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/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | rw [Set.not_mem_compl_iff] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑x ∉ sᶜ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑x ∉ sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive | [98, 1] | [147, 40] | exact x.prop | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑x ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : Set.Nontrivial sᶜ
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → Equiv.Perm ↑s := toPerm
k : Equiv.Perm ↑sᶜ
hk_sign : Equiv.Perm.sign k = -1
hks : Equiv.Perm.ofSubtype k • s = s
hminus_one_ne_one : -1 ≠ 1
g : Equiv.Perm ↑s
g' : Equiv.Perm α :=
if Equiv.Perm.sign g = 1 then Equiv.Perm.ofSubtype g else Equiv.Perm.ofSubtype g * Equiv.Perm.ofSubtype k
hsg : Equiv.Perm.sign g = -1
x : ↑s
⊢ ↑x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | let f : s →ₑ[φ] s :=
{ toFun := id
map_smul' := fun ⟨⟨m, hm'⟩, hm⟩ ⟨a, ha⟩ => rfl } | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | have hf : Function.Surjective f := Function.surjective_id | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | apply isPreprimitive_of_surjective_map hf | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥G) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | apply stabilizer.isPreprimitive | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Surjective ⇑f
⊢ Set.Nontrivial sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Surjective ⇑f
⊢ IsPreprimitive ↥(stabilizer (↥(alternatingGroup α)) s) ↑s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | exact hsc | case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Surjective ⇑f
⊢ Set.Nontrivial sᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
φ : ↥(stabilizer (↥(alternatingGroup α)) s) → ↥(stabilizer (↥G) s) :=
fun g => { val := { val := ↑↑g, property := ⋯ }, property := ⋯ }
f : ↑s →ₑ[φ] ↑s := { toFun := id, map_smul' := ⋯ }
hf : Function.Surjective ⇑f
⊢ Set.Nontrivial sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | use! (g : alternatingGroup α) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↥(stabilizer (↥G) s) | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ G
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ?property } ∈ stabilizer (↥G) s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↥(stabilizer (↥G) s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | apply hG | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ G
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ?property } ∈ stabilizer (↥G) s | case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ⋯ } ∈ stabilizer (↥G) s | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ G
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ?property } ∈ stabilizer (↥G) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | rw [Subgroup.mem_inf] | case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ⋯ } ∈ stabilizer (↥G) s | case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s ∧ ↑↑g ∈ alternatingGroup α
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ⋯ } ∈ stabilizer (↥G) s | Please generate a tactic in lean4 to solve the state.
STATE:
case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ⋯ } ∈ stabilizer (↥G) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | constructor | case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s ∧ ↑↑g ∈ alternatingGroup α
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ⋯ } ∈ stabilizer (↥G) s | case property.a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s
case property.a.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ alternatingGroup α
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ⋯ } ∈ stabilizer (↥G) s | Please generate a tactic in lean4 to solve the state.
STATE:
case property.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s ∧ ↑↑g ∈ alternatingGroup α
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ⋯ } ∈ stabilizer (↥G) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | exact g.prop | case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ⋯ } ∈ stabilizer (↥G) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ { val := ↑↑g, property := ⋯ } ∈ stabilizer (↥G) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | let h := g.prop | case property.a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s | case property.a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
h : ↑g ∈ stabilizer (↥(alternatingGroup α)) s := Subtype.prop g
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s | Please generate a tactic in lean4 to solve the state.
