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/PermFibration.lean | arrowAction.mem_stabilizer_iff | [13, 1] | [14, 84] | rw [eq_comm, ← g.comp_symm_eq] | α : Type u_1
ι : Type u_2
p : α → ι
g : Perm α
⊢ g ∈ stabilizer (Perm α) p ↔ p ∘ ⇑g = p | α : Type u_1
ι : Type u_2
p : α → ι
g : Perm α
⊢ g ∈ stabilizer (Perm α) p ↔ p ∘ ⇑g.symm = p | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
ι : Type u_2
p : α → ι
g : Perm α
⊢ g ∈ stabilizer (Perm α) p ↔ p ∘ ⇑g = p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermFibration.lean | arrowAction.mem_stabilizer_iff | [13, 1] | [14, 84] | rfl | α : Type u_1
ι : Type u_2
p : α → ι
g : Perm α
⊢ g ∈ stabilizer (Perm α) p ↔ p ∘ ⇑g.symm = p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
ι : Type u_2
p : α → ι
g : Perm α
⊢ g ∈ stabilizer (Perm α) p ↔ p ∘ ⇑g.symm = p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermFibration.lean | φ_invFun_eq | [18, 1] | [19, 49] | subst h | α : Type u_1
ι : Type u_2
p : α → ι
g : (i : ι) → Perm ↑{a | p a = i}
a : α
i : ι
h : p a = i
⊢ φ_invFun g a = ↑((g i) { val := a, property := h }) | α : Type u_1
ι : Type u_2
p : α → ι
g : (i : ι) → Perm ↑{a | p a = i}
a : α
⊢ φ_invFun g a = ↑((g (p a)) { val := a, property := ⋯ }) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
ι : Type u_2
p : α → ι
g : (i : ι) → Perm ↑{a | p a = i}
a : α
i : ι
h : p a = i
⊢ φ_invFun g a = ↑((g i) { val := a, property := h })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermFibration.lean | φ_invFun_eq | [18, 1] | [19, 49] | rfl | α : Type u_1
ι : Type u_2
p : α → ι
g : (i : ι) → Perm ↑{a | p a = i}
a : α
⊢ φ_invFun g a = ↑((g (p a)) { val := a, property := ⋯ }) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
ι : Type u_2
p : α → ι
g : (i : ι) → Perm ↑{a | p a = i}
a : α
⊢ φ_invFun g a = ↑((g (p a)) { val := a, property := ⋯ })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | MonoidHom.range_isCommutative | [11, 1] | [17, 52] | apply Subgroup.IsCommutative.mk | G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
⊢ Subgroup.IsCommutative (range f) | case is_comm
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
⊢ Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
⊢ Subgroup.IsCommutative (range f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | MonoidHom.range_isCommutative | [11, 1] | [17, 52] | constructor | case is_comm
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
⊢ Std.Commutative fun x x_1 => x * x_1 | case is_comm.comm
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
⊢ ∀ (a b : ↥(range f)), a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
case is_comm
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | MonoidHom.range_isCommutative | [11, 1] | [17, 52] | rintro ⟨_, a, rfl⟩ ⟨_, b, rfl⟩ | case is_comm.comm
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
⊢ ∀ (a b : ↥(range f)), a * b = b * a | case is_comm.comm.mk.intro.mk.intro
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
a b : G
⊢ { val := f a, property := ⋯ } * { val := f b, property := ⋯ } =
{ val := f b, property := ⋯ } * { val := f a, property := ⋯ } | Please generate a tactic in lean4 to solve the state.
STATE:
case is_comm.comm
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
⊢ ∀ (a b : ↥(range f)), a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | MonoidHom.range_isCommutative | [11, 1] | [17, 52] | rw [← Subtype.coe_inj] | case is_comm.comm.mk.intro.mk.intro
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
a b : G
⊢ { val := f a, property := ⋯ } * { val := f b, property := ⋯ } =
{ val := f b, property := ⋯ } * { val := f a, property := ⋯ } | case is_comm.comm.mk.intro.mk.intro
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
a b : G
⊢ ↑({ val := f a, property := ⋯ } * { val := f b, property := ⋯ }) =
↑({ val := f b, property := ⋯ } * { val := f a, property := ⋯ }) | Please generate a tactic in lean4 to solve the state.
