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.OfFixingSubgroupOfSingleton.map_bijective | [610, 1] | [618, 8] | intro x | 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) | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
x : ↥(ofStabilizer M a)
⊢ ∃ a_1, (map M a) a_1 = 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
a : α
⊢ Function.Surjective ⇑(map M a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfSingleton.map_bijective | [610, 1] | [618, 8] | use x | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
x : ↥(ofStabilizer M a)
⊢ ∃ a_1, (map M a) a_1 = x | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
x : ↥(ofStabilizer M a)
⊢ (map M a) x = 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
a : α
x : ↥(ofStabilizer M a)
⊢ ∃ a_1, (map M a) a_1 = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfSingleton.map_bijective | [610, 1] | [618, 8] | rfl | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
a : α
x : ↥(ofStabilizer M a)
⊢ (map M a) 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 : α
x : ↥(ofStabilizer M a)
⊢ (map M a) x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | constructor | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
⊢ Function.Bijective ⇑(map M hst) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
⊢ Function.Injective ⇑(map M hst)
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
⊢ Function.Surjective ⇑(map 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
hst : s = t
⊢ Function.Bijective ⇑(map M hst)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | 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
hst : s = t
⊢ Function.Injective ⇑(map 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
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : (map M hst) { val := x, property := hx } = (map 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
hst : s = t
⊢ Function.Injective ⇑(map M hst)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | 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
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : (map M hst) { val := x, property := hx } = (map 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
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : ↑((map M hst) { val := x, property := hx }) = ↑((map 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.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : (map M hst) { val := x, property := hx } = (map 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.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | simp only [SubMulAction.OfFixingSubgroupOfEq.map_def M hst] 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
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : ↑((map M hst) { val := x, property := hx }) = ↑((map 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
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : x = y
⊢ ↑{ val := x, property := hx } = ↑{ 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
s t : Set α
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : ↑((map M hst) { val := x, property := hx }) = ↑((map 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.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | simp only | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : x = y
⊢ ↑{ 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
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : x = 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
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : x = y
⊢ ↑{ val := x, property := hx } = ↑{ val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | rw [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
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : x = 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
hst : s = t
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
hxy : x = y
⊢ x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | rintro ⟨x, hxt⟩ | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
⊢ Function.Surjective ⇑(map 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
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ ∃ a, (map M hst) a = { val := x, property := hxt } | 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
hst : s = t
⊢ Function.Surjective ⇑(map M hst)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | use ⟨x, ?_⟩ | case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ ∃ a, (map M hst) a = { val := x, property := hxt } | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ (map M hst) { val := x, property := ?w } = { val := x, property := hxt }
case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ x ∈ ofFixingSubgroup M s | 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
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ ∃ a, (map M hst) a = { val := x, property := hxt }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | rfl | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ (map M hst) { val := x, property := ?w } = { val := x, property := hxt }
case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ x ∈ ofFixingSubgroup M s | case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ x ∈ ofFixingSubgroup M 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
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ (map M hst) { val := x, property := ?w } = { val := x, property := hxt }
case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ x ∈ ofFixingSubgroup M s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | rw [hst] | case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ x ∈ ofFixingSubgroup M s | case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ x ∈ ofFixingSubgroup M t | Please generate a tactic in lean4 to solve the state.
