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_suborbit | [534, 1] | [554, 26] | rw [← mem_stabilizer_iff] 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 : ((↑h)⁻¹ * g * ↑k) • a = 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 ∈ stabilizer G 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 : ((↑h)⁻¹ * g * ↑k) • a = a
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | let hk' := hH 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 : (↑h)⁻¹ * g * ↑k ∈ stabilizer G 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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ 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 : (↑h)⁻¹ * g * ↑k ∈ stabilizer G a
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | rw [Subgroup.mul_mem_cancel_right, Subgroup.mul_mem_cancel_left] 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 : (↑h)⁻¹ * g * ↑k ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ 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 ∈ stabilizer G a
hk' : g ∈ H
⊢ g ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g ∈ H
⊢ (↑h)⁻¹ ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ 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 : (↑h)⁻¹ * g * ↑k ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | exact 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 : (↑h)⁻¹ * g * ↑k ∈ stabilizer G a
hk' : g ∈ H
⊢ g ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g ∈ H
⊢ (↑h)⁻¹ ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ H | case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g ∈ H
⊢ (↑h)⁻¹ ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ 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 : (↑h)⁻¹ * g * ↑k ∈ stabilizer G a
hk' : g ∈ H
⊢ g ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g ∈ H
⊢ (↑h)⁻¹ ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | apply Subgroup.inv_mem | case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g ∈ H
⊢ (↑h)⁻¹ ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ H | case intro.intro.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
h k : ↥H
hk : (↑h)⁻¹ * g * ↑k ∈ stabilizer G a
hk' : (↑h)⁻¹ * g ∈ H
⊢ ↑h ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g ∈ H
⊢ (↑h)⁻¹ ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | exact SetLike.coe_mem h | case intro.intro.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
h k : ↥H
hk : (↑h)⁻¹ * g * ↑k ∈ stabilizer G a
hk' : (↑h)⁻¹ * g ∈ H
⊢ ↑h ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ H | case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.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
h k : ↥H
hk : (↑h)⁻¹ * g * ↑k ∈ stabilizer G a
hk' : (↑h)⁻¹ * g ∈ H
⊢ ↑h ∈ H
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | exact SetLike.coe_mem k | case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.h
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 ∈ stabilizer G a
hk' : (↑h)⁻¹ * g * ↑k ∈ H := hH hk
⊢ ↑k ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_suborbit | [534, 1] | [554, 26] | rw [← Subgroup.coe_mk H g this, ← Subgroup.smul_def] | 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
this : g ∈ H
⊢ 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
h : ↥H
hb' : h • a ∈ g • orbit (↥H) a
this : g ∈ H
⊢ { val := g, property := this } • 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
h : ↥H
hb' : h • a ∈ g • orbit (↥H) a
this : g ∈ H
⊢ 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] | apply smul_orbit_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
g : G
h : ↥H
hb' : h • a ∈ g • orbit (↥H) a
this : g ∈ H
⊢ { val := g, property := this } • orbit (↥H) a ⊆ orbit (↥H) 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
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h : ↥H
hb' : h • a ∈ g • orbit (↥H) a
this : g ∈ H
⊢ { val := g, property := this } • orbit (↥H) a ⊆ orbit (↥H) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | intro g hg | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
⊢ stabilizer G a ≤ stabilizer G B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g ∈ stabilizer G a
⊢ g ∈ stabilizer 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
hB : IsBlock G B
a : X
ha : a ∈ B
⊢ stabilizer G a ≤ stabilizer G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | rw [mem_stabilizer_iff] at hg ⊢ | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g ∈ stabilizer G a
⊢ g ∈ stabilizer G B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
⊢ 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
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g ∈ stabilizer G a
⊢ g ∈ stabilizer G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | cases' IsBlock.def_one.mp hB g with h h' | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
⊢ g • B = B | case inl
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h : g • B = B
⊢ g • B = B
case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) 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
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | exact h | case inl
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h : g • B = B
⊢ g • B = B
case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ g • B = B | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ g • B = 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
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h : g • B = B
⊢ g • B = B
case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | exfalso | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ g • B = B | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ False | 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
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | rw [← Set.mem_empty_iff_false a] | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ False | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ a ∈ ∅ | 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
