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/AlternatingMaximal.lean | alternatingGroup.Nat.Combination.isPreprimitive_of_alt | [750, 1] | [797, 13] | simp only [← Finset.coe_compl, Finset.coe_nonempty, ← Finset.card_compl_lt_iff_nonempty,
compl_compl, hs] | case inr.intro.h1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Fintype.card α
hα : Fintype.card α ≠ 2 * n
hα' : 3 ≤ Fintype.card α
this✝¹ : Nontrivial α
h_one_lt : 1 < n
ht : IsPretransitive ↥(alternatingGroup α) ↑(Nat.Combination α n)
this✝ : Nontrivial ↑(Nat.Combination α n)
sn : ↑(Nat.Combination α n)
s : Finset α := ↑sn
hs : s.card = n := Subtype.prop sn
this : stabilizer (↥(alternatingGroup α)) sn = stabilizer ↥(alternatingGroup α) ↑s
⊢ Set.Nonempty (↑s)ᶜ | case inr.intro.h1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Fintype.card α
hα : Fintype.card α ≠ 2 * n
hα' : 3 ≤ Fintype.card α
this✝¹ : Nontrivial α
h_one_lt : 1 < n
ht : IsPretransitive ↥(alternatingGroup α) ↑(Nat.Combination α n)
this✝ : Nontrivial ↑(Nat.Combination α n)
sn : ↑(Nat.Combination α n)
s : Finset α := ↑sn
hs : s.card = n := Subtype.prop sn
this : stabilizer (↥(alternatingGroup α)) sn = stabilizer ↥(alternatingGroup α) ↑s
⊢ n < Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.h1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Fintype.card α
hα : Fintype.card α ≠ 2 * n
hα' : 3 ≤ Fintype.card α
this✝¹ : Nontrivial α
h_one_lt : 1 < n
ht : IsPretransitive ↥(alternatingGroup α) ↑(Nat.Combination α n)
this✝ : Nontrivial ↑(Nat.Combination α n)
sn : ↑(Nat.Combination α n)
s : Finset α := ↑sn
hs : s.card = n := Subtype.prop sn
this : stabilizer (↥(alternatingGroup α)) sn = stabilizer ↥(alternatingGroup α) ↑s
⊢ Set.Nonempty (↑s)ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.Nat.Combination.isPreprimitive_of_alt | [750, 1] | [797, 13] | exact hn | case inr.intro.h1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Fintype.card α
hα : Fintype.card α ≠ 2 * n
hα' : 3 ≤ Fintype.card α
this✝¹ : Nontrivial α
h_one_lt : 1 < n
ht : IsPretransitive ↥(alternatingGroup α) ↑(Nat.Combination α n)
this✝ : Nontrivial ↑(Nat.Combination α n)
sn : ↑(Nat.Combination α n)
s : Finset α := ↑sn
hs : s.card = n := Subtype.prop sn
this : stabilizer (↥(alternatingGroup α)) sn = stabilizer ↥(alternatingGroup α) ↑s
⊢ n < Fintype.card α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.h1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Fintype.card α
hα : Fintype.card α ≠ 2 * n
hα' : 3 ≤ Fintype.card α
this✝¹ : Nontrivial α
h_one_lt : 1 < n
ht : IsPretransitive ↥(alternatingGroup α) ↑(Nat.Combination α n)
this✝ : Nontrivial ↑(Nat.Combination α n)
sn : ↑(Nat.Combination α n)
s : Finset α := ↑sn
hs : s.card = n := Subtype.prop sn
this : stabilizer (↥(alternatingGroup α)) sn = stabilizer ↥(alternatingGroup α) ↑s
⊢ n < Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.Nat.Combination.isPreprimitive_of_alt | [750, 1] | [797, 13] | simp only [Set.ncard_coe_Finset, hs] | case inr.intro.hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Fintype.card α
hα : Fintype.card α ≠ 2 * n
hα' : 3 ≤ Fintype.card α
this✝¹ : Nontrivial α
h_one_lt : 1 < n
ht : IsPretransitive ↥(alternatingGroup α) ↑(Nat.Combination α n)
this✝ : Nontrivial ↑(Nat.Combination α n)
sn : ↑(Nat.Combination α n)
s : Finset α := ↑sn
hs : s.card = n := Subtype.prop sn
this : stabilizer (↥(alternatingGroup α)) sn = stabilizer ↥(alternatingGroup α) ↑s
⊢ Fintype.card α ≠ 2 * Set.ncard ↑s | case inr.intro.hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Fintype.card α
hα : Fintype.card α ≠ 2 * n
hα' : 3 ≤ Fintype.card α
this✝¹ : Nontrivial α
h_one_lt : 1 < n
ht : IsPretransitive ↥(alternatingGroup α) ↑(Nat.Combination α n)
this✝ : Nontrivial ↑(Nat.Combination α n)
sn : ↑(Nat.Combination α n)
s : Finset α := ↑sn
hs : s.card = n := Subtype.prop sn
this : stabilizer (↥(alternatingGroup α)) sn = stabilizer ↥(alternatingGroup α) ↑s
⊢ Fintype.card α ≠ 2 * n | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Fintype.card α
hα : Fintype.card α ≠ 2 * n
hα' : 3 ≤ Fintype.card α
this✝¹ : Nontrivial α
h_one_lt : 1 < n
ht : IsPretransitive ↥(alternatingGroup α) ↑(Nat.Combination α n)
this✝ : Nontrivial ↑(Nat.Combination α n)
sn : ↑(Nat.Combination α n)
s : Finset α := ↑sn
hs : s.card = n := Subtype.prop sn
this : stabilizer (↥(alternatingGroup α)) sn = stabilizer ↥(alternatingGroup α) ↑s
⊢ Fintype.card α ≠ 2 * Set.ncard ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.Nat.Combination.isPreprimitive_of_alt | [750, 1] | [797, 13] | exact hα | case inr.intro.hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Fintype.card α
hα : Fintype.card α ≠ 2 * n
hα' : 3 ≤ Fintype.card α
this✝¹ : Nontrivial α
h_one_lt : 1 < n
ht : IsPretransitive ↥(alternatingGroup α) ↑(Nat.Combination α n)
this✝ : Nontrivial ↑(Nat.Combination α n)
sn : ↑(Nat.Combination α n)
s : Finset α := ↑sn