STATE:
case property.a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | rw [mem_stabilizer_iff] at h ⊢ | case property.a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
h : ↑g ∈ stabilizer (↥(alternatingGroup α)) s := Subtype.prop g
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s | case property.a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
h : ↑g • s = s
⊢ ↑↑g • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case property.a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
h : ↑g ∈ stabilizer (↥(alternatingGroup α)) s := Subtype.prop g
⊢ ↑↑g ∈ stabilizer (Equiv.Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | exact h | case property.a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
h : ↑g • s = s
⊢ ↑↑g • s = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case property.a.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
h : ↑g • s = s
⊢ ↑↑g • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/StabilizerPrimitive.lean | alternatingGroup.stabilizer.isPreprimitive' | [150, 1] | [168, 12] | exact SetLike.coe_mem _ | case property.a.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ alternatingGroup α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case property.a.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hsc : Set.Nontrivial sᶜ
G : Subgroup (Equiv.Perm α)
hG : stabilizer (Equiv.Perm α) s ⊓ alternatingGroup α ≤ G
g : ↥(stabilizer (↥(alternatingGroup α)) s)
⊢ ↑↑g ∈ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.one_lt_encard_iff_nontrivial | [31, 1] | [38, 28] | unfold Set.Nontrivial | α : Type u_1
s : Set α
⊢ 1 < encard s ↔ Set.Nontrivial s | α : Type u_1
s : Set α
⊢ 1 < encard s ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Set α
⊢ 1 < encard s ↔ Set.Nontrivial s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.one_lt_encard_iff_nontrivial | [31, 1] | [38, 28] | rw [Set.one_lt_encard_iff] | α : Type u_1
s : Set α
⊢ 1 < encard s ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y | α : Type u_1
s : Set α
⊢ (∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b) ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Set α
⊢ 1 < encard s ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.one_lt_encard_iff_nontrivial | [31, 1] | [38, 28] | constructor | α : Type u_1
s : Set α
⊢ (∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b) ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y | case mp
α : Type u_1
s : Set α
⊢ (∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b) → ∃ x ∈ s, ∃ y ∈ s, x ≠ y
case mpr
α : Type u_1
s : Set α
⊢ (∃ x ∈ s, ∃ y ∈ s, x ≠ y) → ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Set α
⊢ (∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b) ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.one_lt_encard_iff_nontrivial | [31, 1] | [38, 28] | all_goals {
rintro ⟨a, b, c, d, e⟩
exact ⟨a, c, b, d, e⟩ } | case mp
α : Type u_1
s : Set α
⊢ (∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b) → ∃ x ∈ s, ∃ y ∈ s, x ≠ y
case mpr
α : Type u_1
s : Set α
⊢ (∃ x ∈ s, ∃ y ∈ s, x ≠ y) → ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
s : Set α
⊢ (∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b) → ∃ x ∈ s, ∃ y ∈ s, x ≠ y
case mpr
α : Type u_1
s : Set α
⊢ (∃ x ∈ s, ∃ y ∈ s, x ≠ y) → ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.one_lt_encard_iff_nontrivial | [31, 1] | [38, 28] | rintro ⟨a, b, c, d, e⟩ | case mpr
α : Type u_1
s : Set α
⊢ (∃ x ∈ s, ∃ y ∈ s, x ≠ y) → ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b | case mpr.intro.intro.intro.intro
α : Type u_1
s : Set α
a : α
b : a ∈ s
c : α
d : c ∈ s
e : a ≠ c
⊢ ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
s : Set α
⊢ (∃ x ∈ s, ∃ y ∈ s, x ≠ y) → ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.one_lt_encard_iff_nontrivial | [31, 1] | [38, 28] | exact ⟨a, c, b, d, e⟩ | case mpr.intro.intro.intro.intro
α : Type u_1
s : Set α
a : α
b : a ∈ s
c : α
d : c ∈ s
e : a ≠ c
⊢ ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro
α : Type u_1
s : Set α
a : α
b : a ∈ s
c : α
d : c ∈ s
e : a ≠ c
⊢ ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.one_lt_ncard_iff_nontrivial | [44, 1] | [47, 21] | rw [← Set.one_lt_encard_iff_nontrivial, ← Set.Finite.cast_ncard_eq (toFinite s),
Nat.one_lt_cast] | α : Type u_1
s : Set α
inst✝ : Finite ↑s
⊢ 1 < ncard s ↔ Set.Nontrivial s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Set α
inst✝ : Finite ↑s
⊢ 1 < ncard s ↔ Set.Nontrivial s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.subsingleton_iff_encard_le_one | [54, 1] | [56, 88] | rw [← Set.not_nontrivial_iff, not_iff_comm, not_le, Set.one_lt_encard_iff_nontrivial] | α✝ : Type u_1
α : Type u_2
B : Set α
⊢ Set.Subsingleton B ↔ encard B ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
α : Type u_2
B : Set α
