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/Iwasawa.lean | commutator_le_iwasawa | [55, 1] | [69, 93] | intro x | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
⊢ ∀ (i : α), IwaS.T i ≤ N ⊔ IwaS.T a | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a x : α
⊢ IwaS.T x ≤ N ⊔ IwaS.T a | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
⊢ ∀ (i : α), IwaS.T i ≤ N ⊔ IwaS.T a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | commutator_le_iwasawa | [55, 1] | [69, 93] | obtain ⟨g, rfl⟩ := MulAction.exists_smul_eq N a x | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a x : α
⊢ IwaS.T x ≤ N ⊔ IwaS.T a | case intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
⊢ IwaS.T (g • a) ≤ N ⊔ IwaS.T a | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a x : α
⊢ IwaS.T x ≤ N ⊔ IwaS.T a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | commutator_le_iwasawa | [55, 1] | [69, 93] | rw [Subgroup.smul_def, IwaS.is_conj g a] | case intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
⊢ IwaS.T (g • a) ≤ N ⊔ IwaS.T a | case intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
⊢ MulAut.conj ↑g • IwaS.T a ≤ N ⊔ IwaS.T a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
⊢ IwaS.T (g • a) ≤ N ⊔ IwaS.T a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | commutator_le_iwasawa | [55, 1] | [69, 93] | rintro _ ⟨k, hk, rfl⟩ | case intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
⊢ MulAut.conj ↑g • IwaS.T a ≤ N ⊔ IwaS.T a | case intro.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
k : M
hk : k ∈ ↑(IwaS.T a)
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut M) M) (MulAut.conj ↑g)) k ∈ N ⊔ IwaS.T a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
⊢ MulAut.conj ↑g • IwaS.T a ≤ N ⊔ IwaS.T a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | commutator_le_iwasawa | [55, 1] | [69, 93] | have hg' : ↑g ∈ N ⊔ IwaS.T a := Subgroup.mem_sup_left (Subtype.mem g) | case intro.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
k : M
hk : k ∈ ↑(IwaS.T a)
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut M) M) (MulAut.conj ↑g)) k ∈ N ⊔ IwaS.T a | case intro.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
k : M
hk : k ∈ ↑(IwaS.T a)
hg' : ↑g ∈ N ⊔ IwaS.T a
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut M) M) (MulAut.conj ↑g)) k ∈ N ⊔ IwaS.T a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
k : M
hk : k ∈ ↑(IwaS.T a)
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut M) M) (MulAut.conj ↑g)) k ∈ N ⊔ IwaS.T a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | commutator_le_iwasawa | [55, 1] | [69, 93] | have hk' : k ∈ N ⊔ IwaS.T a := Subgroup.mem_sup_right hk | case intro.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
k : M
hk : k ∈ ↑(IwaS.T a)
hg' : ↑g ∈ N ⊔ IwaS.T a
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut M) M) (MulAut.conj ↑g)) k ∈ N ⊔ IwaS.T a | case intro.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
k : M
hk : k ∈ ↑(IwaS.T a)
hg' : ↑g ∈ N ⊔ IwaS.T a
hk' : k ∈ N ⊔ IwaS.T a
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut M) M) (MulAut.conj ↑g)) k ∈ N ⊔ IwaS.T a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
k : M
hk : k ∈ ↑(IwaS.T a)
hg' : ↑g ∈ N ⊔ IwaS.T a
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut M) M) (MulAut.conj ↑g)) k ∈ N ⊔ IwaS.T a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | commutator_le_iwasawa | [55, 1] | [69, 93] | exact (N ⊔ IwaS.T a).mul_mem ((N ⊔ IwaS.T a).mul_mem hg' hk') ((N ⊔ IwaS.T a).inv_mem hg') | case intro.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
k : M
hk : k ∈ ↑(IwaS.T a)
hg' : ↑g ∈ N ⊔ IwaS.T a
hk' : k ∈ N ⊔ IwaS.T a
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut M) M) (MulAut.conj ↑g)) k ∈ N ⊔ IwaS.T a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_qprim : IsQuasipreprimitive M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
hNX : MulAction.fixedPoints (↥N) α ≠ ⊤
is_transN : MulAction.IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
g : ↥N
k : M
hk : k ∈ ↑(IwaS.T a)
hg' : ↑g ∈ N ⊔ IwaS.T a
hk' : k ∈ N ⊔ IwaS.T a
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut M) M) (MulAut.conj ↑g)) k ∈ N ⊔ IwaS.T a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | is_simple_iwasawa | [73, 1] | [86, 78] | apply IsSimpleGroup.mk | M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
⊢ IsSimpleGroup M | case eq_bot_or_eq_top_of_normal
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
⊢ ∀ (H : Subgroup M), Subgroup.Normal H → H = ⊥ ∨ H = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
⊢ IsSimpleGroup M
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | is_simple_iwasawa | [73, 1] | [86, 78] | intro N nN | case eq_bot_or_eq_top_of_normal
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
⊢ ∀ (H : Subgroup M), Subgroup.Normal H → H = ⊥ ∨ H = ⊤ | case eq_bot_or_eq_top_of_normal
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
⊢ N = ⊥ ∨ N = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case eq_bot_or_eq_top_of_normal
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
⊢ ∀ (H : Subgroup M), Subgroup.Normal H → H = ⊥ ∨ H = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | is_simple_iwasawa | [73, 1] | [86, 78] | cases or_iff_not_imp_left.mpr (commutator_le_iwasawa is_qprim IwaS nN) with
| inl h =>
refine' Or.inl (N.eq_bot_iff_forall.mpr fun n hn => _)
