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/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card | [128, 1] | [138, 6] | rw [two_mul_card_alternatingGroup, Fintype.card_perm, hα4] | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 * Fintype.card ↥(alternatingGroup α) = 2 * 12
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 ≠ 0 | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ Nat.factorial 4 = 2 * 12
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 * Fintype.card ↥(alternatingGroup α) = 2 * 12
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card | [128, 1] | [138, 6] | all_goals
norm_num | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ Nat.factorial 4 = 2 * 12
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 ≠ 0 | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ Nat.factorial 4 = 24 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ Nat.factorial 4 = 2 * 12
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card | [128, 1] | [138, 6] | rfl | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ Nat.factorial 4 = 24 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ Nat.factorial 4 = 24
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card | [128, 1] | [138, 6] | rw [← Fintype.one_lt_card_iff_nontrivial, hα4] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Nontrivial α | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ 1 < 4 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Nontrivial α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card | [128, 1] | [138, 6] | norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ 1 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ 1 < 4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card | [128, 1] | [138, 6] | norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_card | [142, 1] | [145, 6] | rw [Sylow.card_eq_multiplicity, ← Nat.factors_count_eq, A4_card α hα4] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↥↑S = 4 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ 2 ^ List.count 2 (Nat.factors 12) = 4 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↥↑S = 4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_card | [142, 1] | [145, 6] | rfl | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ 2 ^ List.count 2 (Nat.factors 12) = 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ 2 ^ List.count 2 (Nat.factors 12) = 4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | apply Set.eq_of_subset_of_card_le | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ S.carrier = {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} | case hsub
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ S.carrier ⊆ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↑{g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} ≤ Fintype.card ↑S.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ S.carrier = {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | change _ ≤ Fintype.card S | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↑{g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} ≤ Fintype.card ↑S.carrier | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↑{g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} ≤ Fintype.card ↥↑S | Please generate a tactic in lean4 to solve the state.
STATE:
case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↑{g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} ≤ Fintype.card ↑S.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | rw [A4_sylow_card α hα4 S] | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↑{g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} ≤ Fintype.card ↥↑S | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↑{g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} ≤ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↑{g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} ≤ Fintype.card ↥↑S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | apply le_trans (Fintype.card_subtype_or _ _) | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↑{g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} ≤ 4 | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card { x // Equiv.Perm.cycleType ↑x = 0 } + Fintype.card { x // Equiv.Perm.cycleType ↑x = {2, 2} } ≤ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↑{g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} ≤ 4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | rw [Fintype.card_subtype] | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card { x // Equiv.Perm.cycleType ↑x = 0 } + Fintype.card { x // Equiv.Perm.cycleType ↑x = {2, 2} } ≤ 4 | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (Finset.filter (fun x => Equiv.Perm.cycleType ↑x = 0) Finset.univ).card +
Fintype.card { x // Equiv.Perm.cycleType ↑x = {2, 2} } ≤
4 | Please generate a tactic in lean4 to solve the state.
STATE:
case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card { x // Equiv.Perm.cycleType ↑x = 0 } + Fintype.card { x // Equiv.Perm.cycleType ↑x = {2, 2} } ≤ 4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | rw [Fintype.card_subtype] | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (Finset.filter (fun x => Equiv.Perm.cycleType ↑x = 0) Finset.univ).card +
Fintype.card { x // Equiv.Perm.cycleType ↑x = {2, 2} } ≤
4 | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (Finset.filter (fun x => Equiv.Perm.cycleType ↑x = 0) Finset.univ).card +
(Finset.filter (fun x => Equiv.Perm.cycleType ↑x = {2, 2}) Finset.univ).card ≤
4 | Please generate a tactic in lean4 to solve the state.