⊢ Set.Subsingleton B ↔ encard B ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.subsingleton_iff_ncard_le_one | [58, 1] | [60, 89] | rw [← Set.not_nontrivial_iff, not_iff_comm, not_le, ← Set.one_lt_ncard_iff_nontrivial] | α✝ : Type u_1
α : Type u_2
inst✝ : Finite α
B : Set α
⊢ Set.Subsingleton B ↔ ncard B ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
α : Type u_2
inst✝ : Finite α
B : Set α
⊢ Set.Subsingleton B ↔ ncard B ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.eq_top_iff_ncard | [62, 1] | [71, 37] | rw [top_eq_univ, ← Set.compl_empty_iff, ← Set.ncard_eq_zero] | α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ B = ⊤ ↔ ncard B = Fintype.card α | α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 ↔ ncard B = Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ B = ⊤ ↔ ncard B = Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.eq_top_iff_ncard | [62, 1] | [71, 37] | rw [← Nat.card_eq_fintype_card] | α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 ↔ ncard B = Fintype.card α | α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 ↔ ncard B = Nat.card α | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 ↔ ncard B = Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.eq_top_iff_ncard | [62, 1] | [71, 37] | rw [← Set.ncard_add_ncard_compl B] | α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 ↔ ncard B = Nat.card α | α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 ↔ ncard B = ncard B + ncard Bᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 ↔ ncard B = Nat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.eq_top_iff_ncard | [62, 1] | [71, 37] | constructor | α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 ↔ ncard B = ncard B + ncard Bᶜ | case mp
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 → ncard B = ncard B + ncard Bᶜ
case mpr
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard B = ncard B + ncard Bᶜ → ncard Bᶜ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 ↔ ncard B = ncard B + ncard Bᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.eq_top_iff_ncard | [62, 1] | [71, 37] | intro H | case mp
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 → ncard B = ncard B + ncard Bᶜ | case mp
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
H : ncard Bᶜ = 0
⊢ ncard B = ncard B + ncard Bᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard Bᶜ = 0 → ncard B = ncard B + ncard Bᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.eq_top_iff_ncard | [62, 1] | [71, 37] | rw [H, add_zero] | case mp
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
H : ncard Bᶜ = 0
⊢ ncard B = ncard B + ncard Bᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
H : ncard Bᶜ = 0
⊢ ncard B = ncard B + ncard Bᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.eq_top_iff_ncard | [62, 1] | [71, 37] | intro H | case mpr
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard B = ncard B + ncard Bᶜ → ncard Bᶜ = 0 | case mpr
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
H : ncard B = ncard B + ncard Bᶜ
⊢ ncard Bᶜ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
⊢ ncard B = ncard B + ncard Bᶜ → ncard Bᶜ = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.eq_top_iff_ncard | [62, 1] | [71, 37] | exact Nat.add_left_cancel H.symm | case mpr
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
H : ncard B = ncard B + ncard Bᶜ
⊢ ncard Bᶜ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α✝ : Type u_1
α : Type u_2
inst✝ : Fintype α
B : Set α
H : ncard B = ncard B + ncard Bᶜ
⊢ ncard Bᶜ = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.add_eq_add_iff | [73, 1] | [76, 27] | rw [WithTop.add_right_cancel_iff] | α : Type u_1
c : ℕ∞
m n : ℕ
⊢ c + ↑m = ↑n + ↑m ↔ c = ↑n | α : Type u_1
c : ℕ∞
m n : ℕ
⊢ ↑m ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
c : ℕ∞
m n : ℕ
⊢ c + ↑m = ↑n + ↑m ↔ c = ↑n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.add_eq_add_iff | [73, 1] | [76, 27] | exact WithTop.coe_ne_top | α : Type u_1
c : ℕ∞
m n : ℕ
⊢ ↑m ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
c : ℕ∞
m n : ℕ
⊢ ↑m ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.add_one_eq_coe_succ_iff | [83, 1] | [85, 81] | rw [← Nat.cast_one, Nat.succ_eq_add_one, Nat.cast_add, WithTop.add_eq_add_iff] | α : Type u_1
c : ℕ∞
n : ℕ
⊢ c + 1 = ↑(Nat.succ n) ↔ c = ↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
c : ℕ∞
n : ℕ
⊢ c + 1 = ↑(Nat.succ n) ↔ c = ↑n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.add_coe_lt_add_iff | [91, 1] | [94, 27] | rw [WithTop.add_lt_add_iff_right] | α : Type u_1
c : ℕ∞
m n : ℕ
⊢ c + ↑m < ↑n + ↑m ↔ c < ↑n | α : Type u_1
c : ℕ∞
m n : ℕ
⊢ ↑m ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
c : ℕ∞
m n : ℕ
⊢ c + ↑m < ↑n + ↑m ↔ c < ↑n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.add_coe_lt_add_iff | [91, 1] | [94, 27] | exact WithTop.coe_ne_top | α : Type u_1
c : ℕ∞
m n : ℕ
⊢ ↑m ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