STATE:
case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ x ∈ ofFixingSubgroup M s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.map_bijective | [643, 1] | [656, 14] | exact hxt | case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ x ∈ ofFixingSubgroup M t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
x : α
hxt : x ∈ ofFixingSubgroup M t
⊢ x ∈ ofFixingSubgroup M t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.toMap_bijective | [659, 1] | [670, 13] | constructor | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
⊢ Function.Bijective (MulActionHom.toMap (map M hst)) | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
⊢ Function.Injective (MulActionHom.toMap (map M hst))
case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
⊢ Function.Surjective (MulActionHom.toMap (map 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
hst : s = t
⊢ Function.Bijective (MulActionHom.toMap (map M hst))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.toMap_bijective | [659, 1] | [670, 13] | rintro ⟨g, hg⟩ ⟨k, hk⟩ hgk | case left
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
⊢ Function.Injective (MulActionHom.toMap (map 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
hst : s = t
g : M
hg : g ∈ fixingSubgroup M s
k : M
hk : k ∈ fixingSubgroup M s
hgk :
MulActionHom.toMap (map M hst) { val := g, property := hg } =
MulActionHom.toMap (map M hst) { val := k, property := hk }
⊢ { val := g, property := hg } = { val := k, property := hk } | 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
hst : s = t
⊢ Function.Injective (MulActionHom.toMap (map M hst))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.toMap_bijective | [659, 1] | [670, 13] | rw [← SetLike.coe_eq_coe] at hgk ⊢ | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
g : M
hg : g ∈ fixingSubgroup M s
k : M
hk : k ∈ fixingSubgroup M s
hgk :
MulActionHom.toMap (map M hst) { val := g, property := hg } =
MulActionHom.toMap (map M hst) { val := k, property := hk }
⊢ { val := g, property := hg } = { val := k, property := hk } | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
g : M
hg : g ∈ fixingSubgroup M s
k : M
hk : k ∈ fixingSubgroup M s
hgk :
↑(MulActionHom.toMap (map M hst) { val := g, property := hg }) =
↑(MulActionHom.toMap (map M hst) { val := k, property := hk })
⊢ ↑{ val := g, property := hg } = ↑{ val := k, property := hk } | 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
hst : s = t
g : M
hg : g ∈ fixingSubgroup M s
k : M
hk : k ∈ fixingSubgroup M s
hgk :
MulActionHom.toMap (map M hst) { val := g, property := hg } =
MulActionHom.toMap (map M hst) { val := k, property := hk }
⊢ { val := g, property := hg } = { val := k, property := hk }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.toMap_bijective | [659, 1] | [670, 13] | simp only | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
g : M
hg : g ∈ fixingSubgroup M s
k : M
hk : k ∈ fixingSubgroup M s
hgk :
↑(MulActionHom.toMap (map M hst) { val := g, property := hg }) =
↑(MulActionHom.toMap (map M hst) { val := k, property := hk })
⊢ ↑{ val := g, property := hg } = ↑{ val := k, property := hk } | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
g : M
hg : g ∈ fixingSubgroup M s
k : M
hk : k ∈ fixingSubgroup M s
hgk :
↑(MulActionHom.toMap (map M hst) { val := g, property := hg }) =
↑(MulActionHom.toMap (map M hst) { val := k, property := hk })
⊢ g = k | 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
hst : s = t
g : M
hg : g ∈ fixingSubgroup M s
k : M
hk : k ∈ fixingSubgroup M s
hgk :
↑(MulActionHom.toMap (map M hst) { val := g, property := hg }) =
↑(MulActionHom.toMap (map M hst) { val := k, property := hk })
⊢ ↑{ val := g, property := hg } = ↑{ val := k, property := hk }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.toMap_bijective | [659, 1] | [670, 13] | exact hgk | case left.mk.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
g : M
hg : g ∈ fixingSubgroup M s
k : M
hk : k ∈ fixingSubgroup M s
hgk :
↑(MulActionHom.toMap (map M hst) { val := g, property := hg }) =
↑(MulActionHom.toMap (map M hst) { val := k, property := hk })
⊢ g = k | 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
hst : s = t
g : M
hg : g ∈ fixingSubgroup M s
k : M
hk : k ∈ fixingSubgroup M s
hgk :
↑(MulActionHom.toMap (map M hst) { val := g, property := hg }) =
↑(MulActionHom.toMap (map M hst) { val := k, property := hk })
⊢ g = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.toMap_bijective | [659, 1] | [670, 13] | rintro ⟨k, hk⟩ | case right
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
⊢ Function.Surjective (MulActionHom.toMap (map 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
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ ∃ a, MulActionHom.toMap (map M hst) a = { val := k, property := hk } | 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
hst : s = t
⊢ Function.Surjective (MulActionHom.toMap (map M hst))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.toMap_bijective | [659, 1] | [670, 13] | use ⟨k, ?_⟩ | case right.mk
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ ∃ a, MulActionHom.toMap (map M hst) a = { val := k, property := hk } | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ MulActionHom.toMap (map M hst) { val := k, property := ?w } = { val := k, property := hk }
case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ k ∈ fixingSubgroup M s | 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
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ ∃ a, MulActionHom.toMap (map M hst) a = { val := k, property := hk }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.toMap_bijective | [659, 1] | [670, 13] | rfl | case h
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ MulActionHom.toMap (map M hst) { val := k, property := ?w } = { val := k, property := hk }
case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ k ∈ fixingSubgroup M s | case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ k ∈ fixingSubgroup M 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
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ MulActionHom.toMap (map M hst) { val := k, property := ?w } = { val := k, property := hk }
case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ k ∈ fixingSubgroup M s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.toMap_bijective | [659, 1] | [670, 13] | rw [hst] | case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ k ∈ fixingSubgroup M s | case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ k ∈ fixingSubgroup M t | Please generate a tactic in lean4 to solve the state.
