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/Jordan.lean | eq_s2_of_nontrivial | [751, 1] | [763, 11] | exact (2 : ℕ).factorial | case h.a.refine'_1
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ ℕ
case h.a.refine'_2
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial (Fintype.card α) ≤ ?h.a.refine'_1
case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ ?h.a.refine'_1 ≤ Fintype.card ↥G | case h.a.refine'_2
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial (Fintype.card α) ≤ Nat.factorial 2
case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial 2 ≤ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a.refine'_1
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ ℕ
case h.a.refine'_2
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial (Fintype.card α) ≤ ?h.a.refine'_1
case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ ?h.a.refine'_1 ≤ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | eq_s2_of_nontrivial | [751, 1] | [763, 11] | exact Nat.factorial_le hα | case h.a.refine'_2
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial (Fintype.card α) ≤ Nat.factorial 2
case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial 2 ≤ Fintype.card ↥G | case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial 2 ≤ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a.refine'_2
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial (Fintype.card α) ≤ Nat.factorial 2
case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial 2 ≤ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | eq_s2_of_nontrivial | [751, 1] | [763, 11] | rw [Nat.factorial_two] | case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial 2 ≤ Fintype.card ↥G | case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ 2 ≤ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ Nat.factorial 2 ≤ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | eq_s2_of_nontrivial | [751, 1] | [763, 11] | rw [← Fintype.one_lt_card_iff_nontrivial] at hG | case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ 2 ≤ Fintype.card ↥G | case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : 1 < Fintype.card ↥G
⊢ 2 ≤ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : Nontrivial ↥G
⊢ 2 ≤ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | eq_s2_of_nontrivial | [751, 1] | [763, 11] | exact hG | case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : 1 < Fintype.card ↥G
⊢ 2 ≤ Fintype.card ↥G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a.refine'_3
α : Type u_1
inst✝ : Fintype α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≤ 2
hG : 1 < Fintype.card ↥G
⊢ 2 ≤ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | classical
let φ := MulAction.toPermHom K α
let f : α →ₑ[φ] α :=
{ toFun := id
map_smul' := fun k x => rfl }
have hf : Function.Bijective f := Function.bijective_id
suffices Function.Surjective φ by
rw [isMultiplyPretransitive_of_bijective_map_iff this hf]
rw [← hα]
apply Equiv.Perm.isMultiplyPretransitive
rw [← MonoidHom.range_top_iff_surjective]
apply Subgroup.eq_top_of_card_eq
apply le_antisymm
apply Fintype.card_subtype_le
suffices hg : toPerm g ∈ φ.range by
rw [Fintype.card_perm, hα, Nat.factorial_two, Nat.succ_le_iff, Subgroup.one_lt_card_iff_ne_bot]
intro h; apply hga
rw [h, Subgroup.mem_bot] at hg
rw [← MulAction.toPerm_apply, hg, Equiv.Perm.coe_one, id_eq]
use g
rfl | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
⊢ IsMultiplyPretransitive K α 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
⊢ IsMultiplyPretransitive K α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | let φ := MulAction.toPermHom K α | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
⊢ IsMultiplyPretransitive K α 2 | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
⊢ IsMultiplyPretransitive K α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
⊢ IsMultiplyPretransitive K α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | let f : α →ₑ[φ] α :=
{ toFun := id
map_smul' := fun k x => rfl } | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
⊢ IsMultiplyPretransitive K α 2 | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
⊢ IsMultiplyPretransitive K α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
⊢ IsMultiplyPretransitive K α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | have hf : Function.Bijective f := Function.bijective_id | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
⊢ IsMultiplyPretransitive K α 2 | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsMultiplyPretransitive K α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
⊢ IsMultiplyPretransitive K α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | suffices Function.Surjective φ by
rw [isMultiplyPretransitive_of_bijective_map_iff this hf]
rw [← hα]
apply Equiv.Perm.isMultiplyPretransitive | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsMultiplyPretransitive K α 2 | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Function.Surjective ⇑φ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ IsMultiplyPretransitive K α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | rw [← MonoidHom.range_top_iff_surjective] | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Function.Surjective ⇑φ | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ MonoidHom.range φ = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Function.Surjective ⇑φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | apply Subgroup.eq_top_of_card_eq | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ MonoidHom.range φ = ⊤ | case h