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | simp only [disjoint_iff, Set.inf_eq_inter, Set.bot_eq_empty] at h' | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ a ∈ ∅ | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ ∅ | 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
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : Disjoint (g • B) B
⊢ a ∈ ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | rw [← h', Set.mem_inter_iff] | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ ∅ | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ g • B ∧ a ∈ 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
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | constructor | case inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ g • B ∧ a ∈ B | case inr.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ g • B
case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ 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
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ g • B ∧ a ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | rw [← hg] | case inr.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ g • B
case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B | case inr.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ g • a ∈ g • B
case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ g • B
case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | rw [Set.smul_mem_smul_set_iff] | case inr.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ g • a ∈ g • B
case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B | case inr.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B
case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ g • a ∈ g • B
case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | exact ha | case inr.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B
case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B | case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.left
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B
case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block | [558, 1] | [569, 11] | exact ha | case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.right
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g • a = a
h' : g • B ∩ B = ∅
⊢ a ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | ext x | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
⊢ orbit (↥(stabilizer G B)) a = B | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
⊢ x ∈ orbit (↥(stabilizer G B)) a ↔ x ∈ 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
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
⊢ orbit (↥(stabilizer G B)) a = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | constructor | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
⊢ x ∈ orbit (↥(stabilizer G B)) a ↔ x ∈ B | case h.mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
⊢ x ∈ orbit (↥(stabilizer G B)) a → x ∈ B
case h.mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
⊢ x ∈ B → x ∈ orbit (↥(stabilizer G B)) 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
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
⊢ x ∈ orbit (↥(stabilizer G B)) a ↔ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | intro hx | case h.mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
⊢ x ∈ B → x ∈ orbit (↥(stabilizer G B)) a | case h.mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
hx : x ∈ B
⊢ x ∈ orbit (↥(stabilizer G B)) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
⊢ x ∈ B → x ∈ orbit (↥(stabilizer G B)) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | obtain ⟨k, rfl⟩ := exists_smul_eq G a x | case h.mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
hx : x ∈ B
⊢ x ∈ orbit (↥(stabilizer G B)) a | case h.mpr.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
⊢ k • a ∈ orbit (↥(stabilizer G B)) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
hx : x ∈ B
⊢ x ∈ orbit (↥(stabilizer G B)) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | suffices k ∈ stabilizer G B by
exact ⟨⟨k, this⟩, rfl⟩ | case h.mpr.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
⊢ k • a ∈ orbit (↥(stabilizer G B)) a | case h.mpr.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
⊢ k ∈ stabilizer G B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
⊢ k • a ∈ orbit (↥(stabilizer G B)) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | rw [mem_stabilizer_iff] | case h.mpr.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
⊢ k ∈ stabilizer G B | case h.mpr.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
⊢ k • B = B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
⊢ k ∈ stabilizer G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | exact IsBlock.def_mem hB ha hx | case h.mpr.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
⊢ k • B = B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
⊢ k • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | rintro ⟨k, rfl⟩ | case h.mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
⊢ x ∈ orbit (↥(stabilizer G B)) a → x ∈ B | case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
⊢ (fun m => m • a) k ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
⊢ x ∈ orbit (↥(stabilizer G B)) a → x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | let z := mem_stabilizer_iff.mp (SetLike.coe_mem k) | case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
⊢ (fun m => m • a) k ∈ B | case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : ↑k • B = B := mem_stabilizer_iff.mp (SetLike.coe_mem k)
⊢ (fun m => m • a) k ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
