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/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | simp only [Subgroup.coeSubtype, Subgroup.mem_map, Subgroup.mem_top, true_and] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
this : MonoidHom.ker (φ g) = ⊤
⊢ ConjAct.toConjAct x ∈ Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) ⊤ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
this : MonoidHom.ker (φ g) = ⊤
⊢ ∃ x_1, ↑x_1 = ConjAct.toConjAct x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
this : MonoidHom.ker (φ g) = ⊤
⊢ ConjAct.toConjAct x ∈ Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | exact ⟨⟨x, hx⟩, rfl⟩ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
this : MonoidHom.ker (φ g) = ⊤
⊢ ∃ x_1, ↑x_1 = ConjAct.toConjAct x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
this : MonoidHom.ker (φ g) = ⊤
⊢ ∃ x_1, ↑x_1 = ConjAct.toConjAct x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [MonoidHom.mem_ker, ← Subgroup.mem_bot, ← this, MonoidHom.mem_range] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
this : MonoidHom.range (φ g) = ⊥
⊢ y ∈ MonoidHom.ker (φ g) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
this : MonoidHom.range (φ g) = ⊥
⊢ ∃ x, (φ g) x = (φ g) y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
this : MonoidHom.range (φ g) = ⊥
⊢ y ∈ MonoidHom.ker (φ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | exact ⟨y, rfl⟩ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
this : MonoidHom.range (φ g) = ⊥
⊢ ∃ x, (φ g) x = (φ g) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
this : MonoidHom.range (φ g) = ⊥
⊢ ∃ x, (φ g) x = (φ g) y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.has_trivial_blocks | [121, 1] | [122, 92] | apply h.has_trivial_blocks' | G : Type u_1
X : Type u_2
inst✝ : SMul G X
h : IsPreprimitive G X
B : Set X
hB : IsBlock G B
⊢ Set.Subsingleton B ∨ B = ⊤ | G : Type u_1
X : Type u_2
inst✝ : SMul G X
h : IsPreprimitive G X
B : Set X
hB : IsBlock G B
⊢ IsBlock G B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝ : SMul G X
h : IsPreprimitive G X
B : Set X
hB : IsBlock G B
⊢ Set.Subsingleton B ∨ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.has_trivial_blocks | [121, 1] | [122, 92] | exact hB | G : Type u_1
X : Type u_2
inst✝ : SMul G X
h : IsPreprimitive G X
B : Set X
hB : IsBlock G B
⊢ IsBlock G B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝ : SMul G X
h : IsPreprimitive G X
B : Set X
hB : IsBlock G B
⊢ IsBlock G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.on_subsingleton | [125, 1] | [137, 41] | have : IsPretransitive G X := by
apply IsPretransitive.mk
intro x y
use Classical.arbitrary G
rw [eq_iff_true_of_subsingleton]
trivial | G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
⊢ IsPreprimitive G X | G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
⊢ IsPreprimitive G X | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
⊢ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.on_subsingleton | [125, 1] | [137, 41] | apply IsPreprimitive.mk | G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
⊢ IsPreprimitive G X | case has_trivial_blocks'
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
⊢ ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
⊢ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.on_subsingleton | [125, 1] | [137, 41] | intro B _ | case has_trivial_blocks'
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
⊢ ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B | case has_trivial_blocks'
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
B : Set X
a✝ : IsBlock G B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
⊢ ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.on_subsingleton | [125, 1] | [137, 41] | left | case has_trivial_blocks'
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
B : Set X
a✝ : IsBlock G B
⊢ IsTrivialBlock B | case has_trivial_blocks'.h
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
B : Set X
a✝ : IsBlock G B
⊢ Set.Subsingleton B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
B : Set X
a✝ : IsBlock G B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.on_subsingleton | [125, 1] | [137, 41] | exact Set.subsingleton_of_subsingleton | case has_trivial_blocks'.h
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
B : Set X
a✝ : IsBlock G B
⊢ Set.Subsingleton B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.h
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
this : IsPretransitive G X
B : Set X
a✝ : IsBlock G B
⊢ Set.Subsingleton B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.on_subsingleton | [125, 1] | [137, 41] | apply IsPretransitive.mk | G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
⊢ IsPretransitive G X | case exists_smul_eq
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
⊢ ∀ (x y : X), ∃ g, g • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
⊢ IsPretransitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.on_subsingleton | [125, 1] | [137, 41] | intro x y | case exists_smul_eq
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
⊢ ∀ (x y : X), ∃ g, g • x = y | case exists_smul_eq
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
x y : X
⊢ ∃ g, g • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
⊢ ∀ (x y : X), ∃ g, g • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.on_subsingleton | [125, 1] | [137, 41] | use Classical.arbitrary G | case exists_smul_eq
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
x y : X
⊢ ∃ g, g • x = y | case h
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
x y : X
⊢ Classical.arbitrary G • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
x y : X
⊢ ∃ g, g • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.on_subsingleton | [125, 1] | [137, 41] | rw [eq_iff_true_of_subsingleton] | case h