STATE:
case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (Finset.filter (fun x => Equiv.Perm.cycleType ↑x = 0) Finset.univ).card +
Fintype.card { x // Equiv.Perm.cycleType ↑x = {2, 2} } ≤
4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | simp only [OnCycleFactors.AlternatingGroup.card_of_cycleType, hα4] | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (Finset.filter (fun x => Equiv.Perm.cycleType ↑x = 0) Finset.univ).card +
(Finset.filter (fun x => Equiv.Perm.cycleType ↑x = {2, 2}) Finset.univ).card ≤
4 | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ((if (Multiset.sum 0 ≤ 4 ∧ ∀ a ∈ 0, 2 ≤ a) ∧ Even (Multiset.sum 0 + Multiset.card 0) then
Nat.factorial 4 /
(Nat.factorial (4 - Multiset.sum 0) *
(Multiset.prod 0 *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n 0)) (Multiset.dedup 0))))
else 0) +
if (Multiset.sum {2, 2} ≤ 4 ∧ ∀ a ∈ {2, 2}, 2 ≤ a) ∧ Even (Multiset.sum {2, 2} + Multiset.card {2, 2}) then
Nat.factorial 4 /
(Nat.factorial (4 - Multiset.sum {2, 2}) *
(Multiset.prod {2, 2} *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n {2, 2})) (Multiset.dedup {2, 2}))))
else 0) ≤
4 | Please generate a tactic in lean4 to solve the state.
STATE:
case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (Finset.filter (fun x => Equiv.Perm.cycleType ↑x = 0) Finset.univ).card +
(Finset.filter (fun x => Equiv.Perm.cycleType ↑x = {2, 2}) Finset.univ).card ≤
4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | norm_num | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ((if (Multiset.sum 0 ≤ 4 ∧ ∀ a ∈ 0, 2 ≤ a) ∧ Even (Multiset.sum 0 + Multiset.card 0) then
Nat.factorial 4 /
(Nat.factorial (4 - Multiset.sum 0) *
(Multiset.prod 0 *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n 0)) (Multiset.dedup 0))))
else 0) +
if (Multiset.sum {2, 2} ≤ 4 ∧ ∀ a ∈ {2, 2}, 2 ≤ a) ∧ Even (Multiset.sum {2, 2} + Multiset.card {2, 2}) then
Nat.factorial 4 /
(Nat.factorial (4 - Multiset.sum {2, 2}) *
(Multiset.prod {2, 2} *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n {2, 2})) (Multiset.dedup {2, 2}))))
else 0) ≤
4 | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (Nat.factorial 4 / Nat.factorial 4 + if Even 6 then Nat.factorial 4 / 8 else 0) ≤ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ((if (Multiset.sum 0 ≤ 4 ∧ ∀ a ∈ 0, 2 ≤ a) ∧ Even (Multiset.sum 0 + Multiset.card 0) then
Nat.factorial 4 /
(Nat.factorial (4 - Multiset.sum 0) *
(Multiset.prod 0 *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n 0)) (Multiset.dedup 0))))
else 0) +
if (Multiset.sum {2, 2} ≤ 4 ∧ ∀ a ∈ {2, 2}, 2 ≤ a) ∧ Even (Multiset.sum {2, 2} + Multiset.card {2, 2}) then
Nat.factorial 4 /
(Nat.factorial (4 - Multiset.sum {2, 2}) *
(Multiset.prod {2, 2} *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n {2, 2})) (Multiset.dedup {2, 2}))))
else 0) ≤
4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | rfl | case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (Nat.factorial 4 / Nat.factorial 4 + if Even 6 then Nat.factorial 4 / 8 else 0) ≤ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hcard
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (Nat.factorial 4 / Nat.factorial 4 + if Even 6 then Nat.factorial 4 / 8 else 0) ≤ 4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | intro k hk | case hsub
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ S.carrier ⊆ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} | case hsub
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
⊢ k ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} | Please generate a tactic in lean4 to solve the state.