STATE:
case is_comm.comm.mk.intro.mk.intro
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
a b : G
⊢ { val := f a, property := ⋯ } * { val := f b, property := ⋯ } =
{ val := f b, property := ⋯ } * { val := f a, property := ⋯ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | MonoidHom.range_isCommutative | [11, 1] | [17, 52] | simp only [Submonoid.coe_mul, ← map_mul, hG.comm] | case is_comm.comm.mk.intro.mk.intro
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
a b : G
⊢ ↑({ val := f a, property := ⋯ } * { val := f b, property := ⋯ }) =
↑({ val := f b, property := ⋯ } * { val := f a, property := ⋯ }) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case is_comm.comm.mk.intro.mk.intro
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
hG : Std.Commutative fun x x_1 => x * x_1
a b : G
⊢ ↑({ val := f a, property := ⋯ } * { val := f b, property := ⋯ }) =
↑({ val := f b, property := ⋯ } * { val := f a, property := ⋯ })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.perm_is_nontrivial | [20, 1] | [22, 85] | rw [← Fintype.one_lt_card_iff_nontrivial, Fintype.card_perm, Nat.one_lt_factorial] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
⊢ 1 < Fintype.card α ↔ Nontrivial (Perm α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
⊢ 1 < Fintype.card α ↔ Nontrivial (Perm α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | by_contra h | G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
⊢ ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b | G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
h : ¬∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
⊢ ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | apply not_lt.mpr hG | G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
h : ¬∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ False | G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
h : ¬∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ 2 < Fintype.card G | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
h : ¬∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | push_neg at h | G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
h : ¬∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ 2 < Fintype.card G | G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
h : ∃ a b, a ≠ 1 ∧ b ≠ 1 ∧ a ≠ b
⊢ 2 < Fintype.card G | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
h : ¬∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ 2 < Fintype.card G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | obtain ⟨a, b, ha1, hb1, hab⟩ := h | G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
h : ∃ a b, a ≠ 1 ∧ b ≠ 1 ∧ a ≠ b
⊢ 2 < Fintype.card G | case intro.intro.intro.intro
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
a b : G
ha1 : a ≠ 1
hb1 : b ≠ 1
hab : a ≠ b
⊢ 2 < Fintype.card G | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
h : ∃ a b, a ≠ 1 ∧ b ≠ 1 ∧ a ≠ b
⊢ 2 < Fintype.card G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | rw [Fintype.two_lt_card_iff] | case intro.intro.intro.intro
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
a b : G
ha1 : a ≠ 1
hb1 : b ≠ 1
hab : a ≠ b
⊢ 2 < Fintype.card G | case intro.intro.intro.intro
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
a b : G
ha1 : a ≠ 1
hb1 : b ≠ 1
hab : a ≠ b
⊢ ∃ a b c, a ≠ b ∧ a ≠ c ∧ b ≠ c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
a b : G
ha1 : a ≠ 1
hb1 : b ≠ 1
hab : a ≠ b
⊢ 2 < Fintype.card G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | exact ⟨a, b, 1, hab, ha1, hb1⟩ | case intro.intro.intro.intro
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
a b : G
ha1 : a ≠ 1
hb1 : b ≠ 1
hab : a ≠ b
⊢ ∃ a b c, a ≠ b ∧ a ≠ c ∧ b ≠ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
a b : G
ha1 : a ≠ 1
hb1 : b ≠ 1
hab : a ≠ b
⊢ ∃ a b c, a ≠ b ∧ a ≠ c ∧ b ≠ c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | constructor | G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ Std.Commutative fun x x_1 => x * x_1 | case comm
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ ∀ (a b : G), a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | intro a b | case comm
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ ∀ (a b : G), a * b = b * a | case comm
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
⊢ a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
case comm
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
⊢ ∀ (a b : G), a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | cases' dec_em (a = 1) with ha ha | case comm
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
⊢ a * b = b * a | case comm.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : a = 1
⊢ a * b = b * a
case comm.inr
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
⊢ a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
case comm
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
⊢ a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | cases' dec_em (b = 1) with hb hb | case comm.inr
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
⊢ a * b = b * a | case comm.inr.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
hb : b = 1
⊢ a * b = b * a
case comm.inr.inr
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
hb : ¬b = 1
⊢ a * b = b * a | Please generate a tactic in lean4 to solve the state.
STATE:
case comm.inr
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
⊢ a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | rw [this a b ha hb] | case comm.inr.inr
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
hb : ¬b = 1
⊢ a * b = b * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case comm.inr.inr
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
hb : ¬b = 1
⊢ a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | rw [ha] | case comm.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : a = 1
⊢ a * b = b * a | case comm.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : a = 1
⊢ 1 * b = b * 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case comm.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : a = 1
⊢ a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | simp only [one_mul, mul_one] | case comm.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : a = 1
⊢ 1 * b = b * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case comm.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : a = 1
⊢ 1 * b = b * 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | rw [hb] | case comm.inr.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
hb : b = 1
⊢ a * b = b * a | case comm.inr.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
hb : b = 1
⊢ a * 1 = 1 * a | Please generate a tactic in lean4 to solve the state.