hs : s.card = n := Subtype.prop sn
this : stabilizer (↥(alternatingGroup α)) sn = stabilizer ↥(alternatingGroup α) ↑s
⊢ Fintype.card α ≠ 2 * n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
h_one_le : 1 ≤ n
hn : n < Fintype.card α
hα : Fintype.card α ≠ 2 * n
hα' : 3 ≤ Fintype.card α
this✝¹ : Nontrivial α
h_one_lt : 1 < n
ht : IsPretransitive ↥(alternatingGroup α) ↑(Nat.Combination α n)
this✝ : Nontrivial ↑(Nat.Combination α n)
sn : ↑(Nat.Combination α n)
s : Finset α := ↑sn
hs : s.card = n := Subtype.prop sn
this : stabilizer (↥(alternatingGroup α)) sn = stabilizer ↥(alternatingGroup α) ↑s
⊢ Fintype.card α ≠ 2 * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.stabilizer_compl | [44, 1] | [52, 29] | have : ∀ s : Set α, stabilizer G s ≤ stabilizer G (sᶜ) :=
by
intro s g h
rw [mem_stabilizer_iff, smul_compl_set, mem_stabilizer_iff.1 h] | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
⊢ stabilizer G sᶜ = stabilizer G s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
this : ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ
⊢ stabilizer G sᶜ = stabilizer G s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
⊢ stabilizer G sᶜ = stabilizer G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.stabilizer_compl | [44, 1] | [52, 29] | refine' le_antisymm _ (this _) | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
this : ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ
⊢ stabilizer G sᶜ = stabilizer G s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
this : ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ
⊢ stabilizer G sᶜ ≤ stabilizer G s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
this : ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ
⊢ stabilizer G sᶜ = stabilizer G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.stabilizer_compl | [44, 1] | [52, 29] | convert this _ | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
this : ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ
⊢ stabilizer G sᶜ ≤ stabilizer G s | case h.e'_4.h.e'_5
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
this : ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ
⊢ s = sᶜᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
this : ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ
⊢ stabilizer G sᶜ ≤ stabilizer G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.stabilizer_compl | [44, 1] | [52, 29] | exact (compl_compl _).symm | case h.e'_4.h.e'_5
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
this : ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ
⊢ s = sᶜᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_5
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
this : ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ
⊢ s = sᶜᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.stabilizer_compl | [44, 1] | [52, 29] | intro s g h | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
⊢ ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s✝ s : Set α
g : G
h : g ∈ stabilizer G s
⊢ g ∈ stabilizer G sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
⊢ ∀ (s : Set α), stabilizer G s ≤ stabilizer G sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.stabilizer_compl | [44, 1] | [52, 29] | rw [mem_stabilizer_iff, smul_compl_set, mem_stabilizer_iff.1 h] | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s✝ s : Set α
g : G
h : g ∈ stabilizer G s
⊢ g ∈ stabilizer G sᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s✝ s : Set α
g : G
h : g ∈ stabilizer G s
⊢ g ∈ stabilizer G sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | constructor | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
⊢ H ≤ stabilizer G s ↔ ∀ g ∈ H, g • s ⊆ s | case mp
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
⊢ H ≤ stabilizer G s → ∀ g ∈ H, g • s ⊆ s
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
⊢ (∀ g ∈ H, g • s ⊆ s) → H ≤ stabilizer G s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
⊢ H ≤ stabilizer G s ↔ ∀ g ∈ H, g • s ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | intro hyp | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
⊢ (∀ g ∈ H, g • s ⊆ s) → H ≤ stabilizer G s | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
⊢ H ≤ stabilizer G s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
⊢ (∀ g ∈ H, g • s ⊆ s) → H ≤ stabilizer G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | intro g hg | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
⊢ H ≤ stabilizer G s | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ g ∈ stabilizer G s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
⊢ H ≤ stabilizer G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | rw [mem_stabilizer_iff] | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ g ∈ stabilizer G s | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ g • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ g ∈ stabilizer G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | apply subset_antisymm | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ g • s = s | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ g • s ⊆ s