STATE:
case hsub
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ S.carrier ⊆ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | simp only [Set.mem_setOf_eq] | case hsub
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
⊢ k ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} | case hsub
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
⊢ Equiv.Perm.cycleType ↑k = 0 ∨ Equiv.Perm.cycleType ↑k = {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
case hsub
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
⊢ k ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | obtain ⟨n, hn⟩ := (IsPGroup.iff_orderOf.mp S.isPGroup') ⟨k, hk⟩ | case hsub
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
⊢ Equiv.Perm.cycleType ↑k = 0 ∨ Equiv.Perm.cycleType ↑k = {2, 2} | case hsub.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
n : ℕ
hn : orderOf { val := k, property := hk } = 2 ^ n
⊢ Equiv.Perm.cycleType ↑k = 0 ∨ Equiv.Perm.cycleType ↑k = {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
case hsub
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
⊢ Equiv.Perm.cycleType ↑k = 0 ∨ Equiv.Perm.cycleType ↑k = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | apply mem_V4_of_order_two_pow α hα4 (↑k) k.prop n | case hsub.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
n : ℕ
hn : orderOf { val := k, property := hk } = 2 ^ n
⊢ Equiv.Perm.cycleType ↑k = 0 ∨ Equiv.Perm.cycleType ↑k = {2, 2} | case hsub.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
n : ℕ
hn : orderOf { val := k, property := hk } = 2 ^ n
⊢ orderOf ↑k ∣ 2 ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case hsub.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
n : ℕ
hn : orderOf { val := k, property := hk } = 2 ^ n
⊢ Equiv.Perm.cycleType ↑k = 0 ∨ Equiv.Perm.cycleType ↑k = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | rw [← orderOf_submonoid ⟨k, hk⟩, Subgroup.coe_mk] at hn | case hsub.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
n : ℕ
hn : orderOf { val := k, property := hk } = 2 ^ n
⊢ orderOf ↑k ∣ 2 ^ n | case hsub.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
n : ℕ
hn : orderOf k = 2 ^ n
⊢ orderOf ↑k ∣ 2 ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case hsub.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
n : ℕ
hn : orderOf { val := k, property := hk } = 2 ^ n
⊢ orderOf ↑k ∣ 2 ^ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_sylow_carrier | [149, 1] | [168, 6] | rw [orderOf_submonoid, hn] | case hsub.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
n : ℕ
hn : orderOf k = 2 ^ n
⊢ orderOf ↑k ∣ 2 ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hsub.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : k ∈ S.carrier
n : ℕ
hn : orderOf k = 2 ^ n
⊢ orderOf ↑k ∣ 2 ^ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | apply le_antisymm | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ V4 α = ↑S | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ V4 α ≤ ↑S
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ↑S ≤ V4 α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ V4 α = ↑S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | simp only [V4] | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ V4 α ≤ ↑S | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.closure {g | Equiv.Perm.cycleType ↑g = {2, 2}} ≤ ↑S | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ V4 α ≤ ↑S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | rw [Subgroup.closure_le] | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.closure {g | Equiv.Perm.cycleType ↑g = {2, 2}} ≤ ↑S | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ {g | Equiv.Perm.cycleType ↑g = {2, 2}} ⊆ ↑↑S | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.closure {g | Equiv.Perm.cycleType ↑g = {2, 2}} ≤ ↑S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | intro g hg | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ {g | Equiv.Perm.cycleType ↑g = {2, 2}} ⊆ ↑↑S | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ g ∈ ↑↑S | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ {g | Equiv.Perm.cycleType ↑g = {2, 2}} ⊆ ↑↑S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | rw [SetLike.mem_coe, ← Subgroup.mem_carrier, A4_sylow_carrier α hα4 S, Set.mem_setOf_eq] | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ g ∈ ↑↑S | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ g ∈ ↑↑S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | apply Or.intro_right | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2} | case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ Equiv.Perm.cycleType ↑g = {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | exact hg | case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ Equiv.Perm.cycleType ↑g = {2, 2} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ Equiv.Perm.cycleType ↑g = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | intro k | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ↑S ≤ V4 α | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ k ∈ ↑S → k ∈ V4 α | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ↑S ≤ V4 α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | rw [← Subgroup.mem_carrier, A4_sylow_carrier α hα4 S, Set.mem_setOf_eq, or_imp] | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ k ∈ ↑S → k ∈ V4 α | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ (Equiv.Perm.cycleType ↑k = 0 → k ∈ V4 α) ∧ (Equiv.Perm.cycleType ↑k = {2, 2} → k ∈ V4 α) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ k ∈ ↑S → k ∈ V4 α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | constructor | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ (Equiv.Perm.cycleType ↑k = 0 → k ∈ V4 α) ∧ (Equiv.Perm.cycleType ↑k = {2, 2} → k ∈ V4 α) | case a.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ Equiv.Perm.cycleType ↑k = 0 → k ∈ V4 α