STATE:
case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ k ∈ fixingSubgroup M s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/SubMulActions.lean | SubMulAction.OfFixingSubgroupOfEq.toMap_bijective | [659, 1] | [670, 13] | exact hk | case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ k ∈ fixingSubgroup M t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case w
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
s t : Set α
hst : s = t
k : M
hk : k ∈ fixingSubgroup M t
⊢ k ∈ fixingSubgroup M t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer | [28, 1] | [37, 19] | rw [mem_stabilizer_of_finite_iff_smul_le] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ ofSubtype g ∈ stabilizer (Perm α) s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ ofSubtype g • s ⊆ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ ofSubtype g ∈ stabilizer (Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer | [28, 1] | [37, 19] | intro x | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ ofSubtype g • s ⊆ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
x : α
⊢ x ∈ ofSubtype g • s → x ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ ofSubtype g • s ⊆ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer | [28, 1] | [37, 19] | rw [Set.mem_smul_set] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
x : α
⊢ x ∈ ofSubtype g • s → x ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
x : α
⊢ (∃ y ∈ s, ofSubtype g • y = x) → x ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
x : α
⊢ x ∈ ofSubtype g • s → x ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer | [28, 1] | [37, 19] | rintro ⟨y, hy, rfl⟩ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
x : α
⊢ (∃ y ∈ s, ofSubtype g • y = x) → x ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
y : α
hy : y ∈ s
⊢ ofSubtype g • y ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
x : α
⊢ (∃ y ∈ s, ofSubtype g • y = x) → x ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer | [28, 1] | [37, 19] | simp only [Equiv.Perm.smul_def] | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
y : α
hy : y ∈ s
⊢ ofSubtype g • y ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
y : α
hy : y ∈ s
⊢ (ofSubtype g) y ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
y : α
hy : y ∈ s
⊢ ofSubtype g • y ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer | [28, 1] | [37, 19] | rw [Equiv.Perm.ofSubtype_apply_of_mem g hy] | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
y : α
hy : y ∈ s
⊢ (ofSubtype g) y ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
y : α
hy : y ∈ s
⊢ ↑(g { val := y, property := hy }) ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
y : α
hy : y ∈ s
⊢ (ofSubtype g) y ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer | [28, 1] | [37, 19] | refine' Subtype.mem _ | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
y : α
hy : y ∈ s
⊢ ↑(g { val := y, property := hy }) ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
y : α
hy : y ∈ s
⊢ ↑(g { val := y, property := hy }) ∈ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer | [28, 1] | [37, 19] | exact s.toFinite | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑s
⊢ Set.Finite s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer' | [40, 1] | [55, 19] | rw [mem_stabilizer_of_finite_iff_smul_le] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
⊢ ofSubtype g ∈ stabilizer (Perm α) s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
⊢ ofSubtype g • s ⊆ s
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
⊢ Set.Finite s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
⊢ ofSubtype g ∈ stabilizer (Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer' | [40, 1] | [55, 19] | exact s.toFinite | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
⊢ Set.Finite s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
⊢ Set.Finite s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer' | [40, 1] | [55, 19] | intro x | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
⊢ ofSubtype g • s ⊆ s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
x : α
⊢ x ∈ ofSubtype g • s → x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
⊢ ofSubtype g • s ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer' | [40, 1] | [55, 19] | rw [Set.mem_smul_set] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
x : α
⊢ x ∈ ofSubtype g • s → x ∈ s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
x : α
⊢ (∃ y ∈ s, ofSubtype g • y = x) → x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
x : α
⊢ x ∈ ofSubtype g • s → x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer' | [40, 1] | [55, 19] | rintro ⟨y, hy, rfl⟩ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
x : α
⊢ (∃ y ∈ s, ofSubtype g • y = x) → x ∈ s | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
y : α
hy : y ∈ s
⊢ ofSubtype g • y ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
x : α
⊢ (∃ y ∈ s, ofSubtype g • y = x) → x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer' | [40, 1] | [55, 19] | simp only [Equiv.Perm.smul_def] | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
y : α
hy : y ∈ s
⊢ ofSubtype g • y ∈ s | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