⊢ (fun m => m • a) k ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | rw [← Subgroup.smul_def] at z | case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : ↑k • B = B := mem_stabilizer_iff.mp (SetLike.coe_mem k)
⊢ (fun m => m • a) k ∈ B | case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : k • B = B
⊢ (fun m => m • a) k ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : ↑k • B = B := mem_stabilizer_iff.mp (SetLike.coe_mem k)
⊢ (fun m => m • a) k ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | let zk : k • a ∈ k • B := Set.smul_mem_smul_set_iff.mpr ha | case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : k • B = B
⊢ (fun m => m • a) k ∈ B | case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : k • B = B
zk : k • a ∈ k • B := Set.smul_mem_smul_set_iff.mpr ha
⊢ (fun m => m • a) k ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : k • B = B
⊢ (fun m => m • a) k ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | rw [z] at zk | case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : k • B = B
zk : k • a ∈ k • B := Set.smul_mem_smul_set_iff.mpr ha
⊢ (fun m => m • a) k ∈ B | case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : k • B = B
zk : k • a ∈ B
⊢ (fun m => m • a) k ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : k • B = B
zk : k • a ∈ k • B := Set.smul_mem_smul_set_iff.mpr ha
⊢ (fun m => m • a) k ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | exact zk | case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : k • B = B
zk : k • a ∈ B
⊢ (fun m => m • a) k ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : ↥(stabilizer G B)
z : k • B = B
zk : k • a ∈ B
⊢ (fun m => m • a) k ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.block_of_stabilizer_of_block | [573, 1] | [589, 33] | exact ⟨⟨k, this⟩, rfl⟩ | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
this : k ∈ stabilizer G B
⊢ k • a ∈ orbit (↥(stabilizer G B)) 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
htGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
k : G
hx : k • a ∈ B
this : k ∈ stabilizer G B
⊢ k • a ∈ orbit (↥(stabilizer G B)) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | ext g | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
⊢ stabilizer G (orbit (↥H) a) = H | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
⊢ g ∈ stabilizer G (orbit (↥H) a) ↔ g ∈ H | 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
⊢ stabilizer G (orbit (↥H) a) = H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | constructor | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
⊢ g ∈ stabilizer G (orbit (↥H) a) ↔ g ∈ H | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
⊢ g ∈ stabilizer G (orbit (↥H) a) → g ∈ H
case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
⊢ g ∈ H → g ∈ stabilizer G (orbit (↥H) 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
⊢ g ∈ stabilizer G (orbit (↥H) a) ↔ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | intro hg | case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
⊢ g ∈ H → g ∈ stabilizer G (orbit (↥H) a) | case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ g ∈ stabilizer G (orbit (↥H) 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
⊢ g ∈ H → g ∈ stabilizer G (orbit (↥H) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | rw [mem_stabilizer_iff] | case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ g ∈ stabilizer G (orbit (↥H) a) | case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ g • orbit (↥H) a = orbit (↥H) 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ g ∈ stabilizer G (orbit (↥H) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | rw [← Subgroup.coe_mk H g hg, ← Subgroup.smul_def] | case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ g • orbit (↥H) a = orbit (↥H) a | case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ { val := g, property := hg } • orbit (↥H) a = orbit (↥H) 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ g • orbit (↥H) a = orbit (↥H) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | apply smul_orbit | case h.mpr
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ { val := g, property := hg } • orbit (↥H) a = orbit (↥H) a | no goals | 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ H
⊢ { val := g, property := hg } • orbit (↥H) a = orbit (↥H) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | intro hg | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
⊢ g ∈ stabilizer G (orbit (↥H) a) → g ∈ H | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ stabilizer G (orbit (↥H) a)
⊢ g ∈ H | 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
⊢ g ∈ stabilizer G (orbit (↥H) a) → g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | rw [mem_stabilizer_iff] at hg | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ stabilizer G (orbit (↥H) a)
⊢ g ∈ H | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
⊢ g ∈ H | 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ stabilizer G (orbit (↥H) a)
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | suffices g • a ∈ orbit H a by
rw [mem_orbit_iff] at this
obtain ⟨k, hk⟩ := this
rw [← Subgroup.mul_mem_cancel_left H (SetLike.coe_mem k⁻¹)]
rw [smul_eq_iff_eq_inv_smul] at hk
apply hH
rw [mem_stabilizer_iff]; rw [MulAction.mul_smul]
rw [← Subgroup.smul_def]; exact hk.symm | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
⊢ g ∈ H | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
⊢ g • a ∈ orbit (↥H) 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | rw [← hg] | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
⊢ g • a ∈ orbit (↥H) a | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
⊢ g • a ∈ g • orbit (↥H) 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
⊢ g • a ∈ orbit (↥H) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | simp only [Set.smul_mem_smul_set_iff, mem_orbit_self] | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