case a.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ Equiv.Perm.cycleType ↑k = {2, 2} → k ∈ V4 α | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ (Equiv.Perm.cycleType ↑k = 0 → k ∈ V4 α) ∧ (Equiv.Perm.cycleType ↑k = {2, 2} → k ∈ V4 α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | intro hk | case a.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ Equiv.Perm.cycleType ↑k = 0 → k ∈ V4 α | case a.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = 0
⊢ k ∈ V4 α | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ Equiv.Perm.cycleType ↑k = 0 → k ∈ V4 α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | suffices hk : k = 1 by
rw [hk]; exact Subgroup.one_mem _ | case a.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = 0
⊢ k ∈ V4 α | case a.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = 0
⊢ k = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = 0
⊢ k ∈ V4 α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | rw [← Subtype.coe_inj, Subgroup.coe_one, ← Equiv.Perm.cycleType_eq_zero, hk] | case a.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = 0
⊢ k = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = 0
⊢ k = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | rw [hk] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk✝ : Equiv.Perm.cycleType ↑k = 0
hk : k = 1
⊢ k ∈ V4 α | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk✝ : Equiv.Perm.cycleType ↑k = 0
hk : k = 1
⊢ 1 ∈ V4 α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk✝ : Equiv.Perm.cycleType ↑k = 0
hk : k = 1
⊢ k ∈ V4 α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | exact Subgroup.one_mem _ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk✝ : Equiv.Perm.cycleType ↑k = 0
hk : k = 1
⊢ 1 ∈ V4 α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk✝ : Equiv.Perm.cycleType ↑k = 0
hk : k = 1
⊢ 1 ∈ V4 α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | intro hk | case a.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ Equiv.Perm.cycleType ↑k = {2, 2} → k ∈ V4 α | case a.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ k ∈ V4 α | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
⊢ Equiv.Perm.cycleType ↑k = {2, 2} → k ∈ V4 α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | simp only [V4] | case a.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ k ∈ V4 α | case a.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ k ∈ Subgroup.closure {g | Equiv.Perm.cycleType ↑g = {2, 2}} | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ k ∈ V4 α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | apply Subgroup.subset_closure | case a.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ k ∈ Subgroup.closure {g | Equiv.Perm.cycleType ↑g = {2, 2}} | case a.right.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ k ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}} | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ k ∈ Subgroup.closure {g | Equiv.Perm.cycleType ↑g = {2, 2}}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | simp only [Set.mem_setOf_eq] | case a.right.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ k ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}} | case a.right.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ Equiv.Perm.cycleType ↑k = {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ k ∈ {g | Equiv.Perm.cycleType ↑g = {2, 2}}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_unique_sylow | [172, 1] | [192, 15] | exact hk | case a.right.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ Equiv.Perm.cycleType ↑k = {2, 2} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
k : ↥(alternatingGroup α)
hk : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ Equiv.Perm.cycleType ↑k = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card_two_sylow_eq_one | [196, 1] | [204, 34] | rw [Fintype.card_eq_one_iff] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Fintype.card (Sylow 2 ↥(alternatingGroup α)) = 1 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ ∃ x, ∀ (y : Sylow 2 ↥(alternatingGroup α)), y = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Fintype.card (Sylow 2 ↥(alternatingGroup α)) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card_two_sylow_eq_one | [196, 1] | [204, 34] | obtain ⟨S : Sylow 2 (alternatingGroup α)⟩ := Sylow.nonempty (G := alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ ∃ x, ∀ (y : Sylow 2 ↥(alternatingGroup α)), y = x | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ∃ x, ∀ (y : Sylow 2 ↥(alternatingGroup α)), y = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ ∃ x, ∀ (y : Sylow 2 ↥(alternatingGroup α)), y = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card_two_sylow_eq_one | [196, 1] | [204, 34] | use S | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ∃ x, ∀ (y : Sylow 2 ↥(alternatingGroup α)), y = x | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ∀ (y : Sylow 2 ↥(alternatingGroup α)), y = S | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ∃ x, ∀ (y : Sylow 2 ↥(alternatingGroup α)), y = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card_two_sylow_eq_one | [196, 1] | [204, 34] | intro T | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ∀ (y : Sylow 2 ↥(alternatingGroup α)), y = S | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S T : Sylow 2 ↥(alternatingGroup α)