y : α
hy : y ∈ s
⊢ (ofSubtype g) y ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
y : α
hy : y ∈ s
⊢ ofSubtype g • y ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer' | [40, 1] | [55, 19] | rw [Equiv.Perm.ofSubtype_apply_of_not_mem g (Set.not_mem_compl_iff.mpr hy)] | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
y : α
hy : y ∈ s
⊢ (ofSubtype g) y ∈ s | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
y : α
hy : y ∈ s
⊢ y ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
y : α
hy : y ∈ s
⊢ (ofSubtype g) y ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.ofSubtype_mem_stabilizer' | [40, 1] | [55, 19] | exact hy | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
y : α
hy : y ∈ s
⊢ y ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
g : Perm ↑sᶜ
y : α
hy : y ∈ s
⊢ y ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | let φ : stabilizer (Equiv.Perm α) s → Equiv.Perm s := MulAction.toPerm | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | let f : s →ₑ[φ] s :=
{ toFun := id
map_smul' := fun g x => by
simp only [id_eq, Perm.smul_def, toPerm_apply, φ] } | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | have hf : Function.Bijective f := Function.bijective_id | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | rw [isPreprimitive_of_bijective_map_iff _ hf] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive (Perm ↑s) ↑s
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Function.Surjective φ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | exact Equiv.Perm.isPreprimitive s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive (Perm ↑s) ↑s
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Function.Surjective φ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Function.Surjective φ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive (Perm ↑s) ↑s
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Function.Surjective φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | intro g | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Function.Surjective φ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
⊢ ∃ a, φ a = g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Function.Surjective φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | use! Equiv.Perm.ofSubtype g | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
⊢ ∃ a, φ a = g | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
⊢ ofSubtype g ∈ stabilizer (Perm α) s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
⊢ φ { val := ofSubtype g, property := ?property } = g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
⊢ ∃ a, φ a = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | simp only [id_eq, Perm.smul_def, toPerm_apply, φ] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
g : ↥(stabilizer (Perm α) s)
x : ↑s
⊢ id (g • x) = φ g • id x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
g : ↥(stabilizer (Perm α) s)
x : ↑s
⊢ id (g • x) = φ g • id x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | apply ofSubtype_mem_stabilizer | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
⊢ ofSubtype g ∈ stabilizer (Perm α) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
⊢ ofSubtype g ∈ stabilizer (Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | ext ⟨x, hx⟩ | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
⊢ φ { val := ofSubtype g, property := ⋯ } = g | case h.H.mk.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
x : α
hx : x ∈ s
⊢ ↑((φ { val := ofSubtype g, property := ⋯ }) { val := x, property := hx }) = ↑(g { val := x, property := hx }) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
⊢ φ { val := ofSubtype g, property := ⋯ } = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | change Equiv.Perm.ofSubtype g • x = _ | case h.H.mk.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
x : α
hx : x ∈ s
⊢ ↑((φ { val := ofSubtype g, property := ⋯ }) { val := x, property := hx }) = ↑(g { val := x, property := hx }) | case h.H.mk.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
x : α
hx : x ∈ s
⊢ ofSubtype g • x = ↑(g { val := x, property := hx }) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.mk.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
x : α
hx : x ∈ s
⊢ ↑((φ { val := ofSubtype g, property := ⋯ }) { val := x, property := hx }) = ↑(g { val := x, property := hx })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | simp only [Equiv.Perm.smul_def] | case h.H.mk.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
x : α
hx : x ∈ s
⊢ ofSubtype g • x = ↑(g { val := x, property := hx }) | case h.H.mk.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
x : α
hx : x ∈ s
⊢ (ofSubtype g) x = ↑(g { val := x, property := hx }) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.mk.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
x : α
hx : x ∈ s
⊢ ofSubtype g • x = ↑(g { val := x, property := hx })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.stabilizer_isPreprimitive | [58, 1] | [77, 43] | rw [Equiv.Perm.ofSubtype_apply_of_mem] | case h.H.mk.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
x : α
hx : x ∈ s
