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.le_of_isPreprimitive | [290, 1] | [308, 59] | intro hG' | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ IsPreprimitive (↥G) α → alternatingGroup α ≤ G | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
⊢ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ IsPreprimitive (↥G) α → alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.le_of_isPreprimitive | [290, 1] | [308, 59] | obtain ⟨g, hg3, hg⟩ := has_three_cycle_of_stabilizer s hα | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
⊢ alternatingGroup α ≤ G | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
⊢ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.le_of_isPreprimitive | [290, 1] | [308, 59] | apply jordan_three_cycle hG' hg3 | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ alternatingGroup α ≤ G | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.le_of_isPreprimitive | [290, 1] | [308, 59] | apply hG | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ g ∈ G | case intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.le_of_isPreprimitive | [290, 1] | [308, 59] | simp only [Subgroup.mem_inf, hg, true_and_iff] | case intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α | case intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ g ∈ alternatingGroup α | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.le_of_isPreprimitive | [290, 1] | [308, 59] | exact Equiv.Perm.IsThreeCycle.mem_alternatingGroup hg3 | case intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ g ∈ alternatingGroup α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
hG' : IsPreprimitive (↥G) α
g : Equiv.Perm α
hg3 : Equiv.Perm.IsThreeCycle g
hg : g ∈ stabilizer (Equiv.Perm α) s
⊢ g ∈ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply IsPreprimitive.mk | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
⊢ IsPreprimitive (↥G) α | case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
⊢ ∀ {B : Set α}, IsBlock (↥G) B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
⊢ IsPreprimitive (↥G) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro B hB | case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
⊢ ∀ {B : Set α}, IsBlock (↥G) B → IsTrivialBlock B | case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
⊢ ∀ {B : Set α}, IsBlock (↥G) B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | unfold IsTrivialBlock | case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
⊢ IsTrivialBlock B | case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
⊢ Set.Subsingleton B ∨ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [or_iff_not_imp_left] | case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
⊢ Set.Subsingleton B ∨ B = ⊤ | case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
⊢ ¬Set.Subsingleton B → B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
⊢ Set.Subsingleton B ∨ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro hB' | case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
⊢ ¬Set.Subsingleton B → B = ⊤ | case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
⊢ ¬Set.Subsingleton B → B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | obtain ⟨a, ha, ha'⟩ := Set.not_subset_iff_exists_mem_not_mem.mp
fun h => hB' ((hB_not_le_sc B hB) h) | case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
⊢ B = ⊤ | case has_trivial_blocks'.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∉ sᶜ
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [Set.not_mem_compl_iff] at ha' | case has_trivial_blocks'.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∉ sᶜ
⊢ B = ⊤ | case has_trivial_blocks'.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∉ sᶜ
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | obtain ⟨b, hb, hb'⟩ := Set.not_subset_iff_exists_mem_not_mem.mp
fun h => hB' ((hB_not_le_s B hB) h) | case has_trivial_blocks'.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
⊢ B = ⊤ | case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∉ s
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [← Set.mem_compl_iff] at hb' | case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∉ s
⊢ B = ⊤ | case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∉ s
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [eq_top_iff] | case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
⊢ B = ⊤ | case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
⊢ ⊤ ≤ B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro x _ | case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
⊢ ⊤ ≤ B | case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
⊢ ⊤ ≤ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | obtain ⟨b, hb⟩ := h1.nonempty | case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
⊢ x ∈ B | case has_trivial_blocks'.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | obtain ⟨⟨g, hg⟩, hgbx : g • b = x⟩ := exists_smul_eq G b x | case has_trivial_blocks'.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
⊢ x ∈ B | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | suffices g • B = B by
rw [← hgbx, ← this, Set.smul_mem_smul_set_iff]
exact hsc_le_B hb | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ x ∈ B | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ g • B = B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply or_iff_not_imp_right.mp (IsBlock.def_one.mp hB ⟨g, hg⟩) | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ g • B = B | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ ¬Disjoint ({ val := g, property := hg } • B) B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [Set.not_disjoint_iff_nonempty_inter] | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ ¬Disjoint ({ val := g, property := hg } • B) B | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.Nonempty ({ val := g, property := hg } • B ∩ B) | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ ¬Disjoint ({ val := g, property := hg } • B) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | change (g • B ∩ B).Nonempty | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.Nonempty ({ val := g, property := hg } • B ∩ B) | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.Nonempty (g • B ∩ B) | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.Nonempty ({ val := g, property := hg } • B ∩ B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply Set.ncard_pigeonhole | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.Nonempty (g • B ∩ B) | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Fintype.card α < Set.ncard (g • B) + Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.Nonempty (g • B ∩ B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [← Nat.card_eq_fintype_card, ← Set.ncard_add_ncard_compl s] | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Fintype.card α < Set.ncard (g • B) + Set.ncard B | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.ncard s + Set.ncard sᶜ < Set.ncard (g • B) + Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Fintype.card α < Set.ncard (g • B) + Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply Nat.lt_of_lt_of_le | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.ncard s + Set.ncard sᶜ < Set.ncard (g • B) + Set.ncard B | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.ncard s + Set.ncard sᶜ < ?has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.m
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ ?has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.m ≤ Set.ncard (g • B) + Set.ncard B
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.m
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ ℕ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.ncard s + Set.ncard sᶜ < Set.ncard (g • B) + Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply Nat.add_le_add | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.ncard sᶜ + Set.ncard sᶜ ≤ Set.ncard (g • B) + Set.ncard B | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.a.h₁
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.ncard sᶜ ≤ Set.ncard (g • B)
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.a.h₂
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.ncard sᶜ ≤ Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.ncard sᶜ + Set.ncard sᶜ ≤ Set.ncard (g • B) + Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact Set.ncard_le_ncard hsc_le_B | case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.a.h₂
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.ncard sᶜ ≤ Set.ncard B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.intro.intro.intro.intro.intro.intro.mk.h.a.h₂
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
hB_not_le_s : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hB' : ¬Set.Subsingleton B
a : α
ha : a ∈ B
ha' : a ∈ s
b✝ : α
hb✝ : b✝ ∈ B
hb' : b✝ ∈ sᶜ
hsc_le_B : sᶜ ⊆ B
x : α
a✝ : x ∈ ⊤
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ G
hgbx : g • b = x
⊢ Set.ncard sᶜ ≤ Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | have hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G :=
le_trans (le_of_lt hG) inf_le_left | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
⊢ IsPretransitive (↥G) α | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ IsPretransitive (↥G) α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
⊢ IsPretransitive (↥G) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply Equiv.Perm.IsPretransitive.of_partition G s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ IsPretransitive (↥G) α | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ ∀ a ∈ s, ∀ b ∈ s, ∃ g, g • a = b
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ ∀ a ∈ sᶜ, ∀ b ∈ sᶜ, ∃ g, g • a = b
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ stabilizer (↥G) s ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ IsPretransitive (↥G) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro a ha b hb | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ ∀ a ∈ s, ∀ b ∈ s, ∃ g, g • a = b | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
⊢ ∃ g, g • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ ∀ a ∈ s, ∀ b ∈ s, ∃ g, g • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | obtain ⟨g, hg, H⟩ := moves_in h4 s a ha b hb | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
⊢ ∃ g, g • a = b | case a.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ ∃ g, g • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
⊢ ∃ g, g • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | use! g | case a.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ ∃ g, g • a = b | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ G
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ?property } • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ ∃ g, g • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact hG' hg | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ G
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ?property } • a = b | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ G
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ?property } • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact H | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ s
b : α
hb : b ∈ s
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro a ha b hb | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ ∀ a ∈ sᶜ, ∀ b ∈ sᶜ, ∃ g, g • a = b | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
⊢ ∃ g, g • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ ∀ a ∈ sᶜ, ∀ b ∈ sᶜ, ∃ g, g • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | obtain ⟨g, hg, H⟩ := moves_in h4 (sᶜ) a ha b hb | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
⊢ ∃ g, g • a = b | case a.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ ∃ g, g • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
⊢ ∃ g, g • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | use! g | case a.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ ∃ g, g • a = b | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ G
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ?property } • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ ∃ g, g • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply hG' | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ G
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ?property } • a = b | case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ G
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ?property } • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [stabilizer_compl] at hg | case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b | case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact hg | case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
