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/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | exact Equiv.Perm.sign_surjective α | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
⊢ Function.Surjective ⇑Equiv.Perm.sign
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
⊢ Function.Surjective ⇑φ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
⊢ 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
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
⊢ Function.Surjective ⇑Equiv.Perm.sign
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
⊢ Function.Surjective ⇑φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | exact MulEquiv.surjective φ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
⊢ Function.Surjective ⇑φ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
⊢ Function.Surjective ⇑φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | rw [hg', ← bne_iff_ne] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg' : s g' = -1
⊢ s g' ≠ 1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg' : s g' = -1
⊢ (-1 != 1) = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg' : s g' = -1
⊢ s g' ≠ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | rfl | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg' : s g' = -1
⊢ (-1 != 1) = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg' : s g' = -1
⊢ (-1 != 1) = true
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | rw [map_one, Equiv.Perm.sign.map_one] | case inr.fixed.intro.h.h.a.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g : Equiv.Perm α
⊢ Equiv.Perm.sign (φ 1) = Equiv.Perm.sign 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.fixed.intro.h.h.a.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g : Equiv.Perm α
⊢ Equiv.Perm.sign (φ 1) = Equiv.Perm.sign 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | intro f x y hxy hf | case inr.fixed.intro.h.h.a.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g : Equiv.Perm α
⊢ ∀ (f : Equiv.Perm α) (x y : α),
x ≠ y →
Equiv.Perm.sign (φ f) = Equiv.Perm.sign f →
Equiv.Perm.sign (φ (Equiv.swap x y * f)) = Equiv.Perm.sign (Equiv.swap x y * f) | case inr.fixed.intro.h.h.a.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ Equiv.Perm.sign (φ (Equiv.swap x y * f)) = Equiv.Perm.sign (Equiv.swap x y * f) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.fixed.intro.h.h.a.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g : Equiv.Perm α
⊢ ∀ (f : Equiv.Perm α) (x y : α),
x ≠ y →
Equiv.Perm.sign (φ f) = Equiv.Perm.sign f →
Equiv.Perm.sign (φ (Equiv.swap x y * f)) = Equiv.Perm.sign (Equiv.swap x y * f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | simp only [map_mul, hf] | case inr.fixed.intro.h.h.a.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ Equiv.Perm.sign (φ (Equiv.swap x y * f)) = Equiv.Perm.sign (Equiv.swap x y * f) | case inr.fixed.intro.h.h.a.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ Equiv.Perm.sign (φ (Equiv.swap x y)) * Equiv.Perm.sign f = Equiv.Perm.sign (Equiv.swap x y) * Equiv.Perm.sign f | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.fixed.intro.h.h.a.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ Equiv.Perm.sign (φ (Equiv.swap x y * f)) = Equiv.Perm.sign (Equiv.swap x y * f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | apply congr_arg₂ _ _ rfl | case inr.fixed.intro.h.h.a.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ Equiv.Perm.sign (φ (Equiv.swap x y)) * Equiv.Perm.sign f = Equiv.Perm.sign (Equiv.swap x y) * Equiv.Perm.sign f | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ Equiv.Perm.sign (φ (Equiv.swap x y)) = Equiv.Perm.sign (Equiv.swap x y) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.fixed.intro.h.h.a.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ Equiv.Perm.sign (φ (Equiv.swap x y)) * Equiv.Perm.sign f = Equiv.Perm.sign (Equiv.swap x y) * Equiv.Perm.sign f
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | revert x y hxy | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ Equiv.Perm.sign (φ (Equiv.swap x y)) = Equiv.Perm.sign (Equiv.swap x y) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ ∀ (x y : α), x ≠ y → Equiv.Perm.sign (φ (Equiv.swap x y)) = Equiv.Perm.sign (Equiv.swap x y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ Equiv.Perm.sign (φ (Equiv.swap x y)) = Equiv.Perm.sign (Equiv.swap x y)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | by_contra h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ ∀ (x y : α), x ≠ y → Equiv.Perm.sign (φ (Equiv.swap x y)) = Equiv.Perm.sign (Equiv.swap x y) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
h : ¬∀ (x y : α), x ≠ y → Equiv.Perm.sign (φ (Equiv.swap x y)) = Equiv.Perm.sign (Equiv.swap x y)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
⊢ ∀ (x y : α), x ≠ y → Equiv.Perm.sign (φ (Equiv.swap x y)) = Equiv.Perm.sign (Equiv.swap x y)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | push_neg at h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
h : ¬∀ (x y : α), x ≠ y → Equiv.Perm.sign (φ (Equiv.swap x y)) = Equiv.Perm.sign (Equiv.swap x y)
⊢ False | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
h : ∃ x y, x ≠ y ∧ Equiv.Perm.sign (φ (Equiv.swap x y)) ≠ Equiv.Perm.sign (Equiv.swap x y)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
h : ¬∀ (x y : α), x ≠ y → Equiv.Perm.sign (φ (Equiv.swap x y)) = Equiv.Perm.sign (Equiv.swap x y)
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | obtain ⟨a, b, hab, hk⟩ := h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
