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/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_coe | [338, 1] | [341, 6] | rfl | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
⊢ ↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx }) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
⊢ ↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx }) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | constructor | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Bijective ⇑(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Injective ⇑(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s)
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Surjective ⇑(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Bijective ⇑(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | rintro ⟨x, hx⟩ ⟨y, hy⟩ h | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Injective ⇑(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy } | Please generate a tactic in lean4 to solve the state.
STATE:
case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Injective ⇑(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | simp only [Subtype.mk_eq_mk] | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy } | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | rw [← SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_coe M hx] | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ x = y | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ ↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx }) = y | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | rw [← SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_coe M hy] | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ ↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx }) = y | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ ↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx }) =
↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }) | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ ↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx }) = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | rw [h] | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ ↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx }) =
↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
y : α
hy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
h :
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx } =
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy }
⊢ ↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := x, property := hx }) =
↑↑((equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := y, property := hy })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | rintro ⟨⟨x, hx1⟩, hx2⟩ | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Surjective ⇑(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
⊢ ∃ a_1,
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) a_1 =
{ val := { val := x, property := hx1 }, property := hx2 } | Please generate a tactic in lean4 to solve the state.
STATE:
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Surjective ⇑(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | refine' ⟨⟨x, _⟩, rfl⟩ | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
⊢ ∃ a_1,
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) a_1 =
{ val := { val := x, property := hx1 }, property := hx2 } | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
⊢ x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s)) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
⊢ ∃ a_1,
(equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) a_1 =
{ val := { val := x, property := hx1 }, property := hx2 }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | rw [SubMulAction.mem_ofFixingSubgroup_iff] | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
⊢ x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s)) | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
⊢ x ∉ insert a (Subtype.val '' s) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
⊢ x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | intro h | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
⊢ x ∉ insert a (Subtype.val '' s) | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
h : x ∈ insert a (Subtype.val '' s)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
⊢ x ∉ insert a (Subtype.val '' s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | cases Set.mem_insert_iff.mp h with
| inl h' => rw [SubMulAction.mem_ofStabilizer_iff] at hx1 ; exact hx1 h'
| inr h' =>
rw [SubMulAction.mem_ofFixingSubgroup_iff] at hx2
apply hx2
obtain ⟨x1, hx1', rfl⟩ := h'
simp only [SetLike.eta]
exact hx1' | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
h : x ∈ insert a (Subtype.val '' s)
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
h : x ∈ insert a (Subtype.val '' s)
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | rw [SubMulAction.mem_ofStabilizer_iff] at hx1 | case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
h : x ∈ insert a (Subtype.val '' s)
h' : x = a
⊢ False | case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1✝ : x ∈ ofStabilizer M a
hx1 : x ≠ a
hx2 : { val := x, property := hx1✝ } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
h : x ∈ insert a (Subtype.val '' s)
h' : x = a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
h : x ∈ insert a (Subtype.val '' s)
h' : x = a
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | exact hx1 h' | case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1✝ : x ∈ ofStabilizer M a
hx1 : x ≠ a
hx2 : { val := x, property := hx1✝ } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
h : x ∈ insert a (Subtype.val '' s)
h' : x = a
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1✝ : x ∈ ofStabilizer M a
hx1 : x ≠ a
hx2 : { val := x, property := hx1✝ } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
h : x ∈ insert a (Subtype.val '' s)
h' : x = a
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | rw [SubMulAction.mem_ofFixingSubgroup_iff] at hx2 | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
h : x ∈ insert a (Subtype.val '' s)
h' : x ∈ Subtype.val '' s
⊢ False | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∉ s
h : x ∈ insert a (Subtype.val '' s)
h' : x ∈ Subtype.val '' s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∈ ofFixingSubgroup (↥(stabilizer M a)) s
h : x ∈ insert a (Subtype.val '' s)
h' : x ∈ Subtype.val '' s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | apply hx2 | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∉ s
h : x ∈ insert a (Subtype.val '' s)