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card ↥(MonoidHom.range φ) = Fintype.card (Equiv.Perm α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ MonoidHom.range φ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | apply le_antisymm | case h
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card ↥(MonoidHom.range φ) = Fintype.card (Equiv.Perm α) | case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card ↥(MonoidHom.range φ) ≤ Fintype.card (Equiv.Perm α)
case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥(MonoidHom.range φ) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card ↥(MonoidHom.range φ) = Fintype.card (Equiv.Perm α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | apply Fintype.card_subtype_le | case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card ↥(MonoidHom.range φ) ≤ Fintype.card (Equiv.Perm α)
case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥(MonoidHom.range φ) | case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥(MonoidHom.range φ) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card ↥(MonoidHom.range φ) ≤ Fintype.card (Equiv.Perm α)
case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥(MonoidHom.range φ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | suffices hg : toPerm g ∈ φ.range by
rw [Fintype.card_perm, hα, Nat.factorial_two, Nat.succ_le_iff, Subgroup.one_lt_card_iff_ne_bot]
intro h; apply hga
rw [h, Subgroup.mem_bot] at hg
rw [← MulAction.toPerm_apply, hg, Equiv.Perm.coe_one, id_eq] | case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥(MonoidHom.range φ) | case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ toPerm g ∈ MonoidHom.range φ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥(MonoidHom.range φ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | use g | case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ toPerm g ∈ MonoidHom.range φ | case h
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ φ g = toPerm g | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ toPerm g ∈ MonoidHom.range φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | rfl | case h
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ φ g = toPerm g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
⊢ φ g = toPerm g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | rw [isMultiplyPretransitive_of_bijective_map_iff this hf] | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
this : Function.Surjective ⇑φ
⊢ IsMultiplyPretransitive K α 2 | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
this : Function.Surjective ⇑φ
⊢ IsMultiplyPretransitive (Equiv.Perm α) α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
this : Function.Surjective ⇑φ
⊢ IsMultiplyPretransitive K α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | rw [← hα] | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
this : Function.Surjective ⇑φ
⊢ IsMultiplyPretransitive (Equiv.Perm α) α 2 | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
this : Function.Surjective ⇑φ
⊢ IsMultiplyPretransitive (Equiv.Perm α) α (Fintype.card α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
this : Function.Surjective ⇑φ
⊢ IsMultiplyPretransitive (Equiv.Perm α) α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | apply Equiv.Perm.isMultiplyPretransitive | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
this : Function.Surjective ⇑φ
⊢ IsMultiplyPretransitive (Equiv.Perm α) α (Fintype.card α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
this : Function.Surjective ⇑φ
⊢ IsMultiplyPretransitive (Equiv.Perm α) α (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | rw [Fintype.card_perm, hα, Nat.factorial_two, Nat.succ_le_iff, Subgroup.one_lt_card_iff_ne_bot] | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥(MonoidHom.range φ) | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
⊢ MonoidHom.range φ ≠ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥(MonoidHom.range φ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | intro h | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
⊢ MonoidHom.range φ ≠ ⊥ | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
h : MonoidHom.range φ = ⊥
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
⊢ MonoidHom.range φ ≠ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | apply hga | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
h : MonoidHom.range φ = ⊥
⊢ False | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
h : MonoidHom.range φ = ⊥
⊢ g • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
h : MonoidHom.range φ = ⊥
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | rw [h, Subgroup.mem_bot] at hg | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
h : MonoidHom.range φ = ⊥
⊢ g • a = a | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g = 1