STATE:
case comm.inr.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
hb : b = 1
⊢ a * b = b * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Monoid.isCommutative_of_fintype_card_le_2 | [25, 1] | [41, 33] | simp only [one_mul, mul_one] | case comm.inr.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
hb : b = 1
⊢ a * 1 = 1 * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case comm.inr.inl
G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Fintype G
inst✝ : Monoid G
hG : Fintype.card G ≤ 2
this : ∀ (a b : G), a ≠ 1 → b ≠ 1 → a = b
a b : G
ha : ¬a = 1
hb : b = 1
⊢ a * 1 = 1 * a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | cases' Nat.lt_or_ge 2 (Fintype.card α) with hα hα | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ Fintype.card α ≤ 2 | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ Fintype.card α ≤ 2
case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ Fintype.card α ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ Fintype.card α ≤ 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | rw [← not_lt] | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ Fintype.card α ≤ 2 | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ ¬2 < Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ Fintype.card α ≤ 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | simp only [hα, not_true_eq_false, iff_false] | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ ¬2 < Fintype.card α | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
⊢ ¬Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ ¬2 < Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | intro h | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
⊢ ¬Std.Commutative fun x x_1 => x * x_1 | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
h : Std.Commutative fun x x_1 => x * x_1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
⊢ ¬Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | rw [Fintype.two_lt_card_iff] at hα | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
h : Std.Commutative fun x x_1 => x * x_1
⊢ False | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : ∃ a b c, a ≠ b ∧ a ≠ c ∧ b ≠ c
h : Std.Commutative fun x x_1 => x * x_1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 < Fintype.card α
h : Std.Commutative fun x x_1 => x * x_1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | obtain ⟨a, b, c, hab, hac, hbc⟩ := hα | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : ∃ a b c, a ≠ b ∧ a ≠ c ∧ b ≠ c
h : Std.Commutative fun x x_1 => x * x_1
⊢ False | case inl.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : ∃ a b c, a ≠ b ∧ a ≠ c ∧ b ≠ c
h : Std.Commutative fun x x_1 => x * x_1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | apply hbc | case inl.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ False | case inl.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ b = c | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | convert Equiv.ext_iff.mp (h.comm (Equiv.swap a c) (Equiv.swap a b)) a | case inl.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ b = c | case h.e'_2
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ b = (swap a c * swap a b) a
case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ c = (swap a b * swap a c) a | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ b = c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | rw [coe_mul, Function.comp_apply,
Equiv.swap_apply_left, Equiv.swap_apply_of_ne_of_ne hab.symm hbc] | case h.e'_2
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ b = (swap a c * swap a b) a
case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ c = (swap a b * swap a c) a | case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ c = (swap a b * swap a c) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ b = (swap a c * swap a b) a
case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ c = (swap a b * swap a c) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | rw [coe_mul, Function.comp_apply,
Equiv.swap_apply_left, Equiv.swap_apply_of_ne_of_ne hac.symm hbc.symm] | case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ c = (swap a b * swap a c) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
h : Std.Commutative fun x x_1 => x * x_1
a b c : α
hab : a ≠ b
hac : a ≠ c
hbc : b ≠ c
⊢ c = (swap a b * swap a c) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | simp only [hα, iff_true] | case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ Fintype.card α ≤ 2 | case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ (Std.Commutative fun x x_1 => x * x_1) ↔ Fintype.card α ≤ 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | apply Monoid.isCommutative_of_fintype_card_le_2 | case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ Std.Commutative fun x x_1 => x * x_1 | case inr.hG
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ Fintype.card (Perm α) ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | rw [← Nat.factorial_two, Fintype.card_perm] | case inr.hG
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ Fintype.card (Perm α) ≤ 2 | case inr.hG
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ Nat.factorial (Fintype.card α) ≤ Nat.factorial 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hG
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ Fintype.card (Perm α) ≤ 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/GroupTheory/Subgroup/Basic.lean | Equiv.Perm.isCommutative_iff | [44, 1] | [62, 30] | exact Nat.factorial_le hα | case inr.hG
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ Nat.factorial (Fintype.card α) ≤ Nat.factorial 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hG
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 2 ≥ Fintype.card α
⊢ Nat.factorial (Fintype.card α) ≤ Nat.factorial 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.image_inclusion | [58, 1] | [69, 8] | ext a | M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