⊢ (ofSubtype g) x = ↑(g { val := x, property := hx }) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H.mk.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (Perm α) s) → Perm ↑s := toPerm
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
g : Perm ↑s
x : α
hx : x ∈ s
⊢ (ofSubtype g) x = ↑(g { val := x, property := hx })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.Equiv.Perm.Stabilizer.is_preprimitive' | [80, 1] | [91, 34] | let φ : stabilizer (Equiv.Perm α) s → stabilizer G s := fun g =>
⟨⟨g, hG g.prop⟩, mem_stabilizer_iff.mp g.prop⟩ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.Equiv.Perm.Stabilizer.is_preprimitive' | [80, 1] | [91, 34] | let f : s →ₑ[φ] s :=
{ toFun := id
map_smul' := fun ⟨m, hm⟩ x => by
simp only [id_eq, ← Subtype.coe_inj, SMul.smul_stabilizer_def,
Perm.smul_def, Submonoid.mk_smul] } | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.Equiv.Perm.Stabilizer.is_preprimitive' | [80, 1] | [91, 34] | have : Function.Surjective f := Function.surjective_id | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
this : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.Equiv.Perm.Stabilizer.is_preprimitive' | [80, 1] | [91, 34] | apply isPreprimitive_of_surjective_map this | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
this : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
this : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
this : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.Equiv.Perm.Stabilizer.is_preprimitive' | [80, 1] | [91, 34] | apply stabilizer_isPreprimitive | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
this : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
this : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (Perm α) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | Equiv.Perm.Equiv.Perm.Stabilizer.is_preprimitive' | [80, 1] | [91, 34] | simp only [id_eq, ← Subtype.coe_inj, SMul.smul_stabilizer_def,
Perm.smul_def, Submonoid.mk_smul] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
x✝ : ↥(stabilizer (Perm α) s)
x : ↑s
m : Perm α
hm : m ∈ stabilizer (Perm α) s
⊢ id ({ val := m, property := hm } • x) = φ { val := m, property := hm } • id x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s ≤ G
φ : ↥(stabilizer (Perm α) s) → ↥(stabilizer (↥G) s) := fun g => { val := { val := ↑g, property := ⋯ }, property := ⋯ }
x✝ : ↥(stabilizer (Perm α) s)
x : ↑s
m : Perm α
hm : m ∈ stabilizer (Perm α) s
⊢ id ({ val := m, property := hm } • x) = φ { val := m, property := hm } • id x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | let φ : stabilizer (alternatingGroup α) s → Equiv.Perm s := MulAction.toPerm | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | suffices hφ : Function.Surjective φ by
let f : s →ₑ[φ] s := {
toFun := id
map_smul' := fun ⟨g, hg⟩ ⟨x, hx⟩ => by
simp only [id.def, Equiv.Perm.smul_def]
rw [toPerm_apply] }
have hf : Function.Bijective f := Function.bijective_id
rw [isPreprimitive_of_bijective_map_iff hφ hf]
exact Equiv.Perm.isPreprimitive s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
⊢ Function.Surjective φ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | obtain ⟨a, ha, b, hb, hab⟩ := hs | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
⊢ ∃ k, Equiv.Perm.sign k = -1 | case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ ∃ k, Equiv.Perm.sign k = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
⊢ ∃ k, Equiv.Perm.sign k = -1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | use Equiv.swap ⟨a, ha⟩ ⟨b, hb⟩ | case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ ∃ k, Equiv.Perm.sign k = -1 | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ Equiv.Perm.sign (Equiv.swap { val := a, property := ha } { val := b, property := hb }) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ ∃ k, Equiv.Perm.sign k = -1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | rw [Equiv.Perm.sign_swap _] | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ Equiv.Perm.sign (Equiv.swap { val := a, property := ha } { val := b, property := hb }) = -1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ { val := a, property := ha } ≠ { val := b, property := hb } | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ Equiv.Perm.sign (Equiv.swap { val := a, property := ha } { val := b, property := hb }) = -1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | rw [← Function.Injective.ne_iff Subtype.coe_injective] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ { val := a, property := ha } ≠ { val := b, property := hb } | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ ↑{ val := a, property := ha } ≠ ↑{ val := b, property := hb } | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ { val := a, property := ha } ≠ { val := b, property := hb }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | simp only [Subtype.coe_mk] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ ↑{ val := a, property := ha } ≠ ↑{ val := b, property := hb } | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ a ≠ b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ ↑{ val := a, property := ha } ≠ ↑{ val := b, property := hb }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | exact hab | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ a ≠ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ a ≠ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | let f : s →ₑ[φ] s := {
