mulStab.card - ↑(s ∩ a • ∅.mulStab * (s ∩ a • ∅.mulStab * (t ∩ b • ∅.mulStab)).mulS... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.m... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | simp | case inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C : Finset α
a✝ b✝ a b : α
s t : Finset α
has : a ∈ s
hsC : Disjoint t (a • ∅.mulStab)
hst : (s ∩ a • ∅.mulStab * (t ∩ b • ∅.mulStab)).mulStab ⊆ ∅.mulStab
⊢ ↑∅.mulStab.card - ↑(s ∩ a • ∅.mulStab * (s ∩ a • ∅.mulStab * (t ∩ b • ∅.mulStab)).mulS... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C : Finset α
a✝ b✝ a b : α
s t : Finset α
has : a ∈ s
hsC : Disjoint t (a • ∅.mulStab)
hst : (s ∩ a • ∅.mulStab * (t ∩ b • ∅.mulStab)).mulStab ⊆ ∅.mulStab
⊢ ↑∅.mulStab.card - ↑(s ∩ a... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | rw [card_sdiff
(subset_trans (mul_subset_mul_left hst)
(subset_trans (mul_subset_mul_right inter_subset_right) _)),
card_smul_finset, Int.ofNat_sub] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulS... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab).card... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | apply le_trans (card_le_card (mul_subset_mul_left hst)) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab).card... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab * C.mulStab).card ≤ C.mulStab.card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulS... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | apply
le_trans (card_le_card inter_mul_subset)
(le_of_le_of_eq (card_le_card inter_subset_right) _) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab * C.mulStab).card ≤ C.mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (a • C.mulStab * C.mulStab).card = C.mulStab.card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulS... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | rw [smul_mul_assoc, mulStab_mul_mulStab, card_smul_finset] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (a • C.mulStab * C.mulStab).card = C.mulStab.card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (a • C.mulStab ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | simp only [smul_mul_assoc, mulStab_mul_mulStab, Subset.rfl] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mulStab * C.mulStab ⊆ a • C.mulStab | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mulStab *... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | rw [← Int.ofNat_sub (card_le_card (mul_subset_mul_left hst)), ←
card_sdiff (mul_subset_mul_left hst)] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑(a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulSt... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑(a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulSt... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑(a • C.mulStab... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | norm_cast | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑(a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulSt... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulSta... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑(a • C.mulStab... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | apply card_le_card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulSta... | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (a • C.mulStab ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | refine' fun x hx => mem_sdiff.mpr ⟨_, _⟩ | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.... | case a.refine'_1
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mu... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | have hx' := (mem_sdiff.mp hx).2 | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • ... | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | contrapose! hx' | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • ... | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | obtain ⟨y, hyst, d, hd, hxyd⟩ := mem_mul.mp hx' | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • ... | case a.refine'_2.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | obtain ⟨c, hc, hcx⟩ := mem_smul_finset.mp (mem_sdiff.mp hx).1 | case a.refine'_2.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | rw [← hcx, ← eq_mul_inv_iff_mul_eq] at hxyd | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | have hyC : y ∈ a • C.mulStab := by
rw [hxyd, smul_mul_assoc, smul_mem_smul_finset_iff, ← mulStab_mul_mulStab]
apply mul_mem_mul hc ((mem_mulStab hC).mpr (inv_smul_eq_iff.mpr _))
exact Eq.symm ((mem_mulStab hC).mp (hst hd)) | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | replace hyst : y ∈ s := by
apply or_iff_not_imp_right.mp (mem_union.mp hyst)
contrapose! hsC
exact not_disjoint_iff.mpr ⟨y, hsC, hyC⟩ | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | rw [eq_mul_inv_iff_mul_eq, hcx] at hxyd | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | rw [← hxyd] | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | exact mul_mem_mul (mem_inter.mpr ⟨hyst, hyC⟩) hd | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulS... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | apply smul_finset_subset_smul (mem_union_left t has) (mem_sdiff.mp hx).1 | case a.refine'_1