apply is_faithful.eq_of_smul_eq_smul
intro x
rw [one_smul]
exact Set.eq_univ_iff_forall.mp h x ⟨n, hn⟩
| inr h => exact Or.inr (top_le_iff.mp (le_trans (ge_of_eq is_perfect) h)) | case eq_bot_or_eq_top_of_normal
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
⊢ N = ⊥ ∨ N = ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case eq_bot_or_eq_top_of_normal
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
⊢ N = ⊥ ∨ N = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | is_simple_iwasawa | [73, 1] | [86, 78] | refine' Or.inl (N.eq_bot_iff_forall.mpr fun n hn => _) | case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
⊢ N = ⊥ ∨ N = ⊤ | case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
⊢ n = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
⊢ N = ⊥ ∨ N = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | is_simple_iwasawa | [73, 1] | [86, 78] | apply is_faithful.eq_of_smul_eq_smul | case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
⊢ n = 1 | case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
⊢ ∀ (a : α), n • a = 1 • a | Please generate a tactic in lean4 to solve the state.
STATE:
case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
⊢ n = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | is_simple_iwasawa | [73, 1] | [86, 78] | intro x | case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
⊢ ∀ (a : α), n • a = 1 • a | case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
x : α
⊢ n • x = 1 • x | Please generate a tactic in lean4 to solve the state.
STATE:
case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
⊢ ∀ (a : α), n • a = 1 • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | is_simple_iwasawa | [73, 1] | [86, 78] | rw [one_smul] | case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
x : α
⊢ n • x = 1 • x | case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
x : α
⊢ n • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
x : α
⊢ n • x = 1 • x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | is_simple_iwasawa | [73, 1] | [86, 78] | exact Set.eq_univ_iff_forall.mp h x ⟨n, hn⟩ | case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
x : α
⊢ n • x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case eq_bot_or_eq_top_of_normal.inl
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : MulAction.fixedPoints (↥N) α = ⊤
n : M
hn : n ∈ N
x : α
⊢ n • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Iwasawa.lean | is_simple_iwasawa | [73, 1] | [86, 78] | exact Or.inr (top_le_iff.mp (le_trans (ge_of_eq is_perfect) h)) | case eq_bot_or_eq_top_of_normal.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : commutator M ≤ N
⊢ N = ⊥ ∨ N = ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case eq_bot_or_eq_top_of_normal.inr
M : Type u_2
inst✝¹ : Group M
α : Type u_1
inst✝ : MulAction M α
is_nontrivial : Nontrivial M
is_perfect : commutator M = ⊤
is_qprim : IsQuasipreprimitive M α
is_faithful : FaithfulSMul M α
IwaS : IwasawaStructure M α
N : Subgroup M
nN : Subgroup.Normal N
h : commutator M ≤ N
⊢ N = ⊥ ∨ N = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/ULift.lean | ULift.surjective_iff_surjective | [37, 1] | [39, 80] | rw [← Equiv.comp_surjective, comp_uLift_eq_uLift_comp, Equiv.surjective_comp] | α : Type u
β : Type v
f : α → β
⊢ Iff (Function.Surjective.{max (u_1 + 1) (u + 1), max (u_2 + 1) (v + 1)} (ULift.map.{u, v, u_1, u_2} f))
(Function.Surjective.{u + 1, v + 1} f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
β : Type v
f : α → β
⊢ Iff (Function.Surjective.{max (u_1 + 1) (u + 1), max (u_2 + 1) (v + 1)} (ULift.map.{u, v, u_1, u_2} f))
(Function.Surjective.{u + 1, v + 1} f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/ULift.lean | ULift.injective_iff_injective | [42, 1] | [44, 78] | rw [← Equiv.comp_injective, comp_uLift_eq_uLift_comp, Equiv.injective_comp] | α : Type u
β : Type v
f : α → β
⊢ Iff (Function.Injective.{max (u_1 + 1) (u + 1), max (u_2 + 1) (v + 1)} (ULift.map.{u, v, u_1, u_2} f))
(Function.Injective.{u + 1, v + 1} f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
β : Type v
f : α → β
⊢ Iff (Function.Injective.{max (u_1 + 1) (u + 1), max (u_2 + 1) (v + 1)} (ULift.map.{u, v, u_1, u_2} f))
(Function.Injective.{u + 1, v + 1} f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/ULift.lean | ULift.bijective_iff_bijective | [47, 1] | [49, 78] | rw [← Equiv.comp_bijective, comp_uLift_eq_uLift_comp, Equiv.bijective_comp] | α : Type u
β : Type v
f : α → β
⊢ Iff (Function.Bijective.{max (u_1 + 1) (u + 1), max (u_2 + 1) (v + 1)} (ULift.map.{u, v, u_1, u_2} f))
(Function.Bijective.{u + 1, v + 1} f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
β : Type v
f : α → β
⊢ Iff (Function.Bijective.{max (u_1 + 1) (u + 1), max (u_2 + 1) (v + 1)} (ULift.map.{u, v, u_1, u_2} f))
(Function.Bijective.{u + 1, v + 1} f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | constructor | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ IsPretransitive G X ↔ ∀ (x : X), ∃ g, g • a = x | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ IsPretransitive G X → ∀ (x : X), ∃ g, g • a = x
case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (∀ (x : X), ∃ g, g • a = x) → IsPretransitive G X | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ IsPretransitive G X ↔ ∀ (x : X), ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | intro hG x | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ IsPretransitive G X → ∀ (x : X), ∃ g, g • a = x | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : IsPretransitive G X
