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/Blocks.lean | MulAction.IsBlock.mk_notempty | [86, 1] | [93, 29] | intro g' | case h.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g : G
⊢ ∀ (a : G), g • B = a • B ∨ Disjoint (g • B) (a • B) ↔ g • B ∩ a • B ≠ ∅ → g • B = a • B | case h.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) ↔ g • B ∩ g' • B ≠ ∅ → g • B = g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g : G
⊢ ∀ (a : G), g • B = a • B ∨ Disjoint (g • B) (a • B) ↔ g • B ∩ a • B ≠ ∅ → g • B = a • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty | [86, 1] | [93, 29] | rw [Set.disjoint_iff_inter_eq_empty] | case h.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) ↔ g • B ∩ g' • B ≠ ∅ → g • B = g' • B | case h.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g g' : G
⊢ g • B = g' • B ∨ g • B ∩ g' • B = ∅ ↔ g • B ∩ g' • B ≠ ∅ → g • B = g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) ↔ g • B ∩ g' • B ≠ ∅ → g • B = g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty | [86, 1] | [93, 29] | exact or_iff_not_imp_right | case h.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g g' : G
⊢ g • B = g' • B ∨ g • B ∩ g' • B = ∅ ↔ g • B ∩ g' • B ≠ ∅ → g • B = g' • B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g g' : G
⊢ g • B = g' • B ∨ g • B ∩ g' • B = ∅ ↔ g • B ∩ g' • B ≠ ∅ → g • B = g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_fixed | [97, 1] | [102, 21] | rw [IsBlock.def] | G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
⊢ IsBlock G B | G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
⊢ ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
⊢ IsBlock G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_fixed | [97, 1] | [102, 21] | intro g g' | G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
⊢ ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B) | G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
⊢ ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_fixed | [97, 1] | [102, 21] | apply Or.intro_left | G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | case h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
g g' : G
⊢ g • B = g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_fixed | [97, 1] | [102, 21] | rw [hfB g, hfB g'] | case h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
g g' : G
⊢ g • B = g' • B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
g g' : G
⊢ g • B = g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.bot_IsBlock | [108, 1] | [112, 51] | rw [IsBlock.def] | G : Type u_1
X : Type u_2
inst✝ : SMul G X
⊢ IsBlock G ⊥ | G : Type u_1
X : Type u_2
inst✝ : SMul G X
⊢ ∀ (g g' : G), g • ⊥ = g' • ⊥ ∨ Disjoint (g • ⊥) (g' • ⊥) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝ : SMul G X
⊢ IsBlock G ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.bot_IsBlock | [108, 1] | [112, 51] | intro g g' | G : Type u_1
X : Type u_2
inst✝ : SMul G X
⊢ ∀ (g g' : G), g • ⊥ = g' • ⊥ ∨ Disjoint (g • ⊥) (g' • ⊥) | G : Type u_1
X : Type u_2
inst✝ : SMul G X
g g' : G
⊢ g • ⊥ = g' • ⊥ ∨ Disjoint (g • ⊥) (g' • ⊥) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝ : SMul G X
⊢ ∀ (g g' : G), g • ⊥ = g' • ⊥ ∨ Disjoint (g • ⊥) (g' • ⊥)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.bot_IsBlock | [108, 1] | [112, 51] | apply Or.intro_left | G : Type u_1
X : Type u_2
inst✝ : SMul G X
g g' : G
⊢ g • ⊥ = g' • ⊥ ∨ Disjoint (g • ⊥) (g' • ⊥) | case h
G : Type u_1
X : Type u_2
inst✝ : SMul G X
g g' : G
⊢ g • ⊥ = g' • ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝ : SMul G X
g g' : G
⊢ g • ⊥ = g' • ⊥ ∨ Disjoint (g • ⊥) (g' • ⊥)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.bot_IsBlock | [108, 1] | [112, 51] | simp only [Set.bot_eq_empty, Set.smul_set_empty] | case h
G : Type u_1
X : Type u_2
inst✝ : SMul G X
g g' : G
⊢ g • ⊥ = g' • ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
X : Type u_2
inst✝ : SMul G X
g g' : G
⊢ g • ⊥ = g' • ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.singleton_IsBlock | [117, 1] | [122, 11] | rw [IsBlock.def] | G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
⊢ IsBlock G {a} | G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
⊢ ∀ (g g' : G), g • {a} = g' • {a} ∨ Disjoint (g • {a}) (g' • {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
⊢ IsBlock G {a}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.singleton_IsBlock | [117, 1] | [122, 11] | intro g g' | G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
⊢ ∀ (g g' : G), g • {a} = g' • {a} ∨ Disjoint (g • {a}) (g' • {a}) | G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
g g' : G
⊢ g • {a} = g' • {a} ∨ Disjoint (g • {a}) (g' • {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
⊢ ∀ (g g' : G), g • {a} = g' • {a} ∨ Disjoint (g • {a}) (g' • {a})