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | rw [hxyd, smul_mul_assoc, smul_mem_smul_finset_iff, ← mulStab_mul_mulStab] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ ... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | apply mul_mem_mul hc ((mem_mulStab hC).mpr (inv_smul_eq_iff.mpr _)) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ ... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | exact Eq.symm ((mem_mulStab hC).mp (hst hd)) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | apply or_iff_not_imp_right.mp (mem_union.mp hyst) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ ... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | contrapose! hsC | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ ... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab)
hx' : x ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hsC : Disjoint t (a • C.mulStab)
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.disjoint_mul_sub_card_le | [170, 1] | [218, 55] | exact not_disjoint_iff.mpr ⟨y, hsC, hyC⟩ | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab)
hx' : x ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b✝ a b : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ (s ∩ a • C.mulStab ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | obtain rfl | hC := C.eq_empty_or_nonempty | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab).card -
↑(t ∩ a • C.mulStab * (s... | case inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C : Finset α
a✝ b a : α
s t : Finset α
has : a ∈ s
hst : (s ∩ a • ∅.mulStab * (t ∩ a • ∅.mulStab)).mulStab ⊆ ∅.mulStab
⊢ ↑∅.mulStab.card - ↑(s ∩ a • ∅.mulStab * (s ∩ a • ∅.mulStab * (t ∩ a • ∅.mulStab)).mulStab).card -
↑(t ∩ a • ∅.mulSta... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ a... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | simp | case inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C : Finset α
a✝ b a : α
s t : Finset α
has : a ∈ s
hst : (s ∩ a • ∅.mulStab * (t ∩ a • ∅.mulStab)).mulStab ⊆ ∅.mulStab
⊢ ↑∅.mulStab.card - ↑(s ∩ a • ∅.mulStab * (s ∩ a • ∅.mulStab * (t ∩ a • ∅.mulStab)).mulStab).card -
↑(t ∩ a • ∅.mulSta... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C : Finset α
a✝ b a : α
s t : Finset α
has : a ∈ s
hst : (s ∩ a • ∅.mulStab * (t ∩ a • ∅.mulStab)).mulStab ⊆ ∅.mulStab
⊢ ↑∅.mulStab.card - ↑(s ∩ a • ∅.mulStab * (s ∩ a • ∅.mulStab * ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | rw [card_sdiff, Int.ofNat_sub (card_le_card _), card_smul_finset] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab).card -
↑(t ∩ a ... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab).card -
↑(t ∩ a ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | all_goals
apply subset_trans (mul_subset_mul_left hst)
rw [← union_inter_distrib_right]
refine' subset_trans (mul_subset_mul_right inter_subset_right) _
simp only [smul_mul_assoc, mulStab_mul_mulStab, Subset.rfl] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ a • C.mulStab
α :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | rw [union_mul, le_sub_iff_add_le] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab).card -
↑(t ∩ a ... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab).card -
↑(t ∩ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | apply le_trans (add_le_add_left (Int.ofNat_le.mpr $ card_union_le _ _) _) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab).card -
↑(t ∩ ... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab).card -
↑(t ∩ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | norm_num | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab).card -
↑(t ∩ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑C.mulStab.card - ↑(s ∩ a • C.mulStab * (s ∩ a • C.... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | apply subset_trans (mul_subset_mul_left hst) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ a • C.mulStab | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * C.mulStab ⊆ a • C.mulStab | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | rw [← union_inter_distrib_right] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * C.mulStab ⊆ a • C.mulStab | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∪ t) ∩ a • C.mulStab * C.mulStab ⊆ a • C.mulStab | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * C.mulStab... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | refine' subset_trans (mul_subset_mul_right inter_subset_right) _ | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∪ t) ∩ a • C.mulStab * C.mulStab ⊆ a • C.mulStab | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mulStab * C.mulStab ⊆ a • C.mulStab | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (s ∪ t) ∩ a • C.mulStab * C.mulStab ⊆ a • C.mulStab... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | simp only [smul_mul_assoc, mulStab_mul_mulStab, Subset.rfl] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mulStab * C.mulStab ⊆ a • C.mulStab | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mulStab * C.mulStab ⊆ a • C.mulStab
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | rw [← Int.ofNat_sub (card_le_card (mul_subset_mul_left hst)), ←
card_sdiff (mul_subset_mul_left hst)] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑(a • C.mulStab \
((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab))... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑(a • C.mulStab \