x : X
⊢ ∃ g, g • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ IsPretransitive G X → ∀ (x : X), ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | let hG_eq := hG.exists_smul_eq | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : IsPretransitive G X
x : X
⊢ ∃ g, g • a = x | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : IsPretransitive G X
x : X
hG_eq : ∀ (x y : X), ∃ g, g • x = y := exists_smul_eq
⊢ ∃ g, g • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : IsPretransitive G X
x : X
⊢ ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | exact hG_eq a x | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : IsPretransitive G X
x : X
hG_eq : ∀ (x y : X), ∃ g, g • x = y := exists_smul_eq
⊢ ∃ g, g • a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : IsPretransitive G X
x : X
hG_eq : ∀ (x y : X), ∃ g, g • x = y := exists_smul_eq
⊢ ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | intro hG | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (∀ (x : X), ∃ g, g • a = x) → IsPretransitive G X | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ IsPretransitive G X | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (∀ (x : X), ∃ g, g • a = x) → IsPretransitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | apply IsPretransitive.mk | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ IsPretransitive G X | case mpr.exists_smul_eq
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ ∀ (x y : X), ∃ g, g • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ IsPretransitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | intro x y | case mpr.exists_smul_eq
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ ∀ (x y : X), ∃ g, g • x = y | case mpr.exists_smul_eq
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
⊢ ∃ g, g • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.exists_smul_eq
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ ∀ (x y : X), ∃ g, g • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | obtain ⟨g, hx⟩ := hG x | case mpr.exists_smul_eq
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
⊢ ∃ g, g • x = y | case mpr.exists_smul_eq.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
⊢ ∃ g, g • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.exists_smul_eq
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
⊢ ∃ g, g • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | obtain ⟨h, hy⟩ := hG y | case mpr.exists_smul_eq.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
⊢ ∃ g, g • x = y | case mpr.exists_smul_eq.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ ∃ g, g • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.exists_smul_eq.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
⊢ ∃ g, g • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | use h * g⁻¹ | case mpr.exists_smul_eq.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ ∃ g, g • x = y | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.exists_smul_eq.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ ∃ g, g • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | rw [← hx] | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • x = y | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • g • a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | rw [smul_smul, inv_mul_cancel_right] | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • g • a = y | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ h • a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • g • a = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base_iff | [35, 1] | [47, 13] | exact hy | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ h • a = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ h • a = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base | [50, 1] | [58, 11] | apply IsPretransitive.mk | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ IsPretransitive G X | case exists_smul_eq
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ ∀ (x y : X), ∃ g, g • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ IsPretransitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base | [50, 1] | [58, 11] | intro x y | case exists_smul_eq
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ ∀ (x y : X), ∃ g, g • x = y | case exists_smul_eq
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
⊢ ∃ g, g • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
⊢ ∀ (x y : X), ∃ g, g • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base | [50, 1] | [58, 11] | obtain ⟨g, hx⟩ := hG x | case exists_smul_eq
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
⊢ ∃ g, g • x = y | case exists_smul_eq.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
⊢ ∃ g, g • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
⊢ ∃ g, g • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base | [50, 1] | [58, 11] | obtain ⟨h, hy⟩ := hG y | case exists_smul_eq.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
⊢ ∃ g, g • x = y | case exists_smul_eq.intro.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ ∃ g, g • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