case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ s ⊆ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ g • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | exact hyp g hg | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ g • s ⊆ s
case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ s ⊆ g • s | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ s ⊆ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ g • s ⊆ s
case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ s ⊆ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | intro x hx | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ s ⊆ g • s | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ x ∈ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
⊢ s ⊆ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | use g⁻¹ • x | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ x ∈ g • s | case h
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ g⁻¹ • x ∈ s ∧ (fun x => g • x) (g⁻¹ • x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ x ∈ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | constructor | case h
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ g⁻¹ • x ∈ s ∧ (fun x => g • x) (g⁻¹ • x) = x | case h.left
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ g⁻¹ • x ∈ s
case h.right
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ (fun x => g • x) (g⁻¹ • x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ g⁻¹ • x ∈ s ∧ (fun x => g • x) (g⁻¹ • x) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | apply hyp g⁻¹ (inv_mem hg) | case h.left
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ g⁻¹ • x ∈ s
case h.right
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ (fun x => g • x) (g⁻¹ • x) = x | case h.left.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ g⁻¹ • x ∈ g⁻¹ • s
case h.right
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ (fun x => g • x) (g⁻¹ • x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ g⁻¹ • x ∈ s
case h.right
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ (fun x => g • x) (g⁻¹ • x) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | simp only [Set.smul_mem_smul_set_iff, hx] | case h.left.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ g⁻¹ • x ∈ g⁻¹ • s
case h.right
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ (fun x => g • x) (g⁻¹ • x) = x | case h.right
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ (fun x => g • x) (g⁻¹ • x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ g⁻¹ • x ∈ g⁻¹ • s
case h.right
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ (fun x => g • x) (g⁻¹ • x) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | simp only [smul_inv_smul] | case h.right
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ (fun x => g • x) (g⁻¹ • x) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : ∀ g ∈ H, g • s ⊆ s
g : G
hg : g ∈ H
x : α
hx : x ∈ s
⊢ (fun x => g • x) (g⁻¹ • x) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | intro hyp g hg | case mp
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
⊢ H ≤ stabilizer G s → ∀ g ∈ H, g • s ⊆ s | case mp
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : H ≤ stabilizer G s
g : G
hg : g ∈ H
⊢ g • s ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
⊢ H ≤ stabilizer G s → ∀ g ∈ H, g • s ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | apply Eq.subset | case mp
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : H ≤ stabilizer G s
g : G
hg : g ∈ H
⊢ g • s ⊆ s | case mp.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : H ≤ stabilizer G s
g : G
hg : g ∈ H
⊢ g • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : H ≤ stabilizer G s
g : G
hg : g ∈ H
⊢ g • s ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | rw [← mem_stabilizer_iff] | case mp.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : H ≤ stabilizer G s
g : G
hg : g ∈ H
⊢ g • s = s | case mp.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : H ≤ stabilizer G s
g : G
hg : g ∈ H
⊢ g ∈ stabilizer G s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : H ≤ stabilizer G s
g : G
hg : g ∈ H
⊢ g • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.le_stabilizer_iff_smul_le | [83, 1] | [99, 28] | exact hyp hg | case mp.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : H ≤ stabilizer G s
g : G
hg : g ∈ H
⊢ g ∈ stabilizer G s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
H : Subgroup G
hyp : H ≤ stabilizer G s
g : G
hg : g ∈ H
⊢ g ∈ stabilizer G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | haveI : Fintype s := Set.Finite.fintype hs | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
