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/Blocks.lean | MulAction.IsBlock.of_subgroup_of_conjugate | [390, 1] | [408, 35] | simp only [inv_mul_cancel_right] | case intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
⊢ (g * h * g⁻¹ * g) • B = (g * h) • B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
⊢ (g * h * g⁻¹ * g) • B = (g * h) • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.of_subgroup_of_conjugate | [390, 1] | [408, 35] | simp only [this] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
⊢ h' • g • B = g • B ∨ Disjoint (h' • g • B) (g • B) | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
⊢ g • h • B = g • B ∨ Disjoint (g • h • B) (g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
⊢ h' • g • B = g • B ∨ Disjoint (h' • g • B) (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.of_subgroup_of_conjugate | [390, 1] | [408, 35] | cases' IsBlock.def_one.mp hB ⟨h, hH⟩ with heq hdis | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
⊢ g • h • B = g • B ∨ Disjoint (g • h • B) (g • B) | case inl
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
heq : { val := h, property := hH } • B = B
⊢ g • h • B = g • B ∨ Disjoint (g • h • B) (g • B)
case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
hdis : Disjoint ({ val := h, property := hH } • B) B
⊢ g • h • B = g • B ∨ Disjoint (g • h • B) (g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
⊢ g • h • B = g • B ∨ Disjoint (g • h • B) (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.of_subgroup_of_conjugate | [390, 1] | [408, 35] | left | case inl
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
heq : { val := h, property := hH } • B = B
⊢ g • h • B = g • B ∨ Disjoint (g • h • B) (g • B) | case inl.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
heq : { val := h, property := hH } • B = B
⊢ g • h • B = g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
heq : { val := h, property := hH } • B = B
⊢ g • h • B = g • B ∨ Disjoint (g • h • B) (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.of_subgroup_of_conjugate | [390, 1] | [408, 35] | congr | case inl.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
heq : { val := h, property := hH } • B = B
⊢ g • h • B = g • B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
heq : { val := h, property := hH } • B = B
⊢ g • h • B = g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.of_subgroup_of_conjugate | [390, 1] | [408, 35] | right | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
hdis : Disjoint ({ val := h, property := hH } • B) B
⊢ g • h • B = g • B ∨ Disjoint (g • h • B) (g • B) | case inr.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
hdis : Disjoint ({ val := h, property := hH } • B) B
⊢ Disjoint (g • h • B) (g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
hdis : Disjoint ({ val := h, property := hH } • B) B
⊢ g • h • B = g • B ∨ Disjoint (g • h • B) (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.of_subgroup_of_conjugate | [390, 1] | [408, 35] | exact Set.disjoint_image_of_injective (MulAction.injective g) hdis | case inr.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
hdis : Disjoint ({ val := h, property := hH } • B) B
⊢ Disjoint (g • h • B) (g • B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
hdis : Disjoint ({ val := h, property := hH } • B) B
⊢ Disjoint (g • h • B) (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.of_subgroup_of_conjugate | [390, 1] | [408, 35] | rw [← this] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : ↑h' • g • B = g • h • B
⊢ h' • g • B = g • h • B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : ↑h' • g • B = g • h • B
⊢ h' • g • B = ↑h' • g • B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : ↑h' • g • B = g • h • B
⊢ h' • g • B = g • h • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.of_subgroup_of_conjugate | [390, 1] | [408, 35] | rfl | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : ↑h' • g • B = g • h • B
⊢ h' • g • B = ↑h' • g • B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : ↑h' • g • B = g • h • B
⊢ h' • g • B = ↑h' • g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_block | [412, 1] | [421, 26] | rw [IsBlock.of_top_iff] at hB ⊢ | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock G B
⊢ IsBlock G (g • B) | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
⊢ IsBlock (↥⊤) (g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock G B
⊢ IsBlock G (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_block | [412, 1] | [421, 26] | suffices Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤ by
rw [← this]
exact IsBlock.of_subgroup_of_conjugate hB g | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
⊢ IsBlock (↥⊤) (g • B) | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
⊢ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
⊢ IsBlock (↥⊤) (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_block | [412, 1] | [421, 26] | suffices ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ by
rw [this]
refine Subgroup.map_comap_eq_self_of_surjective ?_ _