h : ∃ x y, x ≠ y ∧ Equiv.Perm.sign (φ (Equiv.swap x y)) ≠ Equiv.Perm.sign (Equiv.swap x y)
⊢ False | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ Equiv.Perm.sign (Equiv.swap a b)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
h : ∃ x y, x ≠ y ∧ Equiv.Perm.sign (φ (Equiv.swap x y)) ≠ Equiv.Perm.sign (Equiv.swap x y)
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | rw [Equiv.Perm.sign_swap hab] at hk | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ Equiv.Perm.sign (Equiv.swap a b)
⊢ False | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ Equiv.Perm.sign (Equiv.swap a b)
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | let hk := Or.resolve_right (Int.units_eq_one_or (s _)) hk | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
⊢ False | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | apply hg' | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
⊢ False | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
⊢ s g' = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | refine' Equiv.Perm.swap_induction_on g' s.map_one _ | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
⊢ s g' = 1 | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
⊢ ∀ (f : Equiv.Perm α) (x y : α), x ≠ y → s f = 1 → s (Equiv.swap x y * f) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
⊢ s g' = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | intro f x y hxy hf | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
⊢ ∀ (f : Equiv.Perm α) (x y : α), x ≠ y → s f = 1 → s (Equiv.swap x y * f) = 1 | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
⊢ s (Equiv.swap x y * f) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f : Equiv.Perm α
hf : Equiv.Perm.sign (φ f) = Equiv.Perm.sign f
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
⊢ ∀ (f : Equiv.Perm α) (x y : α), x ≠ y → s f = 1 → s (Equiv.swap x y * f) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | rw [s.map_mul, hf, mul_one] | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
⊢ s (Equiv.swap x y * f) = 1 | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
⊢ s (Equiv.swap x y) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
⊢ s (Equiv.swap x y * f) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | obtain ⟨u, hu⟩ := Equiv.Perm.isConj_swap hxy hab | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
⊢ s (Equiv.swap x y) = 1 | case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
u : (Equiv.Perm α)ˣ
hu : SemiconjBy (↑u) (Equiv.swap x y) (Equiv.swap a b)
⊢ s (Equiv.swap x y) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
⊢ s (Equiv.swap x y) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | apply mul_left_cancel (a := s u) | case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
u : (Equiv.Perm α)ˣ
hu : SemiconjBy (↑u) (Equiv.swap x y) (Equiv.swap a b)
⊢ s (Equiv.swap x y) = 1 | case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
u : (Equiv.Perm α)ˣ
hu : SemiconjBy (↑u) (Equiv.swap x y) (Equiv.swap a b)
⊢ s ↑u * s (Equiv.swap x y) = s ↑u * 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
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
u : (Equiv.Perm α)ˣ
hu : SemiconjBy (↑u) (Equiv.swap x y) (Equiv.swap a b)
⊢ s (Equiv.swap x y) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup_is_characteristic | [37, 1] | [79, 72] | rw [← s.map_mul, SemiconjBy.eq hu, s.map_mul, hk, mul_one, one_mul] | case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
u : (Equiv.Perm α)ˣ
hu : SemiconjBy (↑u) (Equiv.swap x y) (Equiv.swap a b)
⊢ s ↑u * s (Equiv.swap x y) = s ↑u * 1 | no goals | 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
hα : Nontrivial α
φ : Equiv.Perm α ≃* Equiv.Perm α
s : Equiv.Perm α →* ℤˣ := MonoidHom.comp Equiv.Perm.sign (MulEquiv.toMonoidHom φ)
hs : Function.Surjective ⇑s
g' : Equiv.Perm α
hg'✝ : s g' = -1
hg' : s g' ≠ 1
g f✝ : Equiv.Perm α
hf✝ : Equiv.Perm.sign (φ f✝) = Equiv.Perm.sign f✝
a b : α
hab : a ≠ b
hk✝ : Equiv.Perm.sign (φ (Equiv.swap a b)) ≠ -1
hk : s (Equiv.swap a b) = 1 := Or.resolve_right (Int.units_eq_one_or (s (Equiv.swap a b))) hk✝
f : Equiv.Perm α
x y : α
hxy : x ≠ y
hf : s f = 1
u : (Equiv.Perm α)ˣ
hu : SemiconjBy (↑u) (Equiv.swap x y) (Equiv.swap a b)
⊢ s ↑u * s (Equiv.swap x y) = s ↑u * 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | isCommutative_of_prime_order | [83, 1] | [87, 57] | haveI := isCyclic_of_prime_card h | α : Type ?u.14086
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
h : Fintype.card G = p
⊢ Std.Commutative fun x x_1 => x * x_1 | α : Type ?u.14086
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
h : Fintype.card G = p
this : IsCyclic G
⊢ Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.14086
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
h : Fintype.card G = p
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | isCommutative_of_prime_order | [83, 1] | [87, 57] | exact Std.Commutative.mk (IsCyclic.commGroup.mul_comm) | α : Type ?u.14086
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
h : Fintype.card G = p
this : IsCyclic G
⊢ Std.Commutative fun x x_1 => x * x_1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.14086
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
h : Fintype.card G = p
this : IsCyclic G
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup.isCommutative_of_order_three | [90, 1] | [98, 16] | apply @isCommutative_of_prime_order _ _ _ 3 _ | α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
