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/Primitive.lean | isPreprimitive_of_surjective_map | [262, 1] | [272, 13] | apply isTrivialBlock_of_surjective_map hf | case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsTrivialBlock (⇑f '' (⇑f ⁻¹' B)) | case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsTrivialBlock (⇑f ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsTrivialBlock (⇑f '' (⇑f ⁻¹' B))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_surjective_map | [262, 1] | [272, 13] | apply h.has_trivial_blocks | case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsTrivialBlock (⇑f ⁻¹' B) | case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsBlock M (⇑f ⁻¹' B) | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsTrivialBlock (⇑f ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_surjective_map | [262, 1] | [272, 13] | apply IsBlock_preimage | case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsBlock M (⇑f ⁻¹' B) | case has_trivial_blocks'.hB
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsBlock N B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsBlock M (⇑f ⁻¹' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_surjective_map | [262, 1] | [272, 13] | exact hB | case has_trivial_blocks'.hB
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsBlock N B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.hB
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPreprimitive M α
this : IsPretransitive N β
B : Set β
hB : IsBlock N B
⊢ IsBlock N B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | constructor | M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive M α ↔ IsPreprimitive N β | case mp
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive M α → IsPreprimitive N β
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive M α ↔ IsPreprimitive N β
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | apply isPreprimitive_of_surjective_map hf.surjective | case mp
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive M α → IsPreprimitive N β
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive M α → IsPreprimitive N β
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | intro hN | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
⊢ IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | haveI := (isPretransitive.of_bijective_map_iff hφ hf).mpr hN.toIsPretransitive | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
⊢ IsPreprimitive M α | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
⊢ IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | apply IsPreprimitive.mk | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ IsPreprimitive M α | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ ∀ {B : Set α}, IsBlock M B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | intro B hB | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ ∀ {B : Set α}, IsBlock M B → IsTrivialBlock B | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ ∀ {B : Set α}, IsBlock M B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | rw [← Set.preimage_image_eq B hf.injective] | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock B | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f ⁻¹' (⇑f '' B)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | apply isTrivialBlock_of_injective_map hf.injective | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f ⁻¹' (⇑f '' B)) | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f '' B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f ⁻¹' (⇑f '' B))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | apply hN.has_trivial_blocks | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f '' B) | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock N (⇑f '' B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f '' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | apply IsBlock_image f hφ hf.injective | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock N (⇑f '' B) | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock M B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock N (⇑f '' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff | [275, 1] | [288, 15] | exact hB | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock M B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock M B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | constructor | M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive M α ↔ IsPreprimitive N β | case mp
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive M α → IsPreprimitive N β
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive M α ↔ IsPreprimitive N β
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | apply isPreprimitive_of_surjective_map hf.surjective | case mp
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive M α → IsPreprimitive N β
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive M α → IsPreprimitive N β
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | intro hN | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
⊢ IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPreprimitive N β → IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | haveI := (isPretransitive.of_bijective_map_iff hφ hf).mpr hN.toIsPretransitive | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
⊢ IsPreprimitive M α | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
⊢ IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | apply IsPreprimitive.mk | case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ IsPreprimitive M α | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ ∀ {B : Set α}, IsBlock M B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | intro B hB | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ ∀ {B : Set α}, IsBlock M B → IsTrivialBlock B | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