⊢ ∃ g, g • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base | [50, 1] | [58, 11] | use h * g⁻¹ | case exists_smul_eq.intro.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ ∃ g, g • x = y | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq.intro.intro
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ ∃ g, g • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base | [50, 1] | [58, 11] | rw [← hx] | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • x = y | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • g • a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base | [50, 1] | [58, 11] | rw [smul_smul, inv_mul_cancel_right] | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • g • a = y | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ h • a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ (h * g⁻¹) • g • a = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.IsPretransitive.mk_base | [50, 1] | [58, 11] | exact hy | case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ h • a = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
hG : ∀ (x : X), ∃ g, g • a = x
x y : X
g : G
hx : g • a = x
h : G
hy : h • a = y
⊢ h • a = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.orbit.isPretransitive_iff | [62, 1] | [69, 21] | rw [IsPretransitive.mk_base_iff a] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ orbit G a = ⊤ ↔ IsPretransitive G X | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ orbit G a = ⊤ ↔ ∀ (x : X), ∃ g, g • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ orbit G a = ⊤ ↔ IsPretransitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.orbit.isPretransitive_iff | [62, 1] | [69, 21] | rw [Set.ext_iff] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ orbit G a = ⊤ ↔ ∀ (x : X), ∃ g, g • a = x | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (∀ (x : X), x ∈ orbit G a ↔ x ∈ ⊤) ↔ ∀ (x : X), ∃ g, g • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ orbit G a = ⊤ ↔ ∀ (x : X), ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.orbit.isPretransitive_iff | [62, 1] | [69, 21] | apply forall_congr' | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (∀ (x : X), x ∈ orbit G a ↔ x ∈ ⊤) ↔ ∀ (x : X), ∃ g, g • a = x | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∀ (a_1 : X), (a_1 ∈ orbit G a ↔ a_1 ∈ ⊤) ↔ ∃ g, g • a = a_1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ (∀ (x : X), x ∈ orbit G a ↔ x ∈ ⊤) ↔ ∀ (x : X), ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.orbit.isPretransitive_iff | [62, 1] | [69, 21] | intro x | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∀ (a_1 : X), (a_1 ∈ orbit G a ↔ a_1 ∈ ⊤) ↔ ∃ g, g • a = a_1 | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a x : X
⊢ (x ∈ orbit G a ↔ x ∈ ⊤) ↔ ∃ g, g • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a : X
⊢ ∀ (a_1 : X), (a_1 ∈ orbit G a ↔ a_1 ∈ ⊤) ↔ ∃ g, g • a = a_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.orbit.isPretransitive_iff | [62, 1] | [69, 21] | simp_rw [Set.top_eq_univ, Set.mem_univ, iff_true_iff] | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a x : X
⊢ (x ∈ orbit G a ↔ x ∈ ⊤) ↔ ∃ g, g • a = x | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a x : X
⊢ x ∈ orbit G a ↔ ∃ g, g • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a x : X
⊢ (x ∈ orbit G a ↔ x ∈ ⊤) ↔ ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Pretransitive.lean | MulAction.orbit.isPretransitive_iff | [62, 1] | [69, 21] | rw [mem_orbit_iff] | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a x : X
⊢ x ∈ orbit G a ↔ ∃ g, g • a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
a x : X
⊢ x ∈ orbit G a ↔ ∃ g, g • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_is_commutator | [37, 1] | [45, 27] | apply mem_commutatorSet_of_isConj_sq | β : Type ?u.44
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ g ∈ commutatorSet ↥(alternatingGroup α) | case hg
β : Type ?u.44
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ IsConj g (g ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.44
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ g ∈ commutatorSet ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_is_commutator | [37, 1] | [45, 27] | apply alternatingGroup.isThreeCycle_isConj h5 hg | case hg
β : Type ?u.44
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ IsConj g (g ^ 2) | case hg
β : Type ?u.44
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ IsThreeCycle ↑(g ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
β : Type ?u.44
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ IsConj g (g ^ 2)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_is_commutator | [37, 1] | [45, 27] | exact hg.isThreeCycle_sq | case hg
β : Type ?u.44
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ IsThreeCycle ↑(g ^ 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
β : Type ?u.44
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ IsThreeCycle ↑(g ^ 2)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_is_commutator' | [48, 1] | [53, 89] | rw [← Subgroup.coe_mk (alternatingGroup α) g (Equiv.Perm.IsThreeCycle.sign hg)] at hg | β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg : IsThreeCycle g
⊢ ∃ p q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹ | β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
⊢ ∃ p q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg : IsThreeCycle g