h' : x ∈ Subtype.val '' s
⊢ False | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∉ s
h : x ∈ insert a (Subtype.val '' s)
h' : x ∈ Subtype.val '' s
⊢ { val := x, property := hx1 } ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∉ s
h : x ∈ insert a (Subtype.val '' s)
h' : x ∈ Subtype.val '' s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | obtain ⟨x1, hx1', rfl⟩ := h' | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∉ s
h : x ∈ insert a (Subtype.val '' s)
h' : x ∈ Subtype.val '' s
⊢ { val := x, property := hx1 } ∈ s | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x1 : ↥(ofStabilizer M a)
hx1' : x1 ∈ s
hx1 : ↑x1 ∈ ofStabilizer M a
hx2 : { val := ↑x1, property := hx1 } ∉ s
h : ↑x1 ∈ insert a (Subtype.val '' s)
⊢ { val := ↑x1, property := hx1 } ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x : α
hx1 : x ∈ ofStabilizer M a
hx2 : { val := x, property := hx1 } ∉ s
h : x ∈ insert a (Subtype.val '' s)
h' : x ∈ Subtype.val '' s
⊢ { val := x, property := hx1 } ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | simp only [SetLike.eta] | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x1 : ↥(ofStabilizer M a)
hx1' : x1 ∈ s
hx1 : ↑x1 ∈ ofStabilizer M a
hx2 : { val := ↑x1, property := hx1 } ∉ s
h : ↑x1 ∈ insert a (Subtype.val '' s)
⊢ { val := ↑x1, property := hx1 } ∈ s | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x1 : ↥(ofStabilizer M a)
hx1' : x1 ∈ s
hx1 : ↑x1 ∈ ofStabilizer M a
hx2 : { val := ↑x1, property := hx1 } ∉ s
h : ↑x1 ∈ insert a (Subtype.val '' s)
⊢ x1 ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x1 : ↥(ofStabilizer M a)
hx1' : x1 ∈ s
hx1 : ↑x1 ∈ ofStabilizer M a
hx2 : { val := ↑x1, property := hx1 } ∉ s
h : ↑x1 ∈ insert a (Subtype.val '' s)
⊢ { val := ↑x1, property := hx1 } ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective | [343, 1] | [364, 17] | exact hx1' | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x1 : ↥(ofStabilizer M a)
hx1' : x1 ∈ s
hx1 : ↑x1 ∈ ofStabilizer M a
hx2 : { val := ↑x1, property := hx1 } ∉ s
h : ↑x1 ∈ insert a (Subtype.val '' s)
⊢ x1 ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
x1 : ↥(ofStabilizer M a)
hx1' : x1 ∈ s
hx1 : ↑x1 ∈ ofStabilizer M a
hx2 : { val := ↑x1, property := hx1 } ∉ s
h : ↑x1 ∈ insert a (Subtype.val '' s)
⊢ x1 ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | constructor | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Bijective (MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s)) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Injective (MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s))
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Surjective (MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s)) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Bijective (MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | rintro ⟨m, hm⟩ ⟨n, hn⟩ hmn | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Injective (MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s)) | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
n : M
hn : n ∈ fixingSubgroup M (insert a (Subtype.val '' s))
hmn :
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := hm } =
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := n, property := hn }
⊢ { val := m, property := hm } = { val := n, property := hn } | Please generate a tactic in lean4 to solve the state.
STATE:
case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Injective (MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | rw [← SetLike.coe_eq_coe, ← SetLike.coe_eq_coe] at hmn | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
n : M
hn : n ∈ fixingSubgroup M (insert a (Subtype.val '' s))
hmn :
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := hm } =
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := n, property := hn }
⊢ { val := m, property := hm } = { val := n, property := hn } | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
n : M
hn : n ∈ fixingSubgroup M (insert a (Subtype.val '' s))
hmn :
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := hm }) =
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := n, property := hn })
⊢ { val := m, property := hm } = { val := n, property := hn } | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
n : M
hn : n ∈ fixingSubgroup M (insert a (Subtype.val '' s))
hmn :
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := hm } =
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := n, property := hn }
⊢ { val := m, property := hm } = { val := n, property := hn }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | simp only [Subtype.mk_eq_mk] | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
n : M
hn : n ∈ fixingSubgroup M (insert a (Subtype.val '' s))
hmn :
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := hm }) =
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := n, property := hn })
⊢ { val := m, property := hm } = { val := n, property := hn } | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
n : M
hn : n ∈ fixingSubgroup M (insert a (Subtype.val '' s))
hmn :
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := hm }) =
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := n, property := hn })
⊢ m = n | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
n : M
hn : n ∈ fixingSubgroup M (insert a (Subtype.val '' s))
hmn :
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := hm }) =
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := n, property := hn })
⊢ { val := m, property := hm } = { val := n, property := hn }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | exact hmn | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
n : M
hn : n ∈ fixingSubgroup M (insert a (Subtype.val '' s))
hmn :
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := hm }) =
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := n, property := hn })
⊢ m = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
n : M
hn : n ∈ fixingSubgroup M (insert a (Subtype.val '' s))
hmn :
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := hm }) =
↑↑(MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := n, property := hn })
⊢ m = n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | rintro ⟨⟨m, hm⟩, hm'⟩ | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Surjective (MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s)) | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
⊢ ∃ a_1,
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) a_1 =
{ val := { val := m, property := hm }, property := hm' } | Please generate a tactic in lean4 to solve the state.