⊢ Set.range ⇑(inclusion s) = s.carrier | case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ a ∈ Set.range ⇑(inclusion s) ↔ a ∈ s.carrier | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
⊢ Set.range ⇑(inclusion s) = s.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.image_inclusion | [58, 1] | [69, 8] | simp only [Set.mem_range, Subtype.exists, mem_carrier, SetLike.mem_coe] | case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ a ∈ Set.range ⇑(inclusion s) ↔ a ∈ s.carrier | case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ (∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a) ↔ a ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ a ∈ Set.range ⇑(inclusion s) ↔ a ∈ s.carrier
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.image_inclusion | [58, 1] | [69, 8] | constructor | case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ (∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a) ↔ a ∈ s | case h.mp
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ (∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a) → a ∈ s
case h.mpr
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ a ∈ s → ∃ a_2, ∃ (b : a_2 ∈ s), (inclusion s) { val := a_2, property := b } = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ (∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a) ↔ a ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.image_inclusion | [58, 1] | [69, 8] | intro ha | case h.mp
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ (∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a) → a ∈ s | case h.mp
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
ha : ∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a
⊢ a ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ (∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a) → a ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.image_inclusion | [58, 1] | [69, 8] | obtain ⟨a, h, rfl⟩ := ha | case h.mp
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
ha : ∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a
⊢ a ∈ s | case h.mp.intro.intro
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ (inclusion s) { val := a, property := h } ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
ha : ∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a
⊢ a ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.image_inclusion | [58, 1] | [69, 8] | exact h | case h.mp.intro.intro
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ (inclusion s) { val := a, property := h } ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ (inclusion s) { val := a, property := h } ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.image_inclusion | [58, 1] | [69, 8] | intro h | case h.mpr
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ a ∈ s → ∃ a_2, ∃ (b : a_2 ∈ s), (inclusion s) { val := a_2, property := b } = a | case h.mpr
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ ∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
⊢ a ∈ s → ∃ a_2, ∃ (b : a_2 ∈ s), (inclusion s) { val := a_2, property := b } = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.image_inclusion | [58, 1] | [69, 8] | use a | case h.mpr
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ ∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a | case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ ∃ (b : a ∈ s), (inclusion s) { val := a, property := b } = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ ∃ a_1, ∃ (b : a_1 ∈ s), (inclusion s) { val := a_1, property := b } = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.image_inclusion | [58, 1] | [69, 8] | use h | case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ ∃ (b : a ∈ s), (inclusion s) { val := a, property := b } = a | case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ (inclusion s) { val := a, property := h } = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ ∃ (b : a ∈ s), (inclusion s) { val := a, property := b } = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.image_inclusion | [58, 1] | [69, 8] | rfl | case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ (inclusion s) { val := a, property := h } = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_1
N : Type ?u.2410
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
h : a ∈ s
⊢ (inclusion s) { val := a, property := h } = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.inclusion_injective | [71, 1] | [75, 10] | rintro ⟨a, ha⟩ ⟨b, hb⟩ h | M : Type u_1
N : Type ?u.3198
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
⊢ Function.Injective ⇑(inclusion s) | case mk.mk
M : Type u_1
N : Type ?u.3198
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
h : (inclusion s) { val := a, property := ha } = (inclusion s) { val := b, property := hb }
⊢ { val := a, property := ha } = { val := b, property := hb } | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
N : Type ?u.3198
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
⊢ Function.Injective ⇑(inclusion s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.inclusion_injective | [71, 1] | [75, 10] | simp only [Subtype.mk.injEq] | case mk.mk
M : Type u_1
N : Type ?u.3198
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
h : (inclusion s) { val := a, property := ha } = (inclusion s) { val := b, property := hb }
⊢ { val := a, property := ha } = { val := b, property := hb } | case mk.mk
M : Type u_1
N : Type ?u.3198
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
h : (inclusion s) { val := a, property := ha } = (inclusion s) { val := b, property := hb }
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
M : Type u_1
N : Type ?u.3198
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
h : (inclusion s) { val := a, property := ha } = (inclusion s) { val := b, property := hb }