⊢ Std.Commutative fun x x_1 => x * x_1 | α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
⊢ Fintype.card ↥(alternatingGroup α) = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup.isCommutative_of_order_three | [90, 1] | [98, 16] | have hα' : Nontrivial α := by
rw [← Fintype.one_lt_card_iff_nontrivial, hα]
norm_num | α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
⊢ Fintype.card ↥(alternatingGroup α) = 3 | α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ Fintype.card ↥(alternatingGroup α) = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
⊢ Fintype.card ↥(alternatingGroup α) = 3
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup.isCommutative_of_order_three | [90, 1] | [98, 16] | apply mul_right_injective₀ (a := 2) (by norm_num) | α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ Fintype.card ↥(alternatingGroup α) = 3 | case a
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ (fun x => 2 * x) (Fintype.card ↥(alternatingGroup α)) = (fun x => 2 * x) 3 | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ Fintype.card ↥(alternatingGroup α) = 3
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup.isCommutative_of_order_three | [90, 1] | [98, 16] | simp only [two_mul_card_alternatingGroup, Fintype.card_perm, hα] | case a
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ (fun x => 2 * x) (Fintype.card ↥(alternatingGroup α)) = (fun x => 2 * x) 3 | case a
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ Nat.factorial 3 = 2 * 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ (fun x => 2 * x) (Fintype.card ↥(alternatingGroup α)) = (fun x => 2 * x) 3
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup.isCommutative_of_order_three | [90, 1] | [98, 16] | norm_num | case a
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ Nat.factorial 3 = 2 * 3 | case a
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ Nat.factorial 3 = 6 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ Nat.factorial 3 = 2 * 3
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup.isCommutative_of_order_three | [90, 1] | [98, 16] | rfl | case a
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ Nat.factorial 3 = 6 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ Nat.factorial 3 = 6
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup.isCommutative_of_order_three | [90, 1] | [98, 16] | rw [← Fintype.one_lt_card_iff_nontrivial, hα] | α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
⊢ Nontrivial α | α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
⊢ 1 < 3 | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
⊢ Nontrivial α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup.isCommutative_of_order_three | [90, 1] | [98, 16] | norm_num | α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
⊢ 1 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
⊢ 1 < 3
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | alternatingGroup.isCommutative_of_order_three | [90, 1] | [98, 16] | norm_num | α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ 2 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.15286
inst✝³ : Fintype α✝
inst✝² : DecidableEq α✝
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hα : Fintype.card α = 3
hα' : Nontrivial α
⊢ 2 ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | rw [← Nat.Coprime.dvd_mul_right _] | α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ r ∣ p | α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ r ∣ p * ?m.17725
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ ℕ
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ Nat.Coprime r ?m.17725 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ r ∣ p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | rw [Nat.coprime_iff_gcd_eq_one] | α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ Nat.Coprime r (Nat.factorial (p - 1)) | α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ Nat.gcd r (Nat.factorial (p - 1)) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ Nat.Coprime r (Nat.factorial (p - 1))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | by_contra h | α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ Nat.gcd r (Nat.factorial (p - 1)) = 1 | α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ Nat.gcd r (Nat.factorial (p - 1)) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | obtain ⟨l, hl, hl'⟩ := Nat.exists_prime_and_dvd h | α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
⊢ False | case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ Nat.gcd r (Nat.factorial (p - 1))
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | rw [Nat.dvd_gcd_iff, Nat.Prime.dvd_factorial hl] at hl' | case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ Nat.gcd r (Nat.factorial (p - 1))
⊢ False | case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ Nat.gcd r (Nat.factorial (p - 1))
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | apply (lt_iff_not_ge p.pred p).mp (Nat.pred_lt (Nat.Prime.ne_zero hp)) | case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ False | case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ Nat.pred p ≥ p | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | rw [Nat.pred_eq_sub_one] | case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ Nat.pred p ≥ p | case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ p - 1 ≥ p | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ Nat.pred p ≥ p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | rw [ge_iff_le] | case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ p - 1 ≥ p | case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ p ≤ p - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ p - 1 ≥ p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | exact le_trans (hr hl hl'.left) hl'.right | case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ p ≤ p - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h✝ : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