STATE:
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
⊢ Function.Surjective (MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | suffices m ∈ fixingSubgroup M (insert a (Subtype.val '' s)) by
use ⟨m, this⟩
rfl | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
⊢ ∃ a_1,
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) a_1 =
{ val := { val := m, property := hm }, property := hm' } | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
⊢ m ∈ fixingSubgroup M (insert a (Subtype.val '' s)) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
⊢ ∃ a_1,
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) a_1 =
{ val := { val := m, property := hm }, property := hm' }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | rw [mem_fixingSubgroup_iff] | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
⊢ m ∈ fixingSubgroup M (insert a (Subtype.val '' s)) | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
⊢ ∀ y ∈ insert a (Subtype.val '' s), m • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
⊢ m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | intro x hx | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
⊢ ∀ y ∈ insert a (Subtype.val '' s), m • y = y | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx : x ∈ insert a (Subtype.val '' s)
⊢ m • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
⊢ ∀ y ∈ insert a (Subtype.val '' s), m • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | cases' Set.mem_insert_iff.mp hx with hx hx | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx : x ∈ insert a (Subtype.val '' s)
⊢ m • x = x | case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx✝ : x ∈ insert a (Subtype.val '' s)
hx : x = a
⊢ m • x = x
case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx✝ : x ∈ insert a (Subtype.val '' s)
hx : x ∈ Subtype.val '' s
⊢ m • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx : x ∈ insert a (Subtype.val '' s)
⊢ m • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | use ⟨m, this⟩ | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
this : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
⊢ ∃ a_1,
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) a_1 =
{ val := { val := m, property := hm }, property := hm' } | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
this : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
⊢ MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := this } =
{ val := { val := m, property := hm }, property := hm' } | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
this : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
⊢ ∃ a_1,
MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) a_1 =
{ val := { val := m, property := hm }, property := hm' }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | rfl | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
this : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
⊢ MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := this } =
{ val := { val := m, property := hm }, property := hm' } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
this : m ∈ fixingSubgroup M (insert a (Subtype.val '' s))
⊢ MulActionHom.toMap (equivariantMap_ofFixingSubgroup_to_ofStabilizer M a s) { val := m, property := this } =
{ val := { val := m, property := hm }, property := hm' }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | rw [hx] | case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx✝ : x ∈ insert a (Subtype.val '' s)
hx : x = a
⊢ m • x = x | case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx✝ : x ∈ insert a (Subtype.val '' s)
hx : x = a
⊢ m • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx✝ : x ∈ insert a (Subtype.val '' s)
hx : x = a
⊢ m • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | exact mem_stabilizer_iff.mp hm | case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx✝ : x ∈ insert a (Subtype.val '' s)
hx : x = a
⊢ m • a = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx✝ : x ∈ insert a (Subtype.val '' s)
hx : x = a
⊢ m • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | obtain ⟨y, hy, rfl⟩ := (Set.mem_image _ _ _).mp hx | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx✝ : x ∈ insert a (Subtype.val '' s)
hx : x ∈ Subtype.val '' s
⊢ m • x = x | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
⊢ m • ↑y = ↑y | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
x : α
hx✝ : x ∈ insert a (Subtype.val '' s)
hx : x ∈ Subtype.val '' s
⊢ m • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | rw [mem_fixingSubgroup_iff] at hm' | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
⊢ m • ↑y = ↑y | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : ∀ y ∈ s, { val := m, property := hm } • y = y
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
⊢ m • ↑y = ↑y | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : { val := m, property := hm } ∈ fixingSubgroup (↥(stabilizer M a)) s
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
⊢ m • ↑y = ↑y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | let hz := hm' y hy | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : ∀ y ∈ s, { val := m, property := hm } • y = y
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
⊢ m • ↑y = ↑y | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : ∀ y ∈ s, { val := m, property := hm } • y = y
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
hz : { val := m, property := hm } • y = y := hm' y hy