exact MulEquiv.surjective (MulAut.conj g) | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
⊢ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤ | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
⊢ ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
⊢ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_block | [412, 1] | [421, 26] | rw [Subgroup.comap_top] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
⊢ ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
⊢ ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_block | [412, 1] | [421, 26] | rw [← this] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤
⊢ IsBlock (↥⊤) (g • B) | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤
⊢ IsBlock (↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤)) (g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤
⊢ IsBlock (↥⊤) (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_block | [412, 1] | [421, 26] | exact IsBlock.of_subgroup_of_conjugate hB g | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤
⊢ IsBlock (↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤)) (g • B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤
⊢ IsBlock (↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤)) (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_block | [412, 1] | [421, 26] | rw [this] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤
⊢ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤ | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤
⊢ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤) =
Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤
⊢ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_block | [412, 1] | [421, 26] | refine Subgroup.map_comap_eq_self_of_surjective ?_ _ | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤
⊢ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤) =
Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤ | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤
⊢ Function.Surjective ⇑(MulEquiv.toMonoidHom (MulAut.conj g)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤
⊢ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤) =
Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_block | [412, 1] | [421, 26] | exact MulEquiv.surjective (MulAut.conj g) | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤
⊢ Function.Surjective ⇑(MulEquiv.toMonoidHom (MulAut.conj g)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
hB : IsBlock (↥⊤) B
this : ⊤ = Subgroup.comap (MulEquiv.toMonoidHom (MulAut.conj g)) ⊤
⊢ Function.Surjective ⇑(MulEquiv.toMonoidHom (MulAut.conj g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | constructor | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ IsBlockSystem G (Set.range fun g => g • B) | case left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ Setoid.IsPartition (Set.range fun g => g • B)
case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ b ∈ Set.range fun g => g • B, IsBlock G b | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ IsBlockSystem G (Set.range fun g => g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | constructor | case left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ Setoid.IsPartition (Set.range fun g => g • B)
case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ b ∈ Set.range fun g => g • B, IsBlock G b | case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∅ ∉ Set.range fun g => g • B
case left.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ (a : X), ∃! b x, a ∈ b
case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ b ∈ Set.range fun g => g • B, IsBlock G b | Please generate a tactic in lean4 to solve the state.
STATE:
case left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ Setoid.IsPartition (Set.range fun g => g • B)
case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ b ∈ Set.range fun g => g • B, IsBlock G b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | intro b | case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ b ∈ Set.range fun g => g • B, IsBlock G b | case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
b : Set X
⊢ (b ∈ Set.range fun g => g • B) → IsBlock G b | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ b ∈ Set.range fun g => g • B, IsBlock G b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | rintro ⟨g, hg : g • B = b⟩ | case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
b : Set X
⊢ (b ∈ Set.range fun g => g • B) → IsBlock G b | case right.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
b : Set X
g : G
hg : g • B = b
⊢ IsBlock G b | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
b : Set X
⊢ (b ∈ Set.range fun g => g • B) → IsBlock G b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | rw [← hg] | case right.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
b : Set X
g : G
hg : g • B = b
⊢ IsBlock G b | case right.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
b : Set X
g : G
hg : g • B = b
⊢ IsBlock G (g • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