⊢ ∀ {B : Set α}, IsBlock M B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | rw [← Set.preimage_image_eq B hf.injective] | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock B | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f ⁻¹' (⇑f '' B)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | apply isTrivialBlock_of_injective_map hf.injective | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f ⁻¹' (⇑f '' B)) | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f '' B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f ⁻¹' (⇑f '' B))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | apply hN.has_trivial_blocks | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f '' B) | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock N (⇑f '' B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock (⇑f '' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | apply IsBlock_image f hφ hf.injective | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock N (⇑f '' B) | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock M B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock N (⇑f '' B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_bijective_map_iff' | [291, 1] | [305, 15] | exact hB | case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock M B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.has_trivial_blocks'
M : Type u_4
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
N : Type u_3
β : Type u_1
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
hN : IsPreprimitive N β
this : IsPretransitive M α
B : Set α
hB : IsBlock M B
⊢ IsBlock M B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | rw [nontrivial_iff] | G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ Nontrivial { B // a ∈ B ∧ IsBlock G B } | G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ ∃ x y, x ≠ y | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ Nontrivial { B // a ∈ B ∧ IsBlock G B }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | use (Block.boundedOrderOfMem G a).bot | G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ ∃ x y, x ≠ y | case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ ∃ y, ⊥ ≠ y | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ ∃ x y, x ≠ y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | use (Block.boundedOrderOfMem G a).top | case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ ∃ y, ⊥ ≠ y | case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ ⊥ ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ ∃ y, ⊥ ≠ y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | intro h | case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ ⊥ ≠ ⊤ | case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : ⊥ = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
⊢ ⊥ ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | rw [← Subtype.coe_inj] at h | case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : ⊥ = ⊤
⊢ False | case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : ↑⊥ = ↑⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : ⊥ = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | simp only [Block.boundedOrderOfMem.top_eq, Block.boundedOrderOfMem.bot_eq] at h | case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : ↑⊥ = ↑⊤
⊢ False | case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : ↑⊥ = ↑⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | obtain ⟨b, hb⟩ := exists_ne a | case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
⊢ False | case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | apply hb | case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ False | case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | rw [← Set.mem_singleton_iff] | case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b = a | case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b ∈ {a} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | rw [h] | case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b ∈ {a} | case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b ∈ {a}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | rw [Set.top_eq_univ] | case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b ∈ ⊤ | case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b ∈ Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b ∈ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | Block.mem_is_nontrivial_order_of_nontrivial | [339, 1] | [351, 43] | apply Set.mem_univ | case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b ∈ Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
inst✝ : Nontrivial X
a : X
h : {a} = ⊤
b : X
hb : b ≠ a
⊢ b ∈ Set.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | haveI : Nontrivial { B : Set X // a ∈ B ∧ IsBlock G B } :=
Block.mem_is_nontrivial_order_of_nontrivial G a | G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
⊢ IsPreprimitive G X ↔ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B } | G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
⊢ IsPreprimitive G X ↔ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B } | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
⊢ IsPreprimitive G X ↔ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | constructor | G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
⊢ IsPreprimitive G X ↔ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B } | case mp
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
⊢ IsPreprimitive G X → IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
⊢ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B } → IsPreprimitive G X | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
⊢ IsPreprimitive G X ↔ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | intro hGX' | case mp
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
⊢ IsPreprimitive G X → IsSimpleOrder { B // a ∈ B ∧ IsBlock G B } | case mp
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
⊢ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B } | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