⊢ T = S | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ ∀ (y : Sylow 2 ↥(alternatingGroup α)), y = S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card_two_sylow_eq_one | [196, 1] | [204, 34] | rw [Sylow.ext_iff] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S T : Sylow 2 ↥(alternatingGroup α)
⊢ T = S | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S T : Sylow 2 ↥(alternatingGroup α)
⊢ ↑T = ↑S | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S T : Sylow 2 ↥(alternatingGroup α)
⊢ T = S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card_two_sylow_eq_one | [196, 1] | [204, 34] | rw [← V4_is_unique_sylow α hα4 S] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S T : Sylow 2 ↥(alternatingGroup α)
⊢ ↑T = ↑S | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S T : Sylow 2 ↥(alternatingGroup α)
⊢ ↑T = V4 α | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S T : Sylow 2 ↥(alternatingGroup α)
⊢ ↑T = ↑S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card_two_sylow_eq_one | [196, 1] | [204, 34] | rw [V4_is_unique_sylow α hα4 T] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S T : Sylow 2 ↥(alternatingGroup α)
⊢ ↑T = V4 α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S T : Sylow 2 ↥(alternatingGroup α)
⊢ ↑T = V4 α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_characteristic | [208, 1] | [215, 39] | obtain ⟨S : Sylow 2 (alternatingGroup α)⟩ := Sylow.nonempty (G := alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Subgroup.Characteristic (V4 α) | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.Characteristic (V4 α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Subgroup.Characteristic (V4 α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_characteristic | [208, 1] | [215, 39] | rw [V4_is_unique_sylow α hα4 S] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.Characteristic (V4 α) | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.Characteristic ↑S | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.Characteristic (V4 α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_characteristic | [208, 1] | [215, 39] | refine' Sylow.characteristic_of_normal S _ | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.Characteristic ↑S | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.Normal ↑S | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.Characteristic ↑S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_characteristic | [208, 1] | [215, 39] | rw [← Subgroup.normalizer_eq_top] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.Normal ↑S | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.normalizer ↑S = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.Normal ↑S
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_characteristic | [208, 1] | [215, 39] | rw [← Subgroup.index_eq_one] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.normalizer ↑S = ⊤ | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.index (Subgroup.normalizer ↑S) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.normalizer ↑S = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_characteristic | [208, 1] | [215, 39] | rw [← card_sylow_eq_index_normalizer] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.index (Subgroup.normalizer ↑S) = 1 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card (Sylow 2 ↥(alternatingGroup α)) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Subgroup.index (Subgroup.normalizer ↑S) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_characteristic | [208, 1] | [215, 39] | exact A4_card_two_sylow_eq_one α hα4 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card (Sylow 2 ↥(alternatingGroup α)) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card (Sylow 2 ↥(alternatingGroup α)) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_normal | [221, 1] | [224, 17] | haveI := V4_is_characteristic α hα4 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Subgroup.Normal (V4 α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Characteristic (V4 α)
⊢ Subgroup.Normal (V4 α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Subgroup.Normal (V4 α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_normal | [221, 1] | [224, 17] | infer_instance | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Characteristic (V4 α)
⊢ Subgroup.Normal (V4 α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Characteristic (V4 α)