h : ¬Nat.gcd r (Nat.factorial (p - 1)) = 1
l : ℕ
hl : Nat.Prime l
hl' : l ∣ r ∧ l ≤ p - 1
⊢ p ≤ p - 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | rw [Nat.mul_factorial_pred (Nat.Prime.pos hp)] | α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ r ∣ p * ?m.17725 | α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ r ∣ Nat.factorial p | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ r ∣ p * ?m.17725
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | aux_dvd_lemma | [100, 9] | [111, 44] | exact h | α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ r ∣ Nat.factorial p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.17644
inst✝¹ : Fintype α
inst✝ : DecidableEq α
r p : ℕ
hp : Nat.Prime p
h : r ∣ Nat.factorial p
hr : ∀ {l : ℕ}, Nat.Prime l → l ∣ r → p ≤ l
⊢ r ∣ Nat.factorial p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | let f := MulAction.toPermHom G (G ⧸ H) | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
⊢ Normal H | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ Normal H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
⊢ Normal H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | suffices f.ker = H by
rw [← this]
apply MonoidHom.normal_ker f | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ Normal H | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ MonoidHom.ker f = H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ Normal H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | suffices H.normalCore.relindex H = 1 by
rw [← Subgroup.normalCore_eq_ker]
unfold Subgroup.relindex at this
rw [Subgroup.index_eq_one] at this
apply le_antisymm; apply Subgroup.normalCore_le
intro x hx
rw [← Subgroup.coe_mk H x hx, ← Subgroup.mem_subgroupOf, this]
apply Subgroup.mem_top | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ MonoidHom.ker f = H | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ relindex (normalCore H) H = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ MonoidHom.ker f = H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [hHp] | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ index H ≠ 0 | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ p ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ index H ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | exact Nat.Prime.ne_zero hp | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ p ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ p ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [← this] | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : MonoidHom.ker f = H
⊢ Normal H | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : MonoidHom.ker f = H
⊢ Normal (MonoidHom.ker f) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : MonoidHom.ker f = H
⊢ Normal H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | apply MonoidHom.normal_ker f | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : MonoidHom.ker f = H
⊢ Normal (MonoidHom.ker f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : MonoidHom.ker f = H
⊢ Normal (MonoidHom.ker f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [← Subgroup.normalCore_eq_ker] | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : relindex (normalCore H) H = 1
⊢ MonoidHom.ker f = H | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : relindex (normalCore H) H = 1
⊢ normalCore H = H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : relindex (normalCore H) H = 1
⊢ MonoidHom.ker f = H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | unfold Subgroup.relindex at this | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : relindex (normalCore H) H = 1
⊢ normalCore H = H | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index (subgroupOf (normalCore H) H) = 1
⊢ normalCore H = H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : relindex (normalCore H) H = 1
⊢ normalCore H = H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [Subgroup.index_eq_one] at this | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index (subgroupOf (normalCore H) H) = 1
⊢ normalCore H = H | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ normalCore H = H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index (subgroupOf (normalCore H) H) = 1
⊢ normalCore H = H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | apply le_antisymm | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ normalCore H = H | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ normalCore H ≤ H
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H ≤ normalCore H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ normalCore H = H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | apply Subgroup.normalCore_le | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ normalCore H ≤ H
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H ≤ normalCore H | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H ≤ normalCore H | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ normalCore H ≤ H