⊢ m • ↑y = ↑y | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : ∀ y ∈ s, { val := m, property := hm } • y = y
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
⊢ m • ↑y = ↑y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | rw [← SetLike.coe_eq_coe, SubMulAction.val_smul_of_tower] at hz | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : ∀ y ∈ s, { val := m, property := hm } • y = y
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
hz : { val := m, property := hm } • y = y := hm' y hy
⊢ m • ↑y = ↑y | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : ∀ y ∈ s, { val := m, property := hm } • y = y
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
hz : { val := m, property := hm } • ↑y = ↑y
⊢ m • ↑y = ↑y | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : ∀ y ∈ s, { val := m, property := hm } • y = y
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
hz : { val := m, property := hm } • y = y := hm' y hy
⊢ m • ↑y = ↑y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective | [367, 1] | [390, 15] | exact hz | case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : ∀ y ∈ s, { val := m, property := hm } • y = y
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
hz : { val := m, property := hm } • ↑y = ↑y
⊢ m • ↑y = ↑y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(ofStabilizer M a)
m : M
hm : m ∈ stabilizer M a
hm' : ∀ y ∈ s, { val := m, property := hm } • y = y
y : ↥(ofStabilizer M a)
hy : y ∈ s
hx✝ : ↑y ∈ insert a (Subtype.val '' s)
hx : ↑y ∈ Subtype.val '' s
hz : { val := m, property := hm } • ↑y = ↑y
⊢ m • ↑y = ↑y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | ext h | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
⊢ fixingSubgroup M (g • s) = Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s) | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
⊢ h ∈ fixingSubgroup M (g • s) ↔ h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
⊢ fixingSubgroup M (g • s) = Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | constructor | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
⊢ h ∈ fixingSubgroup M (g • s) ↔ h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s) | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
⊢ h ∈ fixingSubgroup M (g • s) → h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s)
case h.mpr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
⊢ h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s) → h ∈ fixingSubgroup M (g • s) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
⊢ h ∈ fixingSubgroup M (g • s) ↔ h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | intro hh | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
⊢ h ∈ fixingSubgroup M (g • s) → h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s) | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
⊢ h ∈ fixingSubgroup M (g • s) → h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | use (MulAut.conj g⁻¹) h | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s) | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ (MulAut.conj g⁻¹) h ∈ ↑(fixingSubgroup M s) ∧ (MulEquiv.toMonoidHom (MulAut.conj g)) ((MulAut.conj g⁻¹) h) = h | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | simp | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ (MulAut.conj g⁻¹) h ∈ ↑(fixingSubgroup M s) ∧ (MulEquiv.toMonoidHom (MulAut.conj g)) ((MulAut.conj g⁻¹) h) = h | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * h * g ∈ fixingSubgroup M s ∧ g⁻¹ * (g * h * g⁻¹) * g = h | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ (MulAut.conj g⁻¹) h ∈ ↑(fixingSubgroup M s) ∧ (MulEquiv.toMonoidHom (MulAut.conj g)) ((MulAut.conj g⁻¹) h) = h
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | constructor | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * h * g ∈ fixingSubgroup M s ∧ g⁻¹ * (g * h * g⁻¹) * g = h | case h.left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * h * g ∈ fixingSubgroup M s
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * h * g ∈ fixingSubgroup M s ∧ g⁻¹ * (g * h * g⁻¹) * g = h
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | rintro ⟨x, hx⟩ | case h.left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * h * g ∈ fixingSubgroup M s
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h | case h.left.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
x : α
hx : x ∈ s
⊢ (g⁻¹ * h * g) • ↑{ val := x, property := hx } = ↑{ val := x, property := hx }
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * h * g ∈ fixingSubgroup M s
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | simp only [Subtype.coe_mk, ← smul_smul] | case h.left.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
x : α
hx : x ∈ s
⊢ (g⁻¹ * h * g) • ↑{ val := x, property := hx } = ↑{ val := x, property := hx }
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h | case h.left.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
x : α
hx : x ∈ s
⊢ g⁻¹ • h • g • x = x
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
x : α
hx : x ∈ s
⊢ (g⁻¹ * h * g) • ↑{ val := x, property := hx } = ↑{ val := x, property := hx }
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | rw [inv_smul_eq_iff] | case h.left.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
x : α
hx : x ∈ s
⊢ g⁻¹ • h • g • x = x
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h | case h.left.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
x : α
hx : x ∈ s
⊢ h • g • x = g • x
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