((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab))... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑(a • C.mulStab \
((s ∩ a • C.mulStab ∪ t... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | norm_cast | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑(a • C.mulStab \
((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab))... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab)).... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ ↑(a • C.mulStab \
((s ∩ a • C.mulStab ∪ t... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | apply card_le_card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab)).... | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulS... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ (a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mu... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | refine' fun x hx => mem_sdiff.mpr ⟨_, _⟩ | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulS... | case a.refine'_1
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
⊢ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | have hx' := (mem_sdiff.mp hx).2 | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t... | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | contrapose! hx' | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t... | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | rw [← union_inter_distrib_right] | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t... | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | obtain ⟨y, hyst, d, hd, hxyd⟩ := mem_mul.mp hx' | case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t... | case a.refine'_2.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) *... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | obtain ⟨c, hc, hcx⟩ := mem_smul_finset.mp (mem_sdiff.mp hx).1 | case a.refine'_2.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) *... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | rw [← hcx, ← eq_mul_inv_iff_mul_eq] at hxyd | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • ... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | have hyC : y ∈ a • C.mulStab := by
rw [hxyd, smul_mul_assoc, smul_mem_smul_finset_iff, ← mulStab_mul_mulStab]
apply mul_mem_mul hc ((mem_mulStab hC).mpr (inv_smul_eq_iff.mpr _))
exact Eq.symm ((mem_mulStab hC).mp (hst hd)) | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • ... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | rw [eq_mul_inv_iff_mul_eq, hcx] at hxyd | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • ... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | rw [← hxyd] | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • ... | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | exact mul_mem_mul (mem_inter.mpr ⟨hyst, hyC⟩) hd | case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | apply smul_finset_subset_smul (mem_union_left t has) (mem_sdiff.mp hx).1 | case a.refine'_1
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | rw [hxyd, smul_mul_assoc, smul_mem_smul_finset_iff, ← mulStab_mul_mulStab] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | apply mul_mem_mul hc ((mem_mulStab hC).mpr (inv_smul_eq_iff.mpr _)) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)... | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.inter_mul_sub_card_le | [220, 1] | [267, 55] | exact Eq.symm ((mem_mulStab hC).mp (hst hd)) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ t ∩ a • C.mulStab) * (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ s' t✝ t' C✝ : Finset α
a✝ b a : α
s t C : Finset α
has : a ∈ s
hst : (s ∩ a • C.mulStab * (t ∩ a • C.mulStab)).mulStab ⊆ C.mulStab
hC : C.Nonempty
x : α
hx : x ∈ a • C.mulStab \ ((s ∩ a • C.mulStab ∪ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.card_mul_add_card_lt | [269, 1] | [277, 20] | rw [← tsub_pos_iff_lt, ← card_sdiff (mul_subset_mul hs ht), card_pos] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hC : C.Nonempty
hs : s' ⊆ s
ht : t' ⊆ t
hCst : C ⊆ s * t
hCst' : Disjoint C (s' * t')
⊢ (s' * t').card < (s * t).card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hC : C.Nonempty
hs : s' ⊆ s
ht : t' ⊆ t
hCst : C ⊆ s * t
hCst' : Disjoint C (s' * t')
⊢ ((s * t) \ (s' * t')).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hC : C.Nonempty
hs : s' ⊆ s
ht : t' ⊆ t
hCst : C ⊆ s * t
hCst' : Disjoint C (s' * t')
⊢ (s' * t').card < (s * t).card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.card_mul_add_card_lt | [269, 1] | [277, 20] | exact hC.mono (subset_sdiff.2 ⟨hCst, hCst'⟩) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hC : C.Nonempty
hs : s' ⊆ s
ht : t' ⊆ t
hCst : C ⊆ s * t
hCst' : Disjoint C (s' * t')
⊢ ((s * t) \ (s' * t')).Nonempty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hC : C.Nonempty
hs : s' ⊆ s
ht : t' ⊆ t
hCst : C ⊆ s * t
hCst' : Disjoint C (s' * t')
⊢ ((s * t) \ (s' * t')).Nonempty
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | set n : ℕ := (s * t).card + s.card with hn | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
n : ℕ := (s * t).card + s.card
hn : n = (s * t).card + s.card
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | clear_value n | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