⊢ IsPreprimitive G X → IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | apply IsSimpleOrder.mk | case mp
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
⊢ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B } | case mp.eq_bot_or_eq_top
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
⊢ ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
⊢ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | rintro ⟨B, haB, hB⟩ | case mp.eq_bot_or_eq_top
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
⊢ ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ | case mp.eq_bot_or_eq_top.mk.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
⊢ { val := B, property := ⋯ } = ⊥ ∨ { val := B, property := ⋯ } = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.eq_bot_or_eq_top
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
⊢ ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | simp only [← Subtype.coe_inj, Subtype.coe_mk] | case mp.eq_bot_or_eq_top.mk.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
⊢ { val := B, property := ⋯ } = ⊥ ∨ { val := B, property := ⋯ } = ⊤ | case mp.eq_bot_or_eq_top.mk.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
⊢ B = ↑⊥ ∨ B = ↑⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.eq_bot_or_eq_top.mk.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
⊢ { val := B, property := ⋯ } = ⊥ ∨ { val := B, property := ⋯ } = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | cases hGX'.has_trivial_blocks hB with
| inl h =>
apply Or.intro_left
change B = ↑(Block.boundedOrderOfMem G a).bot
rw [Block.boundedOrderOfMem.bot_eq]
exact Set.Subsingleton.eq_singleton_of_mem h haB
| inr h =>
apply Or.intro_right
change B = ↑(Block.boundedOrderOfMem G a).top
exact h | case mp.eq_bot_or_eq_top.mk.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
⊢ B = ↑⊥ ∨ B = ↑⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.eq_bot_or_eq_top.mk.intro
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
⊢ B = ↑⊥ ∨ B = ↑⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | apply Or.intro_left | case mp.eq_bot_or_eq_top.mk.intro.inl
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : Set.Subsingleton B
⊢ B = ↑⊥ ∨ B = ↑⊤ | case mp.eq_bot_or_eq_top.mk.intro.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : Set.Subsingleton B
⊢ B = ↑⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.eq_bot_or_eq_top.mk.intro.inl
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : Set.Subsingleton B
⊢ B = ↑⊥ ∨ B = ↑⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | rw [Block.boundedOrderOfMem.bot_eq] | case mp.eq_bot_or_eq_top.mk.intro.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : Set.Subsingleton B
⊢ B = ↑⊥ | case mp.eq_bot_or_eq_top.mk.intro.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : Set.Subsingleton B
⊢ B = {a} | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.eq_bot_or_eq_top.mk.intro.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : Set.Subsingleton B
⊢ B = ↑⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | exact Set.Subsingleton.eq_singleton_of_mem h haB | case mp.eq_bot_or_eq_top.mk.intro.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : Set.Subsingleton B
⊢ B = {a} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.eq_bot_or_eq_top.mk.intro.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : Set.Subsingleton B
⊢ B = {a}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | apply Or.intro_right | case mp.eq_bot_or_eq_top.mk.intro.inr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : B = ⊤
⊢ B = ↑⊥ ∨ B = ↑⊤ | case mp.eq_bot_or_eq_top.mk.intro.inr.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : B = ⊤
⊢ B = ↑⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.eq_bot_or_eq_top.mk.intro.inr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : B = ⊤
⊢ B = ↑⊥ ∨ B = ↑⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | exact h | case mp.eq_bot_or_eq_top.mk.intro.inr.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : B = ⊤
⊢ B = ↑⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.eq_bot_or_eq_top.mk.intro.inr.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
hGX' : IsPreprimitive G X
B : Set X
haB : a ∈ B
hB : IsBlock G B
h : B = ⊤
⊢ B = ↑⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | intro h | case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
⊢ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B } → IsPreprimitive G X | case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
⊢ IsPreprimitive G X | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
⊢ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B } → IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | let h_bot_or_top := h.eq_bot_or_eq_top | case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
⊢ IsPreprimitive G X | case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
⊢ IsPreprimitive G X | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
⊢ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | apply IsPreprimitive.mk_mem a | case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
⊢ IsPreprimitive G X | case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
⊢ ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
⊢ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | intro B haB hB | case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
⊢ ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B | case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
⊢ ∀ (B : Set X), a ∈ B → IsBlock G B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | cases' h_bot_or_top ⟨B, haB, hB⟩ with hB' hB' <;>
simp only [← Subtype.coe_inj, Subtype.coe_mk] at hB' | case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
⊢ IsTrivialBlock B | case mpr.inl
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ IsTrivialBlock B
case mpr.inr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | left | case mpr.inl