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H ≤ normalCore H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | intro x hx | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H ≤ normalCore H | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ x ∈ normalCore H | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H ≤ normalCore H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [← Subgroup.coe_mk H x hx, ← Subgroup.mem_subgroupOf, this] | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ x ∈ normalCore H | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ { val := x, property := hx } ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ x ∈ normalCore H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | apply Subgroup.mem_top | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ { val := x, property := hx } ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ { val := x, property := hx } ∈ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | apply mul_left_injective₀ this | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ relindex (normalCore H) H = 1 | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ (fun a => a * index H) (relindex (normalCore H) H) = (fun a => a * index H) 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ relindex (normalCore H) H = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | dsimp | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ (fun a => a * index H) (relindex (normalCore H) H) = (fun a => a * index H) 1 | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ relindex (normalCore H) H * index H = 1 * index H | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ (fun a => a * index H) (relindex (normalCore H) H) = (fun a => a * index H) 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [Subgroup.relindex_mul_index (Subgroup.normalCore_le H)] | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ relindex (normalCore H) H * index H = 1 * index H | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (normalCore H) = 1 * index H | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ relindex (normalCore H) H * index H = 1 * index H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [one_mul] | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (normalCore H) = 1 * index H | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (normalCore H) = index H | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (normalCore H) = 1 * index H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [Subgroup.normalCore_eq_ker] | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (normalCore H) = index H | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = index H | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (normalCore H) = index H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [hHp] | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = index H | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = p | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = index H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | change f.ker.index = p | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = p | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker f) = p | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | refine' Or.resolve_left (Nat.Prime.eq_one_or_self_of_dvd hp f.ker.index _) _ | case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker f) = p | case a.refine'_1
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker f) ∣ p
case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ¬index (MonoidHom.ker f) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker f) = p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | apply aux_dvd_lemma hp | case a.refine'_1
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker f) ∣ p
case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ¬index (MonoidHom.ker f) = 1 | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker f) ∣ Nat.factorial p
case a.refine'_1.hr
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ∀ {l : ℕ}, Nat.Prime l → l ∣ index (MonoidHom.ker f) → p ≤ l
case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ¬index (MonoidHom.ker f) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker f) ∣ p
case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ¬index (MonoidHom.ker f) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | have hf := Subgroup.index_ker f | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker f) ∣ Nat.factorial p | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Nat.card ↑(Set.range ⇑f)
⊢ index (MonoidHom.ker f) ∣ Nat.factorial p | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker f) ∣ Nat.factorial p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [Nat.card_eq_fintype_card] at hf | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Nat.card ↑(Set.range ⇑f)
⊢ index (MonoidHom.ker f) ∣ Nat.factorial p | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ index (MonoidHom.ker f) ∣ Nat.factorial p | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Nat.card ↑(Set.range ⇑f)