n : ℕ := (s * t).card + s.card
hn : n = (s * t).card + s.card
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
n : ℕ
hn : n = (s * t).card + s.card
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
n : ℕ := (s * t).card + s.card
hn : n = (s * t).card + s.card
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | clear s' t' C a b | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
n : ℕ
hn : n = (s * t).card + s.card
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
n : ℕ
hn : n = (s * t).card + s.card
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
n : ℕ
hn : n = (s * t).card + s.card
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | induction' n using Nat.strong_induction_on with n ih generalizing α | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
n : ℕ
hn : n = (s * t).card + s.card
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card | case h
n : ℕ
ih :
∀ m < n,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
n : ℕ
hn : n = (s * t).card + s.card
⊢ (s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | subst hn | case h
n : ℕ
ih :
∀ m < n,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α... | case h
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
ih :
∀ m < n,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t).mulStab.card
α : Type ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | obtain rfl | hs := s.eq_empty_or_nonempty | case h
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s * t... | case h.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
t : Finset α
ih :
∀ m < (∅ * t).card + ∅.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s *... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulSt... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | obtain rfl | ht := t.eq_empty_or_nonempty | case h.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s... | case h.inr.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
ih :
∀ m < (s * ∅).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).m... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | simp | case h.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
t : Finset α
ih :
∀ m < (∅ * t).card + ∅.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card + (s *... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
t : Finset α
ih :
∀ m < (∅ * t).card + ∅.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mul... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | simp | case h.inr.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
ih :
∀ m < (s * ∅).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
ih :
∀ m < (s * ∅).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | obtain hstab | hstab := ne_or_eq (s * t).mulStab 1 | case h.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).card ... | case h.inr.inr.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | simp only [hstab, mul_one, card_one] at ih ⊢ | case h.inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).c... | case h.inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | replace ih := fun s' t' h => @ih _ h α _ _ s' t' rfl | case h.inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (s * t).mulStab).card + (t * (s * t).mulStab).card ≤ (s * t).c... | case h.inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s' * t').mulStab).card ≤ (s' * t').card +... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ih :
∀ m < (s * t).card + s.card,
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] (s t : Finset α),
m = (s * t).card + s.card →
(s * (... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial | case h.inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s' * t').mulStab).card ≤ (s' * t').card +... | case h.inr.inr.inr.inl.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
t : Finset α
ht : t.Nonempty
a : α
hs : {a}.Nonempty
hstab : ({a} * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < ({a} * t).card + {a}.card →
(s' * (s' * t').mulStab).card + (t' * (s' * t').mulStab).c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mul... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | have : b / a ∉ t.mulStab := by
refine' fun h => hab (Eq.symm (eq_of_div_eq_one _))
replace h := subset_mulStab_mul_right hs h
rw [hstab, mem_one] at h
exact h | case h.inr.inr.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s' * t').mulS... | case h.inr.inr.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s' * t').mulS... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | simp only [mem_mulStab' ht, smul_eq_mul, Classical.not_forall, exists_prop] at this | case h.inr.inr.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s' * t').mulS... | case h.inr.inr.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s' * t').mulS... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | obtain ⟨c, hc, hbac⟩ := this | case h.inr.inr.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s' * t').mulS... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | set t' := (a / c) • t with ht' | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).c... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | clear_value t' | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).c... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | rw [← inv_smul_eq_iff] at ht' | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).c... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | subst ht' | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab).card + (t' * (s... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t' : Finset α