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ IsTrivialBlock B | case mpr.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ Set.Subsingleton B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inl
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | rw [hB'] | case mpr.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ Set.Subsingleton B | case mpr.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ Set.Subsingleton ↑⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ Set.Subsingleton B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | exact Set.subsingleton_singleton | case mpr.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ Set.Subsingleton ↑⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inl.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ Set.Subsingleton ↑⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | right | case mpr.inr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ IsTrivialBlock B | case mpr.inr.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | rw [hB'] | case mpr.inr.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ B = ⊤ | case mpr.inr.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ ↑⊤ = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_iff_isSimpleOrder_blocks | [356, 1] | [381, 27] | rfl | case mpr.inr.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ ↑⊤ = ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr.h
G : Type u_2
inst✝² : Group G
X : Type u_1
inst✝¹ : MulAction G X
htGX : IsPretransitive G X
inst✝ : Nontrivial X
a : X
this : Nontrivial { B // a ∈ B ∧ IsBlock G B }
h : IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
h_bot_or_top : ∀ (a_1 : { B // a ∈ B ∧ IsBlock G B }), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ ↑⊤ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | maximal_stabilizer_iff_preprimitive | [386, 1] | [391, 76] | rw [isPreprimitive_iff_isSimpleOrder_blocks G a, Subgroup.isMaximal_def, ← Set.isSimpleOrder_Ici_iff_isCoatom] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
hnX : Nontrivial X
a : X
⊢ Subgroup.IsMaximal (stabilizer G a) ↔ IsPreprimitive G X | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
hnX : Nontrivial X
a : X
⊢ IsSimpleOrder ↑(Set.Ici (stabilizer G a)) ↔ IsSimpleOrder { B // a ∈ 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
htGX : IsPretransitive G X
hnX : Nontrivial X
a : X
⊢ Subgroup.IsMaximal (stabilizer G a) ↔ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | maximal_stabilizer_iff_preprimitive | [386, 1] | [391, 76] | simp only [isSimpleOrder_iff_isCoatom_bot] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
hnX : Nontrivial X
a : X
⊢ IsSimpleOrder ↑(Set.Ici (stabilizer G a)) ↔ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B } | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
hnX : Nontrivial X
a : X
⊢ IsCoatom ⊥ ↔ IsCoatom ⊥ | 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
htGX : IsPretransitive G X
hnX : Nontrivial X
a : X
⊢ IsSimpleOrder ↑(Set.Ici (stabilizer G a)) ↔ IsSimpleOrder { B // a ∈ B ∧ IsBlock G B }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | maximal_stabilizer_iff_preprimitive | [386, 1] | [391, 76] | rw [← OrderIso.isCoatom_iff (stabilizerBlockEquiv G a), OrderIso.map_bot] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
htGX : IsPretransitive G X
hnX : Nontrivial X
a : X
⊢ IsCoatom ⊥ ↔ IsCoatom ⊥ | 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
htGX : IsPretransitive G X
hnX : Nontrivial X
a : X
⊢ IsCoatom ⊥ ↔ IsCoatom ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | hasMaximalStabilizersOfPreprimitive | [395, 1] | [400, 13] | haveI : IsPretransitive G X := hpGX.toIsPretransitive | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hnX : Nontrivial X
hpGX : IsPreprimitive G X
a : X
⊢ Subgroup.IsMaximal (stabilizer G a) | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hnX : Nontrivial X
hpGX : IsPreprimitive G X
a : X
this : IsPretransitive G X
⊢ Subgroup.IsMaximal (stabilizer G a) | 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
hnX : Nontrivial X
hpGX : IsPreprimitive G X
a : X
⊢ Subgroup.IsMaximal (stabilizer G a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | hasMaximalStabilizersOfPreprimitive | [395, 1] | [400, 13] | rw [maximal_stabilizer_iff_preprimitive] | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hnX : Nontrivial X
hpGX : IsPreprimitive G X
a : X
this : IsPretransitive G X
⊢ Subgroup.IsMaximal (stabilizer G a) | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hnX : Nontrivial X
hpGX : IsPreprimitive G X
a : X
this : IsPretransitive G X
⊢ IsPreprimitive 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
hnX : Nontrivial X
hpGX : IsPreprimitive G X
a : X
this : IsPretransitive G X
⊢ Subgroup.IsMaximal (stabilizer G a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | hasMaximalStabilizersOfPreprimitive | [395, 1] | [400, 13] | exact hpGX | G : Type u_2
inst✝¹ : Group G
X : Type u_1
inst✝ : MulAction G X
hnX : Nontrivial X
hpGX : IsPreprimitive G X
a : X
this : IsPretransitive G X
⊢ IsPreprimitive G X | 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
hnX : Nontrivial X
hpGX : IsPreprimitive G X
a : X
this : IsPretransitive G X
⊢ IsPreprimitive G X
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isnontrivial_of_nontrivial_action | [410, 1] | [420, 11] | apply Or.resolve_left (subsingleton_or_nontrivial α) | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
⊢ Nontrivial α | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
⊢ ¬Subsingleton α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
⊢ Nontrivial α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isnontrivial_of_nontrivial_action | [410, 1] | [420, 11] | intro hα | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
⊢ ¬Subsingleton α | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