⊢ { val := a, property := ha } = { val := b, property := hb }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.inclusion_injective | [71, 1] | [75, 10] | exact h | case mk.mk
M : Type u_1
N : Type ?u.3198
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
h : (inclusion s) { val := a, property := ha } = (inclusion s) { val := b, property := hb }
⊢ a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
M : Type u_1
N : Type ?u.3198
α : Type u_2
inst✝ : SMul M α
s : SubMulAction M α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
h : (inclusion s) { val := a, property := ha } = (inclusion s) { val := b, property := hb }
⊢ a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.add_card_ofStabilizer_eq | [130, 1] | [138, 55] | unfold PartENat.card | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 + PartENat.card ↥(ofStabilizer M a) = PartENat.card α | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 + Cardinal.toPartENat (Cardinal.mk ↥(ofStabilizer M a)) = Cardinal.toPartENat (Cardinal.mk α) | 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
⊢ 1 + PartENat.card ↥(ofStabilizer M a) = PartENat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.add_card_ofStabilizer_eq | [130, 1] | [138, 55] | rw [← Cardinal.mk_sum_compl {a}, map_add] | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 + Cardinal.toPartENat (Cardinal.mk ↥(ofStabilizer M a)) = Cardinal.toPartENat (Cardinal.mk α) | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 + Cardinal.toPartENat (Cardinal.mk ↥(ofStabilizer M a)) =
Cardinal.toPartENat (Cardinal.mk ↑{a}) + Cardinal.toPartENat (Cardinal.mk ↑{a}ᶜ) | 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
⊢ 1 + Cardinal.toPartENat (Cardinal.mk ↥(ofStabilizer M a)) = Cardinal.toPartENat (Cardinal.mk α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.add_card_ofStabilizer_eq | [130, 1] | [138, 55] | congr | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 + Cardinal.toPartENat (Cardinal.mk ↥(ofStabilizer M a)) =
Cardinal.toPartENat (Cardinal.mk ↑{a}) + Cardinal.toPartENat (Cardinal.mk ↑{a}ᶜ) | case e_a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 = Cardinal.toPartENat (Cardinal.mk ↑{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
⊢ 1 + Cardinal.toPartENat (Cardinal.mk ↥(ofStabilizer M a)) =
Cardinal.toPartENat (Cardinal.mk ↑{a}) + Cardinal.toPartENat (Cardinal.mk ↑{a}ᶜ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.add_card_ofStabilizer_eq | [130, 1] | [138, 55] | simp only [Cardinal.mk_fintype, Fintype.card_ofSubsingleton, Nat.cast_one] | case e_a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 = Cardinal.toPartENat (Cardinal.mk ↑{a}) | case e_a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 = Cardinal.toPartENat 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 = Cardinal.toPartENat (Cardinal.mk ↑{a})
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.add_card_ofStabilizer_eq | [130, 1] | [138, 55] | conv_lhs => rw [← Nat.cast_one] | case e_a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 = Cardinal.toPartENat 1 | case e_a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ ↑1 = Cardinal.toPartENat 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ 1 = Cardinal.toPartENat 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.add_card_ofStabilizer_eq | [130, 1] | [138, 55] | apply symm | case e_a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ ↑1 = Cardinal.toPartENat 1 | case e_a.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ Cardinal.toPartENat 1 = ↑1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ ↑1 = Cardinal.toPartENat 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.add_card_ofStabilizer_eq | [130, 1] | [138, 55] | exact Iff.mpr Cardinal.toPartENat_eq_natCast_iff rfl | case e_a.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ Cardinal.toPartENat 1 = ↑1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
⊢ Cardinal.toPartENat 1 = ↑1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | ext m | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
⊢ fixingSubgroup M (insert a ((fun x => ↑x) '' s)) =
Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s) | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
⊢ m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s)) ↔
m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer 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 ↥(SubMulAction.ofStabilizer M a)
⊢ fixingSubgroup M (insert a ((fun x => ↑x) '' s)) =
Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | constructor | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
⊢ m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s)) ↔
m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s) | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
⊢ m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s)) →
m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s)
case h.mpr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
⊢ m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s) →
m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' 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
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
⊢ m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s)) ↔
m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | intro hm | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
⊢ m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s)) →
m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s) | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s))
⊢ m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) 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
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
⊢ m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s)) →
m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | rw [mem_fixingSubgroup_iff] at hm | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s))