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.singleton_IsBlock | [117, 1] | [122, 11] | simp only [Set.smul_set_singleton, Set.singleton_eq_singleton_iff, Set.disjoint_singleton, Ne.def] | G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
g g' : G
⊢ g • {a} = g' • {a} ∨ Disjoint (g • {a}) (g' • {a}) | G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
g g' : G
⊢ g • a = g' • a ∨ ¬g • a = g' • a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
g g' : G
⊢ g • {a} = g' • {a} ∨ Disjoint (g • {a}) (g' • {a})
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.singleton_IsBlock | [117, 1] | [122, 11] | apply em | G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
g g' : G
⊢ g • a = g' • a ∨ ¬g • a = g' • a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
g g' : G
⊢ g • a = g' • a ∨ ¬g • a = g' • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.subsingleton_IsBlock | [126, 1] | [129, 67] | cases Set.Subsingleton.eq_empty_or_singleton hB with
| inl h => rw [h]; apply bot_IsBlock
| inr h => obtain ⟨a, ha⟩ := h; rw [ha]; apply singleton_IsBlock | G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
⊢ IsBlock G B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
⊢ IsBlock G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.subsingleton_IsBlock | [126, 1] | [129, 67] | rw [h] | case inl
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
h : B = ∅
⊢ IsBlock G B | case inl
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
h : B = ∅
⊢ IsBlock G ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
h : B = ∅
⊢ IsBlock G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.subsingleton_IsBlock | [126, 1] | [129, 67] | apply bot_IsBlock | case inl
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
h : B = ∅
⊢ IsBlock G ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
h : B = ∅
⊢ IsBlock G ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.subsingleton_IsBlock | [126, 1] | [129, 67] | obtain ⟨a, ha⟩ := h | case inr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
h : ∃ x, B = {x}
⊢ IsBlock G B | case inr.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
a : X
ha : B = {a}
⊢ IsBlock G B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
h : ∃ x, B = {x}
⊢ IsBlock G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.subsingleton_IsBlock | [126, 1] | [129, 67] | rw [ha] | case inr.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
a : X
ha : B = {a}
⊢ IsBlock G B | case inr.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
a : X
ha : B = {a}
⊢ IsBlock G {a} | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
a : X
ha : B = {a}
⊢ IsBlock G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.subsingleton_IsBlock | [126, 1] | [129, 67] | apply singleton_IsBlock | case inr.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
a : X
ha : B = {a}
⊢ IsBlock G {a} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : Set.Subsingleton B
a : X
ha : B = {a}
⊢ IsBlock G {a}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | rw [IsBlock.def] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ IsBlock G B ↔ ∀ (g : G), g • B = B ∨ Disjoint (g • B) B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) ↔ ∀ (g : G), g • B = B ∨ Disjoint (g • B) B | 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
B : Set X
⊢ IsBlock G B ↔ ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | constructor | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) ↔ ∀ (g : G), g • B = B ∨ Disjoint (g • B) B | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) → ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), g • B = B ∨ Disjoint (g • B) B) → ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B) | 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
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) ↔ ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | intro hB g | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) → ∀ (g : G), g • B = B ∨ Disjoint (g • B) B | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g : G
⊢ g • B = B ∨ Disjoint (g • B) B | 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
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) → ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | simpa only [one_smul] using hB g 1 | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g : G
⊢ g • B = B ∨ Disjoint (g • B) B | 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
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g : G
⊢ g • B = B ∨ Disjoint (g • B) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | intro hB | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), g • B = B ∨ Disjoint (g • B) B) → ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B) | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