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
x y : X
⊢ Classical.arbitrary G • x = y | case h
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
x y : X
⊢ True | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
x y : X
⊢ Classical.arbitrary G • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.on_subsingleton | [125, 1] | [137, 41] | trivial | case h
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
x y : X
⊢ True | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
X : Type u_2
inst✝² : SMul G X
inst✝¹ : Nonempty G
inst✝ : Subsingleton X
x y : X
⊢ True
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsTrivialBlock.of_card_le_2 | [140, 1] | [149, 68] | cases' le_or_lt (Fintype.card B) 1 with h1 h1 | G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
⊢ IsTrivialBlock B | case inl
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : Fintype.card ↑B ≤ 1
⊢ IsTrivialBlock B
case inr
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : 1 < Fintype.card ↑B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsTrivialBlock.of_card_le_2 | [140, 1] | [149, 68] | apply Or.intro_left | case inl
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : Fintype.card ↑B ≤ 1
⊢ IsTrivialBlock B | case inl.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : Fintype.card ↑B ≤ 1
⊢ Set.Subsingleton B | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : Fintype.card ↑B ≤ 1
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsTrivialBlock.of_card_le_2 | [140, 1] | [149, 68] | rw [← Set.subsingleton_coe, ← Fintype.card_le_one_iff_subsingleton] | case inl.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : Fintype.card ↑B ≤ 1
⊢ Set.Subsingleton B | case inl.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : Fintype.card ↑B ≤ 1
⊢ Fintype.card ↑B ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : Fintype.card ↑B ≤ 1
⊢ Set.Subsingleton B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsTrivialBlock.of_card_le_2 | [140, 1] | [149, 68] | exact h1 | case inl.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : Fintype.card ↑B ≤ 1
⊢ Fintype.card ↑B ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : Fintype.card ↑B ≤ 1
⊢ Fintype.card ↑B ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsTrivialBlock.of_card_le_2 | [140, 1] | [149, 68] | apply Or.intro_right | case inr
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : 1 < Fintype.card ↑B
⊢ IsTrivialBlock B | case inr.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : 1 < Fintype.card ↑B
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : 1 < Fintype.card ↑B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsTrivialBlock.of_card_le_2 | [140, 1] | [149, 68] | rw [Set.top_eq_univ, ← set_fintype_card_eq_univ_iff] | case inr.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : 1 < Fintype.card ↑B
⊢ B = ⊤ | case inr.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : 1 < Fintype.card ↑B
⊢ Fintype.card ↑B = Fintype.card X | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : 1 < Fintype.card ↑B
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsTrivialBlock.of_card_le_2 | [140, 1] | [149, 68] | exact le_antisymm (set_fintype_card_le_univ B) (le_trans hX h1) | case inr.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : 1 < Fintype.card ↑B
⊢ Fintype.card ↑B = Fintype.card X | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
G : Type ?u.3301
X : Type u_1
inst✝ : Fintype X
hX : Fintype.card X ≤ 2
B : Set X
h1 : 1 < Fintype.card ↑B
⊢ Fintype.card ↑B = Fintype.card X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block | [156, 1] | [167, 25] | cases hB with
| inl hB =>
left
apply Set.Subsingleton.image hB
| inr hB =>
apply Or.intro_right
rw [hB, eq_top_iff]
intro x _
rw [Set.mem_smul_set_iff_inv_smul_mem]
exact Set.mem_univ _ | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : IsTrivialBlock B
⊢ IsTrivialBlock (g • B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : IsTrivialBlock B
⊢ IsTrivialBlock (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block | [156, 1] | [167, 25] | left | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : Set.Subsingleton B
⊢ IsTrivialBlock (g • B) | case inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : Set.Subsingleton B
⊢ Set.Subsingleton (g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : Set.Subsingleton B
⊢ IsTrivialBlock (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block | [156, 1] | [167, 25] | apply Set.Subsingleton.image hB | case inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : Set.Subsingleton B
⊢ Set.Subsingleton (g • B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : Set.Subsingleton B
⊢ Set.Subsingleton (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block | [156, 1] | [167, 25] | apply Or.intro_right | case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
⊢ IsTrivialBlock (g • B) | case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
⊢ g • B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
⊢ IsTrivialBlock (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block | [156, 1] | [167, 25] | rw [hB, eq_top_iff] | case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
⊢ g • B = ⊤ | case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
⊢ ⊤ ≤ g • ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
⊢ g • B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block | [156, 1] | [167, 25] | intro x _ | case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
⊢ ⊤ ≤ g • ⊤ | case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
x : X
a✝ : x ∈ ⊤
⊢ x ∈ g • ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