⊢ g • a ∈ g • orbit (↥H) a | no goals | 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
⊢ g • a ∈ g • orbit (↥H) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | rw [mem_orbit_iff] at this | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
this : g • a ∈ orbit (↥H) a
⊢ g ∈ H | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
this : ∃ x, x • a = g • a
⊢ g ∈ H | 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
this : g • a ∈ orbit (↥H) a
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | obtain ⟨k, hk⟩ := this | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
this : ∃ x, x • a = g • a
⊢ g ∈ H | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : k • a = g • a
⊢ g ∈ H | 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
this : ∃ x, x • a = g • a
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | rw [← Subgroup.mul_mem_cancel_left H (SetLike.coe_mem k⁻¹)] | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : k • a = g • a
⊢ g ∈ H | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : k • a = g • a
⊢ ↑k⁻¹ * 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : k • a = g • a
⊢ g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | rw [smul_eq_iff_eq_inv_smul] at hk | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : k • a = g • a
⊢ ↑k⁻¹ * g ∈ H | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ ↑k⁻¹ * 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : k • a = g • a
⊢ ↑k⁻¹ * g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | apply hH | case intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ ↑k⁻¹ * g ∈ H | case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ ↑k⁻¹ * g ∈ stabilizer G 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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ ↑k⁻¹ * g ∈ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | rw [mem_stabilizer_iff] | case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ ↑k⁻¹ * g ∈ stabilizer G a | case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ (↑k⁻¹ * g) • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ ↑k⁻¹ * g ∈ stabilizer G a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | rw [MulAction.mul_smul] | case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ (↑k⁻¹ * g) • a = a | case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ ↑k⁻¹ • g • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ (↑k⁻¹ * g) • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | rw [← Subgroup.smul_def] | case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ ↑k⁻¹ • g • a = a | case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ k⁻¹ • g • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ ↑k⁻¹ • g • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.stabilizer_of_block_of_stabilizer | [593, 1] | [610, 19] | exact hk.symm | case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ k⁻¹ • g • a = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g • orbit (↥H) a = orbit (↥H) a
k : ↥H
hk : a = k⁻¹ • g • a
⊢ k⁻¹ • g • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | classical
have := Fintype.ofFinite α
simp only [finsum_eq_sum_of_fintype, Nat.card_eq_fintype_card]
rw [← Fintype.card_sigma]
refine' Fintype.card_congr (Equiv.ofBijective (fun x => x.snd : (Σ a : ↥c, a) → α) _)
constructor
rintro ⟨⟨x, hx⟩, ⟨a, ha : a ∈ x⟩⟩ ⟨⟨y, hy⟩, ⟨b, hb : b ∈ y⟩⟩ hab
dsimp at hab
rw [hab] at ha
rw [Sigma.subtype_ext_iff]
simp only [Subtype.mk_eq_mk, Subtype.coe_mk]
apply And.intro _ hab
refine' ExistsUnique.unique (hc.2 b) _ _
simp only [exists_unique_iff_exists, exists_prop]
exact ⟨hx, ha⟩
simp only [exists_unique_iff_exists, exists_prop]
exact ⟨hy, hb⟩
intro a
obtain ⟨x, ⟨hx, ha : a ∈ x, _⟩, _⟩ := hc.2 a
use ⟨⟨x, hx⟩, ⟨a, ha⟩⟩ | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
⊢ ∑ᶠ (x : ↑c), Nat.card ↑↑x = Nat.card α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
⊢ ∑ᶠ (x : ↑c), Nat.card ↑↑x = Nat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | have := Fintype.ofFinite α | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
⊢ ∑ᶠ (x : ↑c), Nat.card ↑↑x = Nat.card α | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ ∑ᶠ (x : ↑c), Nat.card ↑↑x = Nat.card α | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
⊢ ∑ᶠ (x : ↑c), Nat.card ↑↑x = Nat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | simp only [finsum_eq_sum_of_fintype, Nat.card_eq_fintype_card] | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ ∑ᶠ (x : ↑c), Nat.card ↑↑x = Nat.card α | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ ∑ x : ↑c, Fintype.card ↑↑x = Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ ∑ᶠ (x : ↑c), Nat.card ↑↑x = Nat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | rw [← Fintype.card_sigma] | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ ∑ x : ↑c, Fintype.card ↑↑x = Fintype.card α | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Fintype.card ((x : ↑c) × ↑↑x) = Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ ∑ x : ↑c, Fintype.card ↑↑x = Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | refine' Fintype.card_congr (Equiv.ofBijective (fun x => x.snd : (Σ a : ↥c, a) → α) _) | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Fintype.card ((x : ↑c) × ↑↑x) = Fintype.card α | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Bijective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Fintype.card ((x : ↑c) × ↑↑x) = Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | constructor | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Bijective fun x => ↑x.snd | case left
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Injective fun x => ↑x.snd
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Bijective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | rintro ⟨⟨x, hx⟩, ⟨a, ha : a ∈ x⟩⟩ ⟨⟨y, hy⟩, ⟨b, hb : b ∈ y⟩⟩ hab | case left