b : Set X
g : G
hg : g • B = b
⊢ IsBlock G b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | exact IsBlock_of_block g hB | case right.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
b : Set X
g : G
hg : g • B = b
⊢ IsBlock G (g • B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
b : Set X
g : G
hg : g • B = b
⊢ IsBlock G (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | simp only [Set.mem_range, not_exists] | case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∅ ∉ Set.range fun g => g • B | case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ (x : G), ¬x • B = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∅ ∉ Set.range fun g => g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | intro x hx | case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ (x : G), ¬x • B = ∅ | case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ (x : G), ¬x • B = ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | apply Set.Nonempty.ne_empty hBe | case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ False | case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ B = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | rw [← Set.image_eq_empty] | case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ B = ∅ | case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ ?m.112447 '' B = ∅
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ Type ?u.112444
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ X → ?m.112446 | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ B = ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | exact hx | case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ ?m.112447 '' B = ∅
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ Type ?u.112444
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ X → ?m.112446 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ ?m.112447 '' B = ∅
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ Type ?u.112444
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
x : G
hx : x • B = ∅
⊢ X → ?m.112446
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | intro a | case left.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ (a : X), ∃! b x, a ∈ b | case left.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
a : X
⊢ ∃! b x, a ∈ b | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
⊢ ∀ (a : X), ∃! b x, a ∈ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | obtain ⟨b : X, hb : b ∈ B⟩ := hBe | case left.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
a : X
⊢ ∃! b x, a ∈ b | case left.right.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
⊢ ∃! b x, a ∈ b | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hBe : Set.Nonempty B
a : X
⊢ ∃! b x, a ∈ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | obtain ⟨g, hab⟩ := exists_smul_eq G b a | case left.right.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
⊢ ∃! b x, a ∈ b | case left.right.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ ∃! b x, a ∈ b | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
⊢ ∃! b x, a ∈ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | have hg : a ∈ g • B := by
change a ∈ (fun b => g • b) '' B
rw [Set.mem_image]
use b | case left.right.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ ∃! b x, a ∈ b | case left.right.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ ∃! b x, a ∈ b | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ ∃! b x, a ∈ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | use g • B | case left.right.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ ∃! b x, a ∈ b | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ (fun b => ∃! x, a ∈ b) (g • B) ∧ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ ∃! b x, a ∈ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | constructor | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ (fun b => ∃! x, a ∈ b) (g • B) ∧ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = g • B | case h.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ (fun b => ∃! x, a ∈ b) (g • B)
case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ (fun b => ∃! x, a ∈ b) (g • B) ∧ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | change a ∈ (fun b => g • b) '' B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ a ∈ g • B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ a ∈ (fun b => g • b) '' B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ a ∈ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | rw [Set.mem_image] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ a ∈ (fun b => g • b) '' B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ ∃ x ∈ B, g • x = a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ a ∈ (fun b => g • b) '' B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | use b | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ ∃ x ∈ B, g • x = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