H : g • a = b
⊢ g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact H | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
g : Equiv.Perm α
hg : g ∈ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α
H : g • a = b
⊢ { val := g, property := ⋯ } • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro h | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ stabilizer (↥G) s ≠ ⊤ | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
⊢ stabilizer (↥G) s ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply (lt_iff_le_not_le.mp hG).right | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ False | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [le_inf_iff] | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ stabilizer (Equiv.Perm α) s ∧ G ⊓ alternatingGroup α ≤ alternatingGroup α | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | constructor | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ stabilizer (Equiv.Perm α) s ∧ G ⊓ alternatingGroup α ≤ alternatingGroup α | case a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ stabilizer (Equiv.Perm α) s
case a.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ alternatingGroup α | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ stabilizer (Equiv.Perm α) s ∧ G ⊓ alternatingGroup α ≤ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro g | case a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ stabilizer (Equiv.Perm α) s | case a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
⊢ g ∈ G ⊓ alternatingGroup α → g ∈ stabilizer (Equiv.Perm α) s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ stabilizer (Equiv.Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [Subgroup.mem_inf, mem_stabilizer_iff] | case a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
⊢ g ∈ G ⊓ alternatingGroup α → g ∈ stabilizer (Equiv.Perm α) s | case a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
⊢ g ∈ G ∧ g ∈ alternatingGroup α → g • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
⊢ g ∈ G ⊓ alternatingGroup α → g ∈ stabilizer (Equiv.Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rintro ⟨hg, _⟩ | case a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
⊢ g ∈ G ∧ g ∈ alternatingGroup α → g • s = s | case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ g • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
⊢ g ∈ G ∧ g ∈ alternatingGroup α → g • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [← Subgroup.coe_mk G g hg] | case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ g • s = s | case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ ↑{ val := g, property := hg } • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ g • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | change (⟨g, hg⟩ : ↥G) • s = s | case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ ↑{ val := g, property := hg } • s = s | case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ { val := g, property := hg } • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ ↑{ val := g, property := hg } • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [← mem_stabilizer_iff, h] | case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ { val := g, property := hg } • s = s | case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ { val := g, property := hg } ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ { val := g, property := hg } • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact Subgroup.mem_top _ | case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ { val := g, property := hg } ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
g : Equiv.Perm α
hg : g ∈ G
right✝ : g ∈ alternatingGroup α
⊢ { val := g, property := hg } ∈ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact inf_le_right | case a.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ alternatingGroup α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
hG' : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
h : stabilizer (↥G) s = ⊤
⊢ G ⊓ alternatingGroup α ≤ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro B hB hBsc | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
⊢ ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
⊢ ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | obtain ⟨b, hb⟩ := h1.nonempty | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
⊢ False | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ sᶜ
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [← hBsc] at hb | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ sᶜ
⊢ False | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ sᶜ
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | obtain ⟨a, ha⟩ := h0.nonempty | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
⊢ False | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | obtain ⟨k, hk⟩ := exists_smul_eq G b a | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
⊢ False | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | suffices Set.ncard (B : Set α) ≤ Set.ncard s by
apply Nat.lt_irrefl (Set.ncard B)
apply lt_of_le_of_lt this
simp_rw [hBsc]; exact hα | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ False | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ Set.ncard B ≤ Set.ncard s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [← smul_set_ncard_eq k B] | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ Set.ncard B ≤ Set.ncard s | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ Set.ncard (k • B) ≤ Set.ncard s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ Set.ncard B ≤ Set.ncard s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply Set.ncard_le_ncard (ht := Set.toFinite s) | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ Set.ncard (k • B) ≤ Set.ncard s | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ k • B ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ Set.ncard (k • B) ≤ Set.ncard s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [← Set.disjoint_compl_right_iff_subset, ← hBsc] | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ k • B ⊆ s | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ Disjoint (k • B) B | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ k • B ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply or_iff_not_imp_left.mp (IsBlock.def_one.mp hB k) | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ Disjoint (k • B) B | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ ¬k • B = B | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ Disjoint (k • B) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro h | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ ¬k • B = B | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