⊢ ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B) | 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
B : Set X
⊢ (∀ (g : G), g • B = B ∨ Disjoint (g • B) B) → ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | intro g g' | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
⊢ ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B) | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | 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
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
⊢ ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | cases hB (g'⁻¹ * g) with
| inl h =>
apply Or.intro_left
rw [← inv_inv g', eq_inv_smul_iff, smul_smul]
exact h
| inr h =>
apply Or.intro_right
rw [Set.disjoint_iff] at h ⊢
rintro x ⟨hx, hx'⟩
simp only [Set.mem_empty_iff_false]
suffices g'⁻¹ • x ∈ (g'⁻¹ * g) • B ⊓ B by
apply h this
simp only [Set.inf_eq_inter, Set.mem_inter_iff]
simp only [← Set.mem_smul_set_iff_inv_smul_mem]
rw [← smul_smul]; rw [smul_inv_smul]
exact ⟨hx, hx'⟩ | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | no goals | 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
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | apply Or.intro_left | case mpr.inl
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B = B
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | case mpr.inl.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B = B
⊢ g • B = g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inl
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B = B
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | rw [← inv_inv g', eq_inv_smul_iff, smul_smul] | case mpr.inl.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B = B
⊢ g • B = g' • B | case mpr.inl.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B = B
⊢ (g'⁻¹ * g) • B = B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inl.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B = B
⊢ g • B = g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | exact h | case mpr.inl.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B = B
⊢ (g'⁻¹ * g) • B = B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inl.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B = B
⊢ (g'⁻¹ * g) • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | apply Or.intro_right | case mpr.inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : Disjoint ((g'⁻¹ * g) • B) B
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | case mpr.inr.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : Disjoint ((g'⁻¹ * g) • B) B
⊢ Disjoint (g • B) (g' • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : Disjoint ((g'⁻¹ * g) • B) B
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | rw [Set.disjoint_iff] at h ⊢ | case mpr.inr.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : Disjoint ((g'⁻¹ * g) • B) B
⊢ Disjoint (g • B) (g' • B) | case mpr.inr.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
⊢ g • B ∩ g' • B ⊆ ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : Disjoint ((g'⁻¹ * g) • B) B
⊢ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | rintro x ⟨hx, hx'⟩ | case mpr.inr.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
⊢ g • B ∩ g' • B ⊆ ∅ | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
⊢ g • B ∩ g' • B ⊆ ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | simp only [Set.mem_empty_iff_false] | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ ∅ | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | suffices g'⁻¹ • x ∈ (g'⁻¹ * g) • B ⊓ B by
apply h this | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ False | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ g'⁻¹ • x ∈ (g'⁻¹ * g) • B ⊓ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | simp only [Set.inf_eq_inter, Set.mem_inter_iff] | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ g'⁻¹ • x ∈ (g'⁻¹ * g) • B ⊓ B | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ g'⁻¹ • x ∈ (g'⁻¹ * g) • B ∧ g'⁻¹ • x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ g'⁻¹ • x ∈ (g'⁻¹ * g) • B ⊓ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | simp only [← Set.mem_smul_set_iff_inv_smul_mem] | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ g'⁻¹ • x ∈ (g'⁻¹ * g) • B ∧ g'⁻¹ • x ∈ B | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ g' • (g'⁻¹ * g) • B ∧ x ∈ g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ g'⁻¹ • x ∈ (g'⁻¹ * g) • B ∧ g'⁻¹ • x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | rw [← smul_smul] | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ g' • (g'⁻¹ * g) • B ∧ x ∈ g' • B | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ g' • g'⁻¹ • g • B ∧ x ∈ g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ g' • (g'⁻¹ * g) • B ∧ x ∈ g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | rw [smul_inv_smul] | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ g' • g'⁻¹ • g • B ∧ x ∈ g' • B | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ g • B ∧ x ∈ g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ g' • g'⁻¹ • g • B ∧ x ∈ g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | exact ⟨hx, hx'⟩ | case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ g • B ∧ x ∈ g' • B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