c : ℕ∞
m n : ℕ
⊢ ↑m ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.add_one_lt_coe_succ_iff | [102, 1] | [104, 85] | rw [← Nat.cast_one, Nat.succ_eq_add_one, Nat.cast_add, WithTop.add_coe_lt_add_iff] | α : Type u_1
c : ℕ∞
n : ℕ
⊢ c + 1 < ↑(Nat.succ n) ↔ c < ↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
c : ℕ∞
n : ℕ
⊢ c + 1 < ↑(Nat.succ n) ↔ c < ↑n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.add_coe_le_add_iff | [110, 1] | [112, 61] | exact WithTop.coe_ne_top | α : Type u_1
c : ℕ∞
m n : ℕ
⊢ ↑m ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
c : ℕ∞
m n : ℕ
⊢ ↑m ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.add_one_le_coe_succ_iff | [119, 1] | [121, 85] | rw [← Nat.cast_one, Nat.succ_eq_add_one, Nat.cast_add, WithTop.add_coe_le_add_iff] | α : Type u_1
c : ℕ∞
n : ℕ
⊢ c + 1 ≤ ↑(Nat.succ n) ↔ c ≤ ↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
c : ℕ∞
n : ℕ
⊢ c + 1 ≤ ↑(Nat.succ n) ↔ c ≤ ↑n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.coe_lt_iff_succ_le | [127, 1] | [134, 32] | induction c using WithTop.recTopCoe with
| top =>
simp only [Nat.cast_succ, le_top, iff_true]
apply WithTop.coe_lt_top
| coe m =>
simp only [ENat.some_eq_coe, Nat.cast_lt, Nat.cast_le]
exact Nat.lt_iff_add_one_le | α : Type u_1
n : ℕ
c : ℕ∞
⊢ ↑n < c ↔ ↑(Nat.succ n) ≤ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
n : ℕ
c : ℕ∞
⊢ ↑n < c ↔ ↑(Nat.succ n) ≤ c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.coe_lt_iff_succ_le | [127, 1] | [134, 32] | simp only [Nat.cast_succ, le_top, iff_true] | case top
α : Type u_1
n : ℕ
⊢ ↑n < ⊤ ↔ ↑(Nat.succ n) ≤ ⊤ | case top
α : Type u_1
n : ℕ
⊢ ↑n < ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case top
α : Type u_1
n : ℕ
⊢ ↑n < ⊤ ↔ ↑(Nat.succ n) ≤ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.coe_lt_iff_succ_le | [127, 1] | [134, 32] | apply WithTop.coe_lt_top | case top
α : Type u_1
n : ℕ
⊢ ↑n < ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case top
α : Type u_1
n : ℕ
⊢ ↑n < ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.coe_lt_iff_succ_le | [127, 1] | [134, 32] | simp only [ENat.some_eq_coe, Nat.cast_lt, Nat.cast_le] | case coe
α : Type u_1
n m : ℕ
⊢ ↑n < ↑m ↔ ↑(Nat.succ n) ≤ ↑m | case coe
α : Type u_1
n m : ℕ
⊢ n < m ↔ Nat.succ n ≤ m | Please generate a tactic in lean4 to solve the state.
STATE:
case coe
α : Type u_1
n m : ℕ
⊢ ↑n < ↑m ↔ ↑(Nat.succ n) ≤ ↑m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | WithTop.coe_lt_iff_succ_le | [127, 1] | [134, 32] | exact Nat.lt_iff_add_one_le | case coe
α : Type u_1
n m : ℕ
⊢ n < m ↔ Nat.succ n ≤ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case coe
α : Type u_1
n m : ℕ
⊢ n < m ↔ Nat.succ n ≤ m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Function.Injective.image_iInter_eq | [146, 1] | [149, 63] | rw [Set.InjOn.image_iInter_eq (Set.injOn_of_injective hf _)] | α✝ : Type u_1
α : Type u_3
β : Type u_4
G : Type ?u.14473
X : Type ?u.14476
ι : Sort u_2
κ : ι → Sort ?u.14484
inst✝ : Nonempty ι
f : α → β
hf : Injective f
s : ι → Set α
⊢ f '' ⋂ i, s i = ⋂ i, f '' s i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
α : Type u_3
β : Type u_4
G : Type ?u.14473
X : Type ?u.14476
ι : Sort u_2
κ : ι → Sort ?u.14484
inst✝ : Nonempty ι
f : α → β
hf : Injective f
s : ι → Set α
⊢ f '' ⋂ i, s i = ⋂ i, f '' s i
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | Set.subset_of_eq | [152, 1] | [153, 9] | rw [h] | α✝¹ : Type u_1
α✝ : Type ?u.14727
β : Type ?u.14730
G : Type ?u.14733
X : Type ?u.14736
ι : Sort ?u.14739
κ : ι → Sort ?u.14744
α : Type u_2
s t : Set α
h : s = t
⊢ s ⊆ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝¹ : Type u_1
α✝ : Type ?u.14727
β : Type ?u.14730
G : Type ?u.14733
X : Type ?u.14736
ι : Sort ?u.14739
κ : ι → Sort ?u.14744
α : Type u_2
s t : Set α
h : s = t
⊢ s ⊆ t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | MulAction.smul_set_iInter | [166, 1] | [172, 52] | obtain _ | _ := isEmpty_or_nonempty ι | α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
⊢ a • ⋂ i, s i = ⋂ i, a • s i | case inl
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : IsEmpty ι
⊢ a • ⋂ i, s i = ⋂ i, a • s i
case inr
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : Nonempty ι
⊢ a • ⋂ i, s i = ⋂ i, a • s i | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
⊢ a • ⋂ i, s i = ⋂ i, a • s i
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | MulAction.smul_set_iInter | [166, 1] | [172, 52] | refine' Eq.trans _ (Set.iInter_of_empty _).symm | case inl
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : IsEmpty ι
⊢ a • ⋂ i, s i = ⋂ i, a • s i | case inl
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : IsEmpty ι
⊢ a • ⋂ i, s i = Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : IsEmpty ι
⊢ a • ⋂ i, s i = ⋂ i, a • s i
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | MulAction.smul_set_iInter | [166, 1] | [172, 52] | rw [Set.iInter_of_empty] | case inl