⊢ g ∈ stabilizer G s ↔ g • s ⊆ s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this : Fintype ↑s
⊢ g ∈ stabilizer G s ↔ g • s ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
⊢ g ∈ stabilizer G s ↔ g • s ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | haveI : Fintype (g • s : Set α) := Fintype.ofFinite _ | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this : Fintype ↑s
⊢ g ∈ stabilizer G s ↔ g • s ⊆ s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g ∈ stabilizer G s ↔ g • s ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this : Fintype ↑s
⊢ g ∈ stabilizer G s ↔ g • s ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | rw [mem_stabilizer_iff] | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g ∈ stabilizer G s ↔ g • s ⊆ s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s = s ↔ g • s ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g ∈ stabilizer G s ↔ g • s ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | constructor | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s = s ↔ g • s ⊆ s | case mp
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s = s → g • s ⊆ s
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s ⊆ s → g • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s = s ↔ g • s ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | exact Eq.subset | case mp
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s = s → g • s ⊆ s
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s ⊆ s → g • s = s | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s ⊆ s → g • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s = s → g • s ⊆ s
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s ⊆ s → g • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | rw [← Set.toFinset_inj, ← Set.toFinset_subset_toFinset] | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s ⊆ s → g • s = s | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ Set.toFinset (g • s) ⊆ Set.toFinset s → Set.toFinset (g • s) = Set.toFinset s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ g • s ⊆ s → g • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | intro h | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ Set.toFinset (g • s) ⊆ Set.toFinset s → Set.toFinset (g • s) = Set.toFinset s | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ Set.toFinset (g • s) = Set.toFinset s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
⊢ Set.toFinset (g • s) ⊆ Set.toFinset s → Set.toFinset (g • s) = Set.toFinset s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | apply Finset.eq_of_subset_of_card_le h | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ Set.toFinset (g • s) = Set.toFinset s | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ (Set.toFinset s).card ≤ (Set.toFinset (g • s)).card | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ Set.toFinset (g • s) = Set.toFinset s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | apply le_of_eq | case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ (Set.toFinset s).card ≤ (Set.toFinset (g • s)).card | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ (Set.toFinset s).card = (Set.toFinset (g • s)).card | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ (Set.toFinset s).card ≤ (Set.toFinset (g • s)).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | suffices (g • s).toFinset = Finset.map ⟨_, MulAction.injective g⟩ hs.toFinset by
rw [this, Finset.card_map, Set.toFinite_toFinset] | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ (Set.toFinset s).card = (Set.toFinset (g • s)).card | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ Set.toFinset (g • s) = Finset.map { toFun := fun x => g • x, inj' := ⋯ } (Set.Finite.toFinset hs) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ (Set.toFinset s).card = (Set.toFinset (g • s)).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | rw [← Finset.coe_inj] | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ Set.toFinset (g • s) = Finset.map { toFun := fun x => g • x, inj' := ⋯ } (Set.Finite.toFinset hs) | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ ↑(Set.toFinset (g • s)) = ↑(Finset.map { toFun := fun x => g • x, inj' := ⋯ } (Set.Finite.toFinset hs)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ Set.toFinset (g • s) = Finset.map { toFun := fun x => g • x, inj' := ⋯ } (Set.Finite.toFinset hs)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | simp only [Set.coe_toFinset, Set.toFinite_toFinset, Finset.coe_map, Function.Embedding.coeFn_mk, Set.image_smul] | case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ ↑(Set.toFinset (g • s)) = ↑(Finset.map { toFun := fun x => g • x, inj' := ⋯ } (Set.Finite.toFinset hs)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝ : Fintype ↑s
this : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