x : α
hx : x ∈ s
⊢ g⁻¹ • h • g • x = x
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | simpa only [Subtype.coe_mk] using hh ⟨_, Set.smul_mem_smul_set hx⟩ | case h.left.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
x : α
hx : x ∈ s
⊢ h • g • x = g • x
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h | case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
x : α
hx : x ∈ s
⊢ h • g • x = g • x
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | group | case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
hh : h ∈ fixingSubgroup M (g • s)
⊢ g⁻¹ * (g * h * g⁻¹) * g = h
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | rintro ⟨k, hk, rfl⟩ | case h.mpr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
⊢ h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s) → h ∈ fixingSubgroup M (g • s) | case h.mpr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
⊢ (MulEquiv.toMonoidHom (MulAut.conj g)) k ∈ fixingSubgroup M (g • s) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
h : M
⊢ h ∈ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M s) → h ∈ fixingSubgroup M (g • s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | rintro ⟨x, hx⟩ | case h.mpr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
⊢ (MulEquiv.toMonoidHom (MulAut.conj g)) k ∈ fixingSubgroup M (g • s) | case h.mpr.intro.intro.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
x : α
hx : x ∈ g • s
⊢ (MulEquiv.toMonoidHom (MulAut.conj g)) k • ↑{ val := x, property := hx } = ↑{ val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
⊢ (MulEquiv.toMonoidHom (MulAut.conj g)) k ∈ fixingSubgroup M (g • s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | simp only [MulEquiv.coe_toMonoidHom, MulAut.conj_apply, Subtype.coe_mk, ← smul_smul] | case h.mpr.intro.intro.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
x : α
hx : x ∈ g • s
⊢ (MulEquiv.toMonoidHom (MulAut.conj g)) k • ↑{ val := x, property := hx } = ↑{ val := x, property := hx } | case h.mpr.intro.intro.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
x : α
hx : x ∈ g • s
⊢ g • k • g⁻¹ • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
x : α
hx : x ∈ g • s
⊢ (MulEquiv.toMonoidHom (MulAut.conj g)) k • ↑{ val := x, property := hx } = ↑{ val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | rw [smul_eq_iff_eq_inv_smul] | case h.mpr.intro.intro.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
x : α
hx : x ∈ g • s
⊢ g • k • g⁻¹ • x = x | case h.mpr.intro.intro.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
x : α
hx : x ∈ g • s
⊢ k • g⁻¹ • x = g⁻¹ • x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
x : α
hx : x ∈ g • s
⊢ g • k • g⁻¹ • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_smul_eq_fixingSubgroup_map_conj | [394, 1] | [412, 58] | exact hk ⟨_, Set.mem_smul_set_iff_inv_smul_mem.mp hx⟩ | case h.mpr.intro.intro.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
x : α
hx : x ∈ g • s
⊢ k • g⁻¹ • x = g⁻¹ • x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
g : M
s : Set α
k : M
hk : k ∈ ↑(fixingSubgroup M s)
x : α
hx : x ∈ g • s
⊢ k • g⁻¹ • x = g⁻¹ • x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | constructor | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
⊢ Function.Bijective ⇑(conjMap_ofFixingSubgroup M hst) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
⊢ Function.Injective ⇑(conjMap_ofFixingSubgroup M hst)
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
⊢ Function.Surjective ⇑(conjMap_ofFixingSubgroup M hst) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
⊢ Function.Bijective ⇑(conjMap_ofFixingSubgroup M hst)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
⊢ Function.Injective ⇑(conjMap_ofFixingSubgroup M hst) | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy :
(conjMap_ofFixingSubgroup M hst) { val := x, property := hx } =
(conjMap_ofFixingSubgroup M hst) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy } | Please generate a tactic in lean4 to solve the state.
STATE:
case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
⊢ Function.Injective ⇑(conjMap_ofFixingSubgroup M hst)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | simp only [Subtype.mk_eq_mk] | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy :
(conjMap_ofFixingSubgroup M hst) { val := x, property := hx } =
(conjMap_ofFixingSubgroup M hst) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy } | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy :
(conjMap_ofFixingSubgroup M hst) { val := x, property := hx } =
(conjMap_ofFixingSubgroup M hst) { val := y, property := hy }
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy :
(conjMap_ofFixingSubgroup M hst) { val := x, property := hx } =
(conjMap_ofFixingSubgroup M hst) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | rw [← SetLike.coe_eq_coe] at hxy | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy :
(conjMap_ofFixingSubgroup M hst) { val := x, property := hx } =
(conjMap_ofFixingSubgroup M hst) { val := y, property := hy }
⊢ x = y | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy :
↑((conjMap_ofFixingSubgroup M hst) { val := x, property := hx }) =
↑((conjMap_ofFixingSubgroup M hst) { val := y, property := hy })
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy :
(conjMap_ofFixingSubgroup M hst) { val := x, property := hx } =
(conjMap_ofFixingSubgroup M hst) { val := y, property := hy }