h : MonoidHom.range φ = ⊥
⊢ g • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g ∈ MonoidHom.range φ
h : MonoidHom.range φ = ⊥
⊢ g • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | nontrivial_on_equiv_perm_two | [766, 1] | [789, 6] | rw [← MulAction.toPerm_apply, hg, Equiv.Perm.coe_one, id_eq] | α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g = 1
h : MonoidHom.range φ = ⊥
⊢ g • a = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝² : Fintype α
G : Subgroup (Equiv.Perm α)
K : Type u_1
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Fintype.card α = 2
g : K
a : α
hga : g • a ≠ a
φ : K →* Equiv.Perm α := toPermHom K α
f : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }
hf : Function.Bijective ⇑f
hg : toPerm g = 1
h : MonoidHom.range φ = ⊥
⊢ g • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | obtain ⟨a, _, hgc⟩ := hgc | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
hgc : Equiv.Perm.IsCycle g
⊢ IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
⊢ IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
hgc : Equiv.Perm.IsCycle g
⊢ IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | have hs : ∀ x : α, g • x ≠ x ↔
x ∈ SubMulAction.ofFixingSubgroup G ((↑g.support : Set α)ᶜ) := by
intro x
rw [SubMulAction.mem_ofFixingSubgroup_iff]
simp only [Set.mem_compl_iff, Finset.mem_coe, Equiv.Perm.not_mem_support]
rfl | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
⊢ IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
⊢ IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | suffices ∀ x ∈ SubMulAction.ofFixingSubgroup G ((↑g.support : Set α)ᶜ),
∃ k : fixingSubgroup G ((↑g.support : Set α)ᶜ), x = k • a
by
apply IsPretransitive.mk
rintro ⟨x, hx⟩ ⟨y, hy⟩
obtain ⟨k, hk⟩ := this x hx
obtain ⟨k', hk'⟩ := this y hy
use k' * k⁻¹
rw [← SetLike.coe_eq_coe]
simp only [SetLike.mk_smul_mk]
rw [hk, hk', smul_smul, inv_mul_cancel_right] | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | intro x hx | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∃ k, x = k • a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | have hg' : (⟨g, hg⟩ : ↥G) ∈ fixingSubgroup G ((↑g.support : Set α)ᶜ) :=
by
simp_rw [mem_fixingSubgroup_iff G]
intro y hy
simpa only [Set.mem_compl_iff, Finset.mem_coe, Equiv.Perm.not_mem_support] using hy | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∃ k, x = k • a | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∃ k, x = k • a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∃ k, x = k • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | let g' : fixingSubgroup (↥G) ((↑g.support : Set α)ᶜ) := ⟨(⟨g, hg⟩ : ↥G), hg'⟩ | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∃ k, x = k • a | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
g' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) := { val := { val := g, property := hg }, property := hg' }
⊢ ∃ k, x = k • a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∃ k, x = k • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | obtain ⟨i, hi⟩ := hgc ((hs x).mpr hx) | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
g' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) := { val := { val := g, property := hg }, property := hg' }
⊢ ∃ k, x = k • a | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
g' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) := { val := { val := g, property := hg }, property := hg' }
i : ℤ
hi : (g ^ i) a = x
⊢ ∃ k, x = k • a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
g' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) := { val := { val := g, property := hg }, property := hg' }
⊢ ∃ k, x = k • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | use g' ^ i | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
g' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) := { val := { val := g, property := hg }, property := hg' }
i : ℤ
hi : (g ^ i) a = x
⊢ ∃ k, x = k • a | case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
g' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) := { val := { val := g, property := hg }, property := hg' }
i : ℤ
hi : (g ^ i) a = x
⊢ x = g' ^ i • a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
g' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) := { val := { val := g, property := hg }, property := hg' }
i : ℤ
hi : (g ^ i) a = x
⊢ ∃ k, x = k • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | exact hi.symm | case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
g' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) := { val := { val := g, property := hg }, property := hg' }
i : ℤ
hi : (g ^ i) a = x
⊢ x = g' ^ i • a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
hg' : { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
g' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) := { val := { val := g, property := hg }, property := hg' }
i : ℤ
hi : (g ^ i) a = x
⊢ x = g' ^ i • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | intro x | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
⊢ ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
x : α
⊢ g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