ht : ((a / c)⁻¹ • t').Nonempty
hstab : (s * (a / c)⁻¹ • t').mulStab = 1
ih :
∀ (s' t'_1 : Finset α),
(... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).c... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | rename' t' => t | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t' : Finset α
ht : ((a / c)⁻¹ • t').Nonempty
hstab : (s * (a / c)⁻¹ • t').mulStab = 1
ih :
∀ (s' t'_1 : Finset α),
(... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : ((a / c)⁻¹ • t).Nonempty
hstab : (s * (a / c)⁻¹ • t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t' : Finset α
ht : ((a / c)⁻¹ • t').Nonempty
hstab : (s * (a ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | rw [mem_inv_smul_finset_iff, smul_eq_mul, div_mul_cancel] at hc | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : ((a / c)⁻¹ • t).Nonempty
hstab : (s * (a / c)⁻¹ • t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * ... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : ((a / c)⁻¹ • t).Nonempty
hstab : (s * (a / c)⁻¹ • t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : ((a / c)⁻¹ • t).Nonempty
hstab : (s * (a / ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | rw [div_mul_comm, mem_inv_smul_finset_iff, smul_eq_mul, ← mul_assoc, div_mul_div_cancel',
div_self', one_mul] at hbac | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : ((a / c)⁻¹ • t).Nonempty
hstab : (s * (a / c)⁻¹ • t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * ... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : ((a / c)⁻¹ • t).Nonempty
hstab : (s * (a / c)⁻¹ • t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : ((a / c)⁻¹ • t).Nonempty
hstab : (s * (a / ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | rw [smul_finset_nonempty] at ht | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : ((a / c)⁻¹ • t).Nonempty
hstab : (s * (a / c)⁻¹ • t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * ... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * (a / c)⁻¹ • t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : ((a / c)⁻¹ • t).Nonempty
hstab : (s * (a / ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | simp only [mul_smul_comm, smul_mul_assoc, mulStab_smul, card_smul_finset] at * | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * (a / c)⁻¹ • t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hc : a ∈ t
hbac : b ∉ t
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t'... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * (a / c)⁻¹ • t).mulS... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | have hst : (s ∩ t).Nonempty := ⟨_, mem_inter.2 ⟨ha, hc⟩⟩ | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hc : a ∈ t
hbac : b ∉ t
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t'... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hc : a ∈ t
hbac : b ∉ t
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t'... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hc : a ∈ t
hbac : b ∉ t
hstab : ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | have hsts : s ∩ t ⊂ s :=
⟨inter_subset_left, not_subset.2 ⟨_, hb, fun h => hbac $ inter_subset_right h⟩⟩ | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hc : a ∈ t
hbac : b ∉ t
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t'... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hc : a ∈ t
hbac : b ∉ t
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t'... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hc : a ∈ t
hbac : b ∉ t
hstab : ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | clear! a b | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hc : a ∈ t
hbac : b ∉ t
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t'... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
c : α
t : Finset α
ht : t.Nonempty
hc : a ∈ t
hbac : b ∉ t
hstab : ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | set convergent : Set (Finset α) :=
{C | C ⊆ s * t ∧ (s ∩ t).card + ((s ∪ t) * C.mulStab).card ≤ C.card + C.mulStab.card} | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | have convergent_nonempty : convergent.Nonempty := by
refine' ⟨s ∩ t * (s ∪ t), inter_mul_union_subset, (add_le_add_right (card_le_card $
subset_mul_left _ $ one_mem_mulStab.2 $ hst.mul $ hs.mono subset_union_left) _).trans $
ih (s ∩ t) (s ∪ t) _⟩
exact add_lt_add_of_le_of_lt (card_le_card inter_mul_union_... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Kneser.lean | Finset.mul_kneser | [287, 1] | [530, 54] | let C := argminOn (fun C : Finset α => C.mulStab.card) IsWellFounded.wf _ convergent_nonempty | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab... | case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'.card < (s * t).card + s.card →
(s' * (s' * t').mulStab... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.inr.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
c : α
t : Finset α
ht : t.Nonempty
hstab : (s * t).mulStab = 1
ih :
∀ (s' t' : Finset α),
(s' * t').card + s'... |
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