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Injective fun x => ↑x.snd
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha : a ∈ x
y : Set α
hy : y ∈ c
b : α
hb : b ∈ y
hab :
(fun x => ↑x.snd) { fst := { val := x, property := hx }, snd := { val := a, property := ha } } =
(fun x => ↑x.snd) { fst := { val := y, property := hy }, snd := { val := b, property := hb } }
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha } } =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
case left
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Injective fun x => ↑x.snd
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | dsimp at hab | case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha : a ∈ x
y : Set α
hy : y ∈ c
b : α
hb : b ∈ y
hab :
(fun x => ↑x.snd) { fst := { val := x, property := hx }, snd := { val := a, property := ha } } =
(fun x => ↑x.snd) { fst := { val := y, property := hy }, snd := { val := b, property := hb } }
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha } } =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha : a ∈ x
y : Set α
hy : y ∈ c
b : α
hb : b ∈ y
hab : a = b
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha } } =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha : a ∈ x
y : Set α
hy : y ∈ c
b : α
hb : b ∈ y
hab :
(fun x => ↑x.snd) { fst := { val := x, property := hx }, snd := { val := a, property := ha } } =
(fun x => ↑x.snd) { fst := { val := y, property := hy }, snd := { val := b, property := hb } }
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha } } =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | rw [hab] at ha | case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha : a ∈ x
y : Set α
hy : y ∈ c
b : α
hb : b ∈ y
hab : a = b
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha } } =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha✝ } } =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha : a ∈ x
y : Set α
hy : y ∈ c
b : α
hb : b ∈ y
hab : a = b
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha } } =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | rw [Sigma.subtype_ext_iff] | case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha✝ } } =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha✝ } }.fst =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }.fst ∧
↑{ fst := { val := x, property := hx }, snd := { val := a, property := ha✝ } }.snd =
↑{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }.snd
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha✝ } } =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | simp only [Subtype.mk_eq_mk, Subtype.coe_mk] | case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha✝ } }.fst =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }.fst ∧
↑{ fst := { val := x, property := hx }, snd := { val := a, property := ha✝ } }.snd =
↑{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }.snd
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ x = y ∧ a = b
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ { fst := { val := x, property := hx }, snd := { val := a, property := ha✝ } }.fst =
{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }.fst ∧
↑{ fst := { val := x, property := hx }, snd := { val := a, property := ha✝ } }.snd =
↑{ fst := { val := y, property := hy }, snd := { val := b, property := hb } }.snd
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | apply And.intro _ hab | case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ x = y ∧ a = b
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ x = y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk.mk.mk.mk.mk
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ x = y ∧ a = b
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | refine' ExistsUnique.unique (hc.2 b) _ _ | G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ x = y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case refine'_1
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x_1, b ∈ x
case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x, b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ x = y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | simp only [exists_unique_iff_exists, exists_prop] | case refine'_1
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x_1, b ∈ x
case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x, b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case refine'_1
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ x ∈ c ∧ b ∈ x
case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x, b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x_1, b ∈ x
case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x, b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | exact ⟨hx, ha⟩ | case refine'_1
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ x ∈ c ∧ b ∈ x
case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x, b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x, b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ x ∈ c ∧ b ∈ x
case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x, b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | simp only [exists_unique_iff_exists, exists_prop] | case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x, b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ y ∈ c ∧ b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ ∃! x, b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | exact ⟨hy, hb⟩ | case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ y ∈ c ∧ b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
x : Set α
hx : x ∈ c
a : α
ha✝ : a ∈ x
y : Set α
hy : y ∈ c
b : α
ha : b ∈ x
hb : b ∈ y
hab : a = b
⊢ y ∈ c ∧ b ∈ y
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | intro a | case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd | case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