⊢ ¬k • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply Set.not_mem_empty a | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ False | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [← Set.inter_compl_self s] | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ ∅ | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ s ∩ sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | constructor | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ s ∩ sᶜ | case intro.intro.intro.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ s
case intro.intro.intro.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ s ∩ sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply Nat.lt_irrefl (Set.ncard B) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ False | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ Set.ncard B < Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply lt_of_le_of_lt this | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ Set.ncard B < Set.ncard B | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ Set.ncard s < Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ Set.ncard B < Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | simp_rw [hBsc] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ Set.ncard s < Set.ncard B | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ Set.ncard s < Set.ncard sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ Set.ncard s < Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact hα | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ Set.ncard s < Set.ncard sᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
this : Set.ncard B ≤ Set.ncard s
⊢ Set.ncard s < Set.ncard sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact ha | case intro.intro.intro.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [← hk, ← hBsc, ← h, Set.smul_mem_smul_set_iff] | case intro.intro.intro.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ sᶜ | case intro.intro.intro.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ b ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ a ∈ sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact hb | case intro.intro.intro.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ b ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
B : Set α
hB : IsBlock (↥G) B
hBsc : B = sᶜ
b : α
hb : b ∈ B
a : α
ha : a ∈ s
k : ↥G
hk : k • b = a
h : k • B = B
⊢ b ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro B hB hBsc | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
⊢ ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Subsingleton B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
⊢ ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [← Equiv.Perm.Subtype.image_preimage_of_val hBsc] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Subsingleton B | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Subsingleton (Subtype.val '' (Subtype.val ⁻¹' B)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Subsingleton B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply Set.Subsingleton.image | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Subsingleton (Subtype.val '' (Subtype.val ⁻¹' B)) | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Subsingleton (Subtype.val ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Subsingleton (Subtype.val '' (Subtype.val ⁻¹' B))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | suffices IsTrivialBlock (Subtype.val ⁻¹' B : Set (sᶜ : Set α)) by
apply Or.resolve_right this
intro hB'
apply hB_ne_sc B hB
simp only [Set.top_eq_univ, Set.preimage_eq_univ_iff, Subtype.range_coe_subtype] at hB'
apply Set.Subset.antisymm hBsc hB' | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Subsingleton (Subtype.val ⁻¹' B) | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ IsTrivialBlock (Subtype.val ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Subsingleton (Subtype.val ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | suffices IsPreprimitive (stabilizer G (sᶜ : Set α)) (sᶜ : Set α) by
refine' IsPreprimitive.has_trivial_blocks this _
let φ' : stabilizer G (sᶜ : Set α) → G := Subtype.val
let f' : (sᶜ : Set α) →ₑ[φ'] α := {
toFun := Subtype.val
map_smul' := fun m x => by simp only [SMul.smul_stabilizer_def] }
apply MulAction.IsBlock_preimage f' hB | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ IsTrivialBlock (Subtype.val ⁻¹' B) | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ IsTrivialBlock (Subtype.val ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply stabilizer.isPreprimitive' | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ | case hs.hsc
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Nontrivial sᶜᶜ
case hs.hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply Or.resolve_right this | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
⊢ Set.Subsingleton (Subtype.val ⁻¹' B) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
⊢ ¬Subtype.val ⁻¹' B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
⊢ Set.Subsingleton (Subtype.val ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro hB' | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
⊢ ¬Subtype.val ⁻¹' B = ⊤ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
hB' : Subtype.val ⁻¹' B = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
⊢ ¬Subtype.val ⁻¹' B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply hB_ne_sc B hB | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
hB' : Subtype.val ⁻¹' B = ⊤
⊢ False | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
hB' : Subtype.val ⁻¹' B = ⊤
⊢ B = sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
hB' : Subtype.val ⁻¹' B = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | simp only [Set.top_eq_univ, Set.preimage_eq_univ_iff, Subtype.range_coe_subtype] at hB' | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
hB' : Subtype.val ⁻¹' B = ⊤
⊢ B = sᶜ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