⊢ ¬Subsingleton α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isnontrivial_of_nontrivial_action | [410, 1] | [420, 11] | apply h | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
⊢ False | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
⊢ fixedPoints (↥N) α = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isnontrivial_of_nontrivial_action | [410, 1] | [420, 11] | rw [eq_top_iff] | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
⊢ fixedPoints (↥N) α = ⊤ | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
⊢ ⊤ ≤ fixedPoints (↥N) α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
⊢ fixedPoints (↥N) α = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isnontrivial_of_nontrivial_action | [410, 1] | [420, 11] | intro x hx | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
⊢ ⊤ ≤ fixedPoints (↥N) α | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
x : α
hx : x ∈ ⊤
⊢ x ∈ fixedPoints (↥N) α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
⊢ ⊤ ≤ fixedPoints (↥N) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isnontrivial_of_nontrivial_action | [410, 1] | [420, 11] | rw [mem_fixedPoints] | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
x : α
hx : x ∈ ⊤
⊢ x ∈ fixedPoints (↥N) α | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
x : α
hx : x ∈ ⊤
⊢ ∀ (m : ↥N), m • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
x : α
hx : x ∈ ⊤
⊢ x ∈ fixedPoints (↥N) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isnontrivial_of_nontrivial_action | [410, 1] | [420, 11] | intro g | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
x : α
hx : x ∈ ⊤
⊢ ∀ (m : ↥N), m • x = x | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
x : α
hx : x ∈ ⊤
g : ↥N
⊢ g • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
x : α
hx : x ∈ ⊤
⊢ ∀ (m : ↥N), m • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isnontrivial_of_nontrivial_action | [410, 1] | [420, 11] | rw [subsingleton_iff] at hα | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
x : α
hx : x ∈ ⊤
g : ↥N
⊢ g • x = x | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : ∀ (x y : α), x = y
x : α
hx : x ∈ ⊤
g : ↥N
⊢ g • x = x | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : Subsingleton α
x : α
hx : x ∈ ⊤
g : ↥N
⊢ g • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isnontrivial_of_nontrivial_action | [410, 1] | [420, 11] | apply hα | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : ∀ (x y : α), x = y
x : α
hx : x ∈ ⊤
g : ↥N
⊢ g • x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
N : Subgroup M
h : fixedPoints (↥N) α ≠ ⊤
hα : ∀ (x y : α), x = y
x : α
hx : x ∈ ⊤
g : ↥N
⊢ g • x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | apply IsQuasipreprimitive.mk | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
⊢ IsQuasipreprimitive M α | case pretransitive_of_normal
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
⊢ ∀ {N : Subgroup M}, Subgroup.Normal N → fixedPoints (↥N) α ≠ ⊤ → IsPretransitive (↥N) α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
⊢ IsQuasipreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | intro N hN hNX | case pretransitive_of_normal
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
⊢ ∀ {N : Subgroup M}, Subgroup.Normal N → fixedPoints (↥N) α ≠ ⊤ → IsPretransitive (↥N) α | case pretransitive_of_normal
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
⊢ IsPretransitive (↥N) α | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
⊢ ∀ {N : Subgroup M}, Subgroup.Normal N → fixedPoints (↥N) α ≠ ⊤ → IsPretransitive (↥N) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | have : ∃ x : α, x ∉ fixedPoints N α :=
by
rw [← Set.ne_univ_iff_exists_not_mem, ← Set.top_eq_univ]
exact hNX | case pretransitive_of_normal
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
⊢ IsPretransitive (↥N) α | case pretransitive_of_normal
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
this : ∃ x, x ∉ fixedPoints (↥N) α
⊢ IsPretransitive (↥N) α | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
⊢ IsPretransitive (↥N) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | obtain ⟨a, ha⟩ := this | case pretransitive_of_normal
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
this : ∃ x, x ∉ fixedPoints (↥N) α
⊢ IsPretransitive (↥N) α | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
⊢ IsPretransitive (↥N) α | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
this : ∃ x, x ∉ fixedPoints (↥N) α
⊢ IsPretransitive (↥N) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | rw [← MulAction.orbit.isPretransitive_iff a] | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
⊢ IsPretransitive (↥N) α | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
⊢ orbit (↥N) a = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
⊢ IsPretransitive (↥N) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | apply Or.resolve_left (hGX.has_trivial_blocks (orbit.IsBlock_of_normal hN a)) | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
⊢ orbit (↥N) a = ⊤ | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
⊢ ¬Set.Subsingleton (orbit (↥N) a) | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
⊢ orbit (↥N) a = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | intro h | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
⊢ ¬Set.Subsingleton (orbit (↥N) a) | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
⊢ ¬Set.Subsingleton (orbit (↥N) a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | apply ha | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
⊢ False | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
⊢ a ∈ fixedPoints (↥N) α | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | simp only [mem_fixedPoints] | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
⊢ a ∈ fixedPoints (↥N) α | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
⊢ ∀ (m : ↥N), m • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