⊢ ⊤ ≤ g • ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block | [156, 1] | [167, 25] | rw [Set.mem_smul_set_iff_inv_smul_mem] | case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
x : X
a✝ : x ∈ ⊤
⊢ x ∈ g • ⊤ | case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
x : X
a✝ : x ∈ ⊤
⊢ g⁻¹ • x ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
x : X
a✝ : x ∈ ⊤
⊢ x ∈ g • ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block | [156, 1] | [167, 25] | exact Set.mem_univ _ | case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
x : X
a✝ : x ∈ ⊤
⊢ g⁻¹ • x ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hB : B = ⊤
x : X
a✝ : x ∈ ⊤
⊢ g⁻¹ • x ∈ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block_iff | [170, 1] | [178, 14] | constructor | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock B ↔ IsTrivialBlock (g • B) | case mp
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock B → IsTrivialBlock (g • B)
case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock (g • B) → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock B ↔ IsTrivialBlock (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block_iff | [170, 1] | [178, 14] | exact isTrivialBlock_of_block g | case mp
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock B → IsTrivialBlock (g • B)
case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock (g • B) → IsTrivialBlock B | case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock (g • B) → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock B → IsTrivialBlock (g • B)
case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock (g • B) → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block_iff | [170, 1] | [178, 14] | intro hgB | case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock (g • B) → IsTrivialBlock B | case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hgB : IsTrivialBlock (g • B)
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
⊢ IsTrivialBlock (g • B) → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block_iff | [170, 1] | [178, 14] | rw [← inv_smul_smul g B] | case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hgB : IsTrivialBlock (g • B)
⊢ IsTrivialBlock B | case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hgB : IsTrivialBlock (g • B)
⊢ IsTrivialBlock (g⁻¹ • g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hgB : IsTrivialBlock (g • B)
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block_iff | [170, 1] | [178, 14] | apply isTrivialBlock_of_block | case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hgB : IsTrivialBlock (g • B)
⊢ IsTrivialBlock (g⁻¹ • g • B) | case mpr.hB
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hgB : IsTrivialBlock (g • B)
⊢ IsTrivialBlock (g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hgB : IsTrivialBlock (g • B)
⊢ IsTrivialBlock (g⁻¹ • g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_block_iff | [170, 1] | [178, 14] | exact hgB | case mpr.hB
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hgB : IsTrivialBlock (g • B)
⊢ IsTrivialBlock (g • B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.hB
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
B : Set X
g : G
hgB : IsTrivialBlock (g • B)
⊢ IsTrivialBlock (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | apply IsPreprimitive.mk | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
⊢ IsPreprimitive G X | case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
⊢ ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
⊢ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | intro B hB | case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
⊢ ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B | case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
⊢ ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | cases Set.eq_empty_or_nonempty B with
| inl h => apply Or.intro_left; rw [h]; exact Set.subsingleton_empty
| inr h =>
obtain ⟨b, hb⟩ := h
obtain ⟨g, hg⟩ := exists_smul_eq G b a
rw [isTrivialBlock_of_block_iff g]
refine' H (g • B) _ (IsBlock_of_block g hB)
use b | case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
⊢ IsTrivialBlock B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | apply Or.intro_left | case has_trivial_blocks'.inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ IsTrivialBlock B | case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | rw [h] | case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton B | case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | exact Set.subsingleton_empty | case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | obtain ⟨b, hb⟩ := h | case has_trivial_blocks'.inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
h : Set.Nonempty B
⊢ IsTrivialBlock B | case has_trivial_blocks'.inr.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
h : Set.Nonempty B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | obtain ⟨g, hg⟩ := exists_smul_eq G b a | case has_trivial_blocks'.inr.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
⊢ IsTrivialBlock B | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | rw [isTrivialBlock_of_block_iff g] | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock B | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock (g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | refine' H (g • B) _ (IsBlock_of_block g hB) | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock (g • B) | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ a ∈ g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem | [181, 1] | [193, 10] | use b | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ a ∈ g • B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
htGX : IsPretransitive G X
a : X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ a ∈ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | apply IsPreprimitive.mk | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