⊢ ∃ p q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_is_commutator' | [48, 1] | [53, 89] | obtain ⟨p, q, h⟩ := three_cycle_is_commutator h5 hg | β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
⊢ ∃ p q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹ | case intro.intro
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
p q : ↥(alternatingGroup α)
h : ⁅p, q⁆ = { val := g, property := ⋯ }
⊢ ∃ p q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
⊢ ∃ p q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_is_commutator' | [48, 1] | [53, 89] | use p | case intro.intro
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
p q : ↥(alternatingGroup α)
h : ⁅p, q⁆ = { val := g, property := ⋯ }
⊢ ∃ p q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹ | case h
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
p q : ↥(alternatingGroup α)
h : ⁅p, q⁆ = { val := g, property := ⋯ }
⊢ ∃ q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
p q : ↥(alternatingGroup α)
h : ⁅p, q⁆ = { val := g, property := ⋯ }
⊢ ∃ p q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_is_commutator' | [48, 1] | [53, 89] | use q | case h
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
p q : ↥(alternatingGroup α)
h : ⁅p, q⁆ = { val := g, property := ⋯ }
⊢ ∃ q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹ | case h
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
p q : ↥(alternatingGroup α)
h : ⁅p, q⁆ = { val := g, property := ⋯ }
⊢ g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
p q : ↥(alternatingGroup α)
h : ⁅p, q⁆ = { val := g, property := ⋯ }
⊢ ∃ q, g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_is_commutator' | [48, 1] | [53, 89] | simp only [← Subgroup.coe_mul, ← Subgroup.coe_inv, ← commutatorElement_def, h, coe_mk] | case h
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
p q : ↥(alternatingGroup α)
h : ⁅p, q⁆ = { val := g, property := ⋯ }
⊢ g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
β : Type ?u.1022
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : Perm α
hg✝ : IsThreeCycle g
hg : IsThreeCycle ↑{ val := g, property := ⋯ }
p q : ↥(alternatingGroup α)
h : ⁅p, q⁆ = { val := g, property := ⋯ }
⊢ g = ↑p * ↑q * ↑p⁻¹ * ↑q⁻¹
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_mem_commutator | [56, 1] | [61, 40] | rw [commutator_eq_closure] | β : Type ?u.4463
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ g ∈ _root_.commutator ↥(alternatingGroup α) | β : Type ?u.4463
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ g ∈ closure (commutatorSet ↥(alternatingGroup α)) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.4463
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ g ∈ _root_.commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_mem_commutator | [56, 1] | [61, 40] | apply Subgroup.subset_closure | β : Type ?u.4463
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ g ∈ closure (commutatorSet ↥(alternatingGroup α)) | case a
β : Type ?u.4463
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ g ∈ commutatorSet ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.4463
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ g ∈ closure (commutatorSet ↥(alternatingGroup α))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | three_cycle_mem_commutator | [56, 1] | [61, 40] | exact three_cycle_is_commutator h5 hg | case a
β : Type ?u.4463
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ g ∈ commutatorSet ↥(alternatingGroup α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
β : Type ?u.4463
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
g : ↥(alternatingGroup α)
hg : IsThreeCycle ↑g
⊢ g ∈ commutatorSet ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup_is_perfect | [75, 1] | [82, 46] | suffices closure {b : alternatingGroup α | (b : Perm α).IsThreeCycle} = ⊤ by
rw [eq_top_iff, ← this, Subgroup.closure_le]
intro b hb
exact three_cycle_mem_commutator h5 hb | β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ _root_.commutator ↥(alternatingGroup α) = ⊤ | β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ closure {b | IsThreeCycle ↑b} = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ _root_.commutator ↥(alternatingGroup α) = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup_is_perfect | [75, 1] | [82, 46] | apply Subgroup.closure_closure_coe_preimage | β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ closure {b | IsThreeCycle ↑b} = ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ closure {b | IsThreeCycle ↑b} = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup_is_perfect | [75, 1] | [82, 46] | rw [eq_top_iff, ← this, Subgroup.closure_le] | β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
this : closure {b | IsThreeCycle ↑b} = ⊤
⊢ _root_.commutator ↥(alternatingGroup α) = ⊤ | β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
this : closure {b | IsThreeCycle ↑b} = ⊤
⊢ {b | IsThreeCycle ↑b} ⊆ ↑(_root_.commutator ↥(alternatingGroup α)) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
this : closure {b | IsThreeCycle ↑b} = ⊤
⊢ _root_.commutator ↥(alternatingGroup α) = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup_is_perfect | [75, 1] | [82, 46] | intro b hb | β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
this : closure {b | IsThreeCycle ↑b} = ⊤