a : α
⊢ ∃ a_1, (fun x => ↑x.snd) a_1 = a | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
⊢ Function.Surjective fun x => ↑x.snd
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | obtain ⟨x, ⟨hx, ha : a ∈ x, _⟩, _⟩ := hc.2 a | case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
a : α
⊢ ∃ a_1, (fun x => ↑x.snd) a_1 = a | case right.intro.intro.intro.intro
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
a : α
x : Set α
right✝¹ : ∀ (y : Set α), (fun b => ∃! x, a ∈ b) y → y = x
hx : x ∈ c
ha : a ∈ x
right✝ : ∀ (y : x ∈ c), (fun x_1 => a ∈ x) y → y = hx
⊢ ∃ a_1, (fun x => ↑x.snd) a_1 = a | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
a : α
⊢ ∃ a_1, (fun x => ↑x.snd) a_1 = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Setoid.nat_sum | [652, 1] | [675, 25] | use ⟨⟨x, hx⟩, ⟨a, ha⟩⟩ | case right.intro.intro.intro.intro
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
a : α
x : Set α
right✝¹ : ∀ (y : Set α), (fun b => ∃! x, a ∈ b) y → y = x
hx : x ∈ c
ha : a ∈ x
right✝ : ∀ (y : x ∈ c), (fun x_1 => a ∈ x) y → y = hx
⊢ ∃ a_1, (fun x => ↑x.snd) a_1 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.intro.intro.intro
G : Type ?u.182986
inst✝² : Group G
X : Type ?u.182992
inst✝¹ : MulAction G X
α : Type u_1
inst✝ : Finite α
c : Set (Set α)
hc : Setoid.IsPartition c
this : Fintype α
a : α
x : Set α
right✝¹ : ∀ (y : Set α), (fun b => ∃! x, a ∈ b) y → y = x
hx : x ∈ c
ha : a ∈ x
right✝ : ∀ (y : x ∈ c), (fun x_1 => a ∈ x) y → y = hx
⊢ ∃ a_1, (fun x => ↑x.snd) a_1 = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | have := Fintype.ofFinite X | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Nat.card X | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Nat.card X | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Nat.card X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | rw [← Setoid.nat_sum (IsBlockSystem.of_block hB hB_ne).1] | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Nat.card X | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = ∑ᶠ (x : ↑(Set.range fun g => g • B)), Nat.card ↑↑x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Nat.card X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | simp only [finsum_eq_sum_of_fintype] | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = ∑ᶠ (x : ↑(Set.range fun g => g • B)), Nat.card ↑↑x | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = ∑ x : ↑(Set.range fun g => g • B), Nat.card ↑↑x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = ∑ᶠ (x : ↑(Set.range fun g => g • B)), Nat.card ↑↑x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | simp_rw [Set.Nat.card_coe_set_eq] | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = ∑ x : ↑(Set.range fun g => g • B), Nat.card ↑↑x | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = ∑ x : ↑(Set.range fun g => g • B), Set.ncard ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = ∑ x : ↑(Set.range fun g => g • B), Nat.card ↑↑x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | rw [Finset.sum_congr rfl ?_] | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = ∑ x : ↑(Set.range fun g => g • B), Set.ncard ↑x | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Finset.sum Finset.univ ?m.191744
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ↑(Set.range fun g => g • B) → ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = ?m.191744 x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = ∑ x : ↑(Set.range fun g => g • B), Set.ncard ↑x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | rw [Finset.sum_const] | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Finset.sum Finset.univ ?m.191744
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ↑(Set.range fun g => g • B) → ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = ?m.191744 x | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Finset.univ.card • ?m.191918
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = ?m.191918 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Finset.sum Finset.univ ?m.191744
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ↑(Set.range fun g => g • B) → ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = ?m.191744 x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | rw [mul_comm] | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Finset.univ.card • ?m.191918
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = ?m.191918 | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard (Set.range fun g => g • B) * Set.ncard B = Finset.univ.card • ?m.191918
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = ?m.191918 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard B * Set.ncard (Set.range fun g => g • B) = Finset.univ.card • ?m.191918
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = ?m.191918
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | congr | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard (Set.range fun g => g • B) * Set.ncard B = Finset.univ.card • ?m.191918
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = ?m.191918 | case e_a
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard (Set.range fun g => g • B) = Finset.univ.card
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard (Set.range fun g => g • B) * Set.ncard B = Finset.univ.card • ?m.191918
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ℕ
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = ?m.191918
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | rw [Set.ncard_eq_toFinset_card'] | case e_a
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard (Set.range fun g => g • B) = Finset.univ.card
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B | case e_a
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ (Set.toFinset (Set.range fun g => g • B)).card = Finset.univ.card