⊢ ↑(Set.toFinset (g • s)) = ↑(Finset.map { toFun := fun x => g • x, inj' := ⋯ } (Set.Finite.toFinset hs))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_smul_le | [103, 1] | [117, 117] | rw [this, Finset.card_map, Set.toFinite_toFinset] | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝¹ : Fintype ↑s
this✝ : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
this : Set.toFinset (g • s) = Finset.map { toFun := fun x => g • x, inj' := ⋯ } (Set.Finite.toFinset hs)
⊢ (Set.toFinset s).card = (Set.toFinset (g • s)).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
this✝¹ : Fintype ↑s
this✝ : Fintype ↑(g • s)
h : Set.toFinset (g • s) ⊆ Set.toFinset s
this : Set.toFinset (g • s) = Finset.map { toFun := fun x => g • x, inj' := ⋯ } (Set.Finite.toFinset hs)
⊢ (Set.toFinset s).card = (Set.toFinset (g • s)).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_le_smul | [121, 1] | [124, 37] | rw [← @inv_mem_iff, mem_stabilizer_of_finite_iff_smul_le G s hs] | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
⊢ g ∈ stabilizer G s ↔ s ⊆ g • s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
⊢ g⁻¹ • s ⊆ s ↔ s ⊆ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
⊢ g ∈ stabilizer G s ↔ s ⊆ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.mem_stabilizer_of_finite_iff_le_smul | [121, 1] | [124, 37] | exact Set.subset_set_smul_iff.symm | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
⊢ g⁻¹ • s ⊆ s ↔ s ⊆ g • s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
hs : Set.Finite s
g : G
⊢ g⁻¹ • s ⊆ s ↔ s ⊆ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.fixingSubgroup_le_stabilizer | [127, 1] | [135, 11] | intro k hk | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
⊢ fixingSubgroup G s ≤ stabilizer G s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : k ∈ fixingSubgroup G s
⊢ k ∈ stabilizer G s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
⊢ fixingSubgroup G s ≤ stabilizer G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.fixingSubgroup_le_stabilizer | [127, 1] | [135, 11] | rw [mem_fixingSubgroup_iff] at hk | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : k ∈ fixingSubgroup G s
⊢ k ∈ stabilizer G s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ k ∈ stabilizer G s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : k ∈ fixingSubgroup G s
⊢ k ∈ stabilizer G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.fixingSubgroup_le_stabilizer | [127, 1] | [135, 11] | rw [mem_stabilizer_iff] | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ k ∈ stabilizer G s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ k • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ k ∈ stabilizer G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.fixingSubgroup_le_stabilizer | [127, 1] | [135, 11] | change (fun x => k • x) '' s = s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ k • s = s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ (fun x => k • x) '' s = s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ k • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.fixingSubgroup_le_stabilizer | [127, 1] | [135, 11] | conv_rhs => rw [← Set.image_id s] | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ (fun x => k • x) '' s = s | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ (fun x => k • x) '' s = id '' s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ (fun x => k • x) '' s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.fixingSubgroup_le_stabilizer | [127, 1] | [135, 11] | apply Set.image_congr | G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ (fun x => k • x) '' s = id '' s | case h
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ ∀ a ∈ s, k • a = id a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ (fun x => k • x) '' s = id '' s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.fixingSubgroup_le_stabilizer | [127, 1] | [135, 11] | simp only [id.def] | case h
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ ∀ a ∈ s, k • a = id a | case h
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ ∀ a ∈ s, k • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ ∀ a ∈ s, k • a = id a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Stabilizer.lean | MulAction.fixingSubgroup_le_stabilizer | [127, 1] | [135, 11] | exact hk | case h
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ ∀ a ∈ s, k • a = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
α : Type u_1
inst✝ : MulAction G α
s : Set α
k : G
hk : ∀ y ∈ s, k • y = y
⊢ ∀ a ∈ s, k • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | mem_commutatorSet_of_isConj_sq | [15, 1] | [21, 88] | obtain ⟨h, hg⟩ := hg | G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
hg : IsConj g (g ^ 2)
⊢ g ∈ commutatorSet G | case intro
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ g ∈ commutatorSet G | Please generate a tactic in lean4 to solve the state.