⊢ {b | IsThreeCycle ↑b} ⊆ ↑(_root_.commutator ↥(alternatingGroup α)) | β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
this : closure {b | IsThreeCycle ↑b} = ⊤
b : ↥(alternatingGroup α)
hb : b ∈ {b | IsThreeCycle ↑b}
⊢ b ∈ ↑(_root_.commutator ↥(alternatingGroup α)) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
this : closure {b | IsThreeCycle ↑b} = ⊤
⊢ {b | IsThreeCycle ↑b} ⊆ ↑(_root_.commutator ↥(alternatingGroup α))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup_is_perfect | [75, 1] | [82, 46] | exact three_cycle_mem_commutator h5 hb | β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
this : closure {b | IsThreeCycle ↑b} = ⊤
b : ↥(alternatingGroup α)
hb : b ∈ {b | IsThreeCycle ↑b}
⊢ b ∈ ↑(_root_.commutator ↥(alternatingGroup α)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.5450
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
this : closure {b | IsThreeCycle ↑b} = ⊤
b : ↥(alternatingGroup α)
hb : b ∈ {b | IsThreeCycle ↑b}
⊢ b ∈ ↑(_root_.commutator ↥(alternatingGroup α))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup_is_perfect' | [86, 1] | [89, 28] | rw [← Subgroup.commutator_eq', alternatingGroup_is_perfect h5, Subgroup.map_top_eq_range,
Subgroup.subtype_range] | β : Type ?u.7470
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ ⁅alternatingGroup α, alternatingGroup α⁆ = alternatingGroup α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.7470
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ ⁅alternatingGroup α, alternatingGroup α⁆ = alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup.commutator_group_le | [92, 1] | [99, 20] | rw [commutator_eq_closure, Subgroup.closure_le] | β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ _root_.commutator (Perm α) ≤ alternatingGroup α | β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ commutatorSet (Perm α) ⊆ ↑(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ _root_.commutator (Perm α) ≤ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup.commutator_group_le | [92, 1] | [99, 20] | intro x | β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ commutatorSet (Perm α) ⊆ ↑(alternatingGroup α) | β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
x : Perm α
⊢ x ∈ commutatorSet (Perm α) → x ∈ ↑(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ commutatorSet (Perm α) ⊆ ↑(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup.commutator_group_le | [92, 1] | [99, 20] | rintro ⟨p, q, rfl⟩ | β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
x : Perm α
⊢ x ∈ commutatorSet (Perm α) → x ∈ ↑(alternatingGroup α) | case intro.intro
β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
p q : Perm α
⊢ ⁅p, q⁆ ∈ ↑(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
x : Perm α
⊢ x ∈ commutatorSet (Perm α) → x ∈ ↑(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup.commutator_group_le | [92, 1] | [99, 20] | simp only [SetLike.mem_coe, mem_alternatingGroup, map_commutatorElement,
commutatorElement_eq_one_iff_commute] | case intro.intro
β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
p q : Perm α
⊢ ⁅p, q⁆ ∈ ↑(alternatingGroup α) | case intro.intro
β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
p q : Perm α
⊢ Commute (sign p) (sign q) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
p q : Perm α
⊢ ⁅p, q⁆ ∈ ↑(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup.commutator_group_le | [92, 1] | [99, 20] | apply Commute.all | case intro.intro
β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
p q : Perm α
⊢ Commute (sign p) (sign q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
β : Type ?u.8051
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
p q : Perm α
⊢ Commute (sign p) (sign q)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup.commutator_group_eq | [103, 1] | [107, 38] | apply le_antisymm alternatingGroup.commutator_group_le | β : Type ?u.9373
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ _root_.commutator (Perm α) = alternatingGroup α | β : Type ?u.9373
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ alternatingGroup α ≤ _root_.commutator (Perm α) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.9373
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ _root_.commutator (Perm α) = alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup.commutator_group_eq | [103, 1] | [107, 38] | rw [← alternatingGroup_is_perfect' h5] | β : Type ?u.9373
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ alternatingGroup α ≤ _root_.commutator (Perm α) | β : Type ?u.9373
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ ⁅alternatingGroup α, alternatingGroup α⁆ ≤ _root_.commutator (Perm α) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.9373
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ alternatingGroup α ≤ _root_.commutator (Perm α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Alternating.lean | alternatingGroup.commutator_group_eq | [103, 1] | [107, 38] | exact commutator_mono le_top le_top | β : Type ?u.9373
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ ⁅alternatingGroup α, alternatingGroup α⁆ ≤ _root_.commutator (Perm α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type ?u.9373
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h5 : 5 ≤ Fintype.card α
⊢ ⁅alternatingGroup α, alternatingGroup α⁆ ≤ _root_.commutator (Perm α)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | constructor | α : Type u_1