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ Set.ncard (Set.range fun g => g • B) = Finset.univ.card
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | rw [Finset.card_congr] | case e_a
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ (Set.toFinset (Set.range fun g => g • B)).card = Finset.univ.card
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B | case e_a.f
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ (a : Set X) → a ∈ Set.toFinset (Set.range fun g => g • B) → ↑(Set.range fun g => g • B)
case e_a.h₁
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), ?e_a.f✝ a ha ∈ Finset.univ
case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a b : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)) (hb : b ∈ Set.toFinset (Set.range fun g => g • B)),
?e_a.f✝ a ha = ?e_a.f✝ b hb → a = b
case e_a.h₃
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ b ∈ Finset.univ, ∃ a, ∃ (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), ?e_a.f✝ a ha = b
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ (Set.toFinset (Set.range fun g => g • B)).card = Finset.univ.card
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | exact fun s hs => ⟨s, Set.mem_toFinset.mp hs⟩ | case e_a.f
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ (a : Set X) → a ∈ Set.toFinset (Set.range fun g => g • B) → ↑(Set.range fun g => g • B)
case e_a.h₁
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), ?e_a.f✝ a ha ∈ Finset.univ
case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a b : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)) (hb : b ∈ Set.toFinset (Set.range fun g => g • B)),
?e_a.f✝ a ha = ?e_a.f✝ b hb → a = b
case e_a.h₃
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ b ∈ Finset.univ, ∃ a, ∃ (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), ?e_a.f✝ a ha = b
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B | case e_a.h₁
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), { val := a, property := ⋯ } ∈ Finset.univ
case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a b : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)) (hb : b ∈ Set.toFinset (Set.range fun g => g • B)),
{ val := a, property := ⋯ } = { val := b, property := ⋯ } → a = b
case e_a.h₃
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ b ∈ Finset.univ, ∃ a, ∃ (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), { val := a, property := ⋯ } = b
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.f
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ (a : Set X) → a ∈ Set.toFinset (Set.range fun g => g • B) → ↑(Set.range fun g => g • B)
case e_a.h₁
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), ?e_a.f✝ a ha ∈ Finset.univ
case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a b : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)) (hb : b ∈ Set.toFinset (Set.range fun g => g • B)),
?e_a.f✝ a ha = ?e_a.f✝ b hb → a = b
case e_a.h₃
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ b ∈ Finset.univ, ∃ a, ∃ (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), ?e_a.f✝ a ha = b
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | rintro ⟨x, ⟨g, rfl⟩⟩ _ | G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B | case mk.intro
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
g : G
a✝ : { val := (fun g => g • B) g, property := ⋯ } ∈ Finset.univ
⊢ Set.ncard ↑{ val := (fun g => g • B) g, property := ⋯ } = Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ x ∈ Finset.univ, Set.ncard ↑x = Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | simp only | case mk.intro
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
g : G
a✝ : { val := (fun g => g • B) g, property := ⋯ } ∈ Finset.univ
⊢ Set.ncard ↑{ val := (fun g => g • B) g, property := ⋯ } = Set.ncard B | case mk.intro
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
g : G
a✝ : { val := (fun g => g • B) g, property := ⋯ } ∈ Finset.univ
⊢ Set.ncard (g • B) = Set.ncard B | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
g : G
a✝ : { val := (fun g => g • B) g, property := ⋯ } ∈ Finset.univ
⊢ Set.ncard ↑{ val := (fun g => g • B) g, property := ⋯ } = Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | exact Set.ncard_image_of_injective B (MulAction.injective g) | case mk.intro
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
g : G
a✝ : { val := (fun g => g • B) g, property := ⋯ } ∈ Finset.univ
⊢ Set.ncard (g • B) = Set.ncard B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
g : G
a✝ : { val := (fun g => g • B) g, property := ⋯ } ∈ Finset.univ
⊢ Set.ncard (g • B) = Set.ncard B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | intro s hs | case e_a.h₁
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), { val := a, property := ⋯ } ∈ Finset.univ | case e_a.h₁
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
s : Set X
hs : s ∈ Set.toFinset (Set.range fun g => g • B)
⊢ { val := s, property := ⋯ } ∈ Finset.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h₁
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), { val := a, property := ⋯ } ∈ Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | apply Finset.mem_univ | case e_a.h₁
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
s : Set X
hs : s ∈ Set.toFinset (Set.range fun g => g • B)
⊢ { val := s, property := ⋯ } ∈ Finset.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h₁
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
s : Set X
hs : s ∈ Set.toFinset (Set.range fun g => g • B)
⊢ { val := s, property := ⋯ } ∈ Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | intro s t hs ht hst | case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a b : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)) (hb : b ∈ Set.toFinset (Set.range fun g => g • B)),