⊢ ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | rw [SubMulAction.mem_ofFixingSubgroup_iff] | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
x : α
⊢ g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
x : α
⊢ g • x ≠ x ↔ x ∉ (↑(Equiv.Perm.support g))ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
x : α
⊢ g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | simp only [Set.mem_compl_iff, Finset.mem_coe, Equiv.Perm.not_mem_support] | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
x : α
⊢ g • x ≠ x ↔ x ∉ (↑(Equiv.Perm.support g))ᶜ | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
x : α
⊢ g • x ≠ x ↔ ¬g x = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
x : α
⊢ g • x ≠ x ↔ x ∉ (↑(Equiv.Perm.support g))ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | rfl | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
x : α
⊢ g • x ≠ x ↔ ¬g x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
x : α
⊢ g • x ≠ x ↔ ¬g x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | apply IsPretransitive.mk | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
⊢ IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | case exists_smul_eq
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
⊢ ∀ (x y : ↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)), ∃ g_1, g_1 • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
⊢ IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | rintro ⟨x, hx⟩ ⟨y, hy⟩ | case exists_smul_eq
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
⊢ ∀ (x y : ↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)), ∃ g_1, g_1 • x = y | case exists_smul_eq.mk.mk
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∃ g_1, g_1 • { val := x, property := hx } = { val := y, property := hy } | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
⊢ ∀ (x y : ↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)), ∃ g_1, g_1 • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | obtain ⟨k, hk⟩ := this x hx | case exists_smul_eq.mk.mk
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∃ g_1, g_1 • { val := x, property := hx } = { val := y, property := hy } | case exists_smul_eq.mk.mk.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
⊢ ∃ g_1, g_1 • { val := x, property := hx } = { val := y, property := hy } | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq.mk.mk
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∃ g_1, g_1 • { val := x, property := hx } = { val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | obtain ⟨k', hk'⟩ := this y hy | case exists_smul_eq.mk.mk.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
⊢ ∃ g_1, g_1 • { val := x, property := hx } = { val := y, property := hy } | case exists_smul_eq.mk.mk.intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ ∃ g_1, g_1 • { val := x, property := hx } = { val := y, property := hy } | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq.mk.mk.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
⊢ ∃ g_1, g_1 • { val := x, property := hx } = { val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | use k' * k⁻¹ | case exists_smul_eq.mk.mk.intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ ∃ g_1, g_1 • { val := x, property := hx } = { val := y, property := hy } | case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ (k' * k⁻¹) • { val := x, property := hx } = { val := y, property := hy } | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq.mk.mk.intro.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ ∃ g_1, g_1 • { val := x, property := hx } = { val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | rw [← SetLike.coe_eq_coe] | case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ (k' * k⁻¹) • { val := x, property := hx } = { val := y, property := hy } | case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ ↑((k' * k⁻¹) • { val := x, property := hx }) = ↑{ val := y, property := hy } | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ (k' * k⁻¹) • { val := x, property := hx } = { val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | simp only [SetLike.mk_smul_mk] | case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ ↑((k' * k⁻¹) • { val := x, property := hx }) = ↑{ val := y, property := hy } | case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ (k' * k⁻¹) • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ ↑((k' * k⁻¹) • { val := x, property := hx }) = ↑{ val := y, property := hy }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | rw [hk, hk', smul_smul, inv_mul_cancel_right] | case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ (k' * k⁻¹) • x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
this : ∀ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ, ∃ k, x = k • a
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
k : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk : x = k • a
k' : ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
hk' : y = k' • a