⊢ x ∈ g • B ∧ x ∈ g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def_one | [138, 1] | [160, 22] | apply h this | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
this : g'⁻¹ • x ∈ (g'⁻¹ * g) • B ⊓ B
⊢ False | no goals | 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
B : Set X
hB : ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
g g' : G
h : (g'⁻¹ * g) • B ∩ B ⊆ ∅
x : X
hx : x ∈ g • B
hx' : x ∈ g' • B
this : g'⁻¹ • x ∈ (g'⁻¹ * g) • B ⊓ B
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty_one | [163, 1] | [169, 29] | rw [IsBlock.def_one] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ IsBlock G B ↔ ∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), g • B = B ∨ Disjoint (g • B) B) ↔ ∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B | 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
B : Set X
⊢ IsBlock G B ↔ ∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty_one | [163, 1] | [169, 29] | apply forall_congr' | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), g • B = B ∨ Disjoint (g • B) B) ↔ ∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ ∀ (a : G), a • B = B ∨ Disjoint (a • B) B ↔ a • B ∩ B ≠ ∅ → a • B = B | 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
B : Set X
⊢ (∀ (g : G), g • B = B ∨ Disjoint (g • B) B) ↔ ∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty_one | [163, 1] | [169, 29] | intro g | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ ∀ (a : G), a • B = B ∨ Disjoint (a • B) B ↔ a • B ∩ B ≠ ∅ → a • B = B | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
⊢ g • B = B ∨ Disjoint (g • B) B ↔ g • B ∩ B ≠ ∅ → g • B = B | 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
B : Set X
⊢ ∀ (a : G), a • B = B ∨ Disjoint (a • B) B ↔ a • B ∩ B ≠ ∅ → a • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty_one | [163, 1] | [169, 29] | rw [Set.disjoint_iff_inter_eq_empty] | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
⊢ g • B = B ∨ Disjoint (g • B) B ↔ g • B ∩ B ≠ ∅ → g • B = B | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
⊢ g • B = B ∨ g • B ∩ B = ∅ ↔ g • B ∩ B ≠ ∅ → g • B = B | 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
B : Set X
g : G
⊢ g • B = B ∨ Disjoint (g • B) B ↔ g • B ∩ B ≠ ∅ → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty_one | [163, 1] | [169, 29] | exact or_iff_not_imp_right | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
g : G
⊢ g • B = B ∨ g • B ∩ B = ∅ ↔ g • B ∩ B ≠ ∅ → g • B = B | 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
B : Set X
g : G
⊢ g • B = B ∨ g • B ∩ B = ∅ ↔ g • B ∩ B ≠ ∅ → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | rw [IsBlock.mk_notempty_one] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ IsBlock G B ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | 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
B : Set X
⊢ IsBlock G B ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | simp_rw [← Set.nonempty_iff_ne_empty, Set.nonempty_def] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), (∃ x, x ∈ g • B ∩ B) → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | 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
B : Set X
⊢ (∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | simp_rw [Set.mem_inter_iff] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), (∃ x, x ∈ g • B ∩ B) → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), (∃ x ∈ g • B, x ∈ B) → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | 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
B : Set X
⊢ (∀ (g : G), (∃ x, x ∈ g • B ∩ B) → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | simp_rw [exists_imp] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), (∃ x ∈ g • B, x ∈ B) → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G) (x : X), x ∈ g • B ∧ x ∈ B → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | 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
B : Set X
⊢ (∀ (g : G), (∃ x ∈ g • B, x ∈ B) → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | simp_rw [and_imp] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G) (x : X), x ∈ g • B ∧ x ∈ B → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), ∀ x ∈ g • B, x ∈ B → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | 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
B : Set X