p q : α → Prop
⊢ (∃ x, p x ∨ q x) ↔ (∃ x, p x) ∨ ∃ x, q x | case mp
α : Type u_1
p q : α → Prop
⊢ (∃ x, p x ∨ q x) → (∃ x, p x) ∨ ∃ x, q x
case mpr
α : Type u_1
p q : α → Prop
⊢ ((∃ x, p x) ∨ ∃ x, q x) → ∃ x, p x ∨ q x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
p q : α → Prop
⊢ (∃ x, p x ∨ q x) ↔ (∃ x, p x) ∨ ∃ x, q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | rintro ⟨x, hpx|hqx⟩ | case mp
α : Type u_1
p q : α → Prop
⊢ (∃ x, p x ∨ q x) → (∃ x, p x) ∨ ∃ x, q x | case mp.intro.inl
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ (∃ x, p x) ∨ ∃ x, q x
case mp.intro.inr
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ (∃ x, p x) ∨ ∃ x, q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
p q : α → Prop
⊢ (∃ x, p x ∨ q x) → (∃ x, p x) ∨ ∃ x, q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | left | case mp.intro.inl
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ (∃ x, p x) ∨ ∃ x, q x | case mp.intro.inl.h
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ ∃ x, p x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.inl
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ (∃ x, p x) ∨ ∃ x, q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | use x | case mp.intro.inl.h
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ ∃ x, p x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.inl.h
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ ∃ x, p x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | right | case mp.intro.inr
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ (∃ x, p x) ∨ ∃ x, q x | case mp.intro.inr.h
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ ∃ x, q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.inr
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ (∃ x, p x) ∨ ∃ x, q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | use x | case mp.intro.inr.h
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ ∃ x, q x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.inr.h
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ ∃ x, q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | rintro (⟨x, hpx⟩|⟨x, hqx⟩) | case mpr
α : Type u_1
p q : α → Prop
⊢ ((∃ x, p x) ∨ ∃ x, q x) → ∃ x, p x ∨ q x | case mpr.inl.intro
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ ∃ x, p x ∨ q x
case mpr.inr.intro
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ ∃ x, p x ∨ q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
p q : α → Prop
⊢ ((∃ x, p x) ∨ ∃ x, q x) → ∃ x, p x ∨ q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | use x | case mpr.inl.intro
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ ∃ x, p x ∨ q x | case h
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ p x ∨ q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inl.intro
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ ∃ x, p x ∨ q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | left | case h
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ p x ∨ q x | case h.h
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ p x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ p x ∨ q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | assumption | case h.h
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ p x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
p q : α → Prop
x : α
hpx : p x
⊢ p x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | use x | case mpr.inr.intro
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ ∃ x, p x ∨ q x | case h
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ p x ∨ q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.intro
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ ∃ x, p x ∨ q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | right | case h
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ p x ∨ q x | case h.h
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ q x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ p x ∨ q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_1 | [21, 1] | [35, 19] | assumption | case h.h
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ q x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
p q : α → Prop
x : α
hqx : q x
⊢ q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_2 | [50, 1] | [54, 11] | intro y | f g : ℝ → ℝ
x : ℝ
h : SurjectiveFunction (g ∘ f)
⊢ SurjectiveFunction g | f g : ℝ → ℝ
x : ℝ
h : SurjectiveFunction (g ∘ f)
y : ℝ
⊢ ∃ x, g x = y | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℝ → ℝ
x : ℝ
h : SurjectiveFunction (g ∘ f)
⊢ SurjectiveFunction g
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_2 | [50, 1] | [54, 11] | obtain ⟨x, hx⟩ := h y | f g : ℝ → ℝ
x : ℝ
h : SurjectiveFunction (g ∘ f)
y : ℝ
⊢ ∃ x, g x = y | case intro
f g : ℝ → ℝ
x✝ : ℝ
h : SurjectiveFunction (g ∘ f)
y x : ℝ
hx : (g ∘ f) x = y
⊢ ∃ x, g x = y | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℝ → ℝ
x : ℝ
h : SurjectiveFunction (g ∘ f)
y : ℝ
⊢ ∃ x, g x = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_2 | [50, 1] | [54, 11] | use f x | case intro
f g : ℝ → ℝ
x✝ : ℝ
h : SurjectiveFunction (g ∘ f)
y x : ℝ
hx : (g ∘ f) x = y
⊢ ∃ x, g x = y | case h
f g : ℝ → ℝ
x✝ : ℝ
h : SurjectiveFunction (g ∘ f)
y x : ℝ
hx : (g ∘ f) x = y
⊢ g (f x) = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f g : ℝ → ℝ