⊢ ∃ x ∈ B, g • x = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | simp only [Set.mem_range, exists_apply_eq_apply, exists_unique_iff_exists, exists_true_left] | case h.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ (fun b => ∃! x, a ∈ b) (g • B) | case h.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ a ∈ g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ (fun b => ∃! x, a ∈ b) (g • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | exact hg | case h.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ a ∈ g • B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ a ∈ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | simp only [Set.mem_range, exists_unique_iff_exists, exists_prop, and_imp, forall_exists_index,
forall_apply_eq_imp_iff'] | case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = g • B | case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ ∀ (y : Set X) (x : G), x • B = y → a ∈ y → y = g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | intro B' g' hg' ha | case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ ∀ (y : Set X) (x : G), x • B = y → a ∈ y → y = g • B | case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ B' = g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
⊢ ∀ (y : Set X) (x : G), x • B = y → a ∈ y → y = g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | rw [← hg'] | case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ B' = g • B | case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ g' • B = g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ B' = g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | apply symm | case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ g' • B = g • B | case h.right.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ g • B = g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ g' • B = g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | apply Or.resolve_right (IsBlock.def.mp hB g g') | case h.right.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ g • B = g' • B | case h.right.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ ¬Disjoint (g • B) (g' • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ g • B = g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | rw [Set.not_disjoint_iff] | case h.right.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ ¬Disjoint (g • B) (g' • B) | case h.right.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ ∃ x ∈ g • B, x ∈ g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ ¬Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | use a | case h.right.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ ∃ x ∈ g • B, x ∈ g' • B | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ a ∈ g • B ∧ a ∈ g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ ∃ x ∈ g • B, x ∈ g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | rw [hg'] | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ a ∈ g • B ∧ a ∈ g' • B | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ a ∈ g • B ∧ a ∈ B' | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ a ∈ g • B ∧ a ∈ g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_block | [435, 1] | [466, 41] | exact ⟨hg, ha⟩ | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ a ∈ g • B ∧ a ∈ B' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a b : X
hb : b ∈ B
g : G
hab : g • b = a
hg : a ∈ g • B
B' : Set X
g' : G
hg' : g' • B = B'
ha : a ∈ B'
⊢ a ∈ g • B ∧ a ∈ B'
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | constructor | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
⊢ Setoid.IsPartition (Set.range fun a => orbit G a) | case left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
⊢ ∅ ∉ Set.range fun a => orbit G a
case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
⊢ ∀ (a : X), ∃! b x, a ∈ b | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
⊢ Setoid.IsPartition (Set.range fun a => orbit G a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | intro a | case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
⊢ ∀ (a : X), ∃! b x, a ∈ b | case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∃! b x, a ∈ b | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
⊢ ∀ (a : X), ∃! b x, a ∈ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | use orbit G a | case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∃! b x, a ∈ b | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (fun b => ∃! x, a ∈ b) (orbit G a) ∧ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = orbit G a | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∃! b x, a ∈ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | constructor | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (fun b => ∃! x, a ∈ b) (orbit G a) ∧ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = orbit G a | case h.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (fun b => ∃! x, a ∈ b) (orbit G a)
case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = orbit G a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (fun b => ∃! x, a ∈ b) (orbit G a) ∧ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = orbit G a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | rintro ⟨a, ha : orbit G a = ∅⟩ | case left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