⊢ x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | change g • x = g • y at hxy | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy :
↑((conjMap_ofFixingSubgroup M hst) { val := x, property := hx }) =
↑((conjMap_ofFixingSubgroup M hst) { val := y, property := hy })
⊢ x = y | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : g • x = g • y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy :
↑((conjMap_ofFixingSubgroup M hst) { val := x, property := hx }) =
↑((conjMap_ofFixingSubgroup M hst) { val := y, property := hy })
⊢ x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | apply (MulAction.injective g) hxy | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : g • x = g • y
⊢ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : g • x = g • y
⊢ x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | rintro ⟨x, hx⟩ | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
⊢ Function.Surjective ⇑(conjMap_ofFixingSubgroup M hst) | case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ ∃ a, (conjMap_ofFixingSubgroup M hst) a = { val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
⊢ Function.Surjective ⇑(conjMap_ofFixingSubgroup M hst)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | have hst' : g⁻¹ • t = s := by
apply symm; rw [← inv_smul_eq_iff]; rw [inv_inv]
exact hst | case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ ∃ a, (conjMap_ofFixingSubgroup M hst) a = { val := x, property := hx } | case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ ∃ a, (conjMap_ofFixingSubgroup M hst) a = { val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ ∃ a, (conjMap_ofFixingSubgroup M hst) a = { val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | use (SubMulAction.conjMap_ofFixingSubgroup M hst') ⟨x, hx⟩ | case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ ∃ a, (conjMap_ofFixingSubgroup M hst) a = { val := x, property := hx } | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ (conjMap_ofFixingSubgroup M hst) ((conjMap_ofFixingSubgroup M hst') { val := x, property := hx }) =
{ val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ ∃ a, (conjMap_ofFixingSubgroup M hst) a = { val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | rw [← SetLike.coe_eq_coe] | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ (conjMap_ofFixingSubgroup M hst) ((conjMap_ofFixingSubgroup M hst') { val := x, property := hx }) =
{ val := x, property := hx } | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ ↑((conjMap_ofFixingSubgroup M hst) ((conjMap_ofFixingSubgroup M hst') { val := x, property := hx })) =
↑{ val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ (conjMap_ofFixingSubgroup M hst) ((conjMap_ofFixingSubgroup M hst') { val := x, property := hx }) =
{ val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | change g • g⁻¹ • x = x | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ ↑((conjMap_ofFixingSubgroup M hst) ((conjMap_ofFixingSubgroup M hst') { val := x, property := hx })) =
↑{ val := x, property := hx } | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ g • g⁻¹ • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ ↑((conjMap_ofFixingSubgroup M hst) ((conjMap_ofFixingSubgroup M hst') { val := x, property := hx })) =
↑{ val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | rw [← mul_smul, mul_inv_self, one_smul] | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ g • g⁻¹ • x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
hst' : g⁻¹ • t = s
⊢ g • g⁻¹ • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | apply symm | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ g⁻¹ • t = s | case a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ s = g⁻¹ • t | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ g⁻¹ • t = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | rw [← inv_smul_eq_iff] | case a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ s = g⁻¹ • t | case a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ g⁻¹⁻¹ • s = t | Please generate a tactic in lean4 to solve the state.
STATE:
case a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ s = g⁻¹ • t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | rw [inv_inv] | case a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ g⁻¹⁻¹ • s = t | case a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ g • s = t | Please generate a tactic in lean4 to solve the state.
STATE:
case a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ g⁻¹⁻¹ • s = t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.conjMap_ofFixingSubgroup_bijective | [436, 1] | [452, 44] | exact hst | case a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ g • s = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ g • s = t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | constructor | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
⊢ Function.Bijective ⇑(conjMap M hab) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
⊢ Function.Injective ⇑(conjMap M hab)
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
⊢ Function.Surjective ⇑(conjMap M hab) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
⊢ Function.Bijective ⇑(conjMap M hab)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
⊢ Function.Injective ⇑(conjMap M hab) | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : (conjMap M hab) { val := x, property := hx } = (conjMap M hab) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy } | Please generate a tactic in lean4 to solve the state.