⊢ (∀ (g : G) (x : X), x ∈ g • B ∧ x ∈ B → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | simp_rw [Set.mem_smul_set_iff_inv_smul_mem] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), ∀ x ∈ g • B, x ∈ B → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | 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
B : Set X
⊢ (∀ (g : G), ∀ x ∈ g • B, x ∈ B → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | constructor | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B) → ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B) → ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B | 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
B : Set X
⊢ (∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | intro H g a ha hga | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B) → ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g • B = B | 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
B : Set X
⊢ (∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B) → ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | rw [← eq_inv_smul_iff] | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g • B = B | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ B = g⁻¹ • B | 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
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | apply symm | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ B = g⁻¹ • B | case mp.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g⁻¹ • B = B | 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
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ B = g⁻¹ • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | apply H g⁻¹ a _ ha | case mp.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g⁻¹ • B = B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g⁻¹⁻¹ • a ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g⁻¹ • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | rw [inv_inv] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g⁻¹⁻¹ • a ∈ B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g • a ∈ B | 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
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g⁻¹⁻¹ • a ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | exact hga | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g • a ∈ B | no goals | 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
B : Set X
H : ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
g : G
a : X
ha : a ∈ B
hga : g • a ∈ B
⊢ g • a ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | intro H g a ha hga | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B) → ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
g : G
a : X
ha : g⁻¹ • a ∈ B
hga : a ∈ B
⊢ g • B = B | 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
B : Set X
⊢ (∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B) → ∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | rw [← eq_inv_smul_iff] | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
g : G
a : X
ha : g⁻¹ • a ∈ B
hga : a ∈ B
⊢ g • B = B | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
g : G
a : X
ha : g⁻¹ • a ∈ B
hga : a ∈ B
⊢ B = g⁻¹ • B | 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
B : Set X
H : ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
g : G
a : X
ha : g⁻¹ • a ∈ B
hga : a ∈ B
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | apply symm | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
g : G
a : X
ha : g⁻¹ • a ∈ B
hga : a ∈ B
⊢ B = g⁻¹ • B | case mpr.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
g : G
a : X
ha : g⁻¹ • a ∈ B
hga : a ∈ B
⊢ g⁻¹ • B = B | 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
B : Set X
H : ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
g : G
a : X
ha : g⁻¹ • a ∈ B
hga : a ∈ B
⊢ B = g⁻¹ • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_mem | [174, 1] | [192, 25] | exact H g⁻¹ a hga ha | case mpr.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
g : G
a : X
ha : g⁻¹ • a ∈ B
hga : a ∈ B
⊢ g⁻¹ • B = B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
H : ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
g : G
a : X
ha : g⁻¹ • a ∈ B
hga : a ∈ B
⊢ g⁻¹ • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | constructor | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ IsBlock G B ↔ ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ IsBlock G B → ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B) → IsBlock G B | 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
B : Set X
⊢ IsBlock G B ↔ ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | intro hB g b hb hgb | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ IsBlock G B → ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
g : G
b : X
hb : b ∈ B
hgb : b ∈ g • B
⊢ g • B ≤ B | 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