⊢ (k' * k⁻¹) • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | simp_rw [mem_fixingSubgroup_iff G] | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∀ y ∈ (↑(Equiv.Perm.support g))ᶜ, { val := g, property := hg } • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ { val := g, property := hg } ∈ fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | intro y hy | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∀ y ∈ (↑(Equiv.Perm.support g))ᶜ, { val := g, property := hg } • y = y | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ (↑(Equiv.Perm.support g))ᶜ
⊢ { val := g, property := hg } • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
⊢ ∀ y ∈ (↑(Equiv.Perm.support g))ᶜ, { val := g, property := hg } • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_of_cycle | [792, 1] | [822, 28] | simpa only [Set.mem_compl_iff, Finset.mem_coe, Equiv.Perm.not_mem_support] using hy | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ (↑(Equiv.Perm.support g))ᶜ
⊢ { val := g, property := hg } • y = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
g : Equiv.Perm α
hg : g ∈ G
a : α
left✝ : g a ≠ a
hgc : ∀ ⦃y : α⦄, g y ≠ y → Equiv.Perm.SameCycle g a y
hs : ∀ (x : α), g • x ≠ x ↔ x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ
y : α
hy : y ∈ (↑(Equiv.Perm.support g))ᶜ
⊢ { val := g, property := hg } • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Equiv.Perm.IsSwap.cycleType | [825, 1] | [828, 37] | simp only [h.isCycle.cycleType, Equiv.Perm.card_support_eq_two.mpr h] | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Perm α)
inst✝ : DecidableEq α
σ : Perm α
h : IsSwap σ
⊢ Perm.cycleType σ = {2} | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Perm α)
inst✝ : DecidableEq α
σ : Perm α
h : IsSwap σ
⊢ ↑[2] = {2} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Perm α)
inst✝ : DecidableEq α
σ : Perm α
h : IsSwap σ
⊢ Perm.cycleType σ = {2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Equiv.Perm.IsSwap.cycleType | [825, 1] | [828, 37] | simp only [Multiset.coe_singleton] | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Perm α)
inst✝ : DecidableEq α
σ : Perm α
h : IsSwap σ
⊢ ↑[2] = {2} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Perm α)
inst✝ : DecidableEq α
σ : Perm α
h : IsSwap σ
⊢ ↑[2] = {2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Equiv.Perm.IsSwap.orderOf | [831, 1] | [833, 85] | rw [← Equiv.Perm.lcm_cycleType, h.cycleType, Multiset.lcm_singleton, normalize_eq] | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Perm α)
inst✝ : DecidableEq α
σ : Perm α
h : IsSwap σ
⊢ _root_.orderOf σ = 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Perm α)
inst✝ : DecidableEq α
σ : Perm α
h : IsSwap σ
⊢ _root_.orderOf σ = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | cases' Nat.lt_or_ge (Fintype.card α) 3 with hα3 hα3 | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
⊢ G = ⊤ | case inl
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α < 3
⊢ G = ⊤
case inr
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
⊢ G = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
⊢ G = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | obtain ⟨n, hn⟩ := Nat.exists_eq_add_of_le hα3 | case inr
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
⊢ G = ⊤ | case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = 3 + n
⊢ G = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
⊢ G = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [add_comm] at hn | case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = 3 + n
⊢ G = ⊤ | case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ G = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = 3 + n
⊢ G = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | have hsc : Set.ncard ((g.support)ᶜ : Set α) = n.succ := by
apply Nat.add_left_cancel
rw [Set.ncard_add_ncard_compl, Nat.card_eq_fintype_card, Set.ncard_coe_Finset, Equiv.Perm.card_support_eq_two.mpr h2g, add_comm, hn] | case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ G = ⊤ | case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ G = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ G = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply IsMultiplyPretransitive.eq_top_of_is_full_minus_one_pretransitive | case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ G = ⊤ | case inr.intro.hmt
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ IsMultiplyPretransitive (↥G) α (Fintype.card α - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ G = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply IsMultiplyPreprimitive.toIsMultiplyPretransitive | case inr.intro.hmt
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ IsMultiplyPretransitive (↥G) α (Fintype.card α - 1) | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ IsMultiplyPretransitive (↥G) α (Fintype.card α - 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | have hn' : Fintype.card α - 1 = 1 + n.succ := by
rw [hn, add_comm 1]
simp only [ge_iff_le, Nat.succ_sub_succ_eq_sub, nonpos_iff_eq_zero, add_eq_zero, and_false, tsub_zero] | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 1) | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [hn'] | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 1) | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (1 + Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | refine' isMultiplyPreprimitive_jordan hG hsc _ _ | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (1 + Nat.succ n) | case inr.intro.hmt.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ 1 + Nat.succ n < Fintype.card α