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : IsEmpty ι
⊢ a • ⋂ i, s i = Set.univ | case inl
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : IsEmpty ι
⊢ a • Set.univ = Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : IsEmpty ι
⊢ a • ⋂ i, s i = Set.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | MulAction.smul_set_iInter | [166, 1] | [172, 52] | exact Set.smul_set_univ | case inl
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : IsEmpty ι
⊢ a • Set.univ = Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : IsEmpty ι
⊢ a • Set.univ = Set.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | MulAction.smul_set_iInter | [166, 1] | [172, 52] | exact (MulAction.injective _).image_iInter_eq _ | case inr
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : Nonempty ι
⊢ a • ⋂ i, s i = ⋂ i, a • s i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.17549
X : Type ?u.17552
ι : Sort u_4
κ : ι → Sort ?u.17560
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : ι → Set β
h✝ : Nonempty ι
⊢ a • ⋂ i, s i = ⋂ i, a • s i
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Set.lean | MulAction.smul_set_iInter₂ | [175, 1] | [177, 30] | simp_rw [smul_set_iInter] | α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.20290
X : Type ?u.20293
ι : Sort u_4
κ : ι → Sort u_5
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : (i : ι) → κ i → Set β
⊢ a • ⋂ i, ⋂ j, s i j = ⋂ i, ⋂ j, a • s i j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
α : Type u_3
β : Type u_2
G : Type ?u.20290
X : Type ?u.20293
ι : Sort u_4
κ : ι → Sort u_5
inst✝¹ : Group α
inst✝ : MulAction α β
a : α
s : (i : ι) → κ i → Set β
⊢ a • ⋂ i, ⋂ j, s i j = ⋂ i, ⋂ j, a • s i j
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | constructor | G : Type u_1
inst✝ : Group G
K : Subgroup G
⊢ IsMaximal K ↔ K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤ | case mp
G : Type u_1
inst✝ : Group G
K : Subgroup G
⊢ IsMaximal K → K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
case mpr
G : Type u_1
inst✝ : Group G
K : Subgroup G
⊢ (K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤) → IsMaximal K | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
K : Subgroup G
⊢ IsMaximal K ↔ K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | intro hK | case mp
G : Type u_1
inst✝ : Group G
K : Subgroup G
⊢ IsMaximal K → K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤ | case mp
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
⊢ K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
K : Subgroup G
⊢ IsMaximal K → K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | constructor | case mp
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
⊢ K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤ | case mp.left
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
⊢ K ≠ ⊤
case mp.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
⊢ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
⊢ K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | exact hK.ne_top | case mp.left
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
⊢ K ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
⊢ K ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | intro H g hKH hgK hgH | case mp.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
⊢ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤ | case mp.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ H = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
⊢ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | apply (isMaximal_def.1 hK).2 | case mp.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ H = ⊤ | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K < H | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ H = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | rw [← Ne.le_iff_lt] | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K < H | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K ≤ H
case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K ≠ H | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K < H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | exact hKH | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K ≤ H
case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K ≠ H | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K ≠ H | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K ≤ H
case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K ≠ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | rw [Ne.def] | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K ≠ H | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ ¬K = H | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ K ≠ H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | intro z | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ ¬K = H | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