STATE:
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
hg : IsConj g (g ^ 2)
⊢ g ∈ commutatorSet G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | mem_commutatorSet_of_isConj_sq | [15, 1] | [21, 88] | use ↑h | case intro
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ g ∈ commutatorSet G | case h
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ ∃ g₂, ⁅↑h, g₂⁆ = g | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ g ∈ commutatorSet G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | mem_commutatorSet_of_isConj_sq | [15, 1] | [21, 88] | use g | case h
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ ∃ g₂, ⁅↑h, g₂⁆ = g | case h
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ ⁅↑h, g⁆ = g | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ ∃ g₂, ⁅↑h, g₂⁆ = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | mem_commutatorSet_of_isConj_sq | [15, 1] | [21, 88] | rw [commutatorElement_def, hg] | case h
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ ⁅↑h, g⁆ = g | case h
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ g ^ 2 * ↑h * (↑h)⁻¹ * g⁻¹ = g | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ ⁅↑h, g⁆ = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | mem_commutatorSet_of_isConj_sq | [15, 1] | [21, 88] | simp only [IsUnit.mul_inv_cancel_right, Units.isUnit, mul_inv_eq_iff_eq_mul, pow_two] | case h
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ g ^ 2 * ↑h * (↑h)⁻¹ * g⁻¹ = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G✝ : Type ?u.8
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
g : G
h : Gˣ
hg : SemiconjBy (↑h) g (g ^ 2)
⊢ g ^ 2 * ↑h * (↑h)⁻¹ * g⁻¹ = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | Subgroup.map_top_eq_range | [24, 1] | [26, 52] | simp only [map_eq_range_iff, codisjoint_top_left] | G✝ : Type ?u.1324
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Group H
inst✝ : Group G
f : H →* G
⊢ map f ⊤ = MonoidHom.range f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G✝ : Type ?u.1324
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Group H
inst✝ : Group G
f : H →* G
⊢ map f ⊤ = MonoidHom.range f
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | Subgroup.map_commutator_eq | [28, 1] | [30, 81] | rw [_root_.commutator_def, Subgroup.map_commutator, Subgroup.map_top_eq_range] | G✝ : Type ?u.2272
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Group H
inst✝ : Group G
f : H →* G
⊢ map f (_root_.commutator H) = ⁅MonoidHom.range f, MonoidHom.range f⁆ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G✝ : Type ?u.2272
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Group H
inst✝ : Group G
f : H →* G
⊢ map f (_root_.commutator H) = ⁅MonoidHom.range f, MonoidHom.range f⁆
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | Subgroup.commutator_eq' | [33, 1] | [35, 47] | simp only [map_commutator_eq, subtype_range] | G✝ : Type ?u.3265
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ map (Subgroup.subtype H) (_root_.commutator ↥H) = ⁅H, H⁆ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G✝ : Type ?u.3265
inst✝¹ : Group G✝
G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ map (Subgroup.subtype H) (_root_.commutator ↥H) = ⁅H, H⁆
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | surj_to_comm | [45, 1] | [52, 22] | obtain ⟨a', ha'⟩ := is_surj a | G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
⊢ a * b = b * a | case intro
G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
a' : G
ha' : φ a' = a
⊢ a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
⊢ a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | surj_to_comm | [45, 1] | [52, 22] | obtain ⟨b', hb'⟩ := is_surj b | case intro
G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
a' : G
ha' : φ a' = a
⊢ a * b = b * a | case intro.intro
G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
a' : G
ha' : φ a' = a
b' : G
hb' : φ b' = b
⊢ a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
a' : G
ha' : φ a' = a
⊢ a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | surj_to_comm | [45, 1] | [52, 22] | simp only [← ha', ← hb', ← map_mul] | case intro.intro
G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
a' : G
ha' : φ a' = a
b' : G
hb' : φ b' = b
⊢ a * b = b * a | case intro.intro
G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
a' : G
ha' : φ a' = a
b' : G
hb' : φ b' = b
⊢ φ (a' * b') = φ (b' * a') | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
a' : G
ha' : φ a' = a
b' : G
hb' : φ b' = b
⊢ a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | surj_to_comm | [45, 1] | [52, 22] | rw [is_comm.comm] | case intro.intro
G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
a' : G
ha' : φ a' = a
b' : G
hb' : φ b' = b
⊢ φ (a' * b') = φ (b' * a') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
G✝ : Type ?u.3685
inst✝² : Group G✝
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
φ : G →ₙ* H
is_surj : Function.Surjective ⇑φ
is_comm : Std.Commutative fun x x_1 => x * x_1
a b : H
a' : G
ha' : φ a' = a
b' : G
hb' : φ b' = b
⊢ φ (a' * b') = φ (b' * a')
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | constructor | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ _root_.commutator G ≤ N | case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
⊢ (Std.Commutative fun x x_1 => x * x_1) → _root_.commutator G ≤ N
case mpr
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
⊢ _root_.commutator G ≤ N → Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ _root_.commutator G ≤ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | intro hcomm | case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
⊢ (Std.Commutative fun x x_1 => x * x_1) → _root_.commutator G ≤ N | case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
⊢ _root_.commutator G ≤ N | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
⊢ (Std.Commutative fun x x_1 => x * x_1) → _root_.commutator G ≤ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | rw [commutator_eq_normalClosure] | case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
⊢ _root_.commutator G ≤ N | case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
⊢ normalClosure (commutatorSet G) ≤ N | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
⊢ _root_.commutator G ≤ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | rw [← Subgroup.normalClosure_subset_iff] | case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