case inr.intro.hmt.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (1 + Nat.succ n)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | have : IsPretransitive _ _ := by
apply isPretransitive_of_cycle hg
exact Equiv.Perm.IsSwap.isCycle h2g | case inr.intro.hmt.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | case inr.intro.hmt.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply isPreprimitive_of_prime | case inr.intro.hmt.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | case inr.intro.hmt.h.refine'_2.hp
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ Nat.Prime (Fintype.card ↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | convert Nat.prime_two | case inr.intro.hmt.h.refine'_2.hp
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ Nat.Prime (Fintype.card ↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)) | case h.e'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ Fintype.card ↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) = 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h.refine'_2.hp
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ Nat.Prime (Fintype.card ↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [Fintype.card_subtype] | case h.e'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ Fintype.card ↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) = 2 | case h.e'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ (Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ).card = 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ Fintype.card ↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [← Equiv.Perm.card_support_eq_two.mpr h2g] | case h.e'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ (Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ).card = 2 | case h.e'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ (Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ).card =
(Equiv.Perm.support g).card | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ (Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ).card = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply congr_arg | case h.e'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ (Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ).card =
(Equiv.Perm.support g).card | case h.e'_1.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ =
Equiv.Perm.support g | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ (Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ).card =
(Equiv.Perm.support g).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | ext x | case h.e'_1.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ =
Equiv.Perm.support g | case h.e'_1.h.a
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
x : α
⊢ x ∈ Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ ↔
x ∈ Equiv.Perm.support g | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ =
Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | simp only [SubMulAction.mem_ofFixingSubgroup_iff, Set.mem_compl_iff, Finset.mem_coe,
Equiv.Perm.mem_support, ne_eq, not_not, Finset.mem_univ, forall_true_left,
Finset.mem_filter, true_and] | case h.e'_1.h.a
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
x : α
⊢ x ∈ Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ ↔
x ∈ Equiv.Perm.support g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h.a
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
x : α
⊢ x ∈ Finset.filter (fun x => x ∈ SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) Finset.univ ↔
x ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [Nat.lt_succ_iff] at hα3 | case inl
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α < 3
⊢ G = ⊤ | case inl
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ G = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α < 3
⊢ G = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply Subgroup.eq_top_of_card_eq | case inl
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ G = ⊤ | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Fintype.card ↥G = Fintype.card (Equiv.Perm α) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ G = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply le_antisymm (Fintype.card_subtype_le _) | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Fintype.card ↥G = Fintype.card (Equiv.Perm α) | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Fintype.card ↥G = Fintype.card (Equiv.Perm α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [Fintype.card_equiv (Equiv.cast rfl)] | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥G | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Nat.factorial (Fintype.card α) ≤ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Fintype.card (Equiv.Perm α) ≤ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | refine' le_trans (Nat.factorial_le hα3) _ | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Nat.factorial (Fintype.card α) ≤ Fintype.card ↥G | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Nat.factorial 2 ≤ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Nat.factorial (Fintype.card α) ≤ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [Nat.factorial_two] | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Nat.factorial 2 ≤ Fintype.card ↥G | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ 2 ≤ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ Nat.factorial 2 ≤ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply Nat.le_of_dvd Fintype.card_pos | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ 2 ≤ Fintype.card ↥G | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ 2 ∣ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ 2 ≤ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [← h2g.orderOf, orderOf_submonoid ⟨g, hg⟩] | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ 2 ∣ Fintype.card ↥G | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ orderOf { val := g, property := hg } ∣ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ 2 ∣ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | exact orderOf_dvd_card | case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ orderOf { val := g, property := hg } ∣ Fintype.card ↥G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≤ 2
⊢ orderOf { val := g, property := hg } ∣ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply Nat.add_left_cancel | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n | case a
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ ?n + Set.ncard (↑(Equiv.Perm.support g))ᶜ = ?n + Nat.succ n
case n
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ ℕ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [Set.ncard_add_ncard_compl, Nat.card_eq_fintype_card, Set.ncard_coe_Finset, Equiv.Perm.card_support_eq_two.mpr h2g, add_comm, hn] | case a
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ ?n + Set.ncard (↑(Equiv.Perm.support g))ᶜ = ?n + Nat.succ n
case n
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ ℕ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ ?n + Set.ncard (↑(Equiv.Perm.support g))ᶜ = ?n + Nat.succ n
case n
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
⊢ ℕ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [hn, add_comm 1] | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ Fintype.card α - 1 = 1 + Nat.succ n | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ n + 3 - 1 = Nat.succ n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ Fintype.card α - 1 = 1 + Nat.succ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | simp only [ge_iff_le, Nat.succ_sub_succ_eq_sub, nonpos_iff_eq_zero, add_eq_zero, and_false, tsub_zero] | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ n + 3 - 1 = Nat.succ n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
⊢ n + 3 - 1 = Nat.succ n + 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | rw [← hn'] | case inr.intro.hmt.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ 1 + Nat.succ n < Fintype.card α | case inr.intro.hmt.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ Fintype.card α - 1 < Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ 1 + Nat.succ n < Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply Nat.sub_lt _ (by norm_num) | case inr.intro.hmt.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ Fintype.card α - 1 < Fintype.card α | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ 0 < Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ Fintype.card α - 1 < Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply lt_of_lt_of_le (by norm_num) hα3 | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ 0 < Fintype.card α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ 0 < Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | norm_num | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ 0 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ 0 < 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | norm_num | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ 0 < 3
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | apply isPretransitive_of_cycle hg | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsPretransitive ?m.318640 ?m.318641 | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ Equiv.Perm.IsCycle g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ IsPretransitive ?m.318640 ?m.318641
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_swap | [837, 1] | [879, 33] | exact Equiv.Perm.IsSwap.isCycle h2g | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ Equiv.Perm.IsCycle g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h2g : Equiv.Perm.IsSwap g
hg : g ∈ G
hα3 : Fintype.card α ≥ 3
n : ℕ
hn : Fintype.card α = n + 3
hsc : Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
hn' : Fintype.card α - 1 = 1 + Nat.succ n
⊢ Equiv.Perm.IsCycle g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | cases' Nat.lt_or_ge (Fintype.card α) 4 with hα4 hα4 | α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
⊢ alternatingGroup α ≤ G | case inl
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α < 4
⊢ alternatingGroup α ≤ G
case inr
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
⊢ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