B : Set X
⊢ IsBlock G B → ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | rw [Set.le_iff_subset, Set.set_smul_subset_iff,
IsBlock.def_mem hB hb (Set.mem_smul_set_iff_inv_smul_mem.mp hgb)] | case mp
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : IsBlock G B
g : G
b : X
hb : b ∈ B
hgb : b ∈ g • B
⊢ g • B ≤ B | 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
B : Set X
hB : IsBlock G B
g : G
b : X
hb : b ∈ B
hgb : b ∈ g • B
⊢ g • B ≤ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | rw [IsBlock.mk_notempty_one] | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B) → IsBlock G B | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B) → ∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B | 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
B : Set X
⊢ (∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B) → IsBlock G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | intro hB g hg | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
⊢ (∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B) → ∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : g • B ∩ B ≠ ∅
⊢ g • B = B | 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
B : Set X
⊢ (∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B) → ∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | rw [← Set.nonempty_iff_ne_empty] at hg | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : g • B ∩ B ≠ ∅
⊢ g • B = B | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
⊢ g • B = B | 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
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : g • B ∩ B ≠ ∅
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | obtain ⟨b : X, hb' : b ∈ g • B, hb : b ∈ B⟩ := Set.nonempty_def.mp hg | case mpr
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
⊢ g • B = B | case mpr.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ g • B = B | 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
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | apply le_antisymm | case mpr.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ g • B = B | case mpr.intro.intro.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ g • B ≤ B
case mpr.intro.intro.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ B ≤ g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | suffices g⁻¹ • B ≤ B by
rw [Set.le_iff_subset] at this ⊢
rw [← inv_inv g, ← Set.set_smul_subset_iff]; exact this | case mpr.intro.intro.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ B ≤ g • B | case mpr.intro.intro.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ g⁻¹ • B ≤ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ B ≤ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | exact
hB (Set.mem_smul_set_iff_inv_smul_mem.mp hb') (Set.smul_mem_smul_set_iff.mpr hb) | case mpr.intro.intro.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ g⁻¹ • B ≤ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ g⁻¹ • B ≤ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | exact hB hb hb' | case mpr.intro.intro.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ g • B ≤ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
⊢ g • B ≤ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | rw [Set.le_iff_subset] at this ⊢ | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
this : g⁻¹ • B ≤ B
⊢ B ≤ g • B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
this : g⁻¹ • B ⊆ B
⊢ B ⊆ g • B | 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
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
this : g⁻¹ • B ≤ B
⊢ B ≤ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | rw [← inv_inv g, ← Set.set_smul_subset_iff] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
this : g⁻¹ • B ⊆ B
⊢ B ⊆ g • B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
this : g⁻¹ • B ⊆ B
⊢ g⁻¹ • B ⊆ B | 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
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
this : g⁻¹ • B ⊆ B
⊢ B ⊆ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_subset | [201, 1] | [219, 87] | exact this | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
this : g⁻¹ • B ⊆ B
⊢ g⁻¹ • B ⊆ B | no goals | 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
B : Set X
hB : ∀ {g : G} {b : X}, b ∈ B → b ∈ g • B → g • B ≤ B
g : G
hg : Set.Nonempty (g • B ∩ B)
b : X
hb' : b ∈ g • B
hb : b ∈ B
this : g⁻¹ • B ⊆ B
⊢ g⁻¹ • B ⊆ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_invariant | [223, 1] | [232, 45] | rw [IsBlock.def_one] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
⊢ IsBlock G B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
⊢ ∀ (g : G), g • B = B ∨ Disjoint (g • B) B | 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
B : Set X
hfB : IsInvariantBlock G B