⊢ IsPreprimitive G X | case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
⊢ ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
⊢ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | intro B hB | case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
⊢ ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B | case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
⊢ ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | cases Set.eq_empty_or_nonempty B with
| inl h => left; rw [h]; exact Set.subsingleton_empty
| inr h =>
obtain ⟨b, hb⟩ := h
obtain ⟨g, hg⟩ := exists_smul_eq G b a
rw [isTrivialBlock_of_block_iff g]
refine' H (g • B) _ (IsBlock_of_block g hB)
use b | case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
⊢ IsTrivialBlock B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | apply IsPretransitive.mk_base a | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
⊢ IsPretransitive G X | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
⊢ ∀ (x : X), ∃ g, g • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
⊢ IsPretransitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | cases' H (orbit G a) (mem_orbit_self a) (IsBlock_of_orbit a) with H H | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
⊢ ∀ (x : X), ∃ g, g • a = x | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : Set.Subsingleton (orbit G a)
⊢ ∀ (x : X), ∃ g, g • a = x
case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = ⊤
⊢ ∀ (x : X), ∃ g, g • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
⊢ ∀ (x : X), ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | exfalso | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : Set.Subsingleton (orbit G a)
⊢ ∀ (x : X), ∃ g, g • a = x | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : Set.Subsingleton (orbit G a)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : Set.Subsingleton (orbit G a)
⊢ ∀ (x : X), ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | apply ha | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : Set.Subsingleton (orbit G a)
⊢ False | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : Set.Subsingleton (orbit G a)
⊢ a ∈ fixedPoints G X | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : Set.Subsingleton (orbit G a)
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | rw [Set.subsingleton_iff_singleton (mem_orbit_self a)] at H | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : Set.Subsingleton (orbit G a)
⊢ a ∈ fixedPoints G X | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
⊢ a ∈ fixedPoints G X | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : Set.Subsingleton (orbit G a)
⊢ a ∈ fixedPoints G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | simp only [mem_fixedPoints] | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
⊢ a ∈ fixedPoints G X | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
⊢ ∀ (m : G), m • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
⊢ a ∈ fixedPoints G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | intro g | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
⊢ ∀ (m : G), m • a = a | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
⊢ ∀ (m : G), m • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | rw [← Set.mem_singleton_iff] | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a = a | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a ∈ {a} | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | rw [← H] | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a ∈ {a} | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a ∈ orbit G a | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a ∈ {a}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | exact mem_orbit a g | case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a ∈ orbit G a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a ∈ orbit G a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | intro x | case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = ⊤
⊢ ∀ (x : X), ∃ g, g • a = x | case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = ⊤
x : X
⊢ ∃ g, g • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = ⊤
⊢ ∀ (x : X), ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | rw [← MulAction.mem_orbit_iff, H] | case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = ⊤
x : X
⊢ ∃ g, g • a = x | case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = ⊤
x : X
⊢ x ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = ⊤
x : X
⊢ ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | exact Set.mem_univ x | case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = ⊤
x : X
⊢ x ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = ⊤
x : X
⊢ x ∈ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | left | case has_trivial_blocks'.inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ IsTrivialBlock B | case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inl
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | rw [h] | case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton B | case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | exact Set.subsingleton_empty | case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inl.h
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
h : B = ∅
⊢ Set.Subsingleton ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | obtain ⟨b, hb⟩ := h | case has_trivial_blocks'.inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
h : Set.Nonempty B
⊢ IsTrivialBlock B | case has_trivial_blocks'.inr.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
h : Set.Nonempty B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | obtain ⟨g, hg⟩ := exists_smul_eq G b a | case has_trivial_blocks'.inr.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