⊢ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | obtain ⟨n, hn⟩ := Nat.exists_eq_add_of_le hα4 | case inr
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
⊢ alternatingGroup α ≤ G | case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = 4 + n
⊢ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
⊢ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | rw [add_comm] at hn | case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = 4 + n
⊢ alternatingGroup α ≤ G | case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
⊢ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = 4 + n
⊢ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | apply IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
⊢ alternatingGroup α ≤ G | case inr.intro.hmt
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
⊢ IsMultiplyPretransitive (↥G) α (Fintype.card α - 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
⊢ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | apply IsMultiplyPreprimitive.toIsMultiplyPretransitive | case inr.intro.hmt
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
⊢ IsMultiplyPretransitive (↥G) α (Fintype.card α - 2) | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
⊢ IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | have hn' : Fintype.card α - 2 = 1 + n.succ := by
simp [hn, add_comm 1] | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 2) | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 2)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | rw [hn'] | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 2) | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (1 + Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (Fintype.card α - 2)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | refine' isMultiplyPreprimitive_jordan hG _ _ _ | case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (1 + Nat.succ n) | case inr.intro.hmt.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ Set α
case inr.intro.hmt.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ Set.ncard ?inr.intro.hmt.h.refine'_1 = Nat.succ n
case inr.intro.hmt.h.refine'_3
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ 1 + Nat.succ n < Fintype.card α
case inr.intro.hmt.h.refine'_4
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsPreprimitive ↥(fixingSubgroup ↥G ?inr.intro.hmt.h.refine'_1)
↥(SubMulAction.ofFixingSubgroup ↥G ?inr.intro.hmt.h.refine'_1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsMultiplyPreprimitive (↥G) α (1 + Nat.succ n)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | exact (g.supportᶜ : Set α) | case inr.intro.hmt.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ Set α
case inr.intro.hmt.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ Set.ncard ?inr.intro.hmt.h.refine'_1 = Nat.succ n
case inr.intro.hmt.h.refine'_3
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ 1 + Nat.succ n < Fintype.card α
case inr.intro.hmt.h.refine'_4
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsPreprimitive ↥(fixingSubgroup ↥G ?inr.intro.hmt.h.refine'_1)
↥(SubMulAction.ofFixingSubgroup ↥G ?inr.intro.hmt.h.refine'_1) | case inr.intro.hmt.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ Set.ncard (↑(Equiv.Perm.support g))ᶜ = Nat.succ n
case inr.intro.hmt.h.refine'_3
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ 1 + Nat.succ n < Fintype.card α
case inr.intro.hmt.h.refine'_4
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ Set α
case inr.intro.hmt.h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ Set.ncard ?inr.intro.hmt.h.refine'_1 = Nat.succ n
case inr.intro.hmt.h.refine'_3
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ 1 + Nat.succ n < Fintype.card α
case inr.intro.hmt.h.refine'_4
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsPreprimitive ↥(fixingSubgroup ↥G ?inr.intro.hmt.h.refine'_1)
↥(SubMulAction.ofFixingSubgroup ↥G ?inr.intro.hmt.h.refine'_1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | have : IsPretransitive _ _ := by
apply isPretransitive_of_cycle hg
exact Equiv.Perm.IsThreeCycle.isCycle h3g | case inr.intro.hmt.h.refine'_4
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | case inr.intro.hmt.h.refine'_4
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h.refine'_4
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | jordan_three_cycle | [883, 1] | [930, 47] | apply isPreprimitive_of_prime | case inr.intro.hmt.h.refine'_4
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ) | case inr.intro.hmt.h.refine'_4.hp
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ Nat.Prime (Fintype.card ↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.hmt.h.refine'_4
α : Type u_1
inst✝¹ : Fintype α
G : Subgroup (Equiv.Perm α)
inst✝ : DecidableEq α
hG : IsPreprimitive (↥G) α
g : Equiv.Perm α
h3g : Equiv.Perm.IsThreeCycle g
hg : g ∈ G
hα4 : Fintype.card α ≥ 4
n : ℕ
hn : Fintype.card α = n + 4
hn' : Fintype.card α - 2 = 1 + Nat.succ n
this :
IsPretransitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
⊢ IsPreprimitive ↥(fixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
↥(SubMulAction.ofFixingSubgroup (↥G) (↑(Equiv.Perm.support g))ᶜ)
TACTIC:
|
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