⊢ IsBlock G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_invariant | [223, 1] | [232, 45] | intro g | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
⊢ ∀ (g : G), g • B = B ∨ Disjoint (g • B) B | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ g • B = B ∨ Disjoint (g • B) B | 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
B : Set X
hfB : IsInvariantBlock G B
⊢ ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_invariant | [223, 1] | [232, 45] | apply Or.intro_left | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ g • B = B ∨ Disjoint (g • B) B | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ g • B = B | 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
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ g • B = B ∨ Disjoint (g • B) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_invariant | [223, 1] | [232, 45] | apply le_antisymm | case h
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ g • B = B | case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ g • B ≤ B
case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ B ≤ g • B | 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
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_invariant | [223, 1] | [232, 45] | exact hfB g | case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ g • B ≤ B
case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ B ≤ g • B | case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ B ≤ g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ g • B ≤ B
case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ B ≤ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_invariant | [223, 1] | [232, 45] | intro x hx | case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ B ≤ g • B | case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ x ∈ g • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
⊢ B ≤ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_invariant | [223, 1] | [232, 45] | rw [Set.mem_smul_set_iff_inv_smul_mem] | case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ x ∈ g • B | case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ g⁻¹ • x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ x ∈ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_invariant | [223, 1] | [232, 45] | apply hfB g⁻¹ | case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ g⁻¹ • x ∈ B | case h.a.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ g⁻¹ • x ∈ g⁻¹ • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ g⁻¹ • x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_invariant | [223, 1] | [232, 45] | rw [Set.smul_mem_smul_set_iff] | case h.a.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ g⁻¹ • x ∈ g⁻¹ • B | case h.a.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ g⁻¹ • x ∈ g⁻¹ • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_invariant | [223, 1] | [232, 45] | exact hx | case h.a.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ x ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a.a
G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
B : Set X
hfB : IsInvariantBlock G B
g : G
x : X
hx : x ∈ B
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_orbit | [236, 1] | [239, 28] | apply IsBlock_of_fixed | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
⊢ IsBlock G (orbit G a) | case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
⊢ IsFixedBlock G (orbit G a) | 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
⊢ IsBlock G (orbit G a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_orbit | [236, 1] | [239, 28] | intro g | case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
⊢ IsFixedBlock G (orbit G a) | case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
g : G
⊢ g • orbit G a = orbit G a | Please generate a tactic in lean4 to solve the state.
STATE:
case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
⊢ IsFixedBlock G (orbit G a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_of_orbit | [236, 1] | [239, 28] | apply smul_orbit | case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
g : G
⊢ g • orbit G a = orbit G a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
g : G
⊢ g • orbit G a = orbit G a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.top_IsBlock | [245, 1] | [249, 49] | apply IsBlock_of_fixed | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
⊢ IsBlock G ⊤ | case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
⊢ IsFixedBlock G ⊤ | 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
⊢ IsBlock G ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.top_IsBlock | [245, 1] | [249, 49] | intro g | case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
⊢ IsFixedBlock G ⊤ | case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
g : G