toFun := id
map_smul' := fun ⟨g, hg⟩ ⟨x, hx⟩ => by
simp only [id.def, Equiv.Perm.smul_def]
rw [toPerm_apply] } | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | have hf : Function.Bijective f := Function.bijective_id | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | rw [isPreprimitive_of_bijective_map_iff hφ hf] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive (Equiv.Perm ↑s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | exact Equiv.Perm.isPreprimitive s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive (Equiv.Perm ↑s) ↑s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsPreprimitive (Equiv.Perm ↑s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | simp only [id.def, Equiv.Perm.smul_def] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
x✝¹ : ↥(stabilizer (↥(alternatingGroup α)) s)
x✝ : ↑s
g : ↥(alternatingGroup α)
hg : g ∈ stabilizer (↥(alternatingGroup α)) s
x : α
hx : x ∈ s
⊢ id ({ val := g, property := hg } • { val := x, property := hx }) =
φ { val := g, property := hg } • id { val := x, property := hx } | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
x✝¹ : ↥(stabilizer (↥(alternatingGroup α)) s)
x✝ : ↑s
g : ↥(alternatingGroup α)
hg : g ∈ stabilizer (↥(alternatingGroup α)) s
x : α
hx : x ∈ s
⊢ { val := g, property := hg } • { val := x, property := hx } =
(φ { val := g, property := hg }) { val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
x✝¹ : ↥(stabilizer (↥(alternatingGroup α)) s)
x✝ : ↑s
g : ↥(alternatingGroup α)
hg : g ∈ stabilizer (↥(alternatingGroup α)) s
x : α
hx : x ∈ s
⊢ id ({ val := g, property := hg } • { val := x, property := hx }) =
φ { val := g, property := hg } • id { val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | rw [toPerm_apply] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
x✝¹ : ↥(stabilizer (↥(alternatingGroup α)) s)
x✝ : ↑s
g : ↥(alternatingGroup α)
hg : g ∈ stabilizer (↥(alternatingGroup α)) s
x : α
hx : x ∈ s
⊢ { val := g, property := hg } • { val := x, property := hx } =
(φ { val := g, property := hg }) { val := x, property := hx } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
hφ : Function.Surjective φ
x✝¹ : ↥(stabilizer (↥(alternatingGroup α)) s)
x✝ : ↑s
g : ↥(alternatingGroup α)
hg : g ∈ stabilizer (↥(alternatingGroup α)) s
x : α
hx : x ∈ s
⊢ { val := g, property := hg } • { val := x, property := hx } =
(φ { val := g, property := hg }) { val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | obtain ⟨k, hk_sign⟩ := this | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
this : ∃ k, Equiv.Perm.sign k = -1
⊢ Function.Surjective φ | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
⊢ Function.Surjective φ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
this : ∃ k, Equiv.Perm.sign k = -1
⊢ Function.Surjective φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | have hks : Equiv.Perm.ofSubtype k • s = s := by
rw [← mem_stabilizer_iff]
exact Equiv.Perm.ofSubtype_mem_stabilizer' s k | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
⊢ Function.Surjective φ | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
⊢ Function.Surjective φ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
⊢ Function.Surjective φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | have hminus_one_ne_one : (-1 : Units ℤ) ≠ 1 := Ne.symm (units_ne_neg_self 1) | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
⊢ Function.Surjective φ | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
⊢ Function.Surjective φ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
⊢ Function.Surjective φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | intro g | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
⊢ Function.Surjective φ | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
⊢ ∃ a, φ a = g | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
⊢ Function.Surjective φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | let g' := if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
⊢ ∃ a, φ a = g | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ ∃ a, φ a = g | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
⊢ ∃ a, φ a = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | use! g' | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ ∃ a, φ a = g | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ g' ∈ alternatingGroup α
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ { val := g', property := ?property } ∈ stabilizer (↥(alternatingGroup α)) s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ?property }, property := ?property } = g | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ ∃ a, φ a = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | rw [Equiv.Perm.mem_alternatingGroup] | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ g' ∈ alternatingGroup α
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ { val := g', property := ?property } ∈ stabilizer (↥(alternatingGroup α)) s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ?property }, property := ?property } = g | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ Equiv.Perm.sign g' = 1