⊢ m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s) | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) 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
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s))
⊢ m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | rw [Subgroup.mem_map] | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s) | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ ∃ x ∈ fixingSubgroup (↥(stabilizer M a)) s, (Subgroup.subtype (stabilizer M a)) x = m | 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
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | suffices hm' : m ∈ stabilizer M a by
use ⟨m, hm'⟩
simp only [Subgroup.coeSubtype, and_true]
rw [mem_fixingSubgroup_iff]
rintro ⟨y, hy⟩ hy'
simp only [SetLike.mk_smul_mk, Subtype.mk.injEq]
change m • y = y
apply hm
apply Set.mem_insert_of_mem
use ⟨y, hy⟩ | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ ∃ x ∈ fixingSubgroup (↥(stabilizer M a)) s, (Subgroup.subtype (stabilizer M a)) x = m | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ m ∈ stabilizer M a | 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
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ ∃ x ∈ fixingSubgroup (↥(stabilizer M a)) s, (Subgroup.subtype (stabilizer M a)) x = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | simp only [mem_stabilizer_iff] | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ m ∈ stabilizer M a | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ m • a = a | 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
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ m ∈ stabilizer M a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | apply hm | case h.mp
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ m • a = a | case h.mp.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ a ∈ insert a ((fun x => ↑x) '' 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
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ m • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | apply Set.mem_insert a | case h.mp.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ a ∈ insert a ((fun x => ↑x) '' s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
⊢ a ∈ insert a ((fun x => ↑x) '' s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | use ⟨m, hm'⟩ | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ ∃ x ∈ fixingSubgroup (↥(stabilizer M a)) s, (Subgroup.subtype (stabilizer M a)) x = m | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ { val := m, property := hm' } ∈ fixingSubgroup (↥(stabilizer M a)) s ∧
(Subgroup.subtype (stabilizer M a)) { val := m, property := hm' } = m | 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 ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ ∃ x ∈ fixingSubgroup (↥(stabilizer M a)) s, (Subgroup.subtype (stabilizer M a)) x = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | simp only [Subgroup.coeSubtype, and_true] | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ { val := m, property := hm' } ∈ fixingSubgroup (↥(stabilizer M a)) s ∧
(Subgroup.subtype (stabilizer M a)) { val := m, property := hm' } = m | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ { val := m, property := hm' } ∈ fixingSubgroup (↥(stabilizer M a)) 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
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ { val := m, property := hm' } ∈ fixingSubgroup (↥(stabilizer M a)) s ∧
(Subgroup.subtype (stabilizer M a)) { val := m, property := hm' } = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | rw [mem_fixingSubgroup_iff] | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ { val := m, property := hm' } ∈ fixingSubgroup (↥(stabilizer M a)) s | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ ∀ y ∈ s, { val := m, property := hm' } • y = y | 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 ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ { val := m, property := hm' } ∈ fixingSubgroup (↥(stabilizer M a)) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | rintro ⟨y, hy⟩ hy' | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ ∀ y ∈ s, { val := m, property := hm' } • y = y | case h.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ { val := m, property := hm' } • { val := y, property := hy } = { val := y, property := hy } | 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 ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
⊢ ∀ y ∈ s, { val := m, property := hm' } • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | simp only [SetLike.mk_smul_mk, Subtype.mk.injEq] | case h.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ { val := m, property := hm' } • { val := y, property := hy } = { val := y, property := hy } | case h.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ { val := m, property := hm' } • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ { val := m, property := hm' } • { val := y, property := hy } = { val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | change m • y = y | case h.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ { val := m, property := hm' } • y = y | case h.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ m • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ { val := m, property := hm' } • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | apply hm | case h.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ m • y = y | case h.mk.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ y ∈ insert a ((fun x => ↑x) '' s) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ m • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | apply Set.mem_insert_of_mem | case h.mk.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ y ∈ insert a ((fun x => ↑x) '' s) | case h.mk.a.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ y ∈ (fun x => ↑x) '' s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ y ∈ insert a ((fun x => ↑x) '' s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | use ⟨y, hy⟩ | case h.mk.a.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ y ∈ (fun x => ↑x) '' s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.a.a