⊢ index (MonoidHom.ker f) ∣ Nat.factorial p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [hf, ← hHp] | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ index (MonoidHom.ker f) ∣ Nat.factorial p | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ Fintype.card ↑(Set.range ⇑f) ∣ Nat.factorial (index H) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ index (MonoidHom.ker f) ∣ Nat.factorial p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | unfold Subgroup.index | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ Fintype.card ↑(Set.range ⇑f) ∣ Nat.factorial (index H) | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ Fintype.card ↑(Set.range ⇑f) ∣ Nat.factorial (Nat.card (G ⧸ H)) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ Fintype.card ↑(Set.range ⇑f) ∣ Nat.factorial (index H)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [Nat.card_eq_fintype_card, ← Fintype.card_perm] | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ Fintype.card ↑(Set.range ⇑f) ∣ Nat.factorial (Nat.card (G ⧸ H)) | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ Fintype.card ↑(Set.range ⇑f) ∣ Fintype.card (Equiv.Perm (G ⧸ H)) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ Fintype.card ↑(Set.range ⇑f) ∣ Nat.factorial (Nat.card (G ⧸ H))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | apply f.range.card_subgroup_dvd_card | case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ Fintype.card ↑(Set.range ⇑f) ∣ Fintype.card (Equiv.Perm (G ⧸ H)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.h
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = Fintype.card ↑(Set.range ⇑f)
⊢ Fintype.card ↑(Set.range ⇑f) ∣ Fintype.card (Equiv.Perm (G ⧸ H))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | intro l hl hl' | case a.refine'_1.hr
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ∀ {l : ℕ}, Nat.Prime l → l ∣ index (MonoidHom.ker f) → p ≤ l | case a.refine'_1.hr
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
l : ℕ
hl : Nat.Prime l
hl' : l ∣ index (MonoidHom.ker f)
⊢ p ≤ l | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.hr
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ∀ {l : ℕ}, Nat.Prime l → l ∣ index (MonoidHom.ker f) → p ≤ l
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | apply hp' hl | case a.refine'_1.hr
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
l : ℕ
hl : Nat.Prime l
hl' : l ∣ index (MonoidHom.ker f)
⊢ p ≤ l | case a.refine'_1.hr
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
l : ℕ
hl : Nat.Prime l
hl' : l ∣ index (MonoidHom.ker f)
⊢ l ∣ Fintype.card G | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.hr
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
l : ℕ
hl : Nat.Prime l
hl' : l ∣ index (MonoidHom.ker f)
⊢ p ≤ l
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | exact dvd_trans hl' f.ker.index_dvd_card | case a.refine'_1.hr
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
l : ℕ
hl : Nat.Prime l
hl' : l ∣ index (MonoidHom.ker f)
⊢ l ∣ Fintype.card G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_1.hr
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
l : ℕ
hl : Nat.Prime l
hl' : l ∣ index (MonoidHom.ker f)
⊢ l ∣ Fintype.card G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | intro hf | case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ¬index (MonoidHom.ker f) = 1 | case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ¬index (MonoidHom.ker f) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | apply Nat.Prime.ne_one hp | case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ False | case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ p = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [← hHp, Subgroup.index_eq_one, eq_top_iff] | case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ p = 1 | case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ ⊤ ≤ H | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ p = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | apply le_trans _ (Subgroup.normalCore_le H) | case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ ⊤ ≤ H | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ ⊤ ≤ normalCore H | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine'_2
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ ⊤ ≤ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | rw [← eq_top_iff, ← Subgroup.index_eq_one, Subgroup.normalCore_eq_ker] | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ ⊤ ≤ normalCore H | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ ⊤ ≤ normalCore H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_smallest_prime_factor | [114, 1] | [157, 29] | exact hf | α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.18788
inst✝³ : Fintype α
inst✝² : DecidableEq α
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
H : Subgroup G
p : ℕ
hp : Nat.Prime p
hHp : index H = p
hp' : ∀ {l : ℕ}, Nat.Prime l → l ∣ Fintype.card G → p ≤ l
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
hf : index (MonoidHom.ker f) = 1
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | have : Fintype (G ⧸ H) := by
apply fintypeOfNotInfinite _
intro h
apply two_ne_zero (α := ℕ)
rw [← hH]
exact Cardinal.mk_toNat_of_infinite | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
⊢ Normal H | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
⊢ Normal H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
⊢ Normal H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | let f := MulAction.toPermHom G (G ⧸ H) | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
⊢ Normal H | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ Normal H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
⊢ Normal H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | convert MonoidHom.normal_ker f | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ Normal H | case h.e'_3