{ val := a, property := ⋯ } = { val := b, property := ⋯ } → a = b | case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
s t : Set X
hs : s ∈ Set.toFinset (Set.range fun g => g • B)
ht : t ∈ Set.toFinset (Set.range fun g => g • B)
hst : { val := s, property := ⋯ } = { val := t, property := ⋯ }
⊢ s = t | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ (a b : Set X) (ha : a ∈ Set.toFinset (Set.range fun g => g • B)) (hb : b ∈ Set.toFinset (Set.range fun g => g • B)),
{ val := a, property := ⋯ } = { val := b, property := ⋯ } → a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | rw [← Subtype.val_inj] at hst | case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
s t : Set X
hs : s ∈ Set.toFinset (Set.range fun g => g • B)
ht : t ∈ Set.toFinset (Set.range fun g => g • B)
hst : { val := s, property := ⋯ } = { val := t, property := ⋯ }
⊢ s = t | case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
s t : Set X
hs : s ∈ Set.toFinset (Set.range fun g => g • B)
ht : t ∈ Set.toFinset (Set.range fun g => g • B)
hst : ↑{ val := s, property := ⋯ } = ↑{ val := t, property := ⋯ }
⊢ s = t | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
s t : Set X
hs : s ∈ Set.toFinset (Set.range fun g => g • B)
ht : t ∈ Set.toFinset (Set.range fun g => g • B)
hst : { val := s, property := ⋯ } = { val := t, property := ⋯ }
⊢ s = t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | exact hst | case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
s t : Set X
hs : s ∈ Set.toFinset (Set.range fun g => g • B)
ht : t ∈ Set.toFinset (Set.range fun g => g • B)
hst : ↑{ val := s, property := ⋯ } = ↑{ val := t, property := ⋯ }
⊢ s = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h₂
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
s t : Set X
hs : s ∈ Set.toFinset (Set.range fun g => g • B)
ht : t ∈ Set.toFinset (Set.range fun g => g • B)
hst : ↑{ val := s, property := ⋯ } = ↑{ val := t, property := ⋯ }
⊢ s = t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | rintro ⟨b, hb⟩ _ | case e_a.h₃
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ b ∈ Finset.univ, ∃ a, ∃ (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), { val := a, property := ⋯ } = b | case e_a.h₃.mk
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
b : Set X
hb : b ∈ Set.range fun g => g • B
a✝ : { val := b, property := hb } ∈ Finset.univ
⊢ ∃ a, ∃ (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), { val := a, property := ⋯ } = { val := b, property := hb } | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h₃
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
⊢ ∀ b ∈ Finset.univ, ∃ a, ∃ (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), { val := a, property := ⋯ } = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | use b | case e_a.h₃.mk
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
b : Set X
hb : b ∈ Set.range fun g => g • B
a✝ : { val := b, property := hb } ∈ Finset.univ
⊢ ∃ a, ∃ (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), { val := a, property := ⋯ } = { val := b, property := hb } | case h
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
b : Set X
hb : b ∈ Set.range fun g => g • B
a✝ : { val := b, property := hb } ∈ Finset.univ
⊢ ∃ (ha : b ∈ Set.toFinset (Set.range fun g => g • B)), { val := b, property := ⋯ } = { val := b, property := hb } | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h₃.mk
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
b : Set X
hb : b ∈ Set.range fun g => g • B
a✝ : { val := b, property := hb } ∈ Finset.univ
⊢ ∃ a, ∃ (ha : a ∈ Set.toFinset (Set.range fun g => g • B)), { val := a, property := ⋯ } = { val := b, property := hb }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | simp only [Set.mem_toFinset] | case h
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
b : Set X
hb : b ∈ Set.range fun g => g • B
a✝ : { val := b, property := hb } ∈ Finset.univ
⊢ ∃ (ha : b ∈ Set.toFinset (Set.range fun g => g • B)), { val := b, property := ⋯ } = { val := b, property := hb } | case h
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
b : Set X
hb : b ∈ Set.range fun g => g • B
a✝ : { val := b, property := hb } ∈ Finset.univ
⊢ ∃ (_ : b ∈ Set.range fun g => g • B), True | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
b : Set X
hb : b ∈ Set.range fun g => g • B
a✝ : { val := b, property := hb } ∈ Finset.univ
⊢ ∃ (ha : b ∈ Set.toFinset (Set.range fun g => g • B)), { val := b, property := ⋯ } = { val := b, property := hb }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.ncard_block_mul_ncard_orbit_eq | [681, 1] | [709, 63] | use hb | case h
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
b : Set X
hb : b ∈ Set.range fun g => g • B
a✝ : { val := b, property := hb } ∈ Finset.univ
⊢ ∃ (_ : b ∈ Set.range fun g => g • B), True | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝³ : Group G
X : Type u_1
inst✝² : MulAction G X
inst✝¹ : Finite X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB_ne : Set.Nonempty B
this : Fintype X
b : Set X
hb : b ∈ Set.range fun g => g • B
a✝ : { val := b, property := hb } ∈ Finset.univ
⊢ ∃ (_ : b ∈ Set.range fun g => g • B), True
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.is_top_of_large_block | [720, 1] | [745, 41] | letI := Fintype.ofFinite X | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hfX : Finite X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB' : Nat.card X < 2 * Set.ncard B
⊢ B = ⊤ | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hfX : Finite X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB' : Nat.card X < 2 * Set.ncard B
this : Fintype X := Fintype.ofFinite X
⊢ 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
hfX : Finite X
hGX : IsPretransitive G X
B : Set X
hB : IsBlock G B
hB' : Nat.card X < 2 * Set.ncard B
⊢ B = ⊤
TACTIC:
|
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