⊢ Subgroup.Normal (V4 α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_card | [234, 1] | [238, 30] | obtain ⟨S : Sylow 2 (alternatingGroup α)⟩ := Sylow.nonempty (G := alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Fintype.card ↥(V4 α) = 4 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↥(V4 α) = 4 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Fintype.card ↥(V4 α) = 4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_card | [234, 1] | [238, 30] | rw [V4_is_unique_sylow α hα4 S] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↥(V4 α) = 4 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↥↑S = 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↥(V4 α) = 4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_card | [234, 1] | [238, 30] | exact A4_sylow_card α hα4 S | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↥↑S = 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ Fintype.card ↥↑S = 4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.isCommutative_of_exponent_two | [242, 1] | [249, 58] | suffices ∀ g : G, g⁻¹ = g by
constructor
intro a b
rw [← mul_inv_eq_iff_eq_mul, ← mul_inv_eq_one, this, this, ← hG2 (a * b), pow_two,
mul_assoc (a * b) a b] | α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
⊢ Std.Commutative fun x x_1 => x * x_1 | α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
⊢ ∀ (g : G), g⁻¹ = g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.isCommutative_of_exponent_two | [242, 1] | [249, 58] | intro g | α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
⊢ ∀ (g : G), g⁻¹ = g | α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
g : G
⊢ g⁻¹ = g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
⊢ ∀ (g : G), g⁻¹ = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.isCommutative_of_exponent_two | [242, 1] | [249, 58] | rw [← mul_eq_one_iff_inv_eq, ← hG2 g, pow_two] | α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
g : G
⊢ g⁻¹ = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
g : G
⊢ g⁻¹ = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.isCommutative_of_exponent_two | [242, 1] | [249, 58] | constructor | α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
this : ∀ (g : G), g⁻¹ = g
⊢ Std.Commutative fun x x_1 => x * x_1 | case comm
α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
this : ∀ (g : G), g⁻¹ = g
⊢ ∀ (a b : G), a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
this : ∀ (g : G), g⁻¹ = g
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.isCommutative_of_exponent_two | [242, 1] | [249, 58] | intro a b | case comm
α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
this : ∀ (g : G), g⁻¹ = g
⊢ ∀ (a b : G), a * b = b * a | case comm
α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
this : ∀ (g : G), g⁻¹ = g
a b : G
⊢ a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
case comm
α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
this : ∀ (g : G), g⁻¹ = g
⊢ ∀ (a b : G), a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.isCommutative_of_exponent_two | [242, 1] | [249, 58] | rw [← mul_inv_eq_iff_eq_mul, ← mul_inv_eq_one, this, this, ← hG2 (a * b), pow_two,
mul_assoc (a * b) a b] | case comm
α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
this : ∀ (g : G), g⁻¹ = g
a b : G
⊢ a * b = b * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case comm
α : Type ?u.56052
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
hG2 : ∀ (g : G), g ^ 2 = 1
this : ∀ (g : G), g⁻¹ = g
a b : G
⊢ a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_carrier_eq | [252, 1] | [258, 32] | obtain ⟨S : Sylow 2 (alternatingGroup α)⟩ := Sylow.nonempty (G := alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ (V4 α).carrier = {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (V4 α).carrier = {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ (V4 α).carrier = {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_carrier_eq | [252, 1] | [258, 32] | rw [V4_is_unique_sylow α hα4 S] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (V4 α).carrier = {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ S.carrier = {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ (V4 α).carrier = {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_carrier_eq | [252, 1] | [258, 32] | rw [A4_sylow_carrier α hα4 S] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ S.carrier = {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ S.carrier = {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_of_exponent_two | [262, 1] | [273, 13] | rintro ⟨⟨g, hg⟩, hg'⟩ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ ∀ (g : ↥(V4 α)), g ^ 2 = 1 | case mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ V4 α
⊢ { val := { val := g, property := hg }, property := hg' } ^ 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ ∀ (g : ↥(V4 α)), g ^ 2 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_of_exponent_two | [262, 1] | [273, 13] | simp only [← Subtype.coe_inj, SubmonoidClass.mk_pow, Subgroup.coe_mk, Subgroup.coe_one] | case mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ V4 α
⊢ { val := { val := g, property := hg }, property := hg' } ^ 2 = 1 | case mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ V4 α
⊢ g ^ 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ V4 α