⊢ IsTrivialBlock B | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | rw [isTrivialBlock_of_block_iff g] | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock B | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock (g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | refine' H (g • B) _ (IsBlock_of_block g hB) | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock (g • B) | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ a ∈ g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ IsTrivialBlock (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk_mem' | [198, 1] | [220, 10] | use b | case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ a ∈ g • B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr.intro.intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H : ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
this : IsPretransitive G X
B : Set X
hB : IsBlock G B
b : X
hb : b ∈ B
g : G
hg : g • b = a
⊢ a ∈ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | have : ∃ a : X, a ∉ fixedPoints G X := by
by_contra h; push_neg at h ; apply Hnt; rw [eq_top_iff]
intro a _; exact h a | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
⊢ IsPreprimitive G X | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
this : ∃ a, a ∉ fixedPoints G X
⊢ IsPreprimitive G X | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
⊢ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | obtain ⟨a, ha⟩ := this | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
this : ∃ a, a ∉ fixedPoints G X
⊢ IsPreprimitive G X | case intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
a : X
ha : a ∉ fixedPoints G X
⊢ IsPreprimitive G X | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
this : ∃ a, a ∉ fixedPoints G X
⊢ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | apply IsPreprimitive.mk_mem' a ha | case intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
a : X
ha : a ∉ fixedPoints G X
⊢ IsPreprimitive G X | case intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
a : X
ha : a ∉ fixedPoints G X
⊢ ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
a : X
ha : a ∉ fixedPoints G X
⊢ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | intro B _ | case intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
a : X
ha : a ∉ fixedPoints G X
⊢ ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B | case intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
a : X
ha : a ∉ fixedPoints G X
B : Set X
x✝ : a ∈ B
⊢ IsBlock G B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
a : X
ha : a ∉ fixedPoints G X
⊢ ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | exact H B | case intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
a : X
ha : a ∉ fixedPoints G X
B : Set X
x✝ : a ∈ B
⊢ IsBlock G B → IsTrivialBlock B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
a : X
ha : a ∉ fixedPoints G X
B : Set X
x✝ : a ∈ B
⊢ IsBlock G B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | by_contra h | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
⊢ ∃ a, a ∉ fixedPoints G X | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ¬∃ a, a ∉ fixedPoints G X
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
⊢ ∃ a, a ∉ fixedPoints G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | push_neg at h | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ¬∃ a, a ∉ fixedPoints G X
⊢ False | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ¬∃ a, a ∉ fixedPoints G X
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | apply Hnt | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
⊢ False | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
⊢ fixedPoints G X = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | rw [eq_top_iff] | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
⊢ fixedPoints G X = ⊤ | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
⊢ ⊤ ≤ fixedPoints G X | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
⊢ fixedPoints G X = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | intro a _ | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
⊢ ⊤ ≤ fixedPoints G X | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
a : X
a✝ : a ∈ ⊤
⊢ a ∈ fixedPoints G X | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
⊢ ⊤ ≤ fixedPoints G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.mk' | [225, 1] | [233, 23] | exact h a | G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
a : X
a✝ : a ∈ ⊤
⊢ a ∈ fixedPoints G X | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ (B : Set X), IsBlock G B → IsTrivialBlock B
h : ∀ (a : X), a ∈ fixedPoints G X
a : X
a✝ : a ∈ ⊤
⊢ a ∈ fixedPoints G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_surjective_map | [244, 1] | [251, 13] | cases' hB with hB hB | M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : IsTrivialBlock B
⊢ IsTrivialBlock (⇑f '' B) | case inl
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : Set.Subsingleton B
⊢ IsTrivialBlock (⇑f '' B)
case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ IsTrivialBlock (⇑f '' B) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : IsTrivialBlock B
⊢ IsTrivialBlock (⇑f '' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_surjective_map | [244, 1] | [251, 13] | apply Or.intro_left | case inl
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : Set.Subsingleton B
⊢ IsTrivialBlock (⇑f '' B) | case inl.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : Set.Subsingleton B
⊢ Set.Subsingleton (⇑f '' B) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : Set.Subsingleton B
⊢ IsTrivialBlock (⇑f '' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_surjective_map | [244, 1] | [251, 13] | apply Set.Subsingleton.image hB | case inl.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : Set.Subsingleton B