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ { val := g', property := ⋯ } ∈ stabilizer (↥(alternatingGroup α)) s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ⋯ }, property := ?property } = g | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ g' ∈ alternatingGroup α
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ { val := g', property := ?property } ∈ stabilizer (↥(alternatingGroup α)) s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ?property }, property := ?property } = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | rw [mem_stabilizer_iff, Submonoid.mk_smul] | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ { val := g', property := ⋯ } ∈ stabilizer (↥(alternatingGroup α)) s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ⋯ }, property := ?property } = g | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ g' • s = s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ⋯ }, property := ⋯ } = g | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ { val := g', property := ⋯ } ∈ stabilizer (↥(alternatingGroup α)) s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ⋯ }, property := ?property } = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | cases' Int.units_eq_one_or (Equiv.Perm.sign g) with hsg hsg | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ g' • s = s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ⋯ }, property := ⋯ } = g | case property.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ g' • s = s
case property.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ g' • s = s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ⋯ }, property := ⋯ } = g | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ g' • s = s
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ⋯ }, property := ⋯ } = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | dsimp only [id_eq, ite_true, ite_false, eq_mpr_eq_cast, cast_eq, φ] | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ⋯ }, property := ⋯ } = g | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ toPerm { val := { val := g', property := ⋯ }, property := ⋯ } = g | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ φ { val := { val := g', property := ⋯ }, property := ⋯ } = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | cases' Int.units_eq_one_or (Equiv.Perm.sign g) with hsg hsg | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ toPerm { val := { val := g', property := ⋯ }, property := ⋯ } = g | case h.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ toPerm { val := { val := g', property := ⋯ }, property := ⋯ } = g
case h.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ toPerm { val := { val := g', property := ⋯ }, property := ⋯ } = g | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
⊢ toPerm { val := { val := g', property := ⋯ }, property := ⋯ } = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | rw [← mem_stabilizer_iff] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
⊢ Equiv.Perm.ofSubtype k • s = s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
⊢ Equiv.Perm.ofSubtype k ∈ stabilizer (Equiv.Perm α) s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
⊢ Equiv.Perm.ofSubtype k • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | exact Equiv.Perm.ofSubtype_mem_stabilizer' s k | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
⊢ Equiv.Perm.ofSubtype k ∈ stabilizer (Equiv.Perm α) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
⊢ Equiv.Perm.ofSubtype k ∈ stabilizer (Equiv.Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | simp only [g', hsg, eq_self_iff_true, if_true, hminus_one_ne_one, if_false, Equiv.Perm.sign_ofSubtype, Equiv.Perm.sign_mul, mul_neg, mul_one, neg_neg, hsg, hk_sign] | case property.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ Equiv.Perm.sign g' = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case property.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ Equiv.Perm.sign g' = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | simp only [g', hsg, eq_self_iff_true, if_true] | case property.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ g' • s = s | case property.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ Equiv.Perm.ofSubtype g • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case property.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ g' • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | exact Equiv.Perm.ofSubtype_mem_stabilizer s g | case property.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ Equiv.Perm.ofSubtype g • s = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case property.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ Equiv.Perm.ofSubtype g • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | simp only [g', hsg, hminus_one_ne_one, if_false, mul_smul, hks] | case property.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ g' • s = s | case property.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ Equiv.Perm.ofSubtype g • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case property.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ g' • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | exact Equiv.Perm.ofSubtype_mem_stabilizer s g | case property.