⊢ ∅ ∉ Set.range fun a => orbit G a | case left.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
ha : orbit G a = ∅
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
⊢ ∅ ∉ Set.range fun a => orbit G a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | exact Set.Nonempty.ne_empty (MulAction.orbit_nonempty a) ha | case left.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
ha : orbit G a = ∅
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
ha : orbit G a = ∅
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | simp only [Set.mem_range_self, mem_orbit_self, exists_unique_iff_exists, exists_true_left] | case h.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (fun b => ∃! x, a ∈ b) (orbit G a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (fun b => ∃! x, a ∈ b) (orbit G a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | simp only [Set.mem_range, exists_unique_iff_exists, exists_prop, and_imp, forall_exists_index,
forall_apply_eq_imp_iff'] | case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = orbit G a | case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∀ (y : Set X) (x : X), orbit G x = y → a ∈ y → y = orbit G a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∀ (y : Set X), (fun b => ∃! x, a ∈ b) y → y = orbit G a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | rintro B b ⟨rfl⟩ ha | case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∀ (y : Set X) (x : X), orbit G x = y → a ∈ y → y = orbit G a | case h.right.refl
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a b : X
ha : a ∈ orbit G b
⊢ orbit G b = orbit G a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∀ (y : Set X) (x : X), orbit G x = y → a ∈ y → y = orbit G a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | apply symm | case h.right.refl
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a b : X
ha : a ∈ orbit G b
⊢ orbit G b = orbit G a | case h.right.refl.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a b : X
ha : a ∈ orbit G b
⊢ orbit G a = orbit G b | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.refl
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a b : X
ha : a ∈ orbit G b
⊢ orbit G b = orbit G a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | rw [orbit_eq_iff] | case h.right.refl.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a b : X
ha : a ∈ orbit G b
⊢ orbit G a = orbit G b | case h.right.refl.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a b : X
ha : a ∈ orbit G b
⊢ a ∈ orbit G b | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.refl.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a b : X
ha : a ∈ orbit G b
⊢ orbit G a = orbit G b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsPartition.of_orbits | [470, 1] | [483, 13] | exact ha | case h.right.refl.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a b : X
ha : a ∈ orbit G b
⊢ a ∈ orbit G b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.refl.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a b : X
ha : a ∈ orbit G b
⊢ a ∈ orbit G b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | rw [IsBlock.def_one] | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
⊢ IsBlock G (orbit (↥N) a) | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
⊢ ∀ (g : G), g • orbit (↥N) a = orbit (↥N) a ∨ Disjoint (g • orbit (↥N) a) (orbit (↥N) a) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
⊢ IsBlock G (orbit (↥N) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | intro g | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
⊢ ∀ (g : G), g • orbit (↥N) a = orbit (↥N) a ∨ Disjoint (g • orbit (↥N) a) (orbit (↥N) a) | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ g • orbit (↥N) a = orbit (↥N) a ∨ Disjoint (g • orbit (↥N) a) (orbit (↥N) a) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
⊢ ∀ (g : G), g • orbit (↥N) a = orbit (↥N) a ∨ Disjoint (g • orbit (↥N) a) (orbit (↥N) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | suffices g • orbit N a = orbit N (g • a) by rw [this]; apply orbit.equal_or_disjoint | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ g • orbit (↥N) a = orbit (↥N) a ∨ Disjoint (g • orbit (↥N) a) (orbit (↥N) a) | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ g • orbit (↥N) a = orbit (↥N) (g • a) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ g • orbit (↥N) a = orbit (↥N) a ∨ Disjoint (g • orbit (↥N) a) (orbit (↥N) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | change ((fun x : X => g • x) '' Set.range fun k : N => k • a) = Set.range fun k : N => k • g • a | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ g • orbit (↥N) a = orbit (↥N) (g • a) | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ ((fun x => g • x) '' Set.range fun k => k • a) = Set.range fun k => k • g • a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ g • orbit (↥N) a = orbit (↥N) (g • a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | simp