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
hm : ∀ y ∈ insert a ((fun x => ↑x) '' s), m • y = y
hm' : m ∈ stabilizer M a
y : α
hy : y ∈ SubMulAction.ofStabilizer M a
hy' : { val := y, property := hy } ∈ s
⊢ y ∈ (fun x => ↑x) '' s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | rintro ⟨⟨n, hn'⟩, hn, rfl⟩ | case h.mpr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
⊢ m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s) →
m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s)) | case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : { val := n, property := hn' } ∈ ↑(fixingSubgroup (↥(stabilizer M a)) s)
⊢ (Subgroup.subtype (stabilizer M a)) { val := n, property := hn' } ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' 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
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
m : M
⊢ m ∈ Subgroup.map (Subgroup.subtype (stabilizer M a)) (fixingSubgroup (↥(stabilizer M a)) s) →
m ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | simp only [Subgroup.coeSubtype, SetLike.mem_coe, mem_fixingSubgroup_iff] at hn ⊢ | case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : { val := n, property := hn' } ∈ ↑(fixingSubgroup (↥(stabilizer M a)) s)
⊢ (Subgroup.subtype (stabilizer M a)) { val := n, property := hn' } ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s)) | case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
⊢ ∀ y ∈ insert a ((fun x => ↑x) '' s), n • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : { val := n, property := hn' } ∈ ↑(fixingSubgroup (↥(stabilizer M a)) s)
⊢ (Subgroup.subtype (stabilizer M a)) { val := n, property := hn' } ∈ fixingSubgroup M (insert a ((fun x => ↑x) '' s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | intro x hx | case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
⊢ ∀ y ∈ insert a ((fun x => ↑x) '' s), n • y = y | case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ insert a ((fun x => ↑x) '' s)
⊢ n • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
⊢ ∀ y ∈ insert a ((fun x => ↑x) '' s), n • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | rw [Set.mem_insert_iff] at hx | case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ insert a ((fun x => ↑x) '' s)
⊢ n • x = x | case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x = a ∨ x ∈ (fun x => ↑x) '' s
⊢ n • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ insert a ((fun x => ↑x) '' s)
⊢ n • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | cases' hx with hx hx | case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x = a ∨ x ∈ (fun x => ↑x) '' s
⊢ n • x = x | case h.mpr.intro.mk.intro.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x = a
⊢ n • x = x
case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ (fun x => ↑x) '' s
⊢ n • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x = a ∨ x ∈ (fun x => ↑x) '' s
⊢ n • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | . simpa [hx] using hn' | case h.mpr.intro.mk.intro.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x = a
⊢ n • x = x
case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ (fun x => ↑x) '' s
⊢ n • x = x | case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ (fun x => ↑x) '' s
⊢ n • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk.intro.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x = a
⊢ n • x = x
case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ (fun x => ↑x) '' s
⊢ n • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | . simp only [Set.mem_image] at hx
rcases hx with ⟨y, hy, rfl⟩
conv_rhs => rw [← hn y hy] | case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ (fun x => ↑x) '' s
⊢ n • x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ (fun x => ↑x) '' s
⊢ n • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | simpa [hx] using hn' | case h.mpr.intro.mk.intro.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x = a
⊢ n • x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk.intro.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x = a
⊢ n • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | simp only [Set.mem_image] at hx | case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ (fun x => ↑x) '' s
⊢ n • x = x | case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : ∃ x_1 ∈ s, ↑x_1 = x
⊢ n • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : x ∈ (fun x => ↑x) '' s
⊢ n • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | rcases hx with ⟨y, hy, rfl⟩ | case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : ∃ x_1 ∈ s, ↑x_1 = x
⊢ n • x = x | case h.mpr.intro.mk.intro.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
y : ↥(SubMulAction.ofStabilizer M a)
hy : y ∈ s
⊢ n • ↑y = ↑y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk.intro.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
x : α
hx : ∃ x_1 ∈ s, ↑x_1 = x
⊢ n • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | fixingSubgroup_of_insert | [191, 1] | [222, 33] | conv_rhs => rw [← hn y hy] | case h.mpr.intro.mk.intro.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
y : ↥(SubMulAction.ofStabilizer M a)
hy : y ∈ s
⊢ n • ↑y = ↑y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.mk.intro.inr.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
s : Set ↥(SubMulAction.ofStabilizer M a)
n : M
hn' : n ∈ stabilizer M a
hn : ∀ y ∈ s, { val := n, property := hn' } • y = y
y : ↥(SubMulAction.ofStabilizer M a)
hy : y ∈ s
⊢ n • ↑y = ↑y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective | [258, 1] | [267, 8] | constructor | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
⊢ Function.Bijective ⇑(ofFixingSubgroupEmpty_equivariantMap M α) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