STATE:
case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
⊢ Function.Injective ⇑(conjMap M hab)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | simp only [Subtype.mk_eq_mk] | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : (conjMap M hab) { val := x, property := hx } = (conjMap M hab) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy } | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : (conjMap M hab) { val := x, property := hx } = (conjMap M hab) { val := y, property := hy }
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : (conjMap M hab) { val := x, property := hx } = (conjMap M hab) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | rw [← SetLike.coe_eq_coe] at hxy | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : (conjMap M hab) { val := x, property := hx } = (conjMap M hab) { val := y, property := hy }
⊢ x = y | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : ↑((conjMap M hab) { val := x, property := hx }) = ↑((conjMap M hab) { val := y, property := hy })
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : (conjMap M hab) { val := x, property := hx } = (conjMap M hab) { val := y, property := hy }
⊢ x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | change g • x = g • y at hxy | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : ↑((conjMap M hab) { val := x, property := hx }) = ↑((conjMap M hab) { val := y, property := hy })
⊢ x = y | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : g • x = g • y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : ↑((conjMap M hab) { val := x, property := hx }) = ↑((conjMap M hab) { val := y, property := hy })
⊢ x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | apply (MulAction.injective g) hxy | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : g • x = g • y
⊢ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M a
y : α
hy : y ∈ ofStabilizer M a
hxy : g • x = g • y
⊢ x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | rintro ⟨x, hx⟩ | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
⊢ Function.Surjective ⇑(conjMap M hab) | case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ ∃ a_1, (conjMap M hab) a_1 = { val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
⊢ Function.Surjective ⇑(conjMap M hab)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | use (SubMulAction.ofStabilizer.conjMap M (inv_smul_eq_iff.mpr hab.symm)) ⟨x, hx⟩ | case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ ∃ a_1, (conjMap M hab) a_1 = { val := x, property := hx } | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ (conjMap M hab) ((conjMap M ⋯) { val := x, property := hx }) = { val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ ∃ a_1, (conjMap M hab) a_1 = { val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | rw [← SetLike.coe_eq_coe] | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ (conjMap M hab) ((conjMap M ⋯) { val := x, property := hx }) = { val := x, property := hx } | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ ↑((conjMap M hab) ((conjMap M ⋯) { val := x, property := hx })) = ↑{ val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ (conjMap M hab) ((conjMap M ⋯) { val := x, property := hx }) = { val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | change g • g⁻¹ • x = x | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ ↑((conjMap M hab) ((conjMap M ⋯) { val := x, property := hx })) = ↑{ val := x, property := hx } | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ g • g⁻¹ • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ ↑((conjMap M hab) ((conjMap M ⋯) { val := x, property := hx })) = ↑{ val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofStabilizer.conjMap_bijective | [479, 1] | [492, 30] | simp only [smul_inv_smul] | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ g • g⁻¹ • x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a b : α
g : M
hab : g • a = b
x : α
hx : x ∈ ofStabilizer M b
⊢ g • g⁻¹ • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | constructor | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
⊢ Function.Bijective ⇑(map_ofFixingSubgroupUnion M s t) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
⊢ Function.Injective ⇑(map_ofFixingSubgroupUnion M s t)
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
⊢ Function.Surjective ⇑(map_ofFixingSubgroupUnion M s t) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
⊢ Function.Bijective ⇑(map_ofFixingSubgroupUnion M s t)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | intro a b h | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
⊢ Function.Injective ⇑(map_ofFixingSubgroupUnion M s t) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a b : ↥(ofFixingSubgroup M (s ∪ t))
h : (map_ofFixingSubgroupUnion M s t) a = (map_ofFixingSubgroupUnion M s t) b
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
⊢ Function.Injective ⇑(map_ofFixingSubgroupUnion M s t)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | simp only [← SetLike.coe_eq_coe] at h ⊢ | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a b : ↥(ofFixingSubgroup M (s ∪ t))
h : (map_ofFixingSubgroupUnion M s t) a = (map_ofFixingSubgroupUnion M s t) b
⊢ a = b | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a b : ↥(ofFixingSubgroup M (s ∪ t))
h : ↑↑((map_ofFixingSubgroupUnion M s t) a) = ↑↑((map_ofFixingSubgroupUnion M s t) b)
⊢ ↑a = ↑b | Please generate a tactic in lean4 to solve the state.
STATE:
case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a b : ↥(ofFixingSubgroup M (s ∪ t))
h : (map_ofFixingSubgroupUnion M s t) a = (map_ofFixingSubgroupUnion M s t) b
⊢ a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | exact h | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a b : ↥(ofFixingSubgroup M (s ∪ t))
h : ↑↑((map_ofFixingSubgroupUnion M s t) a) = ↑↑((map_ofFixingSubgroupUnion M s t) b)
⊢ ↑a = ↑b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a b : ↥(ofFixingSubgroup M (s ∪ t))
h : ↑↑((map_ofFixingSubgroupUnion M s t) a) = ↑↑((map_ofFixingSubgroupUnion M s t) b)
⊢ ↑a = ↑b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | rintro ⟨⟨a, ha⟩, ha'⟩ | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
⊢ Function.Surjective ⇑(map_ofFixingSubgroupUnion M s t) | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
⊢ ∃ a_1, (map_ofFixingSubgroupUnion M s t) a_1 = { val := { val := a, property := ha }, property := ha' } | Please generate a tactic in lean4 to solve the state.