only [Set.image_smul, Set.smul_set_range] | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ ((fun x => g • x) '' Set.range fun k => k • a) = Set.range fun k => k • g • a | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ (Set.range fun i => g • i • a) = Set.range fun k => k • g • a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ ((fun x => g • x) '' Set.range fun k => k • a) = Set.range fun k => k • g • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | ext | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ (Set.range fun i => g • i • a) = Set.range fun k => k • g • a | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (x✝ ∈ Set.range fun i => g • i • a) ↔ x✝ ∈ Set.range fun k => k • g • a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
⊢ (Set.range fun i => g • i • a) = Set.range fun k => k • g • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | simp only [Set.mem_range] | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (x✝ ∈ Set.range fun i => g • i • a) ↔ x✝ ∈ Set.range fun k => k • g • a | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (∃ y, g • y • a = x✝) ↔ ∃ y, y • g • a = x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (x✝ ∈ Set.range fun i => g • i • a) ↔ x✝ ∈ Set.range fun k => k • g • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | constructor | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (∃ y, g • y • a = x✝) ↔ ∃ y, y • g • a = x✝ | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (∃ y, g • y • a = x✝) → ∃ y, y • g • a = x✝
case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (∃ y, y • g • a = x✝) → ∃ y, g • y • a = x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (∃ y, g • y • a = x✝) ↔ ∃ y, y • g • a = x✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | rw [this] | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
this : g • orbit (↥N) a = orbit (↥N) (g • a)
⊢ g • orbit (↥N) a = orbit (↥N) a ∨ Disjoint (g • orbit (↥N) a) (orbit (↥N) a) | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
this : g • orbit (↥N) a = orbit (↥N) (g • a)
⊢ orbit (↥N) (g • a) = orbit (↥N) a ∨ Disjoint (orbit (↥N) (g • a)) (orbit (↥N) a) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
this : g • orbit (↥N) a = orbit (↥N) (g • a)
⊢ g • orbit (↥N) a = orbit (↥N) a ∨ Disjoint (g • orbit (↥N) a) (orbit (↥N) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | apply orbit.equal_or_disjoint | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
this : g • orbit (↥N) a = orbit (↥N) (g • a)
⊢ orbit (↥N) (g • a) = orbit (↥N) a ∨ Disjoint (orbit (↥N) (g • a)) (orbit (↥N) a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
this : g • orbit (↥N) a = orbit (↥N) (g • a)
⊢ orbit (↥N) (g • a) = orbit (↥N) a ∨ Disjoint (orbit (↥N) (g • a)) (orbit (↥N) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | rintro ⟨⟨k, hk⟩, rfl⟩ | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (∃ y, g • y • a = x✝) → ∃ y, y • g • a = x✝ | case h.mp.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ ∃ y, y • g • a = g • { val := k, property := hk } • a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (∃ y, g • y • a = x✝) → ∃ y, y • g • a = x✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | suffices g * k * g⁻¹ ∈ N by
use ⟨g * k * g⁻¹, this⟩
change (g * k * g⁻¹) • g • a = g • k • a
rw [smul_smul, inv_mul_cancel_right, ← smul_smul] | case h.mp.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ ∃ y, y • g • a = g • { val := k, property := hk } • a | case h.mp.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ g * k * g⁻¹ ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ ∃ y, y • g • a = g • { val := k, property := hk } • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | apply nN.conj_mem | case h.mp.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ g * k * g⁻¹ ∈ N | case h.mp.intro.mk.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ k ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ g * k * g⁻¹ ∈ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | exact hk | case h.mp.intro.mk.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ k ∈ N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.mk.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ k ∈ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | use ⟨g * k * g⁻¹, this⟩ | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g * k * g⁻¹ ∈ N
⊢ ∃ y, y • g • a = g • { val := k, property := hk } • a | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g * k * g⁻¹ ∈ N
⊢ { val := g * k * g⁻¹, property := this } • g • a = g • { val := k, property := hk } • a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g * k * g⁻¹ ∈ N
⊢ ∃ y, y • g • a = g • { val := k, property := hk } • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | change (g * k * g⁻¹) • g • a = g • k • a | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g * k * g⁻¹ ∈ N
⊢ { val := g * k * g⁻¹, property := this } • g • a = g • { val := k, property := hk } • a | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g * k * g⁻¹ ∈ N