⊢ { val := { val := g, property := hg }, property := hg' } ^ 2 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_of_exponent_two | [262, 1] | [273, 13] | rw [← Subgroup.mem_carrier, V4_carrier_eq α hα4] at hg' | case mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ V4 α
⊢ g ^ 2 = 1 | case mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ g ^ 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ V4 α
⊢ g ^ 2 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_of_exponent_two | [262, 1] | [273, 13] | cases' hg' with hg' hg' | case mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ g ^ 2 = 1 | case mk.mk.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType ↑{ val := g, property := hg } = 0
⊢ g ^ 2 = 1
case mk.mk.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType ↑{ val := g, property := hg } = {2, 2}
⊢ g ^ 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ g ^ 2 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_of_exponent_two | [262, 1] | [273, 13] | simp only [Subgroup.coe_mk, Equiv.Perm.cycleType_eq_zero] at hg' | case mk.mk.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType ↑{ val := g, property := hg } = 0
⊢ g ^ 2 = 1 | case mk.mk.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : g = 1
⊢ g ^ 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType ↑{ val := g, property := hg } = 0
⊢ g ^ 2 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_of_exponent_two | [262, 1] | [273, 13] | simp only [hg', one_pow] | case mk.mk.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : g = 1
⊢ g ^ 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : g = 1
⊢ g ^ 2 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_of_exponent_two | [262, 1] | [273, 13] | convert pow_orderOf_eq_one g | case mk.mk.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType ↑{ val := g, property := hg } = {2, 2}
⊢ g ^ 2 = 1 | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType ↑{ val := g, property := hg } = {2, 2}
⊢ 2 = orderOf g | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType ↑{ val := g, property := hg } = {2, 2}
⊢ g ^ 2 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_of_exponent_two | [262, 1] | [273, 13] | simp only [Subgroup.coe_mk] at hg' | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType ↑{ val := g, property := hg } = {2, 2}
⊢ 2 = orderOf g | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType g = {2, 2}
⊢ 2 = orderOf g | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType ↑{ val := g, property := hg } = {2, 2}
⊢ 2 = orderOf g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_of_exponent_two | [262, 1] | [273, 13] | rw [← Equiv.Perm.lcm_cycleType, hg'] | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType g = {2, 2}
⊢ 2 = orderOf g | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType g = {2, 2}
⊢ 2 = Multiset.lcm {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType g = {2, 2}
⊢ 2 = orderOf g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_is_of_exponent_two | [262, 1] | [273, 13] | norm_num | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType g = {2, 2}
⊢ 2 = Multiset.lcm {2, 2} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : Equiv.Perm.cycleType g = {2, 2}
⊢ 2 = Multiset.lcm {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_isCommutative | [277, 1] | [279, 85] | refine' { is_comm := isCommutative_of_exponent_two (V4_is_of_exponent_two α hα4) } | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Subgroup.IsCommutative (V4 α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Subgroup.IsCommutative (V4 α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | constructor | α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ commutator G ≤ H | case mp
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
⊢ (Std.Commutative fun x x_1 => x * x_1) → commutator G ≤ H
case mpr
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
⊢ commutator G ≤ H → Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ commutator G ≤ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | intro h | case mp
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
⊢ (Std.Commutative fun x x_1 => x * x_1) → commutator G ≤ H | case mp
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
⊢ commutator G ≤ H | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
⊢ (Std.Commutative fun x x_1 => x * x_1) → commutator G ≤ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | simp only [commutator_eq_closure, Subgroup.closure_le] | case mp
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
⊢ commutator G ≤ H | case mp
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
⊢ commutatorSet G ⊆ ↑H | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
⊢ commutator G ≤ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | rintro g ⟨g1, g2, rfl⟩ | case mp
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
⊢ commutatorSet G ⊆ ↑H | case mp.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
g1 g2 : G
⊢ ⁅g1, g2⁆ ∈ ↑H | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
⊢ commutatorSet G ⊆ ↑H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | simp only [SetLike.mem_coe, ← QuotientGroup.eq_one_iff] | case mp.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
g1 g2 : G
⊢ ⁅g1, g2⁆ ∈ ↑H | case mp.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
g1 g2 : G