⊢ normalClosure (commutatorSet G) ≤ N | case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
⊢ commutatorSet G ⊆ ↑N | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
⊢ normalClosure (commutatorSet G) ≤ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | intro x hx | case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
⊢ commutatorSet G ⊆ ↑N | case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
x : G
hx : x ∈ commutatorSet G
⊢ x ∈ ↑N | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
⊢ commutatorSet G ⊆ ↑N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | obtain ⟨p, q, rfl⟩ := hx | case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
x : G
hx : x ∈ commutatorSet G
⊢ x ∈ ↑N | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ⁅p, q⁆ ∈ ↑N | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
x : G
hx : x ∈ commutatorSet G
⊢ x ∈ ↑N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | simp only [SetLike.mem_coe] | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ⁅p, q⁆ ∈ ↑N | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ⁅p, q⁆ ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ⁅p, q⁆ ∈ ↑N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | rw [← QuotientGroup.eq_one_iff] | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ⁅p, q⁆ ∈ N | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑⁅p, q⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ⁅p, q⁆ ∈ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | rw [commutatorElement_def] | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑⁅p, q⁆ = 1 | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑(p * q * p⁻¹ * q⁻¹) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑⁅p, q⁆ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | simp only [QuotientGroup.mk_mul, QuotientGroup.mk_inv] | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑(p * q * p⁻¹ * q⁻¹) = 1 | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑p * ↑q * (↑p)⁻¹ * (↑q)⁻¹ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑(p * q * p⁻¹ * q⁻¹) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | rw [← commutatorElement_def] | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑p * ↑q * (↑p)⁻¹ * (↑q)⁻¹ = 1 | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ⁅↑p, ↑q⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑p * ↑q * (↑p)⁻¹ * (↑q)⁻¹ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | rw [commutatorElement_eq_one_iff_mul_comm] | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ⁅↑p, ↑q⁆ = 1 | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑p * ↑q = ↑q * ↑p | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ⁅↑p, ↑q⁆ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | apply hcomm.comm | case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑p * ↑q = ↑q * ↑p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hcomm : Std.Commutative fun x x_1 => x * x_1
p q : G
⊢ ↑p * ↑q = ↑q * ↑p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | intro hGN | case mpr
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
⊢ _root_.commutator G ≤ N → Std.Commutative fun x x_1 => x * x_1 | case mpr
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
⊢ Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
⊢ _root_.commutator G ≤ N → Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | apply Std.Commutative.mk | case mpr
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
⊢ Std.Commutative fun x x_1 => x * x_1 | case mpr.comm
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
⊢ ∀ (a b : G ⧸ N), a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | rintro x' | case mpr.comm
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
⊢ ∀ (a b : G ⧸ N), a * b = b * a | case mpr.comm
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x' : G ⧸ N
⊢ ∀ (b : G ⧸ N), x' * b = b * x' | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
⊢ ∀ (a b : G ⧸ N), a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | obtain ⟨x, rfl⟩ := QuotientGroup.mk'_surjective N x' | case mpr.comm
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x' : G ⧸ N
⊢ ∀ (b : G ⧸ N), x' * b = b * x' | case mpr.comm.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x : G
⊢ ∀ (b : G ⧸ N), (QuotientGroup.mk' N) x * b = b * (QuotientGroup.mk' N) x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x' : G ⧸ N
⊢ ∀ (b : G ⧸ N), x' * b = b * x'
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | intro y' | case mpr.comm.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x : G
⊢ ∀ (b : G ⧸ N), (QuotientGroup.mk' N) x * b = b * (QuotientGroup.mk' N) x | case mpr.comm.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x : G
y' : G ⧸ N
⊢ (QuotientGroup.mk' N) x * y' = y' * (QuotientGroup.mk' N) x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x : G
⊢ ∀ (b : G ⧸ N), (QuotientGroup.mk' N) x * b = b * (QuotientGroup.mk' N) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | obtain ⟨y, rfl⟩ := QuotientGroup.mk'_surjective N y' | case mpr.comm.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x : G
y' : G ⧸ N
⊢ (QuotientGroup.mk' N) x * y' = y' * (QuotientGroup.mk' N) x | case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ (QuotientGroup.mk' N) x * (QuotientGroup.mk' N) y = (QuotientGroup.mk' N) y * (QuotientGroup.mk' N) x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x : G
y' : G ⧸ N
⊢ (QuotientGroup.mk' N) x * y' = y' * (QuotientGroup.mk' N) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | rw [← commutatorElement_eq_one_iff_mul_comm, ← map_commutatorElement] | case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ (QuotientGroup.mk' N) x * (QuotientGroup.mk' N) y = (QuotientGroup.mk' N) y * (QuotientGroup.mk' N) x | case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ (QuotientGroup.mk' N) ⁅x, y⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ (QuotientGroup.mk' N) x * (QuotientGroup.mk' N) y = (QuotientGroup.mk' N) y * (QuotientGroup.mk' N) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | simp only [QuotientGroup.mk'_apply] | case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ (QuotientGroup.mk' N) ⁅x, y⁆ = 1 | case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ↑⁅x, y⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ (QuotientGroup.mk' N) ⁅x, y⁆ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | rw [QuotientGroup.eq_one_iff] | case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ↑⁅x, y⁆ = 1 | case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ↑⁅x, y⁆ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | apply hGN | case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ N | case mpr.comm.intro.intro.a