z : K = H
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
⊢ ¬K = H
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | rw [z] at hgK | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
z : K = H
⊢ False | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ H
hgH : g ∈ H
z : K = H
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ K
hgH : g ∈ H
z : K = H
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | exact hgK hgH | case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ H
hgH : g ∈ H
z : K = H
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.a
G : Type u_1
inst✝ : Group G
K : Subgroup G
hK : IsMaximal K
H : Subgroup G
g : G
hKH : K ≤ H
hgK : g ∉ H
hgH : g ∈ H
z : K = H
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | rintro ⟨hG, hmax⟩ | case mpr
G : Type u_1
inst✝ : Group G
K : Subgroup G
⊢ (K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤) → IsMaximal K | case mpr.intro
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ IsMaximal K | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_1
inst✝ : Group G
K : Subgroup G
⊢ (K ≠ ⊤ ∧ ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤) → IsMaximal K
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | constructor | case mpr.intro
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ IsMaximal K | case mpr.intro.out
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ IsCoatom K | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ IsMaximal K
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | constructor | case mpr.intro.out
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ IsCoatom K | case mpr.intro.out.left
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ K ≠ ⊤
case mpr.intro.out.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ ∀ (b : Subgroup G), K < b → b = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.out
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ IsCoatom K
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | assumption | case mpr.intro.out.left
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ K ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.out.left
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ K ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | intro H hKH | case mpr.intro.out.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ ∀ (b : Subgroup G), K < b → b = ⊤ | case mpr.intro.out.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
H : Subgroup G
hKH : K < H
⊢ H = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.out.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
⊢ ∀ (b : Subgroup G), K < b → b = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | obtain ⟨g, hgH, hgK⟩ := Set.exists_of_ssubset hKH | case mpr.intro.out.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
H : Subgroup G
hKH : K < H
⊢ H = ⊤ | case mpr.intro.out.right.intro.intro
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
H : Subgroup G
hKH : K < H
g : G
hgH : g ∈ ↑H
hgK : g ∉ ↑K
⊢ H = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.out.right
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
H : Subgroup G
hKH : K < H
⊢ H = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MaximalSubgroups.lean | Subgroup.isMaximal_iff | [50, 1] | [69, 44] | exact hmax H g (le_of_lt hKH) hgK hgH | case mpr.intro.out.right.intro.intro
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
H : Subgroup G
hKH : K < H
g : G
hgH : g ∈ ↑H
hgK : g ∉ ↑K
⊢ H = ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.out.right.intro.intro
G : Type u_1
inst✝ : Group G
K : Subgroup G
hG : K ≠ ⊤
hmax : ∀ (H : Subgroup G) (g : G), K ≤ H → g ∉ K → g ∈ H → H = ⊤
H : Subgroup G
hKH : K < H
g : G
hgH : g ∈ ↑H
hgK : g ∉ ↑K
⊢ H = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.Combination.mem_iff | [33, 1] | [36, 31] | unfold Combination | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s : Finset α
⊢ s ∈ Combination α n ↔ s.card = n | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s : Finset α
⊢ s ∈ {s | s.card = n} ↔ s.card = n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s : Finset α
⊢ s ∈ Combination α n ↔ s.card = n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.Combination.mem_iff | [33, 1] | [36, 31] | simp only [Set.mem_setOf_eq] | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s : Finset α
⊢ s ∈ {s | s.card = n} ↔ s.card = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s : Finset α
⊢ s ∈ {s | s.card = n} ↔ s.card = n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.Combination.eq_iff_finset_subset | [49, 1] | [57, 24] | constructor | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
⊢ s = t ↔ ↑s ⊆ ↑t | case mp
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
⊢ s = t → ↑s ⊆ ↑t
case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
⊢ ↑s ⊆ ↑t → s = t | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