⊢ (g * k * g⁻¹) • g • a = g • k • a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g * k * g⁻¹ ∈ N
⊢ { val := g * k * g⁻¹, property := this } • g • a = g • { val := k, property := hk } • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | rw [smul_smul, inv_mul_cancel_right, ← smul_smul] | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g * k * g⁻¹ ∈ N
⊢ (g * k * g⁻¹) • g • a = g • k • a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g * k * g⁻¹ ∈ N
⊢ (g * k * g⁻¹) • g • a = g • k • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | rintro ⟨⟨k, hk⟩, rfl⟩ | case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (∃ y, y • g • a = x✝) → ∃ y, g • y • a = x✝ | case h.mpr.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ ∃ y, g • y • a = { val := k, property := hk } • g • a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g : G
x✝ : X
⊢ (∃ y, y • g • a = x✝) → ∃ y, g • y • a = x✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | suffices g⁻¹ * k * g ∈ N by
use ⟨g⁻¹ * k * g, this⟩
change g • (g⁻¹ * k * g) • a = k • g • a
simp only [← mul_assoc, ← smul_smul, smul_inv_smul, inv_inv] | case h.mpr.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ ∃ y, g • y • a = { val := k, property := hk } • g • a | case h.mpr.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ g⁻¹ * k * g ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ ∃ y, g • y • a = { val := k, property := hk } • g • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | convert nN.conj_mem k hk g⁻¹ | case h.mpr.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ g⁻¹ * k * g ∈ N | case h.e'_4.h.e'_6
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ g = g⁻¹⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ g⁻¹ * k * g ∈ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | rw [inv_inv] | case h.e'_4.h.e'_6
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ g = g⁻¹⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_6
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
⊢ g = g⁻¹⁻¹
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | use ⟨g⁻¹ * k * g, this⟩ | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g⁻¹ * k * g ∈ N
⊢ ∃ y, g • y • a = { val := k, property := hk } • g • a | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g⁻¹ * k * g ∈ N
⊢ g • { val := g⁻¹ * k * g, property := this } • a = { val := k, property := hk } • g • a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g⁻¹ * k * g ∈ N
⊢ ∃ y, g • y • a = { val := k, property := hk } • g • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | change g • (g⁻¹ * k * g) • a = k • g • a | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g⁻¹ * k * g ∈ N
⊢ g • { val := g⁻¹ * k * g, property := this } • a = { val := k, property := hk } • g • a | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g⁻¹ * k * g ∈ N
⊢ g • (g⁻¹ * k * g) • a = k • g • a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g⁻¹ * k * g ∈ N
⊢ g • { val := g⁻¹ * k * g, property := this } • a = { val := k, property := hk } • g • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.orbit.IsBlock_of_normal | [489, 1] | [511, 17] | simp only [← mul_assoc, ← smul_smul, smul_inv_smul, inv_inv] | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g⁻¹ * k * g ∈ N
⊢ g • (g⁻¹ * k * g) • a = k • g • a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
g k : G
hk : k ∈ N
this : g⁻¹ * k * g ∈ N
⊢ g • (g⁻¹ * k * g) • a = k • g • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_normal | [514, 1] | [519, 39] | constructor | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
⊢ IsBlockSystem G (Set.range fun a => orbit (↥N) a) | case left
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
⊢ Setoid.IsPartition (Set.range fun a => orbit (↥N) a)
case right
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
⊢ ∀ b ∈ Set.range fun a => orbit (↥N) a, IsBlock G b | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
⊢ IsBlockSystem G (Set.range fun a => orbit (↥N) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_normal | [514, 1] | [519, 39] | apply IsPartition.of_orbits | case left
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
⊢ Setoid.IsPartition (Set.range fun a => orbit (↥N) a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
⊢ Setoid.IsPartition (Set.range fun a => orbit (↥N) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_normal | [514, 1] | [519, 39] | intro b | case right
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
⊢ ∀ b ∈ Set.range fun a => orbit (↥N) a, IsBlock G b | case right
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
b : Set X
⊢ (b ∈ Set.range fun a => orbit (↥N) a) → IsBlock G b | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
⊢ ∀ b ∈ Set.range fun a => orbit (↥N) a, IsBlock G b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_normal | [514, 1] | [519, 39] | rintro ⟨a, rfl⟩ | case right
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
b : Set X
⊢ (b ∈ Set.range fun a => orbit (↥N) a) → IsBlock G b | case right.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