⊢ Function.Injective ⇑(ofFixingSubgroupEmpty_equivariantMap M α)
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
⊢ Function.Surjective ⇑(ofFixingSubgroupEmpty_equivariantMap M α) | 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
⊢ Function.Bijective ⇑(ofFixingSubgroupEmpty_equivariantMap M α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective | [258, 1] | [267, 8] | rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
⊢ Function.Injective ⇑(ofFixingSubgroupEmpty_equivariantMap M α) | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
x : α
hx : x ∈ ofFixingSubgroup M ∅
y : α
hy : y ∈ ofFixingSubgroup M ∅
hxy :
(ofFixingSubgroupEmpty_equivariantMap M α) { val := x, property := hx } =
(ofFixingSubgroupEmpty_equivariantMap M α) { 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
⊢ Function.Injective ⇑(ofFixingSubgroupEmpty_equivariantMap M α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective | [258, 1] | [267, 8] | 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
x : α
hx : x ∈ ofFixingSubgroup M ∅
y : α
hy : y ∈ ofFixingSubgroup M ∅
hxy :
(ofFixingSubgroupEmpty_equivariantMap M α) { val := x, property := hx } =
(ofFixingSubgroupEmpty_equivariantMap M α) { 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
x : α
hx : x ∈ ofFixingSubgroup M ∅
y : α
hy : y ∈ ofFixingSubgroup M ∅
hxy :
(ofFixingSubgroupEmpty_equivariantMap M α) { val := x, property := hx } =
(ofFixingSubgroupEmpty_equivariantMap M α) { 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
x : α
hx : x ∈ ofFixingSubgroup M ∅
y : α
hy : y ∈ ofFixingSubgroup M ∅
hxy :
(ofFixingSubgroupEmpty_equivariantMap M α) { val := x, property := hx } =
(ofFixingSubgroupEmpty_equivariantMap M α) { 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.ofFixingSubgroupEmpty_equivariantMap_bijective | [258, 1] | [267, 8] | exact hxy | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
x : α
hx : x ∈ ofFixingSubgroup M ∅
y : α
hy : y ∈ ofFixingSubgroup M ∅
hxy :
(ofFixingSubgroupEmpty_equivariantMap M α) { val := x, property := hx } =
(ofFixingSubgroupEmpty_equivariantMap M α) { val := y, property := hy }
⊢ 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
x : α
hx : x ∈ ofFixingSubgroup M ∅
y : α
hy : y ∈ ofFixingSubgroup M ∅
hxy :
(ofFixingSubgroupEmpty_equivariantMap M α) { val := x, property := hx } =
(ofFixingSubgroupEmpty_equivariantMap M α) { val := y, property := hy }
⊢ x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective | [258, 1] | [267, 8] | intro x | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
⊢ Function.Surjective ⇑(ofFixingSubgroupEmpty_equivariantMap M α) | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
x : α
⊢ ∃ a, (ofFixingSubgroupEmpty_equivariantMap M α) a = x | 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
⊢ Function.Surjective ⇑(ofFixingSubgroupEmpty_equivariantMap M α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective | [258, 1] | [267, 8] | use ⟨x, (SubMulAction.mem_ofFixingSubgroup_iff M).mp (Set.not_mem_empty x)⟩ | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
x : α
⊢ ∃ a, (ofFixingSubgroupEmpty_equivariantMap M α) a = x | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
x : α
⊢ (ofFixingSubgroupEmpty_equivariantMap M α) { val := x, property := ⋯ } = x | 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
x : α
⊢ ∃ a, (ofFixingSubgroupEmpty_equivariantMap M α) a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective | [258, 1] | [267, 8] | rfl | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
x : α
⊢ (ofFixingSubgroupEmpty_equivariantMap M α) { val := x, property := ⋯ } = 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
x : α
⊢ (ofFixingSubgroupEmpty_equivariantMap M α) { val := x, property := ⋯ } = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.of_fixingSubgroupEmpty_mapScalars_surjective | [270, 1] | [276, 72] | intro g | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
⊢ Function.Surjective ⇑(Subgroup.subtype (fixingSubgroup M ∅)) | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
⊢ ∃ a, (Subgroup.subtype (fixingSubgroup M ∅)) a = g | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
⊢ Function.Surjective ⇑(Subgroup.subtype (fixingSubgroup M ∅))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.of_fixingSubgroupEmpty_mapScalars_surjective | [270, 1] | [276, 72] | suffices g ∈ fixingSubgroup M (∅ : Set α) by
exact ⟨⟨g, this⟩, rfl⟩ | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
⊢ ∃ a, (Subgroup.subtype (fixingSubgroup M ∅)) a = g | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
⊢ g ∈ fixingSubgroup M ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
⊢ ∃ a, (Subgroup.subtype (fixingSubgroup M ∅)) a = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.of_fixingSubgroupEmpty_mapScalars_surjective | [270, 1] | [276, 72] | rw [mem_fixingSubgroup_iff] | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
⊢ g ∈ fixingSubgroup M ∅ | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
⊢ ∀ y ∈ ∅, g • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
⊢ g ∈ fixingSubgroup M ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.of_fixingSubgroupEmpty_mapScalars_surjective | [270, 1] | [276, 72] | simp only [Set.mem_empty_iff_false, IsEmpty.forall_iff, implies_true] | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
⊢ ∀ y ∈ ∅, g • y = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
⊢ ∀ y ∈ ∅, g • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.of_fixingSubgroupEmpty_mapScalars_surjective | [270, 1] | [276, 72] | exact ⟨⟨g, this⟩, rfl⟩ | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
this : g ∈ fixingSubgroup M ∅
⊢ ∃ a, (Subgroup.subtype (fixingSubgroup M ∅)) a = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
g : M
this : g ∈ fixingSubgroup M ∅
⊢ ∃ a, (Subgroup.subtype (fixingSubgroup M ∅)) a = g
TACTIC:
|
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