⊢ g • ⊤ = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
⊢ IsFixedBlock G ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.top_IsBlock | [245, 1] | [249, 49] | simp only [Set.top_eq_univ, Set.smul_set_univ] | case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
g : G
⊢ g • ⊤ = ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hfB
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
g : G
⊢ g • ⊤ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Subgroup.IsBlock | [255, 1] | [258, 44] | rw [IsBlock.def_one] | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
B : Set X
hfB : MulAction.IsBlock G B
⊢ MulAction.IsBlock (↥H) B | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
B : Set X
hfB : MulAction.IsBlock G B
⊢ ∀ (g : ↥H), g • B = B ∨ Disjoint (g • B) B | 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
H : Subgroup G
B : Set X
hfB : MulAction.IsBlock G B
⊢ MulAction.IsBlock (↥H) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Subgroup.IsBlock | [255, 1] | [258, 44] | rintro ⟨g, _⟩ | G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
B : Set X
hfB : MulAction.IsBlock G B
⊢ ∀ (g : ↥H), g • B = B ∨ Disjoint (g • B) B | case mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
B : Set X
hfB : MulAction.IsBlock G B
g : G
property✝ : g ∈ H
⊢ { val := g, property := property✝ } • B = B ∨ Disjoint ({ val := g, property := property✝ } • B) B | 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
H : Subgroup G
B : Set X
hfB : MulAction.IsBlock G B
⊢ ∀ (g : ↥H), g • B = B ∨ Disjoint (g • B) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.Subgroup.IsBlock | [255, 1] | [258, 44] | simpa only using IsBlock.def_one.mp hfB g | case mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
B : Set X
hfB : MulAction.IsBlock G B
g : G
property✝ : g ∈ H
⊢ { val := g, property := property✝ } • B = B ∨ Disjoint ({ val := g, property := property✝ } • B) B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
H : Subgroup G
B : Set X
hfB : MulAction.IsBlock G B
g : G
property✝ : g ∈ H
⊢ { val := g, property := property✝ } • B = B ∨ Disjoint ({ val := g, property := property✝ } • B) B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_preimage | [261, 1] | [271, 40] | rw [IsBlock.def_one] | G : Type u_4
inst✝³ : Group G
X : Type u_3
inst✝² : MulAction G X
H : Type u_1
Y : Type u_2
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
⊢ IsBlock H (⇑j ⁻¹' B) | G : Type u_4
inst✝³ : Group G
X : Type u_3
inst✝² : MulAction G X
H : Type u_1
Y : Type u_2
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
⊢ ∀ (g : H), g • ⇑j ⁻¹' B = ⇑j ⁻¹' B ∨ Disjoint (g • ⇑j ⁻¹' B) (⇑j ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_4
inst✝³ : Group G
X : Type u_3
inst✝² : MulAction G X
H : Type u_1
Y : Type u_2
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
⊢ IsBlock H (⇑j ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_preimage | [261, 1] | [271, 40] | intro g | G : Type u_4
inst✝³ : Group G
X : Type u_3
inst✝² : MulAction G X
H : Type u_1
Y : Type u_2
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
⊢ ∀ (g : H), g • ⇑j ⁻¹' B = ⇑j ⁻¹' B ∨ Disjoint (g • ⇑j ⁻¹' B) (⇑j ⁻¹' B) | G : Type u_4
inst✝³ : Group G
X : Type u_3
inst✝² : MulAction G X
H : Type u_1
Y : Type u_2
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
g : H
⊢ g • ⇑j ⁻¹' B = ⇑j ⁻¹' B ∨ Disjoint (g • ⇑j ⁻¹' B) (⇑j ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_4
inst✝³ : Group G
X : Type u_3
inst✝² : MulAction G X
H : Type u_1
Y : Type u_2
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
⊢ ∀ (g : H), g • ⇑j ⁻¹' B = ⇑j ⁻¹' B ∨ Disjoint (g • ⇑j ⁻¹' B) (⇑j ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock_preimage | [261, 1] | [271, 40] | cases' IsBlock.def_one.mp hB (φ g) with heq hdis | G : Type u_4
inst✝³ : Group G
X : Type u_3
inst✝² : MulAction G X
H : Type u_1
Y : Type u_2
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
g : H
⊢ g • ⇑j ⁻¹' B = ⇑j ⁻¹' B ∨ Disjoint (g • ⇑j ⁻¹' B) (⇑j ⁻¹' B) | case inl
G : Type u_4
inst✝³ : Group G
X : Type u_3
inst✝² : MulAction G X
H : Type u_1
Y : Type u_2
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
g : H
heq : φ g • B = B
⊢ g • ⇑j ⁻¹' B = ⇑j ⁻¹' B ∨ Disjoint (g • ⇑j ⁻¹' B) (⇑j ⁻¹' B)
case inr
G : Type u_4
inst✝³ : Group G
X : Type u_3
inst✝² : MulAction G X
H : Type u_1
Y : Type u_2
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
g : H
hdis : Disjoint (φ g • B) B
⊢ g • ⇑j ⁻¹' B = ⇑j ⁻¹' B ∨ Disjoint (g • ⇑j ⁻¹' B) (⇑j ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_4
inst✝³ : Group G
X : Type u_3
inst✝² : MulAction G X
H : Type u_1
Y : Type u_2
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
g : H
⊢ g • ⇑j ⁻¹' B = ⇑j ⁻¹' B ∨ Disjoint (g • ⇑j ⁻¹' B) (⇑j ⁻¹' B)
TACTIC:
|
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