STATE:
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
⊢ Function.Surjective ⇑(map_ofFixingSubgroupUnion M s t)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | suffices a ∈ ofFixingSubgroup M (s ∪ t) by
exact ⟨⟨a, this⟩, rfl⟩ | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
⊢ ∃ a_1, (map_ofFixingSubgroupUnion M s t) a_1 = { val := { val := a, property := ha }, property := ha' } | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
⊢ a ∈ ofFixingSubgroup M (s ∪ t) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
⊢ ∃ a_1, (map_ofFixingSubgroupUnion M s t) a_1 = { val := { val := a, property := ha }, property := ha' }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | intro hy | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
⊢ a ∈ ofFixingSubgroup M (s ∪ t) | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
⊢ a ∈ ofFixingSubgroup M (s ∪ t)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | cases' (Set.mem_union a s t).mp hy with h h | case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
⊢ False | case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ s
⊢ False
case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | exact ⟨⟨a, this⟩, rfl⟩ | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
this : a ∈ ofFixingSubgroup M (s ∪ t)
⊢ ∃ a_1, (map_ofFixingSubgroupUnion M s t) a_1 = { val := { val := a, property := ha }, property := ha' } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
this : a ∈ ofFixingSubgroup M (s ∪ t)
⊢ ∃ a_1, (map_ofFixingSubgroupUnion M s t) a_1 = { val := { val := a, property := ha }, property := ha' }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | exact ha h | case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ s
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | apply ha' | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ t
⊢ False | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ t
⊢ { val := a, property := ha } ∈ Subtype.val ⁻¹' t | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ t
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | simp only [Set.mem_preimage] | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ t
⊢ { val := a, property := ha } ∈ Subtype.val ⁻¹' t | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ t
⊢ a ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ t
⊢ { val := a, property := ha } ∈ Subtype.val ⁻¹' t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupUnion.map_bijective | [561, 1] | [576, 14] | exact h | case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ t
⊢ a ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk.mk.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : { val := a, property := ha } ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
hy : a ∈ s ∪ t
h : a ∈ t
⊢ a ∈ t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofFixingSubgroup.mapOfInclusion_injective | [592, 1] | [597, 12] | rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : t ⊆ s
⊢ Function.Injective ⇑(mapOfInclusion M hst) | case mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : t ⊆ s
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : (mapOfInclusion M hst) { val := x, property := hx } = (mapOfInclusion M hst) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy } | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : t ⊆ s
⊢ Function.Injective ⇑(mapOfInclusion M hst)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofFixingSubgroup.mapOfInclusion_injective | [592, 1] | [597, 12] | rw [← SetLike.coe_eq_coe] at hxy ⊢ | case mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : t ⊆ s
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : (mapOfInclusion M hst) { val := x, property := hx } = (mapOfInclusion M hst) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy } | case mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : t ⊆ s
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : ↑((mapOfInclusion M hst) { val := x, property := hx }) = ↑((mapOfInclusion M hst) { val := y, property := hy })
⊢ ↑{ val := x, property := hx } = ↑{ val := y, property := hy } | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : t ⊆ s
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : (mapOfInclusion M hst) { val := x, property := hx } = (mapOfInclusion M hst) { val := y, property := hy }
⊢ { val := x, property := hx } = { val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofFixingSubgroup.mapOfInclusion_injective | [592, 1] | [597, 12] | exact hxy | case mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : t ⊆ s
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : ↑((mapOfInclusion M hst) { val := x, property := hx }) = ↑((mapOfInclusion M hst) { val := y, property := hy })
⊢ ↑{ val := x, property := hx } = ↑{ val := y, property := hy } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : t ⊆ s
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : ↑((mapOfInclusion M hst) { val := x, property := hx }) = ↑((mapOfInclusion M hst) { val := y, property := hy })
⊢ ↑{ val := x, property := hx } = ↑{ val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfSingleton.map_bijective | [610, 1] | [618, 8] | constructor | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ Function.Bijective ⇑(map M a) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ Function.Injective ⇑(map M a)
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ Function.Surjective ⇑(map M a) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ Function.Bijective ⇑(map M a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfSingleton.map_bijective | [610, 1] | [618, 8] | intro _ _ hxy | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ Function.Injective ⇑(map M a) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
a₁✝ a₂✝ : ↥(ofFixingSubgroup M {a})
hxy : (map M a) a₁✝ = (map M a) a₂✝
⊢ a₁✝ = a₂✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ Function.Injective ⇑(map M a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfSingleton.map_bijective | [610, 1] | [618, 8] | exact hxy | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
a₁✝ a₂✝ : ↥(ofFixingSubgroup M {a})
hxy : (map M a) a₁✝ = (map M a) a₂✝
⊢ a₁✝ = a₂✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
a₁✝ a₂✝ : ↥(ofFixingSubgroup M {a})
hxy : (map M a) a₁✝ = (map M a) a₂✝
⊢ a₁✝ = a₂✝
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.