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ Equiv.Perm.ofSubtype g • s = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case property.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ Equiv.Perm.ofSubtype g • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | simp only [g', hsg, eq_self_iff_true, if_true, hminus_one_ne_one, if_false] | case h.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ toPerm { val := { val := g', property := ⋯ }, property := ⋯ } = g | case h.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ toPerm { val := { val := Equiv.Perm.ofSubtype g, property := ⋯ }, property := ⋯ } = g | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ toPerm { val := { val := g', property := ⋯ }, property := ⋯ } = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | ext x | case h.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ toPerm { val := { val := Equiv.Perm.ofSubtype g, property := ⋯ }, property := ⋯ } = g | case h.inl.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
x : ↑s
⊢ ↑((toPerm { val := { val := Equiv.Perm.ofSubtype g, property := ⋯ }, property := ⋯ }) x) = ↑(g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
⊢ toPerm { val := { val := Equiv.Perm.ofSubtype g, property := ⋯ }, property := ⋯ } = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | simp only [toPerm_apply, SMul.smul_stabilizer_def, Submonoid.mk_smul, Equiv.Perm.smul_def,
Equiv.Perm.ofSubtype_apply_coe] | case h.inl.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
x : ↑s
⊢ ↑((toPerm { val := { val := Equiv.Perm.ofSubtype g, property := ⋯ }, property := ⋯ }) x) = ↑(g x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inl.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = 1
x : ↑s
⊢ ↑((toPerm { val := { val := Equiv.Perm.ofSubtype g, property := ⋯ }, property := ⋯ }) x) = ↑(g x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | simp only [g', hsg, eq_self_iff_true, if_true, hminus_one_ne_one, if_false] | case h.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ toPerm { val := { val := g', property := ⋯ }, property := ⋯ } = g | case h.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ toPerm { val := { val := Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k, property := ⋯ }, property := ⋯ } = g | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ toPerm { val := { val := g', property := ⋯ }, property := ⋯ } = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | ext x | case h.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ toPerm { val := { val := Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k, property := ⋯ }, property := ⋯ } = g | case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑((toPerm { val := { val := Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k, property := ⋯ }, property := ⋯ }) x) =
↑(g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
⊢ toPerm { val := { val := Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k, property := ⋯ }, property := ⋯ } = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | simp only [toPerm_apply, SMul.smul_stabilizer_def, Submonoid.mk_smul] | case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑((toPerm { val := { val := Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k, property := ⋯ }, property := ⋯ }) x) =
↑(g x) | case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ (Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k) • ↑x = ↑(g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑((toPerm { val := { val := Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k, property := ⋯ }, property := ⋯ }) x) =
↑(g x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | simp only [Equiv.Perm.smul_def, Equiv.Perm.coe_mul, Function.comp_apply] | case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ (Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k) • ↑x = ↑(g x) | case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ (Equiv.Perm.ofSubtype g) ((Equiv.Perm.ofSubtype k) ↑x) = ↑(g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ (Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k) • ↑x = ↑(g x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | rw [Equiv.Perm.ofSubtype_apply_of_not_mem k _] | case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ (Equiv.Perm.ofSubtype g) ((Equiv.Perm.ofSubtype k) ↑x) = ↑(g x) | case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ (Equiv.Perm.ofSubtype g) ↑x = ↑(g x)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑x ∉ sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ (Equiv.Perm.ofSubtype g) ((Equiv.Perm.ofSubtype k) ↑x) = ↑(g x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | exact Equiv.Perm.ofSubtype_apply_coe g x | case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ (Equiv.Perm.ofSubtype g) ↑x = ↑(g x)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑x ∉ sᶜ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑x ∉ sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.H.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ (Equiv.Perm.ofSubtype g) ↑x = ↑(g x)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑x ∉ sᶜ
TACTIC:
|
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