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ H = MonoidHom.ker f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ Normal H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | suffices H.normalCore.relindex H = 1 by
rw [← Subgroup.normalCore_eq_ker]
erw [Subgroup.index_eq_one] at this
apply le_antisymm _ (Subgroup.normalCore_le _)
intro x hx
rw [← Subgroup.coe_mk H x hx, ← Subgroup.mem_subgroupOf, this]
apply Subgroup.mem_top | case h.e'_3
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ H = MonoidHom.ker f | case h.e'_3
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ relindex (normalCore H) H = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ H = MonoidHom.ker f
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply fintypeOfNotInfinite _ | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
⊢ Fintype (G ⧸ H) | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
⊢ ¬Infinite (G ⧸ H) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
⊢ Fintype (G ⧸ H)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | intro h | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
⊢ ¬Infinite (G ⧸ H) | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
h : Infinite (G ⧸ H)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
⊢ ¬Infinite (G ⧸ H)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply two_ne_zero (α := ℕ) | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
h : Infinite (G ⧸ H)
⊢ False | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
h : Infinite (G ⧸ H)
⊢ 2 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
h : Infinite (G ⧸ H)
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [← hH] | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
h : Infinite (G ⧸ H)
⊢ 2 = 0 | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
h : Infinite (G ⧸ H)
⊢ index H = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
h : Infinite (G ⧸ H)
⊢ 2 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | exact Cardinal.mk_toNat_of_infinite | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
h : Infinite (G ⧸ H)
⊢ index H = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
h : Infinite (G ⧸ H)
⊢ index H = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [← Subgroup.normalCore_eq_ker] | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : relindex (normalCore H) H = 1
⊢ H = MonoidHom.ker f | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : relindex (normalCore H) H = 1
⊢ H = normalCore H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : relindex (normalCore H) H = 1
⊢ H = MonoidHom.ker f
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | erw [Subgroup.index_eq_one] at this | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : relindex (normalCore H) H = 1
⊢ H = normalCore H | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H = normalCore H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : relindex (normalCore H) H = 1
⊢ H = normalCore H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply le_antisymm _ (Subgroup.normalCore_le _) | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H = normalCore H | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H ≤ normalCore H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H = normalCore H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | intro x hx | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H ≤ normalCore H | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ x ∈ normalCore H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
⊢ H ≤ normalCore H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [← Subgroup.coe_mk H x hx, ← Subgroup.mem_subgroupOf, this] | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ x ∈ normalCore H | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ { val := x, property := hx } ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ x ∈ normalCore H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply Subgroup.mem_top | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ { val := x, property := hx } ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : subgroupOf (normalCore H) H = ⊤
x : G
hx : x ∈ H
⊢ { val := x, property := hx } ∈ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply mul_left_injective₀ this | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ relindex (normalCore H) H = 1 | case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ (fun a => a * index H) (relindex (normalCore H) H) = (fun a => a * index H) 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ relindex (normalCore H) H = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | dsimp | case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ (fun a => a * index H) (relindex (normalCore H) H) = (fun a => a * index H) 1 | case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ relindex (normalCore H) H * index H = 1 * index H | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ (fun a => a * index H) (relindex (normalCore H) H) = (fun a => a * index H) 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [Subgroup.relindex_mul_index (Subgroup.normalCore_le H), one_mul,
Subgroup.normalCore_eq_ker, hH] | case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ relindex (normalCore H) H * index H = 1 * index H | case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ relindex (normalCore H) H * index H = 1 * index H
TACTIC:
|
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