⊢ ↑⁅g1, g2⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
g1 g2 : G
⊢ ⁅g1, g2⁆ ∈ ↑H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | rw [← QuotientGroup.mk'_apply, map_commutatorElement (QuotientGroup.mk' H) g1 g2] | case mp.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
g1 g2 : G
⊢ ↑⁅g1, g2⁆ = 1 | case mp.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
g1 g2 : G
⊢ ⁅(QuotientGroup.mk' H) g1, (QuotientGroup.mk' H) g2⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
g1 g2 : G
⊢ ↑⁅g1, g2⁆ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | simp only [QuotientGroup.mk'_apply, commutatorElement_eq_one_iff_mul_comm, h.comm] | case mp.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
g1 g2 : G
⊢ ⁅(QuotientGroup.mk' H) g1, (QuotientGroup.mk' H) g2⁆ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : Std.Commutative fun x x_1 => x * x_1
g1 g2 : G
⊢ ⁅(QuotientGroup.mk' H) g1, (QuotientGroup.mk' H) g2⁆ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | intro h | case mpr
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
⊢ commutator G ≤ H → Std.Commutative fun x x_1 => x * x_1 | case mpr
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
⊢ Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
⊢ commutator G ≤ H → Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | constructor | case mpr
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
⊢ Std.Commutative fun x x_1 => x * x_1 | case mpr.comm
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
⊢ ∀ (a b : G ⧸ H), a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | intro a b | case mpr.comm
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
⊢ ∀ (a b : G ⧸ H), a * b = b * a | case mpr.comm
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
a b : G ⧸ H
⊢ a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
⊢ ∀ (a b : G ⧸ H), a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | obtain ⟨g1, rfl⟩ := QuotientGroup.mk'_surjective H a | case mpr.comm
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
a b : G ⧸ H
⊢ a * b = b * a | case mpr.comm.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
b : G ⧸ H
g1 : G
⊢ (QuotientGroup.mk' H) g1 * b = b * (QuotientGroup.mk' H) g1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
a b : G ⧸ H
⊢ a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | obtain ⟨g2, rfl⟩ := QuotientGroup.mk'_surjective H b | case mpr.comm.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
b : G ⧸ H
g1 : G
⊢ (QuotientGroup.mk' H) g1 * b = b * (QuotientGroup.mk' H) g1 | case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ (QuotientGroup.mk' H) g1 * (QuotientGroup.mk' H) g2 = (QuotientGroup.mk' H) g2 * (QuotientGroup.mk' H) g1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
b : G ⧸ H
g1 : G
⊢ (QuotientGroup.mk' H) g1 * b = b * (QuotientGroup.mk' H) g1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | rw [← commutatorElement_eq_one_iff_mul_comm] | case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ (QuotientGroup.mk' H) g1 * (QuotientGroup.mk' H) g2 = (QuotientGroup.mk' H) g2 * (QuotientGroup.mk' H) g1 | case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ⁅(QuotientGroup.mk' H) g1, (QuotientGroup.mk' H) g2⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ (QuotientGroup.mk' H) g1 * (QuotientGroup.mk' H) g2 = (QuotientGroup.mk' H) g2 * (QuotientGroup.mk' H) g1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | rw [← map_commutatorElement _ g1 g2] | case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ⁅(QuotientGroup.mk' H) g1, (QuotientGroup.mk' H) g2⁆ = 1 | case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ (QuotientGroup.mk' H) ⁅g1, g2⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ⁅(QuotientGroup.mk' H) g1, (QuotientGroup.mk' H) g2⁆ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | rw [QuotientGroup.mk'_apply] | case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ (QuotientGroup.mk' H) ⁅g1, g2⁆ = 1 | case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ↑⁅g1, g2⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ (QuotientGroup.mk' H) ⁅g1, g2⁆ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | rw [QuotientGroup.eq_one_iff] | case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ↑⁅g1, g2⁆ = 1 | case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ⁅g1, g2⁆ ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ↑⁅g1, g2⁆ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | apply h | case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ⁅g1, g2⁆ ∈ H | case mpr.comm.intro.intro.a
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ⁅g1, g2⁆ ∈ commutator G | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ⁅g1, g2⁆ ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | apply Subgroup.commutator_mem_commutator | case mpr.comm.intro.intro.a
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ⁅g1, g2⁆ ∈ commutator G | case mpr.comm.intro.intro.a.h₁
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ g1 ∈ ⊤
case mpr.comm.intro.intro.a.h₂
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ g2 ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro.a
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ ⁅g1, g2⁆ ∈ commutator G
TACTIC:
|
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