⊢ s = t ↔ ↑s ⊆ ↑t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.Combination.eq_iff_finset_subset | [49, 1] | [57, 24] | intro h | case mp
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
⊢ s = t → ↑s ⊆ ↑t | case mp
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : s = t
⊢ ↑s ⊆ ↑t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
⊢ s = t → ↑s ⊆ ↑t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.Combination.eq_iff_finset_subset | [49, 1] | [57, 24] | rw [h] | case mp
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : s = t
⊢ ↑s ⊆ ↑t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : s = t
⊢ ↑s ⊆ ↑t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.Combination.eq_iff_finset_subset | [49, 1] | [57, 24] | intro h | case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
⊢ ↑s ⊆ ↑t → s = t | case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : ↑s ⊆ ↑t
⊢ s = t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
⊢ ↑s ⊆ ↑t → s = t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.Combination.eq_iff_finset_subset | [49, 1] | [57, 24] | rw [← Subtype.coe_inj] | case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : ↑s ⊆ ↑t
⊢ s = t | case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : ↑s ⊆ ↑t
⊢ ↑s = ↑t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : ↑s ⊆ ↑t
⊢ s = t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.Combination.eq_iff_finset_subset | [49, 1] | [57, 24] | apply Finset.eq_of_subset_of_card_le h | case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : ↑s ⊆ ↑t
⊢ ↑s = ↑t | case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : ↑s ⊆ ↑t
⊢ (↑t).card ≤ (↑s).card | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : ↑s ⊆ ↑t
⊢ ↑s = ↑t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.Combination.eq_iff_finset_subset | [49, 1] | [57, 24] | rw [s.prop, t.prop] | case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : ↑s ⊆ ↑t
⊢ (↑t).card ≤ (↑s).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
h : ↑s ⊆ ↑t
⊢ (↑t).card ≤ (↑s).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination_mulAction_apply | [81, 1] | [84, 6] | rfl | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n✝ n : ℕ
g : G
s : Finset α
hs : s ∈ Combination α n
⊢ g • s = g • ↑{ val := s, property := hs } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n✝ n : ℕ
g : G
s : Finset α
hs : s ∈ Combination α n
⊢ g • s = g • ↑{ val := s, property := hs }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.smul_ne_iff_hasMem_not_mem | [93, 1] | [101, 38] | simp only [ne_eq, exists_prop] | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • s ≠ t ↔ ∃ a ∈ ↑s, a ∉ g⁻¹ • ↑t | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ ¬g • s = t ↔ ∃ a ∈ ↑s, a ∉ g⁻¹ • ↑t | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • s ≠ t ↔ ∃ a ∈ ↑s, a ∉ g⁻¹ • ↑t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.smul_ne_iff_hasMem_not_mem | [93, 1] | [101, 38] | rw [← Finset.not_subset] | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ ¬g • s = t ↔ ∃ a ∈ ↑s, a ∉ g⁻¹ • ↑t | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ ¬g • s = t ↔ ¬↑s ⊆ g⁻¹ • ↑t | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ ¬g • s = t ↔ ∃ a ∈ ↑s, a ∉ g⁻¹ • ↑t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.smul_ne_iff_hasMem_not_mem | [93, 1] | [101, 38] | rw [not_iff_not] | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ ¬g • s = t ↔ ¬↑s ⊆ g⁻¹ • ↑t | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • s = t ↔ ↑s ⊆ g⁻¹ • ↑t | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ ¬g • s = t ↔ ¬↑s ⊆ g⁻¹ • ↑t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.smul_ne_iff_hasMem_not_mem | [93, 1] | [101, 38] | rw [← Nat.combination_mulAction.coe_apply'] | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • s = t ↔ ↑s ⊆ g⁻¹ • ↑t | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • s = t ↔ ↑s ⊆ ↑(g⁻¹ • t) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • s = t ↔ ↑s ⊆ g⁻¹ • ↑t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.smul_ne_iff_hasMem_not_mem | [93, 1] | [101, 38] | rw [Nat.Combination.eq_iff_finset_subset] | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • s = t ↔ ↑s ⊆ ↑(g⁻¹ • t) | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ ↑(g • s) ⊆ ↑t ↔ ↑s ⊆ ↑(g⁻¹ • t) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • s = t ↔ ↑s ⊆ ↑(g⁻¹ • t)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.smul_ne_iff_hasMem_not_mem | [93, 1] | [101, 38] | simp only [Nat.combination_mulAction.coe_apply', Finset.le_eq_subset] | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ ↑(g • s) ⊆ ↑t ↔ ↑s ⊆ ↑(g⁻¹ • t) | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • ↑s ⊆ ↑t ↔ ↑s ⊆ g⁻¹ • ↑t | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ ↑(g • s) ⊆ ↑t ↔ ↑s ⊆ ↑(g⁻¹ • t)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.smul_ne_iff_hasMem_not_mem | [93, 1] | [101, 38] | exact Finset.smul_finset_subset_iff | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • ↑s ⊆ ↑t ↔ ↑s ⊆ g⁻¹ • ↑t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