hB' : {x | x ∈ sᶜ} ⊆ B
⊢ B = sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
hB' : Subtype.val ⁻¹' B = ⊤
⊢ B = sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply Set.Subset.antisymm hBsc hB' | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
hB' : {x | x ∈ sᶜ} ⊆ B
⊢ B = sᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsTrivialBlock (Subtype.val ⁻¹' B)
hB' : {x | x ∈ sᶜ} ⊆ B
⊢ B = sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | refine' IsPreprimitive.has_trivial_blocks this _ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
⊢ IsTrivialBlock (Subtype.val ⁻¹' B) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
⊢ IsBlock (↥(stabilizer (↥G) sᶜ)) (Subtype.val ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
⊢ IsTrivialBlock (Subtype.val ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | let φ' : stabilizer G (sᶜ : Set α) → G := Subtype.val | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
⊢ IsBlock (↥(stabilizer (↥G) sᶜ)) (Subtype.val ⁻¹' B) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
φ' : ↥(stabilizer (↥G) sᶜ) → ↥G := Subtype.val
⊢ IsBlock (↥(stabilizer (↥G) sᶜ)) (Subtype.val ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
⊢ IsBlock (↥(stabilizer (↥G) sᶜ)) (Subtype.val ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | let f' : (sᶜ : Set α) →ₑ[φ'] α := {
toFun := Subtype.val
map_smul' := fun m x => by simp only [SMul.smul_stabilizer_def] } | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
φ' : ↥(stabilizer (↥G) sᶜ) → ↥G := Subtype.val
⊢ IsBlock (↥(stabilizer (↥G) sᶜ)) (Subtype.val ⁻¹' B) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
φ' : ↥(stabilizer (↥G) sᶜ) → ↥G := Subtype.val
f' : ↑sᶜ →ₑ[φ'] α := { toFun := Subtype.val, map_smul' := ⋯ }
⊢ IsBlock (↥(stabilizer (↥G) sᶜ)) (Subtype.val ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
φ' : ↥(stabilizer (↥G) sᶜ) → ↥G := Subtype.val
⊢ IsBlock (↥(stabilizer (↥G) sᶜ)) (Subtype.val ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply MulAction.IsBlock_preimage f' hB | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
φ' : ↥(stabilizer (↥G) sᶜ) → ↥G := Subtype.val
f' : ↑sᶜ →ₑ[φ'] α := { toFun := Subtype.val, map_smul' := ⋯ }
⊢ IsBlock (↥(stabilizer (↥G) sᶜ)) (Subtype.val ⁻¹' B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
φ' : ↥(stabilizer (↥G) sᶜ) → ↥G := Subtype.val
f' : ↑sᶜ →ₑ[φ'] α := { toFun := Subtype.val, map_smul' := ⋯ }
⊢ IsBlock (↥(stabilizer (↥G) sᶜ)) (Subtype.val ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | simp only [SMul.smul_stabilizer_def] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
φ' : ↥(stabilizer (↥G) sᶜ) → ↥G := Subtype.val
m : ↥(stabilizer (↥G) sᶜ)
x : ↑sᶜ
⊢ ↑(m • x) = φ' m • ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
this : IsPreprimitive ↥(stabilizer (↥G) sᶜ) ↑sᶜ
φ' : ↥(stabilizer (↥G) sᶜ) → ↥G := Subtype.val
m : ↥(stabilizer (↥G) sᶜ)
x : ↑sᶜ
⊢ ↑(m • x) = φ' m • ↑x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [compl_compl] | case hs.hsc
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Nontrivial sᶜᶜ | case hs.hsc
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Nontrivial s | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.hsc
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Nontrivial sᶜᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact h0 | case hs.hsc
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Nontrivial s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.hsc
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ Set.Nontrivial s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | rw [stabilizer_compl] | case hs.hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α ≤ G | case hs.hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ stabilizer (Equiv.Perm α) sᶜ ⊓ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | exact le_trans (le_of_lt hG) inf_le_left | case hs.hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
B : Set α
hB : IsBlock (↥G) B
hBsc : B ⊆ sᶜ
⊢ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | intro B hB hBs | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
⊢ ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
⊢ Set.Subsingleton B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
⊢ ∀ (B : Set α), IsBlock (↥G) B → B ⊆ s → Set.Subsingleton B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | suffices IsPreprimitive (stabilizer G s) (s : Set α) by
refine' IsPreprimitive.has_trivial_blocks this _
let φ' : stabilizer G s → G := Subtype.val
let f' : s →ₑ[φ'] α := {
toFun := Subtype.val
map_smul' := fun ⟨m, _⟩ x => by simp }
apply MulAction.IsBlock_preimage f' hB | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
⊢ IsTrivialBlock (Subtype.val ⁻¹' B) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
⊢ IsTrivialBlock (Subtype.val ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply stabilizer.isPreprimitive' s h1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | case hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
⊢ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | apply le_trans (le_of_lt hG) inf_le_left | case hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
⊢ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
⊢ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/AlternatingMaximal.lean | alternatingGroup.isPreprimitive_of_stabilizer_lt | [413, 2] | [618, 36] | cases' this with hB' hB' | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
this : IsTrivialBlock (Subtype.val ⁻¹' B)
⊢ Set.Subsingleton B | case inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
hB' : Set.Subsingleton (Subtype.val ⁻¹' B)
⊢ Set.Subsingleton B
case inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
hB' : Subtype.val ⁻¹' B = ⊤
⊢ Set.Subsingleton B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
h0 : Set.Nontrivial s
h1 : Set.Nontrivial sᶜ
hα : Set.ncard s < Set.ncard sᶜ
h4 : 4 ≤ Fintype.card α
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α < G ⊓ alternatingGroup α
this✝ : IsPretransitive (↥G) α
hB_ne_sc : ∀ (B : Set α), IsBlock (↥G) B → ¬B = sᶜ
hB_not_le_sc : ∀ (B : Set α), IsBlock (↥G) B → B ⊆ sᶜ → Set.Subsingleton B
B : Set α
hB : IsBlock (↥G) B
hBs : B ⊆ s
this : IsTrivialBlock (Subtype.val ⁻¹' B)
⊢ Set.Subsingleton B
TACTIC:
|
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