x✝ : ℝ
h : SurjectiveFunction (g ∘ f)
y x : ℝ
hx : (g ∘ f) x = y
⊢ ∃ x, g x = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_2 | [50, 1] | [54, 11] | exact hx | case h
f g : ℝ → ℝ
x✝ : ℝ
h : SurjectiveFunction (g ∘ f)
y x : ℝ
hx : (g ∘ f) x = y
⊢ g (f x) = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g : ℝ → ℝ
x✝ : ℝ
h : SurjectiveFunction (g ∘ f)
y x : ℝ
hx : (g ∘ f) x = y
⊢ g (f x) = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_3 | [57, 1] | [66, 6] | constructor | f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction (g ∘ f) ↔ SurjectiveFunction g | case mp
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction (g ∘ f) → SurjectiveFunction g
case mpr
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction g → SurjectiveFunction (g ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction (g ∘ f) ↔ SurjectiveFunction g
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_3 | [57, 1] | [66, 6] | intro hg z | case mpr
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction g → SurjectiveFunction (g ∘ f) | case mpr
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z : ℝ
⊢ ∃ x, (g ∘ f) x = z | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction g → SurjectiveFunction (g ∘ f)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_3 | [57, 1] | [66, 6] | obtain ⟨y, hy⟩ := hg z | case mpr
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z : ℝ
⊢ ∃ x, (g ∘ f) x = z | case mpr.intro
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
⊢ ∃ x, (g ∘ f) x = z | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z : ℝ
⊢ ∃ x, (g ∘ f) x = z
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_3 | [57, 1] | [66, 6] | obtain ⟨x, hx⟩ := hf y | case mpr.intro
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
⊢ ∃ x, (g ∘ f) x = z | case mpr.intro.intro
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
x : ℝ
hx : f x = y
⊢ ∃ x, (g ∘ f) x = z | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
⊢ ∃ x, (g ∘ f) x = z
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_3 | [57, 1] | [66, 6] | use x | case mpr.intro.intro
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
x : ℝ
hx : f x = y
⊢ ∃ x, (g ∘ f) x = z | case h
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
x : ℝ
hx : f x = y
⊢ (g ∘ f) x = z | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
x : ℝ
hx : f x = y
⊢ ∃ x, (g ∘ f) x = z
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_3 | [57, 1] | [66, 6] | rw [← hy, ← hx] | case h
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
x : ℝ
hx : f x = y
⊢ (g ∘ f) x = z | case h
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
x : ℝ
hx : f x = y
⊢ (g ∘ f) x = g (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
x : ℝ
hx : f x = y
⊢ (g ∘ f) x = z
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_3 | [57, 1] | [66, 6] | rfl | case h
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
x : ℝ
hx : f x = y
⊢ (g ∘ f) x = g (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
hg : SurjectiveFunction g
z y : ℝ
hy : g y = z
x : ℝ
hx : f x = y
⊢ (g ∘ f) x = g (f x)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_3 | [57, 1] | [66, 6] | exact exercise2_2 | case mp
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction (g ∘ f) → SurjectiveFunction g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction (g ∘ f) → SurjectiveFunction g
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_4 | [70, 1] | [77, 7] | intro y | f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction fun x => 2 * f (3 * (x + 4)) + 1 | f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
y : ℝ
⊢ ∃ x, (fun x => 2 * f (3 * (x + 4)) + 1) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction fun x => 2 * f (3 * (x + 4)) + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_4 | [70, 1] | [77, 7] | obtain ⟨x, hx⟩ := hf ((y - 1) / 2) | f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
y : ℝ
⊢ ∃ x, (fun x => 2 * f (3 * (x + 4)) + 1) x = y | case intro
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
y x : ℝ
hx : f x = (y - 1) / 2
⊢ ∃ x, (fun x => 2 * f (3 * (x + 4)) + 1) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
y : ℝ
⊢ ∃ x, (fun x => 2 * f (3 * (x + 4)) + 1) x = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_4 | [70, 1] | [77, 7] | use x / 3 - 4 | case intro
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
y x : ℝ
hx : f x = (y - 1) / 2
⊢ ∃ x, (fun x => 2 * f (3 * (x + 4)) + 1) x = y | case h
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
y x : ℝ
hx : f x = (y - 1) / 2
⊢ (fun x => 2 * f (3 * (x + 4)) + 1) (x / 3 - 4) = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
y x : ℝ
hx : f x = (y - 1) / 2
⊢ ∃ x, (fun x => 2 * f (3 * (x + 4)) + 1) x = y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions2.lean | exercise2_4 | [70, 1] | [77, 7] | ring | case h
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
y x : ℝ
hx : f x = (y - 1) / 2
⊢ (fun x => 2 * f (3 * (x + 4)) + 1) (x / 3 - 4) = y | case h
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
y x : ℝ
hx : f x = (y - 1) / 2
⊢ 1 + f x * 2 = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g : ℝ → ℝ
x✝ : ℝ
hf : SurjectiveFunction f
y x : ℝ
hx : f x = (y - 1) / 2
⊢ (fun x => 2 * f (3 * (x + 4)) + 1) (x / 3 - 4) = y
TACTIC:
|
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