⊢ Set.Subsingleton (⇑f '' B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : Set.Subsingleton B
⊢ Set.Subsingleton (⇑f '' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_surjective_map | [244, 1] | [251, 13] | apply Or.intro_right | case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ IsTrivialBlock (⇑f '' B) | case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ ⇑f '' B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ IsTrivialBlock (⇑f '' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_surjective_map | [244, 1] | [251, 13] | rw [hB] | case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ ⇑f '' B = ⊤ | case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ ⇑f '' ⊤ = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ ⇑f '' B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_surjective_map | [244, 1] | [251, 13] | simp only [Set.top_eq_univ, Set.image_univ, Set.range_iff_surjective] | case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ ⇑f '' ⊤ = ⊤ | case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ Function.Surjective ⇑f | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ ⇑f '' ⊤ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_surjective_map | [244, 1] | [251, 13] | exact hf | case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ Function.Surjective ⇑f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = ⊤
⊢ Function.Surjective ⇑f
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_injective_map | [254, 1] | [259, 81] | cases' hB with hB hB | M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : IsTrivialBlock B
⊢ IsTrivialBlock (⇑f ⁻¹' B) | case inl
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : Set.Subsingleton B
⊢ IsTrivialBlock (⇑f ⁻¹' B)
case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ IsTrivialBlock (⇑f ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : IsTrivialBlock B
⊢ IsTrivialBlock (⇑f ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_injective_map | [254, 1] | [259, 81] | apply Or.intro_left | case inl
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : Set.Subsingleton B
⊢ IsTrivialBlock (⇑f ⁻¹' B)
case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ IsTrivialBlock (⇑f ⁻¹' B) | case inl.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : Set.Subsingleton B
⊢ Set.Subsingleton (⇑f ⁻¹' B)
case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ IsTrivialBlock (⇑f ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : Set.Subsingleton B
⊢ IsTrivialBlock (⇑f ⁻¹' B)
case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ IsTrivialBlock (⇑f ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_injective_map | [254, 1] | [259, 81] | exact Set.Subsingleton.preimage hB hf | case inl.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : Set.Subsingleton B
⊢ Set.Subsingleton (⇑f ⁻¹' B)
case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ IsTrivialBlock (⇑f ⁻¹' B) | case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ IsTrivialBlock (⇑f ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : Set.Subsingleton B
⊢ Set.Subsingleton (⇑f ⁻¹' B)
case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ IsTrivialBlock (⇑f ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_injective_map | [254, 1] | [259, 81] | apply Or.intro_right | case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ IsTrivialBlock (⇑f ⁻¹' B) | case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ ⇑f ⁻¹' B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ IsTrivialBlock (⇑f ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_injective_map | [254, 1] | [259, 81] | simp only [hB, Set.top_eq_univ] | case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ ⇑f ⁻¹' B = ⊤ | case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ ⇑f ⁻¹' Set.univ = Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ ⇑f ⁻¹' B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isTrivialBlock_of_injective_map | [254, 1] | [259, 81] | apply Set.preimage_univ | case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ ⇑f ⁻¹' Set.univ = Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = ⊤
⊢ ⇑f ⁻¹' Set.univ = Set.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_surjective_map | [262, 1] | [272, 13] | have : IsPretransitive N β := isPretransitive.of_surjective_map hf h.toIsPretransitive | M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
⊢ IsPreprimitive N β | M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
⊢ IsPreprimitive N β | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
⊢ IsPreprimitive N β
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_surjective_map | [262, 1] | [272, 13] | apply IsPreprimitive.mk | M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
⊢ IsPreprimitive N β | case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
⊢ ∀ {B : Set β}, IsBlock N B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
⊢ IsPreprimitive N β
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_surjective_map | [262, 1] | [272, 13] | intro B hB | case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
⊢ ∀ {B : Set β}, IsBlock N B → IsTrivialBlock B | case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
⊢ ∀ {B : Set β}, IsBlock N B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_surjective_map | [262, 1] | [272, 13] | rw [← Set.image_preimage_eq B hf] | case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsTrivialBlock B | case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsTrivialBlock (⇑f '' (⇑f ⁻¹' B)) | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsTrivialBlock B
TACTIC:
|
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