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ _root_.commutator G | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | rw [commutator_eq_closure] | case mpr.comm.intro.intro.a
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ _root_.commutator G | case mpr.comm.intro.intro.a
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ closure (commutatorSet G) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro.a
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ _root_.commutator G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | apply Subgroup.subset_closure | case mpr.comm.intro.intro.a
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ closure (commutatorSet G) | case mpr.comm.intro.intro.a.a
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ commutatorSet G | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro.a
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ closure (commutatorSet G)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | quotient_comm_contains_commutators_iff | [55, 1] | [81, 43] | exact commutator_mem_commutatorSet x y | case mpr.comm.intro.intro.a.a
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ commutatorSet G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro.a.a
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ commutatorSet G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | rw [← quotient_comm_contains_commutators_iff nN] | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
⊢ _root_.commutator G ≤ N | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
⊢ Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
⊢ _root_.commutator G ≤ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | let φ : H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H) | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
⊢ Std.Commutative fun x x_1 => x * x_1 | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | refine' surj_to_comm φ.toMulHom _ hH.is_comm | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ Std.Commutative fun x x_1 => x * x_1 | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ Function.Surjective ⇑↑φ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | simp only [MulHom.coe_mk, OneHom.toFun_eq_coe, MonoidHom.toOneHom_coe] | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ Function.Surjective ⇑↑φ | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ Function.Surjective ⇑φ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ Function.Surjective ⇑↑φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | rw [← MonoidHom.range_top_iff_surjective] | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ Function.Surjective ⇑φ | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ MonoidHom.range φ = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ Function.Surjective ⇑φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | suffices S_top : φ.range.comap (QuotientGroup.mk' N) = ⊤ by
rw [eq_top_iff]
intro x _
let y := Quotient.out' x
have hy : y ∈ φ.range.comap (QuotientGroup.mk' N) := by rw [S_top]; exact Subgroup.mem_top y
rw [← QuotientGroup.out_eq' x]
exact Subgroup.mem_comap.mp hy | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ MonoidHom.range φ = ⊤ | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ MonoidHom.range φ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | rw [eq_top_iff, ← hHN, sup_le_iff] | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤ | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ N ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ) ∧ H ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | constructor | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ N ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ) ∧ H ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ) | case left
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ N ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
case right
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ H ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ N ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ) ∧ H ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | rw [eq_top_iff] | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
⊢ MonoidHom.range φ = ⊤ | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
⊢ ⊤ ≤ MonoidHom.range φ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
⊢ MonoidHom.range φ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | intro x _ | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
⊢ ⊤ ≤ MonoidHom.range φ | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
⊢ x ∈ MonoidHom.range φ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
⊢ ⊤ ≤ MonoidHom.range φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | let y := Quotient.out' x | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
⊢ x ∈ MonoidHom.range φ | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
⊢ x ∈ MonoidHom.range φ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
⊢ x ∈ MonoidHom.range φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | have hy : y ∈ φ.range.comap (QuotientGroup.mk' N) := by rw [S_top]; exact Subgroup.mem_top y | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
⊢ x ∈ MonoidHom.range φ | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
hy : y ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
⊢ x ∈ MonoidHom.range φ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
⊢ x ∈ MonoidHom.range φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | rw [← QuotientGroup.out_eq' x] | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
hy : y ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
⊢ x ∈ MonoidHom.range φ | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
hy : y ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
⊢ ↑(Quotient.out' x) ∈ MonoidHom.range φ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
hy : y ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
⊢ x ∈ MonoidHom.range φ
TACTIC:
|
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