⊢ IsBlock G ((fun a => orbit (↥N) a) a) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
b : Set X
⊢ (b ∈ Set.range fun a => orbit (↥N) a) → IsBlock G b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlockSystem.of_normal | [514, 1] | [519, 39] | exact orbit.IsBlock_of_normal nN a | case right.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
⊢ IsBlock G ((fun a => orbit (↥N) a) a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
N : Subgroup G
nN : Subgroup.Normal N
a : X
⊢ IsBlock G ((fun a => orbit (↥N) a) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | rw [IsBlock.mk_subset] | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
⊢ IsBlock G (orbit (↥H) a) | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
⊢ ∀ {g : G} {b : X}, b ∈ orbit (↥H) a → b ∈ g • orbit (↥H) a → g • orbit (↥H) a ≤ orbit (↥H) a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
⊢ IsBlock G (orbit (↥H) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | intro g b | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
⊢ ∀ {g : G} {b : X}, b ∈ orbit (↥H) a → b ∈ g • orbit (↥H) a → g • orbit (↥H) a ≤ orbit (↥H) a | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
b : X
⊢ b ∈ orbit (↥H) a → b ∈ g • orbit (↥H) a → g • orbit (↥H) a ≤ orbit (↥H) a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
⊢ ∀ {g : G} {b : X}, b ∈ orbit (↥H) a → b ∈ g • orbit (↥H) a → g • orbit (↥H) a ≤ orbit (↥H) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | rintro ⟨h, rfl⟩ | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
b : X
⊢ b ∈ orbit (↥H) a → b ∈ g • orbit (↥H) a → g • orbit (↥H) a ≤ orbit (↥H) a | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
⊢ (fun m => m • a) h ∈ g • orbit (↥H) a → g • orbit (↥H) a ≤ orbit (↥H) a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
b : X
⊢ b ∈ orbit (↥H) a → b ∈ g • orbit (↥H) a → g • orbit (↥H) a ≤ orbit (↥H) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | simp | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
⊢ (fun m => m • a) h ∈ g • orbit (↥H) a → g • orbit (↥H) a ≤ orbit (↥H) a | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
⊢ h • a ∈ g • orbit (↥H) a → g • orbit (↥H) a ⊆ orbit (↥H) a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
⊢ (fun m => m • a) h ∈ g • orbit (↥H) a → g • orbit (↥H) a ≤ orbit (↥H) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | intro hb' | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
⊢ h • a ∈ g • orbit (↥H) a → g • orbit (↥H) a ⊆ orbit (↥H) a | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
hb' : h • a ∈ g • orbit (↥H) a
⊢ g • orbit (↥H) a ⊆ orbit (↥H) a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
⊢ h • a ∈ g • orbit (↥H) a → g • orbit (↥H) a ⊆ orbit (↥H) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | suffices g ∈ H by
rw [← Subgroup.coe_mk H g this, ← Subgroup.smul_def]
apply smul_orbit_subset | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
hb' : h • a ∈ g • orbit (↥H) a
⊢ g • orbit (↥H) a ⊆ orbit (↥H) a | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
hb' : h • a ∈ g • orbit (↥H) a
⊢ g ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
hb' : h • a ∈ g • orbit (↥H) a
⊢ g • orbit (↥H) a ⊆ orbit (↥H) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | rw [Set.mem_smul_set_iff_inv_smul_mem, Subgroup.smul_def, ← MulAction.mul_smul] at hb' | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
hb' : h • a ∈ g • orbit (↥H) a
⊢ g ∈ H | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
hb' : (g⁻¹ * ↑h) • a ∈ orbit (↥H) a
⊢ g ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
hb' : h • a ∈ g • orbit (↥H) a
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | obtain ⟨k : ↥H, hk⟩ := hb' | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
hb' : (g⁻¹ * ↑h) • a ∈ orbit (↥H) a
⊢ g ∈ H | case intro.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h k : ↥H
hk : (fun m => m • a) k = (g⁻¹ * ↑h) • a
⊢ g ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
hb' : (g⁻¹ * ↑h) • a ∈ orbit (↥H) a
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | simp only at hk | case intro.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h k : ↥H
hk : (fun m => m • a) k = (g⁻¹ * ↑h) • a
⊢ g ∈ H | case intro.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h k : ↥H
hk : k • a = (g⁻¹ * ↑h) • a
⊢ g ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h k : ↥H
hk : (fun m => m • a) k = (g⁻¹ * ↑h) • a
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | rw [MulAction.mul_smul, ← smul_eq_iff_eq_inv_smul, ← inv_inv (h : G), ← smul_eq_iff_eq_inv_smul, ←
MulAction.mul_smul, Subgroup.smul_def, ← MulAction.mul_smul] at hk | case intro.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h k : ↥H
hk : k • a = (g⁻¹ * ↑h) • a
⊢ g ∈ H | case intro.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h k : ↥H
hk : ((↑h)⁻¹ * g * ↑k) • a = a
⊢ g ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h k : ↥H
hk : k • a = (g⁻¹ * ↑h) • a
⊢ g ∈ H
TACTIC:
|
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