s t : ↑(Combination α n)
g : G
⊢ g • ↑s ⊆ ↑t ↔ ↑s ⊆ g⁻¹ • ↑t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.mulAction_faithful | [104, 1] | [146, 22] | have zob : ∃ (a : α), (g • a : α) ≠ a := by
by_contra h
push_neg at h
apply hg
ext a
simpa only using h a | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
⊢ ∃ s, g • s ≠ s | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
zob : ∃ a, g • a ≠ a
⊢ ∃ s, g • s ≠ s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
⊢ ∃ s, g • s ≠ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.mulAction_faithful | [104, 1] | [146, 22] | obtain ⟨a, ha⟩ := zob | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
zob : ∃ a, g • a ≠ a
⊢ ∃ s, g • s ≠ s | case intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
⊢ ∃ s, g • s ≠ s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
zob : ∃ a, g • a ≠ a
⊢ ∃ s, g • s ≠ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.mulAction_faithful | [104, 1] | [146, 22] | have : ({a} : Set α) ⊆ {g • a}ᶜ := by
simp only [Set.subset_compl_singleton_iff, Set.mem_singleton_iff]
exact ha | case intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s | case intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.mulAction_faithful | [104, 1] | [146, 22] | have hα' : n ≤ Set.encard ({g • a}ᶜ) := by
rw [← not_lt, ← WithTop.add_one_lt_coe_succ_iff, not_lt, add_comm,
← Set.encard_singleton (g • a), Set.encard_add_encard_compl, Set.encard_univ]
rw [← PartENat.withTopEquiv_natCast, PartENat.withTopEquiv_le]
exact hα | case intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s | case intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.mulAction_faithful | [104, 1] | [146, 22] | obtain ⟨s, has, has', hs⟩ := Set.exists_superset_subset_encard_eq this (by
rw [Set.encard_singleton, ← Nat.cast_one, Nat.cast_le]
exact hn) hα' | case intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s | case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
s : Set α
has : {a} ⊆ s
has' : s ⊆ {g • a}ᶜ
hs : Set.encard s = ↑n
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.mulAction_faithful | [104, 1] | [146, 22] | use s | case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
s : Set α
has : {a} ⊆ s
has' : s ⊆ {g • a}ᶜ
hs : Set.encard s = ↑n
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s | case h
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
s : Set α
has : {a} ⊆ s
has' : s ⊆ {g • a}ᶜ
hs : Set.encard s = ↑n
⊢ Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
s : Set α
has : {a} ⊆ s
has' : s ⊆ {g • a}ᶜ
hs : Set.encard s = ↑n
⊢ ∃ s, Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.mulAction_faithful | [104, 1] | [146, 22] | constructor | case h
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
s : Set α
has : {a} ⊆ s
has' : s ⊆ {g • a}ᶜ
hs : Set.encard s = ↑n
⊢ Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s | case h.left
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
s : Set α
has : {a} ⊆ s
has' : s ⊆ {g • a}ᶜ
hs : Set.encard s = ↑n
⊢ Set.encard s = ↑n
case h.right
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
s : Set α
has : {a} ⊆ s
has' : s ⊆ {g • a}ᶜ
hs : Set.encard s = ↑n
⊢ a ∈ s ∧ g • a ∉ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
a : α
ha : g • a ≠ a
this : {a} ⊆ {g • a}ᶜ
hα' : ↑n ≤ Set.encard {g • a}ᶜ
s : Set α
has : {a} ⊆ s
has' : s ⊆ {g • a}ᶜ
hs : Set.encard s = ↑n
⊢ Set.encard s = ↑n ∧ a ∈ s ∧ g • a ∉ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.mulAction_faithful | [104, 1] | [146, 22] | by_contra h | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
⊢ ∃ a, g • a ≠ a | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
h : ¬∃ a, g • a ≠ a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
⊢ ∃ a, g • a ≠ a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.mulAction_faithful | [104, 1] | [146, 22] | push_neg at h | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
h : ¬∃ a, g • a ≠ a
⊢ False | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
h : ∀ (a : α), g • a = a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
h : ¬∃ a, g • a ≠ a
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MulActionCombination.lean | Nat.combination.mulAction_faithful | [104, 1] | [146, 22] | apply hg | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
h : ∀ (a : α), g • a = a
⊢ False | G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
h : ∀ (a : α), g • a = a
⊢ toPerm g = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : MulAction G α
n : ℕ
hn : 1 ≤ n
hα : ↑(succ n) ≤ PartENat.card α
g : G
hg : toPerm g ≠ 1
h : ∀ (a : α), g • a = a
⊢ False
TACTIC:
|
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