⊢ a ∈ fixedPoints (↥N) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | intro n | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
⊢ ∀ (m : ↥N), m • a = a | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
⊢ n • a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
⊢ ∀ (m : ↥N), m • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | rw [← Set.mem_singleton_iff] | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
⊢ n • a = a | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
⊢ n • a ∈ {a} | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
⊢ n • a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | suffices orbit N a = {a} by rw [← this]; use n | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
⊢ n • a ∈ {a} | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
⊢ orbit (↥N) a = {a} | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
⊢ n • a ∈ {a}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | rw [← Set.ne_univ_iff_exists_not_mem, ← Set.top_eq_univ] | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
⊢ ∃ x, x ∉ fixedPoints (↥N) α | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
⊢ fixedPoints (↥N) α ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
⊢ ∃ x, x ∉ fixedPoints (↥N) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | exact hNX | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
⊢ fixedPoints (↥N) α ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
⊢ fixedPoints (↥N) α ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | rw [← this] | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
this : orbit (↥N) a = {a}
⊢ n • a ∈ {a} | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
this : orbit (↥N) a = {a}
⊢ n • a ∈ orbit (↥N) a | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
this : orbit (↥N) a = {a}
⊢ n • a ∈ {a}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | use n | M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
this : orbit (↥N) a = {a}
⊢ n • a ∈ orbit (↥N) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
this : orbit (↥N) a = {a}
⊢ n • a ∈ orbit (↥N) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | ext b | case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
⊢ orbit (↥N) a = {a} | case pretransitive_of_normal.intro.h
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
b : α
⊢ b ∈ orbit (↥N) a ↔ b ∈ {a} | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal.intro
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
⊢ orbit (↥N) a = {a}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | IsPreprimitive.isQuasipreprimitive | [426, 1] | [442, 77] | rw [Set.Subsingleton.eq_singleton_of_mem h (MulAction.mem_orbit_self a)] | case pretransitive_of_normal.intro.h
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
b : α
⊢ b ∈ orbit (↥N) a ↔ b ∈ {a} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pretransitive_of_normal.intro.h
M : Type u_1
inst✝¹ : Group M
α : Type u_2
inst✝ : MulAction M α
hGX : IsPreprimitive M α
N : Subgroup M
hN : Subgroup.Normal N
hNX : fixedPoints (↥N) α ≠ ⊤
a : α
ha : a ∉ fixedPoints (↥N) α
h : Set.Subsingleton (orbit (↥N) a)
n : ↥N
b : α
⊢ b ∈ orbit (↥N) a ↔ b ∈ {a}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_prime | [583, 1] | [603, 38] | apply IsPreprimitive.mk | M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
⊢ IsPreprimitive M α | case has_trivial_blocks'
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
⊢ ∀ {B : Set α}, IsBlock M B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
⊢ IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_prime | [583, 1] | [603, 38] | intro B hB | case has_trivial_blocks'
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
⊢ ∀ {B : Set α}, IsBlock M B → IsTrivialBlock B | case has_trivial_blocks'
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
⊢ ∀ {B : Set α}, IsBlock M B → IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_prime | [583, 1] | [603, 38] | cases' Set.subsingleton_or_nontrivial B with hB' hB' | case has_trivial_blocks'
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock B | case has_trivial_blocks'.inl
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
hB' : Set.Subsingleton B
⊢ IsTrivialBlock B
case has_trivial_blocks'.inr
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
hB' : Set.Nontrivial B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_prime | [583, 1] | [603, 38] | apply Or.intro_left | case has_trivial_blocks'.inl
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
hB' : Set.Subsingleton B
⊢ IsTrivialBlock B | case has_trivial_blocks'.inl.h
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
hB' : Set.Subsingleton B
⊢ Set.Subsingleton B | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inl
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
hB' : Set.Subsingleton B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_prime | [583, 1] | [603, 38] | exact hB' | case has_trivial_blocks'.inl.h
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
hB' : Set.Subsingleton B
⊢ Set.Subsingleton B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inl.h
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
hB' : Set.Subsingleton B
⊢ Set.Subsingleton B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Primitive.lean | isPreprimitive_of_prime | [583, 1] | [603, 38] | apply Or.intro_right | case has_trivial_blocks'.inr
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
hB' : Set.Nontrivial B
⊢ IsTrivialBlock B | case has_trivial_blocks'.inr.h
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
hB' : Set.Nontrivial B
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'.inr
M : Type u_2
inst✝⁴ : Group M
α : Type u_1
inst✝³ : MulAction M α
N : Type ?u.75053
β : Type ?u.75056
inst✝² : Group N
inst✝¹ : MulAction N β
inst✝ : Fintype α
hGX : IsPretransitive M α
hp : Nat.Prime (Fintype.card α)
B : Set α
hB : IsBlock M B
hB' : Set.Nontrivial B
⊢ IsTrivialBlock B
TACTIC:
|
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